Colorant Estimation for the Kodak Q60C Target. 95. Colorant Estimation for the Still Life Painting. 97. Colorant Estimation for 105 Mixtures Using Ψ Space. 102.
SPECTRAL-BASED COLOR SEPARATION ALGORITHM DEVELOPMENT FOR MULTIPLE-INK COLOR REPRODUCTION
by Di-Yuan Tzeng B.S. Chinese Culture University, Taipei, Taiwan (1988) M.A. Central Connecticut State University (1994) A dissertation submitted in partial fulfillment of the requirements for the degree of Ph.D. in the Chester F. Carlson Center for Imaging Science of the College of Science Rochester Institute of Technology September 1999
Signature of the Author ____________________________________________________
Accepted by ____________________________________________________ Coordinator, Ph.D. Degree Program Date
CHESTER F. CARLSON CENTER FOR IMAGING SCIENCE COLLEGE OF SCIENCE ROCHESTER INSTITUTE OF TECHNOLOGY ROCHESTER, NY
CERTIFICATE OF APPROVAL
Ph.D. DEGREE DISSERTATION
The Ph.D. Degree Dissertation of Di-Yuan Tzeng has been examined and approved by the dissertation committee as satisfactory for the dissertation requirement for the Ph.D. degree in Imaging Science
Dr. Roy S. Berns, Dissertation Advisor
Dr. Mark D. Fairchild
Dr. Jonathan S. Arney
Mr. Hubert D. Wood
Date
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DISSERTATION RELEASE PERMISSION ROCHESTER INSTITUTE OF TECHNOLOGY ROCHESTER, NY YORK
SPECTRAL-BASED COLOR SEPARATION ALGORITHM DEVELOPMENT FOR MULTIPLE-INK COLOR REPRODUCTION
I, Di-Yuan Tzeng, hereby grant permission to Wallace Memorial Library of R.I.T. to reproduce my dissertation in whole or in part. Any reproduction will not be for commercial use or profit.
Di-Yuan Tzeng
Date
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SPECTRAL-BASED COLOR SEPARATION ALGORITHM DEVELOPMENT FOR MULTIPLE-INK COLOR REPRODUCTION by Di-Yuan Tzeng
Submitted to the Center for Imaging Science in partial fulfill of the requirements for the Ph. D. degree as Rochester Institute of Technology September 1999
ABSTRACT Conventional four-color printing systems are limited by an insufficient number of degrees of freedom for tuning the visible region of the spectrum; as a consequence, they are often limited to metameric color reproductions. That is, color matches defined for a single observer and illuminant (usually CIE illuminant D50 and the 1931 standard observer) are often unstable when viewed under other illuminants or by other observers. For critical color-matching applications, such as catalog sales and artwork reproductions, the results are usually disappointing due to typical uncontrolled lighting and viewing. Furthermore, the existing multiple-ink printing systems, which all focus on expanding color gamut, do not alleviate metamerism since their separation algorithms are trichromatic in nature. The advantage of an increased number of degrees of freedom is not exploited. A research and development program has been initiated at the Rochester Institute of Technology’s Munsell Color Science Laboratory to develop a spectral-based color reproduction system. Research has included multi-spectral acquisition systems and spectral-based printing. The current research is concerned with bridging these analysis and synthesis stages of color reproduction. The goal of the doctoral research was to minimize
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metamerism between originals and their corresponding reproductions, thus best approaching spectral matches. Accomplishing this goal required first estimating the spectral properties of the colorants used to create the original object or set of objects. After the possible colorants were statistically uncovered, they were correlated to an existing ink database for determining an optimal ink set. An algorithm was developed for predicting ink overprints, known as the Neugebauer secondary, tertiary, quaternary primaries, etc., which is the required information for color synthesis using a halftone printing process. Once all the required Neugebauer primaries were determined, a spectralbased printing model minimizing metamerism was derived to calculate the corresponding color separations for each selected ink. The various research components were tested computationally and experimentally. Finally, a DuPont Waterproof® system was used as a representation of halftone printing process to output each color separation in order to test several computational subsystems. This completes the chain of a spectral-based output system. Five modules of this chain were developed and discussed in this dissertation.
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DEDICATION This dissertation is dedicated to my parents, Chien-feng and Ai-Tsu, my beloved wife, Wei-Chien, my dearest son, Andrew Tsu-Jin, my second coming son, Yo-Jin, my younger brother, Chih-Yuan, and my youngest brother, Li-Yuan.
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ACKNOWLEDGEMENTS I would like to thank Dr. Roy S. Berns, my advisor and a good friend, who saw the best of me and helped to make this document and its associated research the single most significant accomplishment of my life. His tireless guidance and inspiring advise has taught me so much more than he realizes. I could not come to this point without his wisdom. I thank my parents for giving me the strength and foundation materially and spiritually that have accompanied me to come to this point. I could not have done this without their unselfish support and so many sleepless nights. I greatly thank them for never giving up on me when I was a wild teenager. I would like to thank my wife, Wei-Chien, for her entire devotion to my family taking full care of our family such that I can fully concentrate on pursuing this academic achievement. I would like to thank my two sons, Andrew Tsu-Jin and Yo-Jin, who all came on the crucial time of my academic life and brought me luck. Wei-chien was pregnant with TsuJin when I had the Ph.D. comprehensive exam and passed it later. Now she is in her fifthmonth of pregnancy with Yo-Jin. The advent of Yo-Jin brought me through my defense toward my Ph.D. degree. I gratefully acknowledge the financial support of the Munsell Color Science Laboratory, the Center for Imaging Science, and E. I. Du Pont de Nemours and Company. I thank Dr. Mark D. Fairchild, Dr. Jonathan S. Arney, and Mr. Hubert D. Wood for their generous efforts monitoring the quality of this doctoral research and their informative advise. I would like to express my appreciation to Dr. Tony Liang and his colleagues at DuPont for their tireless sample preparations to help substantiate this research. Finally, I would like to thank all my friends and colleagues in the Munsell Color Science Laboratory for their innumerable comments, inspirations, and encouragement to this research, especially, Dr. Noboru Ohta; Dr. Ethan Montag; Dr. Francisco Imai; Dr. Peter Burns; Mr. Gus Braun, who is going to be a Ph.D. in a short time; Mr. Dave Wyble, who helped with the proofreading of this dissertation; and Mrs. Colleen Desimone.
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Table of Contents I. INTRODUCTION A. OVERALL PROBLEM B. OVERALL SOLUTION C. SCOPE OF DISSERTATION 1. Colorant Estimation 2. Optimal Ink Selection 3. Ink Overprint Prediction 4. Spectral-Based Six-Color Separation Minimizing Metamerism 5. Multiple-Ink Direct Printing
1 3 7 9 10 12 13
II. BACKGROUND
20 20 21 26 28 32 34 34 43 46 51 57 58 59 61 62
A. B. C. D.
MULTIPLE-INK COLOR SYSTEMS KUBELKA-MUNK TURBID MEDIA THEORY COMPUTER COLORANT MATCHING VECTOR REPRESENTATION FOR COLORANTS Vector Subspace E. LINEAR MODELING TECHNIQUES Principal Component Analysis (PCA) Principal Component Analysis for Color Science Applications Interpretations for Coloration Processing Applications by PCA Multivariate Normality for Effective Data Reduction F. SPECTRAL PRINTING MODELS Basic Assumptions Murray-Davies Theory Dot Gain Neugebauer Theory
III. LINEAR COLORANT MIXING SPACES A. REFLECTION AND ABSORPTION SPACES B. TRANSFORMATION BETWEEN REFLECTANCE AND Φ SPECTRA C. DIMENSIONALITY REDUCTION: NORMAL VERSUS NON-NORMAL POPULATIONS D. VERIFICATIONS E. CONCLUSIONS
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17 18
66 67 70 73 77 85
IV. COLORANT ESTIMATION OF ORIGINAL OBJECTS A. B. C. D.
APPROXIMATELY LINEAR COLORANT MIXING SPACE PRINCIPAL COMPONENT ANALYSIS COLORANT ESTIMATION JUSTIFICATION OF EIGENVECTOR RECONSTRUCTION WITHOUT SAMPLE MEAN E. VERIFICATIONS Testing the Constrained-Rotation Engine by a Virtual Sample Population Colorant Estimation for the Kodak Q60C Target Colorant Estimation for the Still Life Painting Colorant Estimation for 105 Mixtures Using Ψ Space F. CONCLUSIONS
V. OPTIMAL INK-SELECTION
92 93 93 95 97 102 107 109
A. FIRST ORDER INK-SELECTION BY VECTOR CORRELATION B. CONTINUOUS TONE APPROXIMATION C. VERIFICATIONS AND RESULTS Deriving a Linear Color Mixing Space for Continuous Tone Approximation Vector Correlation Analysis in Ψ Space Colorimetric and Spectral Performance by Continuous Tone Approximation D. CONCLUSIONS
VI. SPECTRAL REFLECTANCE PREDICTION OF INK OVERPRINT USING KUBELKA-MUNK TURBIDMEDIA THEORY A. B. C. D. E. F.
86 87 88 90
PRIVIOUS RESEARCH TECHNICAL APPROACH EXPERIMENTAL RESULTS DISCUSSIONS CONCLUSIONS
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110 113 116 116 120 123 127
129 131 132 137 139 144 145
VII. SPECTRAL-BASED SIX-COLOR SEPARATION MINIMIZING METAMERISM A. SIX-COLOR YULE-NIELSEN MODIFIED SPECTRAL NUEGEBAUER EQUATION B. AN ALTERNATIVE APPROACH USING SIX-COLOR HALFTONE PRINTING PROCESS MINIMIZING METAMERISM Subdivision of Six-Color Modeling Forward Four-Color Halftone Spectral Printing Models 1. 2. 3. 4.
The first order forward model Second Order improvement (modeling for ink- and optical-trapping) Alternative second order improvement Modeling by matrix transformation
Proper Four-Color Sub-Model Selection Backward Printing Models for Six-Color Separation Minimizing Metamerism C. EXPERIMENTAL AND VERIFICATION Sample Preparation 1. 2. 3. 4.
Preparation for ramps Preparation of the verification target (5x5x5x5 combinatorial design for mixtures) Sample measurements Accuracy metric
Determining the Yule-Nielsen n-Factor Accuracy for the First Order Six-Color Forward Printing Model Second Order Modification (By Iino and Berns’ Suggestion) Alternative Second Order Modification (Proposed Algorithm) Modeling by Matrix Transformation Six-Color Separation Minimizing Metamerism D. SPECTRAL PERFORMANCE COMPARISONS FOR THREE-, FOUR-, AND SIX-COLOR PRINTING PROCESSES E. CONCLUSIONS
VIII. MULTIPLE-INK DIRECT PRINTING A. VERIFICATIONS B. CONCLUSIONS
IX. CONCLUSIONS, DISCUSSIONS, AND SUGGESTIONS FOR FUTURE RESEARCH
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146 147
151 152 153 153 157 162 168 169 171 174 175 176 176 177 178 178 180 184 189 199 202 206 209 212 212 215
217
X. REFERENCES
228
XI. APPENDICES
236
APPENDIX A : MATLAB PROGRAMS FOR THE “MUNSELL” LIBRARY APPENDIX B : MATLAB PROGRAMS FOR THE COLORANT ESTIMATION SUBSYSTEM APPENDIX C : MATLAB PROGRAMS FOR THE INK SELECTION SUBSYSTEM APPENDIX D : MATLAB PROGRAMS FOR THE INK OVERPRINT PREDICTION SUBSYSTEM APPENDIX E : MATLAB PROGRAMS FOR THE PROPOSED SIX-COLOR FORWARD PRINTING MODEL APPENDIX F : MATLAB PROGRAMS FOR THE SPECTRALBASED SIX-COLOR SEPARATION MINIMIZNG METAMERISM APPENDIX G : THE SIX ESTIMATED COLORANTS FOR THE 105 MIXTURES OF THE POSTER COLORS APPENDIX H : THE ESTIMATED SPECTRAL ABSORPTION AND SCATTERING COEFFICIENTS FOR THE WATERPROOF® CMYRGB PRIMARIES APPENDIX I : THE REFLECTANCE SPECTRA OF THE ORIGINAL GRETAG MACBETH COLOR CHECKER APPENDIX J : THE REFLECTANCE SPECTRA OF THE PREDICTED GRETAG MACBETH COLOR CHECKER BY THE PROPOSED SIX-COLOR SEPARATION ALGORITHM APPENDIX K : THE REFLECTANCE SPECTRA OF THE REPRODUCED GRETAG MACBETH COLOR CHECKER USING DUPONT WATERPROOF® SYSTEM APPENDIX L : THE REFLECTANCE SPECTRA OF THE PREDICTED GRETAG MACBETH COLOR CHECKER USING FUJIX PICTROGRAPH 3000 APPENDIX M : THE REFLECTANCE SPECTRA OF THE PREDICTED GRETAG MACBETH COLOR CHECKER USING KODAK PROFESSIONAL 8670 PS THERMAL PRINTER
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237 248 251 254 271
285 294
295 297
301
305
309
313
APPENDIX N : THE REFLECTANCE SPECTRA OF THE PREDICTED GRETAG MACBETH COLOR CHECKER USING DUPONT WATERPROOF® WITH CMYK PRIMARIES APPENDIX O : THE ACCURACY OF THE GRETAG SPECTROLINO SPECTROPHOTOMETER
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LIST OF TABELS Table 1-1: The number of overprints of a k-primary halftone printing process. Table 2-1: A sample set (shown up to five samples) of absorption measurements from 400 nm to 700 nm at 20 nm intervals. Table 2-2: The principal components as the new coordinates in the eigenvector coordinate system of the samples in Table 2-1. Table 3-1: The colorimetric and spectral accuracy of the multivariate normal and non-normal sample sets reconstructed with different number of dimensions, where Stdev stands for the standard deviation and RMS representing the totalroot mean square error of the reconstructed reflectance spectra. Table 3-2: The percent variance by eigenvector analysis in both reflectance and absorption space of IT8.7/2 reflection target. Table 3-3: The colorimetric and spectral performance of three eigenvector reconstruction in both spaces for an IT8.7/2 reflection target. Table 3-4: The statistical performance by six-eigenvector reconstruction for 141 acrylic colors in R, Ψ and Φ spaces. Table 3-5: The statistical performance by six-eigenvector reconstruction for 105 poster colors in R, Ψ ,and Φ spaces. Table 4-1: The spectral and colorimetric accuracy of the three-eigenvector reconstruction for the Kodak Q60C. Table 4-2: The spectral and colorimetric accuracy of the six-eigenvector reconstruction for the still life painting. Table 4-3: The spectral and colorimetric accuracy of the six estimated colorants for the still life painting. Table 4-4: The colorimetric and spectral accuracy of the six eigenvector reconstruction for the 105 mixtures. Table 5-1: The colorimetric and spectral accuracy for the four SWOP primaries synthesizing the 928 samples of IT8/7.3 target in the proposed linear colorant mixing space. Table 5-2: The correation coefficeints and the chroma of the 18 inks with the five chromatic statistical primaries. Table 5-3: The spectral and colorimetric accuracy of the three optimal ink sets.
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16
41 42
77 78 80 81 82
95 98 100 105
118 122 123
Table 5-4: The Pantone color names of the three optimal ink sets. Table 5-5: The spectral and colorimetric accuracy of the three worst performing ink sets. Table 5-6: The ink combinations of the three worst performing ink sets.
125 125
Table 6-1: The colorimetric accuracy, spectral accuracy, and the statistical thickness for the six primaries. Table 6-2: The colorimetric and spectral accuracy of the 25 overprints.
140 142
Table 7-1: The effective dot areas and the correction scalar, q, of cyan fixed at 50% theoretical dot area by overlapping the secondary magenta ink at various theoretical dot areas (Iino and Berns, 1998). Table 7-2: The determined q scalars by proposed modification and the dot-gain of the primary (magenta) given that the secondary (cyan) is present. Table 7-3: The colorimetric and spectral accuracy of n = 2.2 in predicting all 72 samples of the six primary ramps where Stdev stands for the standard deviation and RMS represents the root-mean-square error in unit of reflectance factor. Table 7-4: The colorimetric and spectral accuracy of the first order forward model in predicting the 6,250 samples of the verification target. Table 7-5: The colorimetric and spectral accuracy in predicting the verification of 6,250 sample by the algorithms suggested by Iino and Berns. Table 7-6: The theoretical dot areas and their correction scalar by Eq. (7-14) without logical correction. Table 7-7: The theoretical dot areas and their correction scalar by Eq. (7-14) with logical correction. Table 7-8: The colorimeric and spectral accuracy of the proposed algorithms in predicting the verification target of 6,250 sample mixtures. Table 7-9: The colorimeric and spectral accuracy of the proposed algorithms modified by Eq. (7-20) in predicting the verification target of 6,250 sample mixtures. Table 7-10: The colorimetric and spectral performance of the verification target predicted by the matrix method. Table 7-11: The predicted theoretical dot areas, colorimetric and spectral errors of the 24 colors in Macbeth color checker where M.I. represents the metamerism index. Table 7-12: The statistical results in predicting the 24 colors using the proposed six-color printing model.
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124
161
164
179 181 186 190 192 193
196 201
203 204
Table 7- 13: The colorimetric and spectral performance of the 22 in gamut colors. Table 7-14: The color difference of the predicted Macbeth Color Checker under illuminant A and F2 by four different printing processes, where the color names corresponding to the bold faced entries are the out of colorimetric gamut colors of each device. Table 8-1: The colorimetric and spectral accuracy of original vs. reproduction and prediction vs. reproduction for the Gretag Macbeth Color Checker. Table 8-2: The statistical colorimetric and spectral accuracy corresponding to the predicted and reproduced Gretag Macbeth Color Checker.
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213 214
LIST OF FIGURES Fig. 1-1: An application of paint catalog sales by the conventional four-color printing. Notice that the paint chip approximately matches the printed paint chip under daylight illuminant and mismatches the printed paint chip under the Incandescent illuminant. Fig. 1-2: The color gamuts projected on the xy chromaticity plane for Pantone Hexachrome, conventional four-color printing process, and CRT. Fig. 1-3: The outline of a multi-spectral color reproduction system. Fig. 1-4: The structure chart of the research development for spectral printing. Fig. 1-5: The structure diagram of colorant estimation. Fig. 1-6: The optimal ink selection scheme for low-error spectral reproduction. Fig. 1-7: The microscopic structure of halftone color formulation of a CMY printing device, where Rλ, color is the spectral reflectance factor of a color appearing in the figure. Fig. 1-8: The structure of spectral-based six-color printing process where YNSN stands for Yule-Nielsen modified spectral Neugebauer equation. Fig. 2-1: The three dimensional vector, P3, in ℜ3. Fig. 2-2: Spectral reflectance curve of the example. Fig. 2-3: Example for a set of three dimensional vectors localized in various vector subspaces. Fig. 2-4: Scatter plot of a two dimensional sample set. Fig. 2-5: The representation of sample set of Figure 4 in the eigenvector coordinate system. Fig. 2-6: The mean (thick line), the first eigenvector (thin line), and the second eigenvector (dashed line) of the 622 daylight measurements (Judd, MacAdam, and Wyszecki, 1964). Fig. 2-7: Spectral distribution of typical daylight at correlated color temperatures 4800 K, 5500 K, 6500 K, 7500 K, and 10,000 K (Judd, MacAdam, and Wyszecki, 1964). Fig. 2-8: Three sets of absorption spectra of cyan, magenta, and yellow dyes (left to right) at eleven different concentrations. Fig. 2-9: The first six eigenvectors obtained for the virtual sample set. (As many as thirty-one eigenvectors can be shown.) Fig. 2-10: The six eigenvectors obtained from an IT8.7/2 reflection target. Fig. 2-11: Scatter plot of the 500 random samples and their eigenvectors from a uniform distribution.
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5 6 8 10 12 13
14 19 29 31 33 35 38
45
46 47 48 50 52
Fig. 2-12: Normal probability plot of absorption coefficients at 500 nm from the virtual normal population obtained by linear combinations of three linearly independent dyes. Fig. 2-13: Normal probability plot of absorption coefficients at 500 nm of the exponential population obtained by linear combinations of three linearly independent dyes. Fig. 2-14: The two-dimensional scatter plot of absorption coefficients at 550 nm vs. 500 nm from the virtual normal population obtained by linear combinations of three independent dyes. Fig. 2-15: The outline of a 2x2 halftone cell. Fig. 2-16: The 3x3 halftone cells covered by three different dot area coverage. Fig. 2-17: Non-ideal halftone dot shapes vary due to the mechanical dot gain effect. Fig. 2-18: The cause of optical dot gain. Fig. 3-1: The possible field of view of a spectrophotometer. Fig. 3-2: The normal plot of simulated reflectance factors at each sample wavelength obtained by linear combinations of six approximately normal distributions. Fig. 3-3:The normal plot of simulated reflectance factors at each sample wavelength generated by linear combinations of six non-normal distributions. Fig. 3-4: The normal plot of marginal distributions of IT8.7/2 reflection target in reflectance space. Fig. 3-5: The Kubelka-Munk inverse transformation, Eq. (3-3), for opaque color. Fig. 3-6: An example of enhanced spectral error after the transformation by Eq. (3-3). Fig. 3-7: The transformation from Ψ to R space by Eq. (3-7). Fig. 4-1: The still life painting of a floral arrangement creating with six opaque colorants. Fig. 4-2: The six eigenvectors obtained from the still life painting. Fig. 4-3: The six acrylic paints used for the computer generated sample population. Fig. 4-4: The all-positive eigenvectors as the estimated dye spectra (thick line) and the local eigenvectors (dotted lines). Fig. 4-5: The six all-positive eigenvectors as the estimated colorants for the still life painting. Fig. 4-6: The estimated colorants (solid lines) and the original colorants (astroidal lines) used for the still life painting.
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55
55
57 58 60 61 62 71
75
76 79 83 84 84
87 89 93 96 99 100
Fig. 4-7: The 105 mixtures created by hand mixing six opaque poster paints. Fig. 4-8: The spectral reflectance factors of the six poster colors used for creating the 105 opaque mixtures. v Fig. 4-9: The offset vector, a , for transforming the 105 mixtures to Ψ space. Fig. 4-10: The six eigenvectors of the 105 mixtures in Ψ space. Fig. 4-11: The reflectance factors of the six original colorants (solid lines) and statistical primaries (dashed lines) derived by a constrained rotation from the six eigenvectors. Fig. 5-1: The SWOP specified process CMYK primaries and paper substrate. Fig. 5-2: The colorimetric and spectral error vs. w values for the empirical transformation, Eq. (5-2), where M.I. represents metamerism index. Fig. 5-3: The four spectra reconstructed with the highest colorimetric and spectral errors based on the linear colorant mixing space where solid line is the measured spectrum and the dashed line is the reconstructed spectrum . Fig. 5-4: The six statistical primaries derived from the 105 mixtures by the colorant estimation module. Fig. 5-5: The Pantone 14 basic colors and the process CMYK as the ink database (Color name order corresponding to each spectrum is from left to right and top to bottom). Fig. 5-6: The three spectral reconstructions of a sample corresponding to the maximum prediction error by the three optimal ink sets. Fig. 5-7: The three spectral predictions of the sample, shown in Fig. 5-6, by the three worst performed ink sets. Fig. 5-8: The two reconstructed spectra by set 23 and set 26 for the sample used as the example in Fig. 5-6. Fig. 6-1: The microscopic structure of color formation by a halftone printing process where Rλ,color represents the spectral reflectance factor of a color appearing in the square area. Fig. 6-2: An ink film applied on black and white contrast paper. Fig. 6-3: The diagram of a three-ink-layer overprint. Fig. 6-4: The six primaries printed on contrast paper. Fig. 6-5: Twenty-five overprints printed on coated paper. Fig. 6-6: The spectral absorption (solid line) and scattering (dashed line multiplied by ten times) curves of the six primaries. Fig. 6-7: The difference spectra between measured and predicted primaries. Fig. 6-8: Histogram of the metamerism indices for prediction of the 25 overprints.
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103 103 104 105
106 117 117
119 120
121 125 126 127
129 133 136 138 138 139 140 142
Fig. 6-9: The difference spectra of four overprints best predicted with high accuracy. Fig. 6-10: The difference spectra of four overprints predicted with relatively low accuracy. Fig. 6-11: The estimated two optical constants for cyan ink and corresponding difference spectrum in units of reflectance factor. Fig. 7-1: The structure chart for the development of six-color separation minimizing metamerism. Fig. 7-2: The structure of a general forward halftone printing model where f( ) is a mathematical function or LUT describing the dot-gain effect and Nλ,n( ) is the function of Yule-Nielsen modified spectral Neugebauer equation. Fig. 7-3: The algorithm structure of determining the Yule-Nielsen n-factor where INV(YNMD) stands for the inverse function, Eq. (7-3), of n-factor corrected spectral Murray-Davies equation. Fig. 7-4: A set of theoretical to effective dot area transfer functions determined from a CMYK halftone printing process. Fig. 7-5: The dot-gain functions of the CMYK ramps given in Fig. 7-4. Fig. 7-6: The family of dot-gain curves of a cyan ramp when magenta ink presents at 0%, 25%, 50%, and 75% fractional dot areas. Fig. 7-7: The dot-gain loci of a cyan ramp when a magenta ink is present at different theoretical dot areas where the locus goes through a3, b3, and c3 are the dot gains esimated by the first-order model, Iino and Berns' algoritms, and the proposed algorithms, respectively. Fig. 7- 8: The functions of dot gain corrrection scalar by Iino and Berns (left) and the proposed (right) algorithms. Fig. 7-9: The structure of the six-color backward spectral printing model using cyan, magenta, yellow, green, orange and black ink as printing primaries. Fig. 7-10: The reflectance spectra of the printed six primaries and substrate. Fig. 7-11: The mean prediction accuracy of all 72 samples of the six primary ramps as function of n-factor. Fig. 7-12: The measured and the predicted reflectance spectra by Eqs. (7-3) and (2-33) using n = 2.2 where the solid lines are measured spectra and the dashed lines are the predicted spectra. Fig. 7-13: The theoretical to effective transfer functions of the six primaries. Fig. 7-14: Histogram of the colorimetric performance using the first-order forward printing model in predicting the 6,250 samples of the verification target.
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Fig. 7-15: The vector plot of the 100 predicted samples with the highest colorimeric errors by the first-order forward model. The vector tail represents the measured coordinate and the vector head represents the prediction. Fig. 7-16: Four example spectra showing under prediction by the first order model. Fig. 7-17: The three example functions of dot-gain correction scalar determined for CMYK sub-model based on Iino and Berns' algorithms. Fig. 7-18: Histogram of the colorimetric error in units of ∆E*94 for the 6,250 samples predicted by Iino and Berns' algorithms. Fig. 7-19: The vector plot of L* vs. a* for the 100 samples used as examples in Fig. 7-15 predicted by Iino and Berns' algorithms. Fig. 7-20: The spectral predictions of the four example samples used as examples in Fig. 7-16 by the Iino and Berns' algorithms. Fig. 7-21: The three example functions of dot-gain correction scalar determined for CMYK sub-model based on proposed algorithms. Fig. 7-22: Histogram of the colorimeric accuracy of the proposed algorithms in predicting the verification target of 6,250 sample mixtures. Fig. 7-23: The vector plot of L* vs. a* for the 100 samples used as examples in Fig. 7-15 predicted by the proposed algorithms. Fig. 7-24: The spectral prediction of the four samples used as examples in Fig. 7-16 by the proposed algorithms. Fig. 7-25: Histogram of the colorimeric accuracy of the proposed algorithms modified by Eq. (7-20) in predicting the verification target of 6,250 sample mixtures. Fig. 7-26: The vector plot of L* vs. a*, L* vs. b*, and b* vs a* for the 300 samples whose colorimetric error is predicted higher than 2.41 units of ∆E*94. Fig. 7-27: The spectral prediction by the proposed algorithms modified by Eq. (7-20) for the four samples used as examples in Fig. 7-16. Fig. 7-28: Histogram of the colorimeric accuracy of the matrix method in predicting the verification target of 6,250 sample mixtures. Fig. 7-29: The four best predicted spectra in terms of metamerism index of the 24 colors. Fig. 7-30: The four worst predicted spectra in terms of metamerism index of the 24 colors. Fig. 8-1: The original, predicted, and reproduced spectral reflectance factors of the six Gretag Macbeth Colors.
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I. INTRODUCTION Conventional graphic reproduction at the analysis stage, in general, uses broad band RGB filters to determine color densities (Dr, Dg, and Db) of an original. It then converts the measured color densities to effective halftone fractional dot areas corresponding to process cyan, magenta, yellow and black (CMYK) inks. This is done by using empirically determined ink tables constructed by exhaustive sampling and measuring prints of a particular printing process (Pobboravsky and Pearson, 1972; Viggiano, 1985). High-end color scanners for the printing industry were initially invented by Hardy and Wurzburg in 1948, known as the Hardy and Wurzburg scanner, and by Murray and Morse in 1941 known as the P.D.I. scanner. The Hardy and Wurzburg scanner was designed with spectral sensitivities equal to the CIE color matching functions, x (λ ), y(λ ), and z (λ ), or linear transformations of them, in order to record tristimulus values of originals and evaluate the amount of CMY inks needed to be delivered onto paper substrate through electronic computing circuits (Hardy and Wurzburg, 1948; Hardy and Dench, 1948). Whereas, the P.D.I. scanner was devised based on the masking theory instead of the Neugebauer theory (Murray and Morse, 1941; Hunt, 1995). The later modified versions such as the Crossfield Diascan, Hell Chromagraph, and Linotype-Paul Linoscan, are generally designed with three narrow-band spectral sensitivities centered around short (450 nm), medium (550 nm), and long (650 nm)
1
wavelength regions throughout the visible spectrum to evaluate the color densities (Dr, Dg, and Db) or CMYK concentrations of an original. These scanners function more closely to densitometers instead of colorimeters since their spectral sensitivities are not linearly related to CIE color matching functions. High-end scanners used in the printing industry, which evaluate color densities (Dr, Dg, and Db) or CMYK concentrations of originals, do not unambiguously record the color information of originals. As a consequence, only a metameric reproduction can be achieved by a printing process if colorants used in reproduction are different from that of original (Berns and Shyu, 1995).
Thus, the
outcome of the pre-press color acquisition together with the four process colors by the conventional printing is intrinsically metameric. Although metameric reproduction by CMYK color reproduction of conventional printing technology accomplishes pleasing results, the color mismatch under illuminants other than that standardized by the printing industry is always problematic for critical color-matching applications such as catalog sales, art-work reproduction, and computeraided design. Nowadays, the demanded quality of color reproduction is skyrocketing. Consumers are more willing to invest extra capital in pursuing high fidelity color reproduction for its appealing results with respect to printing technology. Generally, the terms Hi-Fi, multispectral, and multiple-ink are synonyms. The so called “high fidelity” color reproduction utilizes more than four primary inks to expand device gamuts (Carli
2
and Davis, 1991). It is believed that a larger device gamut has a better chance to match colors from the real world because, theoretically, a system can exactly reproduce the colorimetric values that fall within its color gamut. Previous research by Ostromoukohv suggested a way to enhance chromaticity gamut via heptatone multicolor printing process (Ostromoukhov, 1993).
He employed the Neugebauer equation together with seven
printing primaries to match the colorimetric
information (tristimulus values) of an
original. This approach is a prevalent scheme for state-of-the-art Hi-Fi color systems. However, the colorimetric matching reproduction still suffers from both illuminant and observer metamerism (Grum and Bartleson, 1980). Kohler and Berns pointed out that illuminant metamerism could be reduced by using five or more colored inks based on spectral color reproduction (Kohler and Berns, 1993). The primary goal of this doctoral research is to devise color separation algorithms for multiple-ink printing systems that are capable of reconstructing the spectral information from original objects such that the metamerism is minimized between original objects and their color reproductions.
A. OVERALL PROBLEM
One manifestation of metamerism occurs when two surfaces, each produced with different colorants as judged by their distinct spectral reflectance factors integrate to the same color perception under a given lighting condition and to a different color perception under other illuminants with respect to the human visual system. Metamerism can be used
3
as an advantage for color devices such as CRTs and printers to reproduce trichromatic matches between originals and reproductions under a predefined viewing condition. That is, metamerism enables CRTs and printers to accomplish colorimetric synthesis in a relatively inexpensive fashion since the color match is defined as the identical colorimetric values between originals and their reproductions. However, all sorts of materials are not necessarily made from phosphors and inks. The color synthesis by phosphors and inks do not render color unambiguously when only a metameric color match can be achieved. Ambiguity means that the color match is only defined for a standard viewing environment. Color mismatch, due to uncontrollable lighting conditions, affects the accuracy of color communication (Berns, 1998). Current four-color printing systems are limited to metameric reproduction due to an insufficient number of degrees of freedom. This means that the possible spectral variations from originals can not be synthesized by only four inks given that the originals are likely constituted by more than four distinct colorants. Figure 1-1 exemplifies a catalog sales for a paint retail store. Paint samples were printed on the catalog to convey the chromatic information and try to attract the preference of consumers. A paint chip is placed underneath the catalog to represent the paint sold in a paint store. Standard daylight is assumed as the illumination for paint stores and incandescent illuminant is assumed to be the typical indoor illumination used by general consumers. Once a consumer matches his or her color preference under home illumination from the catalog
4
and purchases the paint based on a visual match by the catalog under the illumination in a paint store, it is possible that the consumer will return the paint after he or she realizes there are mismatches for his or her applications indicating by that the paint matches the printed sample under daylight illuminant and mismatches under the incandescent illuminant. This is a typical problem of metamerism. The paint industry can not just sell paint by catalog because of metameric catalog reproduction.
Daylight illuminant
Incandescent illuminant
Fig. 1-1: An application of paint catalog sales by the conventional four-color printing. Notice that the paint chip approximately matches the printed paint chip under daylight illuminant and mismatches the printed paint chip under the Incandescent illuminant.
Since the insufficient number of degrees of freedom lead to metameric reproduction, the intuitive action for increasing degrees of freedom by adopting two or more inks should alleviate metamerism. Although the current multiple-ink printing
5
systems, such as Pantone Hexachrome, Küppers 7-Color, and DuPont Hyper Color, employ six or more inks for high fidelity color reproduction, their color separation algorithms are still based on trichromatic color reproduction. The increased degrees of freedom are used for expanding device color gamut (Herbert, 1993; Boll, 1994; Stollnitz, Ostromoukhov, and Salesin, 1998; Viggiano, 1998). The problem of metamerism is not alleviated by these systems. The advantage of more number of degrees of freedom is not fully exploited. Figure 1-2 demonstrates the color gamuts, plotted in terms of xy chromaticities, of Hexachrome, RGB monitior, and four-color printing process, respectively.
Hexachrome RGB Monitor
y
Process 4C
x
Fig. 1-2: The color gamuts projected on the xy chromaticity plane for Pantone Hexachrome, conventional four-color printing process, and CRT.
6
Although the comparison should be made by evaluating the three gamut volumes, the projection of each device gamut onto the xy chromaticity plane is shown for that the Hexachrome system has a larger gamut projection relative to RGB monitor and conventional four-color printing.
B. OVERVIEW OF SOLUTION
In order to reduce the degree of metamerism, a spectral approach, as opposed to a trichromatic approach, is the answer for critical color matching applications. The goal of spectral reproduction is to reproduce the best spectral match whose color-match remains constant under most illuminants within a visual tolerance. This ensures that the colors of merchandise reproduced in a catalog, or the colors of an original painting reproduced in an art book, will unambiguously be rendered under various illuminating and viewing conditions. A research and development program has been initiated at the Munsell Color Science Laboratory in Rochester Institute of Technology for developing a spectral-based color reproduction system (Berns, Imai, Burns, and Tzeng, 1998). Research has included multi-spectral acquisition systems (Burns and Berns, 1996; Burns, 1997; Burns and Berns, 1997a; Burns and Berns, 1997b) and spectral-based printing algorithms (Iino and Berns, 1997; Iino and Berns, 1998a; Iino and Berns, 1998b). The current research effort is devoted for the development of the spectral output system which is comprised of an ink-
7
selection module, and the implementation of spectral-based printing algorithms for a multiple-ink printing device, shown as Fig. 1-3.
Multi-Spectral Acquisition
Spectal-Based Printing Fig. 1-3: The outline of a multi-spectral color reproduction system. Since the limitation of four-color reproduction for spectral reconstruction is due to insufficient number of degrees of freedom for adjusting the reproduced spectra, printers need extra primary inks to shape the region of the spectrum that is beyond the limitations of four-color processes. Intuitively, the extra inks extend the capability and accuracy for spectral reproduction, when chosen properly. A properly selected ink set implies that the adoption of a fixed ink set by previously mentioned multiple-ink systems are not sufficiently equipped for spectral reproduction since the spectral variations may be beyond the synthesis capability of those ink sets. In another words, original objects are unlikely
8
created by those ink sets. Thus, the overall solution for multiple-ink printing devices focus on minimizing metamerism by developing spectral-based color separation algorithms which employ a greater number of degrees of freedom as well as the dynamic ink selection to overcome the shortcoming of current multiple-ink systems. It is expected that the greater number of degrees of freedom together with the flexibility of dynamic ink selection can well describe the spectral variations from an arbitrary original object.
C. SCOPE OF DISERTATION
The scope of this dissertation was to develop a spectral-based color separation algorithm for a multiple-ink printing process and to derive a spectral reconstruction scheme based on the spectral Neugebauer equation at the synthesis stage. The current research aims to identify a set of six inks for spectral halftone printing, corresponding to the fact that six-color capabilities are widely available around the world. According to NAFTA statistics, six-ink printing capabilities are available at over seven thousand sites in North America alone. The choice of the number six is based on this production convenience. The research development of spectral six-color printing is divided into modules of colorant estimation (Tzeng and Berns, 1998), optimal ink selection (Tzeng and Berns, 1999a), prediction of ink overprint (Tzeng and Berns, 1999b), spectral-based color separation minimizing metamerism, and direct digital or conventional printing. Structure
9
of the proposed research development is depicted in Fig. 1-4 by assuming that the input of this spectral printing system is a spectral image acquired by a multi-spectral acquisition device.
Ink Selection Algorithm
Colorant Estimation
Spectral Image
Multiple-Ink Direct Digital or Conventional Printing
Ink Overprint Prediction Spectral-Based Color Separation Minimizing Metamerism
Fig. 1-4: The structure chart of the research development for spectral printing. Function of each module is designated as follows: 1. Colorant Estimation To minimize metamerism between spectral inputs and their corresponding reproductions, it is desirable to understand the spectral properties of the colorants possibly utilized for creating the original objects. Then the use of a new set of colorants with similar or identical spectral properties for synthesis should achieve the minimum or zero spectral error. Since, in practice, it is not likely that the colorant information for original
10
objects is available from their creators, the estimation has to resort to statistical approaches. Any set of spectral samples can be captured by a multi-spectral acquisition device or measured via a spectrophotometer or spectroradiometer with finite bandwidth across the visible spectrum. Then the set of spectral samples is distributed in a multidimensional vector space. Principal component analysis (PCA) can provide a gauge to statistically decompose these spectral samples into fewer statistical dimensions. These dimensions dominate the major sample variations of these spectral measurements. In another words, majority of samples vary along the directions of these statistical dimensions.
It can be viewed as if there existed a set of colorants whose spectral
properties coincide with the directions of the major sample variation, therefore, the whole sample variations are the exact combinations of the set of imaginary colorants, which are the eigenvectors derived by PCA. Since the eigenvectors are the only link to the physical colorants which can be used for synthesis, a transformation to a set of all-positive vector representations as the estimated primary colorants is necessary to account for the allpositive spectral properties of real materials. Thus, at the analysis stage, this research will perform statistical analysis via PCA on original images, which are the spectral inputs prior to color separation, followed by a constrained transformation to statistically estimate the possible primaries that can be used for low-error spectral synthesis. That is, the colorant estimation procedure statistically estimates a set of possible primary colorants, whose linear mixtures are the closest
11
approximation for each pixel of spectral inputs. Ideally, if the estimated primary colorants exist in a current ink database, then the use of the exact ink set for halftone printing will yield ideal spectral reproduction. A detailed chart of this module is shown in Fig. 1-5.
2.5
Kubelka-Munk or empirical transformation into a linear color mixing representation
2
K/S
1.5
1
Sampled wavelength
0.5
0 400
450
500
550 Wavele ngth
600
650
700
Principal component analysis
Spectral image The six rotated normalized all-positive eigenvectors
1
1
1
0.5
0.5
0.5
0
0
0
-0.5
-0.5
-0.5
-1 400
500
600
700
-1 400
500
600
700
-1 400
The 5th eigenvector.
1
Constrained transformation 500
600
700
The 6th eigenvector.
1
1
1
0.5
0.5
0.5
0.8
0.6
0.4
0.2
500 600 Wavelength
700
500 600 Wavelength
700
700
700
670
640
0 500 600 Wavelength
610
-1 400
580
-1 400
550
-1 400
520
0 -0.5
490
0 -0.5
460
0 -0.5
430
K/S
The 4th eigenvector.
1.2
The 3rd eigenvector.
(Absorption / Scattering)
The 2nd eigenvector.
400
K/S
The 1st eigenvector.
Wa velength
Six eigenvectors
Statistical primaries
Fig. 1-5: The structure diagram of colorant estimation. 2. Optimal Ink Selection It is not likely to have an ink set with identical spectral properties to the estimated primaries exist in the ink data base. The optimal ink set which minimizes the spectral error between original objects and reproductions could be manufactured by colorant chemists to obtain the desired spectral characteristic. However, the optimal ink set is image
12
dependent. Frequently, it is not time and cost efficient to manufacture an optimal ink set for industrial applications whenever spectral color reproduction is encountered. It is more practical to select an optimal ink set from a large ink database, currently in manufacture. Hence, devising an optimal ink selection method in order to choose the corresponding inks for the best spectral reconstruction is necessary. An optimal ink selection scheme, shown as Fig. 1-6, will be devised based on vector correlation analysis by searching through the existing ink database such as Pantone formula colors to identify the most similar ink set. Th e six ro t a te d no r m ali z ed all - po si tiv e ei gen v e c to rs 1.2
(Absorption / Scattering)
1
Ink selection by vector correlation analysis
0.8 0.6 0.4 0.2
700
670
640
610
580
550
520
490
460
430
400
0 W a ve l e ngth
Statistical primaries
1 0.9 Reflector factor
0.8 0.7
Rhodamine red
0.6
Purple
0.5
Reflex blue
0.4
process black
0.3
process yellow
0.2
process cyan
0.1 700
670
640
610
580
550
520
490
460
430
400
0
Pantone colors
Wavelength
Optimal ink-set
Fig. 1-6: The optimal ink selection scheme for low-error spectral reproduction. 3. Ink Overprint Prediction At the synthesis stage of this research, the chosen optimal ink set will be used as the printing primaries. The spectral expansion of the Neugebauer equation together with
13
the Yule-Nielsen n-factor correction will be designated as the halftone printer model. This model is capable of reconstructing the spectra of an original with high accuracy (Pobboravsky and Pearson, 1972; Rolleston and Balasubramanian, 1993). Consider the color formulation depicted as Fig. 1-7 by a CMY halftone printing device, the overall color stimulus over a printed spot is contributed by the colors appearing in that spot.
Rλ,Y
Rλ,R
Rλ,M
Rλ,G
Rλ,C
Rλ,B
Rλ,White
Rλ,CMY
Fig. 1-7: The microscopic structure of halftone color formulation of a CMY printing device, where Rλ, color is the spectral reflectance factor of a color appearing in the figure. Assuming the spot is circumscribed by the thick line square in Fig. 1-7, then the overall color stimulus over the thick line square in terms of spectral reflectance factor is the linear sum of the spectral reflectance factor of each color inside the square. Apparently, the corresponding modulation is proportional to the percentage of the occurrence of each color inside the square. This percentage of occurrence is quantified by
14
the fractional dot area of each color. These colors inside the square are not just printing primary colors (cyan, magenta, and yellow) but also overprints (red, green, blue, and three-color black). The linear sum is exactly the three-color spectral Neugebauer equation, shown as Eq. (1-1) and its Yule-Nielsen n-factor (Yule and Nielsen, 1951) corrected spectral Neugebauer equation, shown as Eq. (1-2). Rλ = aCRλ,C + aMRλ,M + aYRλ,Y + aRRλ,R + aGRλ,G + aBRλ,B + aCMYRλ,CMY
,
(1-1)
+ (1- aC - aM - aY - aR - aG - aB - aCMY)Rλ,white, Rλ = [aCRλ,C1/n + aMRλ,M1/n + aYRλ,Y1/n + aRRλ,R1/n + aGRλ,G1/n + aBRλ,B1/n + aCMYRλ,CMY1/n ,
(1-2)
+ (1- aC - aM - aY - aR - aG - aB - aCMY)Rλ,white1/n]n, where λ is wavelength, R represents reflectance factor, a is fractional dot area, and the capitalized subscripts represent the color names of primaries, secondary, and tertiary primaries. These primaries are also named the Neugebauer primaries. To utilize the original or n-factor modified spectral Neugebauer equation, it is necessary to posses the knowledge for the spectra of the Neugebauer primaries, i.e., the spectra of secondary, tertiary, quaternary primaries (for four-color halftone printing), and so forth. The spectral reflectance factors of overprints are usually obtained by printing and measuring. Recognizing that with a change of materials (i.e., inks and paper), the spectral reflectance factor on top of the printed samples includes a change in overprints. In order
15
to characterize a halftone printing process, the ramps for primaries and overprints need to be printed and measured when new materials are applied. There are 2 k-k-1 overprints for a k-color printing process. The effort required for printing and measuring of ramps may not be significant when a four-color printing process is encountered since the number of over printings is only eleven. Table 1-1 lists the number of overprints corresponding to three to eight-color printing processes. It shows that as the number of primaries increases linearly, the number of corresponding overprints increases exponentially. The use of analytical prediction of overprints can avoid the necessity of printing and measuring upon changing the ink set selected by the ink-selection procedure. The computation for the entire integrated multiple-ink printing system can be automated without interruption due to the change of ink set and paper. A reasonable prediction process for overprints is to use the Kubelka-Munk turbid media theory for translucent materials to predict the spectra of ink overprints. Once all the spectral information of Neugebauer primaries is compiled, the spectral estimation is enabled by the use of the Yule-Nielsen modified spectral Neugebauer equation. Table 1-1: The number of overprints of a k-primary halftone printing process. Number of primaries 3 4 5 6 7 8
Number of overprints 4 11 26 57 120 247
16
4. Spectral-Based Six-Color Separation Minimizing Metamerism When printing solid inks on top of each other, the ability to print the later wet ink on top of formerly printed wet ink is call ink trapping. Ink trapping failure is caused by the wetness and total amount of ink piling on a printed spot. The SWOP standard constrains the ink trapping limit in terms of total fraction dot area to be 300%. Several printing applications extending this limitation to 340% to 400% depending on the physical and chemical design of the ink materials. Based on this physical limitation of ink trapping, it is impossible to overprint six solid primaries in one location. Hence, the ink limiting is the primary concern of the design of the six color printing process. In addition, consider an input color with an arbitrary spectrum, if the subtractive synthesis requires more than four independent colors then this particular color is of low spectral reflectance factor. Hence, it is able to be approximated by a black and three chromatic inks for its spectral synthesis. Therefore, the ink limiting predefining the six-color printing model is based on ten fourcolor printing sub-models since choosing one black ink and three chromatic ink out of remaining five from the optimal ink set is ten (C(5, 3)). Often, given a color with a known spectrum, the real world problem is to determine the amount of ink, or the percentage of dot areas, which needs to be delivered onto a paper support.
The Yule-Nielsen modified spectral Neugebauer equation is
required to be inverted to solve for the effective dot areas for a known color. Since the analytical inversion of Neugebauer equation is impossible to accomplish, it must be
17
approximated by a numerical approach (Mahy and Delabastita, 1996). For this research development, MATLAB and its optimization toolbox are utilized for inverting the YuleNielsen modified spectral Neugebauer equation (MATLAB, 1996). With the derived effective fractional dot areas, spectral reconstruction of an original will be performed using the forward Yule-Nielsen modified Neugebauer equation at the synthesis stage. The performance of this process will be evaluated in terms of colorimetric and spectral accuracy.
If the minimized spectral error still reveals low
colorimetric accuracy then the post-process of slightly trading in spectral accuracy to exchange for the high colorimetric accuracy under the standard illuminant D50 will be adopted. This will result in the closest possible match across multiple light sources. Hence, the degree of metamerism is reduced. 5. Multiple-Ink Direct Printing The six-color separations are output by a multiple-color proofing system with stochastic screening at 175 LPI (or equivalent) screen frequency to avoid the Moiré pattern. Figure 1-8 shows the processes of color separation minimizing metamerism and multiple-ink direct printing. Accomplishment of the processes described above completes the research of the spectral-based color separation algorithms development for multipleink color reproduction. Detailed technical contents are described in the later chapters.
18
Original painting
Color separation by YNSN
Direct printing or proofing
Spectral reproduction
Fig. 1-8: The structure of spectral-based six-color printing process where YNSN stands for Yule-Nielsen modified spectral Neugebauer equation.
19
II. BACKGROUND Current research developments involve colorant estimation, ink selection, ink overprint prediction, and spectral-based printing minimizing metamerism. The theoretical underpinnings include multiple-ink printing systems, Kubelka-Munk turbid media theory, computer colorant matching, multivariate statistics including principal component analysis, and the Neugebauer theory for modeling the halftone printing process. This chapter provides a review for these topics.
A. MULTIPLE-INK COLOR SYSTEMS
The concept of utilizing multiple inks for color reproduction is not new (Leekley, Cox, and Gordon, 1953). It originated during the 19th century for color enhancement (Friedman, 1978). State-of-the-art multiple-ink color reproduction systems such as Hi-Fi color systems utilize more colors than that of conventional CMYK color reproduction processes. The most well known Hi-Fi color systems are DuPont Hyper Color, Küppers 7Color (Küppers, 1986), K&E (BASF) 7-Color, Fogra 7-Color, and Pantone Hexachrome (Di Bernardo and Matarazzo, 1995; Herbert and Di Bernardo, 1998). DuPont Hyper Color applies the process CMY inks with two different density levels and one process black for color reproduction. Pantone Hexachrome was designed with fluorescent mixtures to form cyan, magenta, yellow, black, orange, and green as the printing
20
primaries. Designing with the usage of fluorescent inks is to drive the reproduction gamut as large as possible. The remaining systems all use CMYK plus RGB for color reproduction. Since present multiple-ink systems are all originally contrived to expand the reproduction color gamut, color matches are still limited to standard ANSI viewing conditions. Nevertheless, multiple-ink systems are quite capable of reproducing highly saturated colors well beyond that of a conventional CMYK printing process (Takaghi, Ozeki, Ogata, and Minato, 1994; Granger, 1996).
B. KUBELKA-MUNK TURBID MEDIA THEORY
Kubelka and Munk examined the reflectance of a material which had a thin layer of colorant in optical contact with its diffuse opaque substrate (Kubelka and Munk, 1931; Kubelka, 1948). They assumed that the layer of colorant could be further divided into a large number of sublayers parallel to the surface of the entire colorant layer.
Then,
sublayers were homogenous with identical optical properties to each other. Assuming the thickness of the entire layer is X, then the thickness of the sublayers is differentially defined as dx. Kubelka and Munk further assumed two diffuse light fluxes i, a downward flux, and j, an upward flux. The magnitude of the downward flux, i is decreased by the absorption and scattering of the colorant sublayer. The effect of the scattering process is to reverse the portion of downward flux, i, to the upward direction toward the surface of
21
the colorant layer. The upward flux, j, is further absorbed and scattered back to the downward direction by the sublayer. Hence, the portion of the upward flux, j, scattered back to the downward direction, needs to be added to the remainder of the downward flux. Continuing in this fashion, the differential equations to account for the downward and upward flux was initially set up as − di = −(S + K)idx + Sjdx
(2-1)
dj = −(S + K)jdx + Sidx ,
(2-2)
and
where K is the absorption coefficient and S is the scattering coefficient. The negative sign in Eqs. (2-1) and (2-2) to account for the downward direction had been defined as the negative direction (Wyszecki and Stiles, 1982). To solve the above differential equation in terms of reflectance factor, R, of the material, let ρ be the ratio of j to i then, from the quotient rule of differentiation (Allen, 1980), dρ d( j / i ) i(dj / dx ) − j( di / dx ) = = . dx dx i2
(2-3)
Substituting Eqs. (2-1) and (2-2) to Eq. (2-3), Eq. (4) is obtained as dρ = S − 2( K + S)ρ + Sρ 2 . dx
(2-4)
Equation (2-4) is a separable first-order differential equation. The boundary conditions are that ρ = Rg when x = 0, and ρ = R when x = X, where Rg is the reflectance factor of a
22
substrate, R is the surface reflectance factor of a material, and X is the thickness of the colorant layer of the corresponding material. Rearranging Eq. (2-4) together with the boundary conditions, it turns out as Eq. (2-5),
∫ dx = ∫ X
R
0
Rg
dρ . S − 2( K + S)ρ + Sρ 2
(2-5)
Solving for Eq. (2-5) in terms of R, the famous Kubelka-Munk equation results in R=
1 − Rg( a − b coth( bSX )) , a − Rg + b coth( bSX )
(2-6)
where a is equal to 1+K/S and b is equal to (a2 - 1)1/2. For an opaque colorant layer, known as the complete hiding case, light flux traveling in the colorant layer keeps being scattered and never reaches the substrate. It is equivalent to treating the thickness X of the entire colorant layer as infinitely large. Another assumption by Kubelka and Munk is that there is no fluorescence in the colorant layer. Based on these assumptions, Eq. (2-6) can be further simplified to Eq. (2-7), R ∞ = 1 + ( K / S) − ( K / S) 2 + 2( K / S) ,
(2-7)
K / S = (1 − R ∞ ) 2 / 2 R ∞ ,
(2-8)
and its inverse, Eq. (2-8),
for the opaque material where R∞ denotes as the surface reflectance factor of an opaque material with thickness.
23
In the application of a transparent layer in optical contact with a highly scattering support, such as the photographic paper, the scattering of the colorant layer is ideally zero.
Therefore, a portion of downward flux passing through the colorant layer is
absorbed and not scattered. The remaining flux reaches the substrate and gets reflected upward. The second absorption proceeds as the upward flux travels again through the colorant layer. Hence, Eq. (2-6) approaches Eq. (2-9) as S approaches zero. That is,
lim R = R g e −2 KX .
(2-9)
S→ 0
For most applications predicting color mixing for a transparent colorant layer in optical contact with an opaque support, the thickness X of the colorant layer is frequently assumed to be unity. Thus, the inverse of Eq. (2-9) in terms of absorption coefficient K is obtain as:
K = −0.5 ln(
R ). Rg
(2-10)
It has been shown that the absorption and scattering coefficients are linearly related to concentration (Allen, 1980; McDonald, 1987; Shan and Gandhi, 1990). Hence, the colorant mixing, mathematically described by Eq. (2-11), for the transparent material in optical contact with an opaque support is simply the mixture of the absorption coefficients normalized to unit concentration, k, of primary colorants modulated by their concentrations. K mix = c1 k 1 + c 2 k 2 + c 3 k 3 +
24
L+c k , i
i
(2-11)
where ci is the concentration of the ith colorant. The reflectance factor of such material can be calculated by Eq. (2-9). For the opaque material, the reflectance factor of color mixture is a function of the mixtures of the absorption coefficient, shown as Eq. (2-12), and the scattering coefficient, shown as Eq. (2-13). K mix = k t + c 1 k 1 + c 2 k 2 +
L+c k , i
(2-12)
i
where t denotes the substrate that suspends the primary colorant inside the colorant layer and kt is the absorption coefficient of the substrate. S mix = s t + c1s1 + c 2 s 2 +
L+c s .
(2-13)
i i
Coefficient s represents the scattering normalized to its unit concentration. Hence the ratio of the absorption to scattering coefficients in the opaque colorant layer is k t + c1 k 1 + c 2 k 2 + K = S mix s t + c 1s 1 + c 2 s 2 +
L+c k L+cs i
i
.
(2-14)
i i
Approaches leading to Eq. (2-14) are known as Kubelka-Munk two constant theory. Consider that when the individual scattering influence contributed by each primary colorant within the colorant layer is far less than the scattering power of the substrate. Equation (2-14) can be simplified to Eq. (2-15), known as Kubleka-Munk single constant theory,
25
k t + c1 k 1 + c 2 k 2 + K = S mix st
L+c k
k k k = + c1 + c 2 + st s1 s2
i
i
L + c ks
.
(2-15)
i
i
The reflectance factor of a mixture of opaque colorants can be calculated by Eq. (2-7).
C. COMPUTER COLORANT MATCHING
Computers have long been used for dye recipe formulation in the textile and paint industries. The early commercial applications were first proposed in 1961 (Anderson, Atherton, and Derbyshire, 1961). Several alternative approaches were later refined in 1963 when computers were practically available. Up-to-date, computer methods for colorant formulation have been vastly developed and published. Allen disclosed a common algorithm for tristimulus matching in the applications of computer colorant formulation (Allen, 1966; 1974). This algorithm is based on Kubelka-Munk turbid theory for a dyed highly scattering opaque substrate. The matrix equation of the single constant theory for the first prediction of concentration vector c corresponding to a matched standard color is defined by Eq. (2-16). c = (TEDΦ Φ)-1 TED(f(a) - f(t))
(2-16)
where T is a matrix of one of the CIE color matching functions, E is a matrix of the spectral power distribution of a CIE standard illuminant, f(a) is a matrix of the linearized function of the reflectance of an opaque sample fabricated by colorant mixtures in optical
26
contact with an opaque substrate, f(t) is a matrix of the linearized function of reflectance of an opaque substrate, D is a diagonal matrix of a derivative weighting function, and Φ is a matrix of the ratios of spectral absorptivities and unit scattering coefficients of three basis colorants. In addition, their matrix forms are shown by the following: x 400 T = y400 z 400
x410 y410 z410
L L L
x700 y700 , E = z700
0
1) φ(2) φ(3) L 0 φ(400 400 400 (1) ( 2 ) ( 3) d 410 L 0 φ410 φ410 φ410 = , M M O M Φ M M M 1) φ(2) φ(3) L d700 φ(700 0 700 700
(t) d f ( R )400 400 ( t ) 0 f ( R )410 D f (t) = = , (t) 0 f ( R ) 700
M
( a) L 0 f ( R )400 E410 L 0 (a) f ( R )( a ) M M O M , f = M410 , ( a) L E700 0 f ( R )700
E 400 0 0
.
0
The linearized function for opaque samples is f(Rλ) = (K/S)λ = (1-Rλ)2/(2Rλ) and the Jacobian suggested by Allen is dλ = (-2Rλ2)/(1- Rλ2). Equation (2-16) is used to estimate concentrations of the matched standard. This is only an approximation. The real solution requires an iterative process by defining the tristimulus and concentration
difference vectors,
∆X ∆t = ∆Y , ∆c = ∆Z
∆c1 ∆c2 , respectively, where ∆t represents the ∆c3
tristimulus difference and ∆c is the concentration difference vector between the standard
27
and the prediction at the current stage of iteration. Both values are bipolar and are interrelated in the iterative correction process as Eq. (2-17), ∆c = (TEDφ φ)-1 ∆t .
(2-17)
The value of ∆t is used as the conditional specification for the iterative process. If ∆t is within a goodness criteria then the iteration stops. If not, the ∆c at current iteration is added to the old concentration to get the new concentration used for the next iteration. This process continues until a goodness criteria is met. (The computer colorant matching technique for the Kubelka-Munk two-constant theory is not shown in this review. Interested reader can refer to Allen’s 1974 publication.)
D. VECTOR REPRESENTATION FOR COLORANTS
Generally speaking, a vector is a description for measurements at several features of an event. For instance, an event can represent a point, P3 = (x, y, z), in a three dimensional Cartesian coordinate system, denoted as ℜ3, where ℜ is the set of real numbers. Since the standard basis vectors (Marsden and Tromba), which are orthonormal, of ℜ3 are i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1), x, y, and z are the three features describing the magnitudes of P3 along the directions of i, j , and k, respectively. (“Orthonormal vectors” are a set of vectors with unit length which are perpendicular to each other. “Basis vectors” are a set of vectors which are not only linearly independent to each other but also span the entire vector space (Anton, 1991).) Although P3 is a
28
coordinate in ℜ3, it also can be viewed as a vector emerging from the origin since P3 can be expressed as a linear combination of the i, j , and k, i.e., P3 = x⋅ i + y⋅ j + z⋅ k .
(2-18)
Figure 2-1 denotes the visualization of a three dimensional vector P3 in ℜ3 where x, y, and z are the scalar magnitudes (or projection length) along the i, j , and k, respectively.
P3 = (x, y, z)
k z
j y
x
(0, 0, 0)
i
Fig. 2-1: The three dimensional vector, P3, in ℜ3. By extension, a point Pn = (x1, x2, x3,…, xn) in an n dimensional Cartesian coordinate system, ℜn , has n feature measurements, x1, x2, x3,…, xn, describing the magnitudes of Pn along the directions of n standard basis vectors in ℜn† , respectively. Hence, for any
The standard basis vectors in ℜn are i1 = (1, 0, 0,…,0), i2 = (0, 1, 0,…,0), i3 = (0, 0, 1,…,0), …, in = (0, 0, 0,…,1).
†
29
coordinate, Pn, treated as a vector in ℜn can be expressed as the linear combination of the standard basis vector in ℜn, i.e., n
Pn = ∑ x j ⋅ i j .
(2-19)
j=1
The basic mathematical operations applied to vectors are vector addition and scalar multiplication. For two vectors, Pn,1 and Pn,2, in ℜn, the vector addition is defined as n
n
n
j =1
j=1
j=1
Pn,1 + Pn , 2 = ∑ x j,1 ⋅ i j + ∑ x j, 2 ⋅ i j = ∑ ( x j,1 + x j, 2 ) ⋅ i j .
(2-20)
Equation (2-20) indicates that the vector addition operates by adding the two vectors componentwise. The scalar multiplication is defined as n
n
j =1
j =1
a ⋅ Pn = a ⋅ ∑ x j ⋅ i j = ∑ ( a ⋅ x j ) ⋅ i j ,
(2-21)
where a is a real number. Equation (2-21) shows that scalar multiplication modulates every component of the vector, Pn, with the same scalar. It is useful to examine whether measured spectral information can be represented as a vector; then all the merits of vector algebra can be used for color science applications. Given a sample with spectral reflectance factor, Rλ, measured within the visible spectrum between 400 nm and 700 nm at 10 nm intervals where λ is a sampled wavelength, Rλ is merely an array of thirty-one numbers; for example, Rλ,yellow = (0.0153, 0.0154, 0.0152, 0.0148, 0.0153, 0.0161, 0.0165, 0.0166, 0.0191, 0.0301, 0.0905, 0.2706, 0.5216, 0.685, 0.7541, 0.7825, 0.8014, 0.8128,
30
0.8202, 0.8284, 0.832, 0.8341, 0.836, 0.8386, 0.8403, 0.8461, 0.8451, 0.8429, 0.8459, 0.849, 0.8491). Rλ,yellow is usually plotted versus measured wavelength, shown in Fig. 2-2. 1 0.9
Reflectance factor
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 400
450
500
550
600
650
700
Wavelength
Fig. 2-2: Spectral reflectance curve of the example. Although Rλ,yellow is sampled discretely at a 10 nm bandpass, it is viewed as a spectral curve for the reason of smooth transition across 10 nm spectral neighborhood for typical materials. Color scientists are used to examining and analyzing the thirty-one component array on such a two dimensional layout of spectral measurements. In a coloration process, various amounts of colorants are used for manufacturing a desired color. The modulated spectral curves might be treated by multiplying the spectral curve of the colorant at 100% of its maximum concentration for any percentage of modulation. Every component of a
31
measured array is multiplied by the same percentage such that, as a consequence, the modulated spectral curves are parallel. Similarly, in color science applications, the previous mentioned event can be a spectral curve such as Rλ,yellow which has thirty-one measured reflectance factors at each sampled wavelength. Measurements at each sampled wavelength specifies a dimension of reflectance variations. Thus, Rλ,yellow can be treated as a vector in ℜ31 with the set of standard basis vectors, i1, i2, i3,…, i31. Modulation by the amount of a colorant is equivalent to scalar multiplication of the basic vector operation. The measured spectral reflectance factors can be further transformed to different representations such as spectral absorption where the vector addition and scalar multiplication are well defined. The determination of the optimal transformation is discussed in a later section. Therefore, adopted vector representations of the spectral information for color objects can further employ vector algebra and linear modeling techniques to analyze originals and synthesize reproductions. Vector Subspace For any set of n dimensional vectors, it is distributed in the n dimensional vector space or sometimes localized in a lower dimensional vector subspace. Figure 2-3 demonstrates the three possible distributions for three sets of three dimensional vectors.
32
1 0.8
1
1
0 .8
0 .8
0 .6
0 .6
0 .4
0 .4
0 .2
0 .2
0 1
0 1
0 .8
0 .8
0.6 0.4 0.2 0 1
0 .6 1 0.5 0.5 0
0
0 .6 1
0 .4 0 .5
0 .2 0 0
1
0 .4 0 .5
0 .2 0 0
Fig. 2-3: Example for a set of three dimensional vectors localized in various vector subspaces. Each set of vectors are normalized and translated into the cube of [0, 1] 3 which represents the universe of ℜ3. The left-most figure shows that the uniform distribution of the three dimensional vectors spanning the entire three dimensional vector space. The middle- and right-most figures indicate that two sets of three dimensional vectors localize in a two and one dimensional vector subspaces, respectively. A set of spectral measurements is probably localized in a lower dimensional vector subspace since the spectral variations are highly correlated across neighborhood spectral regions. The dimensionality of the set of spectral measurements can be analyzed and approximated by principal component analysis.
33
E. LINEAR MODELING TECHNIQUES
Linear modeling techniques based on principle component analysis (PCA) have been vastly applied for estimating the spectral reflectance factors of objects (Jaaskelainen, Parkkien, and Toyooka, 1990; Maloney, 1986; Vrhel, Gershon ,and Iwan, 1994; Vrhel and Trussel, 1992; García-Beltrán, Nieves, Hernández-Andrés, and Romero, 1998; Tajima, 1998). Previous research was aimed at determining a smaller numbers of basis vectors (or eigenvectors) dominating the data variation from a particular set of measurements. These basis vectors are usually an acceptable representation of the original objects. Seemingly, the greater the number of basis vectors, the higher the accuracy of the spectral reconstruction. Applications based on PCA have been generated to characterize multispectral CCD cameras for capturing spectral reflectance factors of a scene (Burns, 1997; Burns and Berns, 1997a; Burns and Berns, 1997b; Burns and Berns, 1996; Haneishi, Hasegawa, Tsumura, and Miyake, 1997). Principal Component Analysis (PCA) Principal component analysis explores the variance-covariance or correlation structure of a sample set in vector form. It primarily serves the purpose of data (or dimensionality) reduction and interpretation (Johnson and Wichern, 1992). Data reduction is accomplished by neglecting the unimportant directions along where samples’ variances are insignificantly small. Since major sample variations are along several significant directions, the number of these directions approximates the dimensionality of the sample
34
set. Figure 2-4 demonstrates a set of two dimensional samples migrating along a significant direction in which i = (1, 0) and j = (0, 1) are two standard basis vectors of the Cartesian coordinate system and e1 and e2 are two orthonormal vectors specifying the directions of a rotated coordinate system.
3
2.5
e1 y axis withstandard basis vector j=(0,1)
2
1.5
e2
1
Sample mean
0.5
0 0
0.5
1 1.5 2 x axis withstandard basis vector i=(1,0)
2.5
Fig. 2-4: Scatter plot of a two dimensional sample set.
35
3
Figure 2-4 indicates strongly that samples vary significantly along direction of e1, whereas, direction of e2 accounts for significantly less variation. The variation along directions of e1 and e2 reveals the covariance structure between samples. Let i and j be the directions for feature measurements in terms of, for example, reflectance factors separately measured at two sampled wavelengths. The corresponding coefficients, x and y, of i and j can be treated as two random variables of a random measurement at the two features, denoted as feature x and feature y, which describe an arbitrary sample. The example set of two dimensional samples in Fig. 2-4 specifies one trend that as feature x of the samples increases, feature y increases, and vise versa. Hence, features x and y are positively correlated. The other trend is that as the feature x increases, the feature y decreases, and vise versa, indicating that features x and y are negatively correlated. The direction of e1 is the weighted average (or weighted sum) direction of x and y, and the direction of e2 is the contrast (or weighted difference) direction of x and y. Observation for algebraic interpretation can be made by finding a line equation of ax + by = c which is parallel to the direction of e1 where a and b are positive real numbers and c is a real number. Thus, a sample along the direction of e1 is exactly described by the weighted sum of features x and y as the equation ax + by = c shows. Similarly, a line equation of dx - ey = f can be found to be parallel to the direction of e2 where d and e are positive real number and f is a real number. Hence, a sample along e2 is exactly described by the difference of weighted features x and y. These justifications are useful for data interpretation depending on the
36
physical meanings of the features x and y (recall that x and y are representing the reflectance factors measured at any two different sampled wavelengths for this example). Since there is less variation along e2, it might be considered as an insignificant direction (or dimension). The entire two dimensional sample set can be approximated by modulating the vector, e1. Thus, data (dimensionality) reduction is achieved by eliminating the dimension with negligible variance. The e1 and e2 are known as the two eigenvectors of the two dimensional sample set. Having samples in Fig. 2-4 represented in the new (or eigenvector) coordinate system whose basis vectors are e1 and e2, then e1 and e2, as opposed to the standard basis vectors in ℜ2, become two new metrics of feature measurements describing each sample. Each sample has been relabeled by the coefficients of the linear combination of e1 and e2. That is, if a sample P = xi + yj = le1 + me2 where x, y, l, and m are real numbers, then P is labeled as (x, y) in the Cartesian coordinate system and labeled as (l, m) in the eigenvector coordinate system. Figure 2-5 depicts the appearance of sample set represented in the eigenvector coordinate system. The sample set in the eigenvector coordinate system is decorrelated, i.e., the l and m features of a sample are uncorrelated. Hence, the modulation of e1 is not affecting the modulation of e2.
37
1
m axis with eigenvector vector e2
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1
-0.5
0 0.5 1 l axis with eigenvector vector e1
1.5
2
2.5
Fig. 2-5: The representation of sample set of Figure 4 in the eigenvector coordinate system.
Given a sample set, V, with q number of samples and n features, the sample variance-covariance matrix of size nxn is calculated as S=
1 q ∑ ( V − V )( Vi − V ) T , q − 1 i =1 i
(2-22)
where V is the mean vector of size nx1 of the sample set. Notice that n features specify that the sample set is in an n dimensional vector space. Diagonal entries of S are variances measured along the standard basis vectors, i1, i2, i3,…, in, in the n dimensional Cartesian
38
coordinate system. S can be further transformed to a diagonal matrix, Λ, for uncovering the variances along the significant directions. Equation (2-23) specifies the matrix diagonalization transformation. Λ = ( E T ) −1 SE −1 ,
(2-23)
where E is an nxn matrix whose n column vectors are the eigenvectors of S. The transformation by Eq. (2-23) decorrelates the sample variance-covariance matrix, S, of a sample set according to the eigenvectors such that the new sample representations are uncorrelated and the corresponding variance are maximized along the basis vectors, e1, e2, e3,…, en, in the eigenvector coordinate system. The off-diagonal terms of Λ are zeros which further symbolize the sample features represented in the eigenvector coordinate system are uncorrelated. The diagonal terms, L1, L2, L3,…, Ln, of Λ, the so called eigenvalues, are the variances measured with respect to the eigenvectors. Eigenvalues of the sample set can be attained from the sample variance-covariance matrix by the characteristic equation, det( L ⋅ I − S ) = 0 ,
(2-24)
where det is the determinant of a square matrix, L is an arbitrary eigenvalue, and I is the nxn identity matrix. There are n eigenvalues solved by Eq. (2-24). With these eigenvalues, their corresponding eigenvectors can be derived by Eq. (2-23). Eigenvalues can be zeros when a set of sample, V, is actually distributed in an mdimensional subspace, where m ≤ n. Eigenvectors whose corresponding eigenvalues are
39
zero explain no variance. Thus, an arbitrary sample, Vsample, in the sample set, V, can be exactly described by a linear combination of the significant m eigenvectors derived from the variance-covariance matrix of V, i.e., m
Vsample = ∑ b i e i ,
(2-25)
i =1
where bi is the coefficient for reconstructing Vsample. In this case, m eigenvectors explain hundred percent of the variance. The percent variance explained by m eigenvectors is calculated by m
Percent Variance =
∑L i =1 n
∑L i =1
i
% .
(2-26)
i
Percent variance can be used as a gauge to estimate the dimensionality of a sample set. Given an n dimensional sample set, if 99.00% (depends on the type of applications and the required accuracy for reconstruction) of total variance is explained by m eigenvectors where m ≤ n then the sample set can be approximately reconstructed by the significant m eigenvectors together with the sample mean vector, V sample mean, i.e., m
Vsample ≅ ∑ b i e i + Vsample i =1
mean
.
(2-27)
Therefore, data reduction is accomplished by an m-dimensional approximation. Principal components are often mistakenly referred to as eigenvectors (Vrhel and Trussel, 1992; Tajima, 1998; Haneishi, Hasegawa, Tsumura, and Miyake, 1997; García-
40
Beltrán, Nieves, Hernández-Andrés , and Romero, 1998). The following is given to clarify the difference between principal components and eigenvectors. Having eigenvectors, e1, e2, e3,…, en, and their corresponding eigenvalues, L1, L2, L3,…, Ln such that L1 ≥ L2 ≥ L3 ≥ … ≥ Ln, principal components are defined as the new coordinates of samples onto eigenvector coordinate system. Hence, there are n principal components for an n dimensional sample set. The ith principal component, Pci, is the collection by projecting samples, V, of size nxp, onto the ith eigenvector, i.e., Pc i = e Ti V ,
(2-28)
where i = 1…n. Hence, the whole sample set is decomposed into n uncorrlated components, Pc1, Pc2, … Pcn, by Eq. (2-28) whose variances are exactly the n eigenvalues, L1, L2, … ,Ln. Assuming the data in Table 2-1 is the measured spectral absorption coefficients, shown up to five samples, sampled between 400 nm and 700 nm at each 20 nm interval. The principal components calculated by Eq. (2-28) are shown in Table 2-2. Table 2-1: A sample set (shown up to five samples) of absorption measurements from 400 nm to 700 nm at 20 nm intervals. Wavelength 400 420 440 460 480 500 520 540
Sample 1 0.569 0.757 0.670 0.642 0.812 1.127 1.439 1.618
Sample 2 0.334 0.556 0.551 0.595 0.814 1.161 1.481 1.619
Sample 3 0.360 0.650 0.693 0.739 0.906 1.186 1.451 1.557
Sample 4 0.663 1.127 1.230 1.267 1.353 1.548 1.763 1.840
41
Sample 5 0.921 1.552 1.732 1.743 1.628 1.509 1.467 1.432
Variance 0.058 0.169 0.248 0.246 0.136 0.042 0.019 0.022
560 580 600 620 640 660 680 700
1.563 1.326 1.222 1.267 1.341 1.377 1.322 1.178
1.463 1.042 0.719 0.577 0.520 0.499 0.469 0.417
1.380 0.944 0.581 0.387 0.281 0.232 0.198 0.166
1.621 1.102 0.679 0.461 0.346 0.294 0.256 0.219
1.246 0.866 0.550 0.384 0.297 0.258 0.228 0.197
0.022 0.031 0.074 0.139 0.201 0.234 0.225 0.182
Table 2-2: The principal components as the new coordinates in the eigenvector coordinate system of the samples in Table 2-1. Principal component Pc1 Pc2 Pc3 Pc4 Pc5 Pc6 Pc7 Pc8 Pc9 Pc10 Pc11 Pc12 Pc13 Pc14 Pc15 Pc16
Sample 1 4.582 0.066 -1.226 0.069 -0.001 -0.012 -0.003 0.021 -0.004 0.001 0.003 -0.001 -0.001 0.000 0.000 0.002
Sample 2 3.063 -1.066 -1.538 0.052 -0.050 -0.018 0.000 0.017 -0.012 0.005 0.003 -0.002 -0.001 0.001 -0.002 0.002
Sample 3 2.725 -1.573 -1.381 -0.016 -0.079 -0.020 0.012 0.024 -0.001 0.003 0.005 0.003 -0.001 -0.001 -0.001 0.001
Sample 4 3.661 -2.348 -1.211 -0.043 -0.051 -0.003 0.005 0.022 0.005 0.004 0.003 0.001 -0.001 0.000 -0.001 0.002
Sample 5 3.701 -2.774 -0.163 -0.072 -0.031 0.007 -0.001 0.022 0.001 0.002 0.003 0.002 0.000 0.000 -0.001 0.001
Variance 0.504 1.245 0.294 0.004 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Variances along standard basis (or absorption measured at each sampled wavelength) and along eigenvectors are listed as the last of columns of Tables 2-1 and 2-2, respectively. Variances along each sampled wavelength are not informative. Attention is specially paid in the eigenvector coordinate system. Large percentage of variances explained by the first three principal components retain as much information as that of Table 2-1. Thus, the sample information is approximately preserved by retaining only three eigenvectors and
42
three principal components, as a consequence, data reduction is achieved. That is, the five samples can be approximated by the linear combination of the first three eigenvectors together with the first three principal components as the coefficients of reconstruction. The further justification to explore the concept of the last sentence. Let E equal the nxn matrix of eigenvectors as the column vectors and Pc equal the nxp matrix of principal components of a sample set, V. The matrix E is known as an orthogonal matrix with properties such that EET = ETE = I,
(2-29)
where I is the nxn identity matrix. Equation (2-28) is rewritten as Pc = ETV.
(2-30)
Multiplying E on sides of Eq. (2-30) yielded EPc = EETV = IV = V.
(2-31)
Hence, an arbitrary sample set, V, can be reconstructed by linear combinations of eigenvectors modulated by their corresponding principal components as claimed. The principal components are not eigenvectors but exactly the coefficients, bi, used in Eqs. (225) and (2-27). Principal Component Analysis for Color Science Applications Color science research employing PCA was published as early as 1954 (Morris and Morrissey, 1954). This early research devised an objective method for determining equivalent neutral densities of color film images. The basic concept was to determine the
43
minimum number of spectral density curves as the basis vectors necessary to account for spectral variability of typical processed film with different exposures. They calculated the three most significant eigenvectors and corresponding eigenvalues based on forty-two patches of Ektachrome film. Then the three eigenvectors were used to spectrally reconstruct the spectral densities of single-layer-coated films (e.g., cyan, or magenta, or yellow dyes). The eigenvector-fitted dye density spectra were considered as the “assumed dyes” which were the closest representations of spectral densities of three dyes accounting for the minimum residual variance from normal manufacturing and processing. Later, Simonds, in a landmark article, exemplified the applications of PCA for photographic and optical response data (Simonds, 1963). He showed a visualization of how a measured response curve was decomposed by six curves, that is, the response curve can be synthesized by a linear combination of the six curves provided by his demonstration. Mathematical basics and numerical examples were also discussed. The spectral distribution of typical daylight as a function of correlated color temperature was determined using PCA (Judd, MacAdam, and Wyszecki, 1964). The compiled six hundred and twenty-two samples of daylight (249 daylight measurements from Rochester, United States, 274 daylight measurements from Enfield, England, and 99 daylight measurements from Ottawa, Canada) were used to derive the mean daylight vector and two eigenvectors of daylight, shown in Fig. 2-6. By Eq. (2-27), the mean and first two eigenvectors were used to reconstruct the daylight distributions, plotted in Fig. 2-
44
7, at correlated color temperatures 4800 K, 5500 K, 6500 K, 7500 K, and 10000 K by specially chosen scalar multiples in Eq. (2-27), bi, for reproducing exactly the same chromaticities of the specified correlated color temperatures.
1400 1200
Relative irradiance
1000 800 600 400 200 0
810
780
750
720
690
660
630
600
570
540
510
480
450
420
390
360
330
300
-200 Wavelength
Fig. 2-6: The mean (thick line), the first eigenvector (thin line), and the second eigenvector (dashed line) of the 622 daylight measurements (Judd, MacAdam, and Wyszecki, 1964). Figure 2-6 reveals three aspects of important information. First, the average daylight is blue-greenish. Second, the first eigenvector indicates that, among six hundred and twenty-two measurements, daylight spectral distributions vary mainly along the yellow-blue direction which corresponds to the sky colors changing from yellowish to bluish during a day. Finally, the second eigenvector explains the greenish to purplish variation of daylight. This green-purple variation, according to the interpretation of Judd et al., may be caused by the water vapor in the sky.
45
1800 1600
10000 K
Relative irradiance
1400 1200 1000 800 7500 K 600 6500 K
400
5500 K
200
4800 K 810
780
750
720
690
660
630
600
570
540
510
480
450
420
390
360
330
300
0 Wavelength
Fig. 2-7: Spectral distribution of typical daylight at correlated color temperatures 4800 K, 5500 K, 6500 K, 7500 K, and 10,000 K (Judd, MacAdam, and Wyszecki, 1964). Interpretations for Coloration Processing Applications by PCA Assuming objects are measured within the visible spectrum between 400 nm and 700 nm at 10 nm intervals, every measured sample is a vector of thirty-one components. Thus, a measured sample set is thirty-one dimensional. Vector representations of colored objects will adopt a type of spectral information whose vector addition and scalar multiplication are defined. Hence, temporarily, the matrix-vector notation, Φ, is employed as the additive and scalar multiplicative spectral information. Consider three sets of spectral absorption spectra of cyan, magenta, and yellow dyes modulated at different concentrations, depicted in Fig. 2-8.
46
3
2 .5
2 .5
2
2
1 .5
1 .5
1
1
0 .5
0 .5
2 .5
A b s o rp t a nc e
2
1 .5
1
0 .5
0 400
500
600
700
0 400
500 600 W a ve le ng th
700
0 400
500
600
700
Fig. 2-8: Three sets of absorption spectra of cyan, magenta, and yellow dyes (left to right) at eleven different concentrations.
A “virtual” sample set, Φλ,mixture can be created by each combination of spectra at six different percentages (0%, 20%, 40% 60%, 80% and 100%) of the three. Thus, the a priori knowledge about this virtual sample set of two hundred and sixteen samples is that it is theoretically distributed in a three dimensional subspace of ℜ31 since it is formed from a combination of three dyes. Hence, visualizing in a thirty-one dimensional space, the underlying variations of three-dye mixtures are along the cyan, magenta, and yellow dimensions as shown in Fig. 2-8. Spectra of cyan, magenta, and yellow dyes at one hundred percent concentration, Φλ,cyan, Φλ,magenta, and Φλ,yellow, respectively, are the basis vectors of the three-dye coordinate system. Spectra of mixtures can not be easily
47
decomposed with respect to Φλ,cyan, Φλ,magenta, and Φλ,yellow if they are initially unknown. Alternatively, if the Φλ,mixture could be scatter plotted in ℜ31 then the appearance of the sample set would have samples varying along one weighted average direction and several contrast directions, as mentioned previously. Since the visualization is impossible for a thirty-one dimensional scatter plot, the feasible alternative is to observe the spectral curves of the eigenvectors, shown in Fig. 2-9, derived from Φλ,mixture (spectral absorption coefficients in this case).
Absorptance
The 1st eigenvector
The 2nd eigenvector 1
1
0.5
0.5
0.5
0
0
0
-0.5
-0.5
-0.5
-1 400
500
600
700
-1 400
The 4th eigenvector
Absorptance
The 3rd eigenvector
1
500
600
700
-1 400
The 5th eigenvector 1
1
0.5
0.5
0.5
0
0
0
-0.5
-0.5
-0.5
500 600 Wavelength
700
-1 400
500 600 Wavelength
600
700
The 6th eigenvector
1
-1 400
500
700
-1 400
500 600 Wavelength
700
Fig. 2-9: The first six eigenvectors obtained for the virtual sample set. (As many as thirtyone eigenvectors can be shown.) The first eigenvector points out the weighted average direction along which the samples are distributed. This implies that there are neutral colors in the Φλ,mixture judged by
48
the appearance of a flat spectrum, i. e., majority of samples are distributed along the direction of weighted-average absoprtion. The second eigenvector reveals that samples vary from short wavelength regions to long wavelength regions indicating the existence of bluish, blue-greenish, yellowish, and reddish samples in Φλ,mixture. Similarly, the third eigenvector describes that samples varying along the middle wavelength regions and the corresponding complementary spectral regions indicates the existence of greenish and purplish color samples in Φλ,mixture. These three eigenvectors explain one hundred percent of total variance and the corresponding directions reveal the color information of samples according to the previous interpretation. The rest of the eigenvectors (the fourth to sixth are plotted to demonstrate their content) are associated with zero variance, that is, no sample varies along such directions. Therefore, the interpretation of them is meaningless. In practice, a measured sample set does not provide the statistical results of exactly the same dimensionality as the known number of colorants used to construct the sample set. For example, Fig. 2-10 depicts the six eigenvectors determined from the IT8.7/2 reflection target, Kodak Q60C for Ektacolor paper, a test target of two hundred and sixty-four color patches sampling the photographic paper’s color gamut (McDowell, 1993).
49
Absorptance
The 1st eigenvector
The 2nd eigenvector 1
1
0.5
0.5
0.5
0
0
0
-0.5
-0.5
-0.5
-1 400
500
600
700
-1 400
The 4th eigenvector
Absorptance
The 3rd eigenvector
1
500
600
700
-1 400
The 5th eigenvector 1
1
0.5
0.5
0.5
0
0
0
-0.5
-0.5
-0.5
500 600 W avelength
700
-1 400
500 600 W avelength
600
700
The 6th eigenvector
1
-1 400
500
700
-1 400
500 600 W avelength
700
Fig. 2-10: The six eigenvectors obtained from an IT8.7/2 reflection target. The first three eigenvectors explain 99.96% of total variance. The interpretation of the first three eigenvectors was described previously. Interest is focused on the extra dimensions of the statistical estimation. As the variance explained by the ith eigenvector decreases, the oscillating appearance of the corresponding eigenvector increases. This can be realized by inspecting the fourth to the ith eigenvectors with nonzero variance. Explanations of the high degree of oscillation can be attributed to noise caused by manufacturing, photographic processing, and spectrophotometric measurements. Since the rapidly oscillating eigenvectors indicate the directions of samples with spectral properties varying rapidly across the neighborhood spectral regions, no such colorants exist in the
50
physical world. Such eigenvectors must describe the noise behavior of the sample set. Fortunately, the noisy behavior only contributes to insignificant statistical results. Thus, the reconstruction of a sample set achieved by the significant eigenvectors fulfills the goal of data (dimensionality) reduction. Multivariate Normality for Effective Data Reduction It is worth speculating on the multivariate normality of a given set of multivariate measurements although it is not a requirement for deriving eigenvector-eigenvalue pairs. Multivariate normality ensures that eigenvectors and the associated eigenvalues derived from measured samples is close to the eigenvector-eigenvalue pairs of the entire population since manufacturing and sampling procedures may not be optimally performed (Johnson and Wichern, 1992a; Anderson, 1984; Anderson, 1963; Grishick, 1939). If a population is multivariate normally distributed then the 99% of the population should distribute inside an ellipse, ellipsoid, and hyper-ellipsoid for two, three, and higher dimensional populations, respectively. Half lengths of the axes of an ellipsoid in an eigenvector coordinate system are directly proportional to the corresponding eigenvalues. If a population is not of ellipsoidal shape then data reduction may not be optimally applied for reconstructing the samples from this population since the non-ellipsoidal shape of the population may imply a bizarre structure among the random variables representing the sample features (in this case, the random variable can be, for instance, the Φλ measured at
51
each sampled wavelength). To give a clearer insight, a set of 500 random samples was generated from a bivariate uniform distribution, plotted in Fig. 2-11.
7 e1
Random variable Y
6 e2
5 4 3 2 1 0 0
1
2
3 4 5 Random variable X
6
Fig. 2-11: Scatter plot of the 500 random samples and their eigenvectors from a uniform distribution.
The corresponding sample variance-covariance matrix and mean were calculated. A 99% confidence region of an ellipse with two eigenvectors given by the very sample variancecovariance matrix and mean to form a bivariate normal distribution are also super imposed on to the 500 random samples in Fig. 2-11. Assuming another set of 500 samples are from
52
a bivariate normal distribution of the same covariance matrix and sample mean, then 99% of them should be located inside the elliptical confidence region. By design, these two sets of samples have the identical eigenvector-eigenvalue pairs. Once data reduction is required by the first eigenvector approximation, the total error of the normal sample set by the first eigenvector approximation is less that of uniform sample set since the total error is proportional to their corresponding areas. When the calculated eigenvectors with their corresponding eigenvalues explaining small variances, which are considered as insignificant, are overlooked, the overlooked eigenvectors may still be important for reconstruction due to not knowing how the samples are distributed. The corresponding eigenvalues can not provide correct information for data (dimensionality) reduction; that is, the estimation of the true dimensionality based on the information of the total explained variance calculated by Eq. (2-26) is not objective. Therefore, the information given by the calculation of percent variance is less informative by non-normal multivariate distributions for the decision of data (dimensionality) reduction. It is required to check on two necessary conditions for the normality of a sample set. First is the normality of the marginal distributions (spectral information measured at each sampled wavelength) since all linear combinations of normal distributions are also normal (Johnson and Wichern, 1992b). The normality of a univariate, often a marginal, distribution can be inspected by the Q-Q plot or gamma plot for a bivariate distribution
53
(Johnson and Wichern, 1992c). The degree of normality can be specified by the correlation coefficient of the Q-Q plot and the Chi-squire distance for univariate and bivariate distributions, respectively. The normal plot procedure provided in MATLAB was utilized for visual inspection. A virtual normal marginal distribution of one thousand samples was generated by linear combinations of the spectral absorption coefficients of three linearly independent colorant vectors. If an examined set is normally distributed then the corresponding normal plot is a straight line. As an example, Fig. 2-12 depicts the normal plot of absorption coefficients at 500 nm. The coefficients of linear combinations were generated from a normal random number generator. The normal plot of the absorption coefficients at each sampled wavelength appears to be straight lines as expected. Non-normal population of one thousand samples, whose normal plot of absorption coefficients at 500 nm is shown as Fig. 2-13, was also obtained by linear combinations of the same colorant vectors as that of Fig. 2-12 whose coefficients came from an exponential distribution. The non-normal marginal distribution reveals curvature in the normal plot. The shape of the sample set with non-normal marginal distributions can not be ellipsoidal in a higher dimensional space by the implications of the curvatures of marginal distributions in the normal plot. It can be expected that most errors yielded of the reconstructed population by PCA are from those samples deviating from the estimated straight line of their corresponding normal plot.
54
0 .9 9 9 0 .9 9 7 0 .9 9 0 .9 8
P r o b a b ilit y
0 .9 5 0 .9 0 0 .7 5 0 .5 0 0 .2 5 0 .1 0 0 .0 5 0 .0 2 0 .0 1 0 .0 0 3 0 .0 0 1 2
4
6 8 10 A b s o r p t io n c o e f f ic ie n t s a t 5 0 0 n m
12
14
Fig. 2-12: Normal probability plot of absorption coefficients at 500 nm from the virtual normal population obtained by linear combinations of three linearly independent dyes.
0 .9 9 9 0 .9 9 7 0 .9 9 0 .9 8 0 .9 5 P r o b a b ilit y
0 .9 0 0 .7 5 0 .5 0 0 .2 5 0 .1 0 0 .0 5 0 .0 2 0 .0 1 0 .0 0 3 0 .0 0 1 0
5
10 15 20 A b s o r p t io n c o e f f ic ie n t s a t 5 0 0 n m
25
30
Fig. 2-13: Normal probability plot of absorption coefficients at 500 nm of the exponential population obtained by linear combinations of three linearly independent dyes.
55
The second condition is determined by whether two dimensional scatter plots of any measurements of any two random variables generate an elliptical appearance. For a set of spectral measurements using sampled between 400 nm and 700 nm at 10 nm interval, there are four hundred and forty-three ( C(31,2)-31=434) scatter plots to be examined. An elliptical appearance of a two dimensional scatter plot reveal the existence of bivariate normality. Figure 2-14 is depicted as an example by scatter plotting the absorption coefficients at 550 nm versus absorption coefficients at 500 nm of the virtual normal sample set. The elliptical appearance of the two dimensional scatter plot suggests that absorption coefficients at 550 nm and 500 nm are bivariate normally distributed. If the first and second conditions are met then univariate and bivariate normality are generated. Whether or not it implies the multivariate normality of an examined sample set, it is difficult to conclude inductively. Without the satisfaction of the first and second conditions, the existence of multivariate normality of the examined sample set should be denied. Non-normal samples can be further transformed to a set of representations with more degrees of normality by logarithmic, power, or polynomial transformations (Johnson and Wichern, 1992d).
56
3
Absorption coefficients at 550 nm
2.5
2
1.5
1
0.5
0 0
0.5
1 1.5 2 Absorption coefficients at 500 nm
2.5
3
Fig. 2-14: The two-dimensional scatter plot of absorption coefficients at 550 nm vs. 500 nm from the virtual normal population obtained by linear combinations of three independent dyes.
F. SPECTRAL PRINTING MODELS
Conventional CMYK halftone printing processes are the most popular color reproduction processes for their relative inexpensive cost and reproduction speed. High speed printing presses generate more than 10,000 prints per hour. A priceless artwork can be massively reproduced in such a fashion at a relatively low cost. Since the goal of this research is to build a six-color output system. A halftone printing process is a natural choice.
57
In this section, the underlying physical phenomena of the color formation of a halftone printing process will be discussed. Mathematical description for the halftone color formation such as Murray-Davies and Neugebauer equations as well as the dot-gain effect introduced by Yule-Nielsen will be shown. Basic Assumptions A halftone cell is designed by laying down a grid of pixels. The overall reflectance factor over a halftone cell is theoretically the average result of the reflectance factor of the pixels which are turned on, termed as on-pixels, and the reflectance factor of the pixels which are off, termed as off-pixels, modulated by the percentage of on-pixels and the percentage of off-pixels. A halftone cell with NxN addressibility is depicted in Fig. 2-15, where N = 2.
i=1
i=2
i=3
i=4
Fig. 2-15: The outline of a 2x2 halftone cell. The calculations for spectral reflectance of a halftone cell equation are based on the following underlying assumptions: 1. The spectral reflectance factor of the primary colorant varies proportionally to that of the primary at 100% area coverage.
58
2. Spectral reflectance factor is additive within the halftone cell. 3. Human eye cannot resolve the halftone cell. These three assumptions lead to the spectral reflectance factor Rλ of a halftone cell, conceptually, being the sum of the spectral reflectance factor of each primary color at 100% area coverage modulated by their corresponding fractional dot area coverage inside the halftone cell. In practice, the analytical description is more complicated than this simple concept. The analytical description of a halftone color formation depends on the type of halftone cell and the alignment of dots. Generally, the so called dot-on-dot type of halftone device can be analytically modeled by Murray-Davies theory, and the traditional halftone dot placement by rotated-screen or stochastic dot formation can be analytically described by Neugebauer theory together with Demichel’s probability model for dot overlap. Murray-Davies Theory Figure 2-16 shows a halftone cell printed by a single ink on an opaque support can be covered with different dot area coverage. It shows the notion of how the spectral reflectance factor over a halftone cell varies with respect to the percentage of the area covered by a single color ink. By the assumptions of additivity inside a halftone cell, the spectral reflectance factor Rλ of a halftone cell printed on paper by a single color ink, can be estimated based on the Murray-Davies equation, R λ = aR λ ,100% + (1 − a )R λ , paper ,
59
(2-32)
where a is the fractional dot area of a single color ink, Rλ,100% is the spectral reflectance factor of the color ink at 100% dot area coverage, and Rλ,paper is the spectral reflectance factor of the paper support (Murray, 1936).
11% area coverage
56% area coverage
100% area coverage
Fig. 2-16: The 3x3 halftone cell covered by three different dot areas. The accuracy of the estimated spectral reflectance factor of a halftone cell, predicted by the Murray-Davies equation, relies on linearity for each ink printed on a paper support. Linearity is defined as that, for each primary, the reflectance factors should be summed up according to their fractional dot areas. That is, if the fraction dot areas of a primary are summed up to a percent, where 0 ≤ a ≤ 100, then the resultant reflectance factor is summed up to a percent of the reflectance factor of the primary at 100% area coverage. Due to the failure of linearity, the Murray-Davies equation can not provide adequate prediction of the estimated spectral reflectance factor inside a halftone cell. The physical phenomena that limit linearity are known as mechanical and optical dot-gain. To model both mechanical and optical dot-gain, the Yule-Nielsen n-factor was employed to modify
60
the Murray-Davies equation (Yule and Nielsen, 1951). The n-factor modified MurrayDavies Equation is shown in Eq. (2-33), where terms in the Eq. (2-33) are similarly defined as those in Eq. (2-32). R λ = [aR 1λ/,n100% + (1 − a )R 1λ/,npaper ]n ,
(2-33)
Dot Gain There are two sources of dot gain, mechanical and optical dot gain. Mechanical dot gain is caused by printing ink submerging and spreading when delivered onto a paper substrate causing the physical dot size to be changed. Figure 2-17 depicts the appearance of mechanical dot gain at a microscopic level.
non-ideal dot shape after mechnical dot gain
Ideal dot shape
Fig. 2-17: Non-ideal halftone dot shapes vary due to the mechanical dot gain effect. Optical dot gain effect is depicted in Fig. 2-18, known as the Yule-Nielsen effect, and originates from light flux that penetrates the ink film, enters the paper substrate, scatters, and finally emerges from non-ink area. Simultaneously, light flux can enter the paper
61
substrate and get scattered, penetrate the ink film, and finally emerge from the ink film. These two physical phenomena cause printing models to predict smaller or larger dot size.
Light
Dot Paper
Paper
Fig. 2-18: The cause of optical dot gain. Several dot gain models have been published to take into account that the dot gain effect is a function of the size of halftone dots, scattering, and the surface reflection of ink films (Arney, Engeldrum, and Zeng, 1995; Wedin and Kruse, 1995; Arney, Arney, and Engeldrum, 1996). Arney et al., (1996) further determined the paper spread function, or the modulation transfer function, by Fourier analysis. Although the Yule-Nielsen n-factor is an empirical parameter, several research efforts have shown that it is capable to maintain high accuracy of model fitting. YuleNielsen n-factor is still widely employed for its simplicity (Viggano, 1985; Balasubramanian, 1995; Iino and Berns, 1997; 1998a; 1998b; Pearson, 1980; Pope, 1989). Neugebauer Theory The Murray-Davies equation is only used for prediction of a single-color halftone cell. It can not predict the spectral reflectance over a multiple-color halftone cell. A multiple-color halftone cell is defined as an area covered by various printer primary inks.
62
Wide varieties of colors are produced by modulating the area coverage of printer primaries in a halftone cell on an opaque medium such as paper. The spectral Neugebauer equation can be used to predict spectral reflectance factor over a multiple-color halftone cell (Neugebauer, 1937). For a CMYK printing process, channels of white, cyan, magenta, yellow, red, green, blue, three-color black, black, black-cyan, black magenta, black yellow, black-red, black-green, black-blue, and four-color black are known as the sixteen Neugebauer primaries. Neugebauer theory assumes that each channel of Neugebauer primary is linear and color halftone dots are placed at random. Thus, the spectral reflectance factor over a halftone cell, printed by a CMYK printer, is calculated by the sum of the fractional dot area of each channel multiplied by the spectral reflectance of the corresponding Neugebauer primary at 100% dot area coverage. The spectral Neugebauer equation is defined as the following: 16
R λ = ∑ a i R λ,i ,
(2-34)
i =1
where Rλ is the spectral reflectance factor of a halftone cell printed on a paper support, Rλ,i is the spectral reflectance factor of the ith Neugebauer primary, and ai is the fractional dot area coverage of the ith Neugebauer primary. Fractional dot area coverage, ai, of the ith Neugebauer primary is calculated by employing the Demichel’s dot-overlap model (Demichel, 1924; Yule, 1967; Kang, 1997). The Demichel model is based on the assumption that dots are delivered on the substrate in
63
a random fashion. Hence, the overlap area of two inks is nothing more than the joint probability of these two inks. There are 24 = 16 combinations to predict the Neugebauer primaries in the four-color case.
Non-overlapping portions are simply the CMYKW
primaries, two-color overlaps are defined as RGB secondary primaries, three-color overlaps are tertiary primaries, and four-color overlap is essentially the unique four-color black. The fractional dot areas for sixteen Neugebauer primaries are calculated as below: White Cyan Magenta Yellow Red Green Blue 3-Color Black Black Black-Cyan Black-Magenta Black-Yellow Black-Red Black-Green Black-Blue 4-Color Black
: : : : : : : : : : : : : : : :
a 1 = (1 - c) (1 - m) (1 - y) (1-k) a2 = c (1 - m) (1 - y) (1-k) a3 = m (1 - c) (1 - y) (1-k) a4 = y (1 - c) (1 - m) (1-k) a5 = y m (1 - c) (1-k) a6 = c y (1 - m) (1-k) a 7 = c m (1 - y) (1-k) a8 = c m y (1-k) a9 = k (1 - c) (1 - m) (1 - y) a10 = c k (1 - m) (1 - y) a11 = m k (1 - c) (1 - y) a12 = y k (1 - c) (1 - m) a13 = y m k (1 - c) a14 = c y k (1 - m) a15 = c m k (1 - y) a16 = c m y k
(2-35)
where c, m, y, and k are fractional dot areas of printer primaries within a halftone cell. As the result of linearity failure caused by mechanical dot-gain, optical dot-gain, or both, the spectral Neugebauer equation can not accurately predicte spectral reflectance over a halftone cell. The spectral Neugebauer equation together with the Yule-Nielsen n-factor
64
again can be employed to improve the model accuracy which is shown as equation (2-36), where terms in Eq. (2-36) are similarly defined as that of (2-34). 16
R λ = [ ∑ a i R 1λ/,ni ]n , i =1
65
(2-36)
III. LINEAR COLORANT MIXING SPACES As discussed in the previous chapter of theoretical background, researchers frequently perform PCA using reflectance factor of color samples requiring reproduction. Often, the number of significant dimensions (basis vectors) exceeds the number of physical parameters, for example, a photographic system requires more than three dimensions for spectral reconstruction. This contradicts the knowledge that photographic materials are manufactured by three known dyes. Apparently, the color synthesis by PCA using reflectance factor as the representation for photographic materials is not optimal. This justification also applies for all analysis and synthesis using PCA on surface colors of all types. Hence, a transformation to account for the real physical dimensions of a set of measurements as well as to agree with the process of an opaque coloration is desirable. Mathematical transformation based on Kubelka-Munk turbid media theory is an obvious choice. The resultant space after Kubelka-Munk transformation is termed as Φ space for simplicity through the rest of this dissertation. Upon our numerous attempts using Φ space for opaque color analysis and synthesis, PCA failed to predict the reconstructed accuracy with constrained number of dimensions. For example, a set of opaque mixtures was created by mixing six opaque paints. Six eigenvectors of the set of opaque mixtures in Φ space frequently fail to spectral reconstruct the low absorptive samples. Such phenomena which are not realizable for physical materials are profound by the negative spectral
66
components as a result of six-eigenvector reconstruction. Since Kubelka-Munk transformation is highly nonlinear, the consequence of inverse transforming the negative spectral components is not interpretable. Our primary goal is to employ or derive a transformation to ensure that the dimensionality of the linear representation of a set of color samples agrees with their physical dimensionality. During the process of searching and deriving the optimal transformation, the fundamentals of PCA were thoroughly reviewed. It was discovered that the multivariate normality of a sample set contributed to the efficiency of data (dimensionality) reduction. This lead us to utility the multivariate normality as a beneficial factor in designing the transformation.
A. REFLECTANCE AND ABSORPTION SPACES
Both reflection and absorption occur when opaque objects are exposed to electromagnetic radiation. If there is no thermal or other energy loss then the total reflected and absorbed energy should be identical to the total incident energy. Quantitatively, there is a relationship between reflectance and absorption according to the law of conservation of energy. Kubelka and Munk initiated the derivations of this relationship (Kubelka and Munk, 1931; Kubelka, 1948). They found reflectance factor is a function of the ratio of absorption coefficient, K, and scattering coefficient, S. The ratio, K/S, is again denoted as Φ for simplicity and (K/S)λ as well as Φλ are denoted as their
67
spectral extension. It is also known that the ratio of the absorption coefficient and scattering coefficient is approximately linear with respect to concentration (Allen, 1966; Allen, 1980; Shah and Gandhi,1990). Colorant formulation can be performed by linear combinations of (K/S)λ of colorants used for synthesis, i. e., n
(K/S) λ,mixture = ∑ c i (k/s) λ,i ,
(3-1)
i =1
where c represents the concentration, k and s are the absorption and scattering coefficients of a colorant normalized to unit concentration. Therefore, the absorption and scattering properties of materials also provide a useful platform for color scientists to perform color analysis and synthesis. Again, the vector space of Eq. (3-1) is denoted as Φ space for compactness of terminology. Although Φλ has the advantageous linear property with respect to concentration, interestingly, there is relatively little research concerned with estimating spectral information by applying linear modeling techniques in Φ space. Φ space related research by Ohta determined the three eigenvectors from measurements of spectral densities of a photography material (Ohta, 1973). He then tried to linearly transform the three eigenvectors to a set of all-positive vectors as the estimation of the underlying real dye spectra. Once the basis density spectra are statistically uncovered, any spectral density measured from the photographic material can be synthesized with high colorimatric and spectral accuracy. Reconstructed density spectra can be further transformed to the representations of spectral transmission or reflectance
68
factors. Berns and Shyu extensively carried out this process to estimate the spectral transmission and reflectance factors of four photographic films (Berns and Shyu, 1995). In order to validate the accuracy of their dye estimates based on PCA, a tristimulus matching algorithm was used to achieve an exact match for CIE illuminant D50 and the 1931 observer (Allen, 1980). CIELAB color difference for illuminant A was used as an metamerism index. Smaller indices indicated better spectral reconstruction. Accuracy for the four photographic films tested, averages and maximum metamerism indices varied from 0.1 to 0.3 ∆E*ab and 0.4 to 1.3 ∆E*ab, respectively. By the inspiration of these two researches, Φ space as opposed to reflectance space might be a better alternative space to approximate natural scenes since Φ is approximately linear with respect to concentration. The superiority between two spaces depends on whether or not the spectral properties (reflectance factor or Φ) forming a scene whose dimensionality distribute in each space agree with their physical dimensionality. A linear colorant mixing space with larger degree of multivariate normality can be further approximated with lower dimensionality. This is a beneficial factor when research applications are frequently confined to use limited number of primary colorants for color synthesis. This research project will be focusing on the comparisons among spaces based on these two factors.
69
B. TRANSFORMATION BETWEEN REFLECTANCE AND Φ SPECTRA
The transformation between reflectance factor and Φ is often based upon KubelkaMunk turbid media theory. Equations (3-2) and (3-3) are used for opaque materials such as acrylic paints and textiles, where the R λ,∞ is the spectral reflectance factor of an opaque material. R λ ,∞ = 1 + Φ λ − Φ λ 2 + 2Φ λ
(3-2)
Φ λ = (1 − R λ ,∞ ) 2 / 2 R λ ,∞
(3-3)
Equations (3-4) and (3-5) are used for transparent color layer in optical contact with an opaque support such as photographic paper,where Rλ, g is the spectral reflectance factor of an opaque support and X is the thickness of the transparent colorant layer. Equation (3-5) is the inverse transformation of Eq. (3-4) by assuming that the thickness, X, is unity.
lim R λ = R λ ,g e −2 Φ λ X S→ 0
lim Φ λ = −0.5 ln( S→ 0
Rλ ) R λ ,g
(3-4)
(3-5)
Kubleka-Munk turbid media theory is based on a two-flux assumption, that is, the light in the colorant layer only become scattered upward or downward. No other directional scattering is assumed. Hence, the Kubleka-Munk transformation itself is an approximation of coloration processes (Van De Hulst, 1980; Nobbs, 1985). Accuracy is
70
quite reasonable for materials with optical characteristics modeled by Eqs. (3-4) and (3-5). However, as this research progressed, it was discovered that the transformation for opaque materials does not always describe the optical properties of mixtures formed by the corresponding coloration. In retrospect, this leads to the violation of the two flux assumption. A real material most frequently scatters light in all directions which causes the failure of Eqs. (3-2) and (3-3). Furthermore, measurement by spectrophotometers with non-optimal aperture sizes causes failure to the linear assumption in Φ space (Tzeng and Berns, 1998a). Concentration is no longer linear with respect to Φλ. Consider a spectrophotometer measuring the surface of a multicolor object. Its field of view may cover several color surfaces as shown in Fig. 3-1.
Spectrophotometer
Sensor plane
Field of view
Color object
Fig. 3-1: The possible field of view of a spectrophotometer. In this case, the reading of the spectrophotometer is the result of spatial averaging inside its field of view. This implies the additive operation has already been performed in
71
reflectance space. Since the Kubelka-Munk opaque transformation is nonlinear, the additivity of colorant vectors is, therefore, not well defined in Φ space. In dealing with the failure of Kubelka-Munk turbid media theory, many more theories utilizing multi-flux methods in solving radiation transfer problem have been published by a number of authors for improving the predicting accuracy (Mudget and Richards, 1971; Mudget and Richards, 1972; Maheu, Letoulouzan and Gouesbet, 1984; Mehta and Shah, 1986a; Mehta and Shah, 1986b). However, these complex models, despite their improved correlation with the true optics of colorant mixtures, usually required considerable parameter optimization in order to result in acceptable accuracy. It seems reasonable to directly derive an empirical transformation. The main concerns for the derivation of an empirical space are: transforming a non-normal population to a near normal population since the normality is a beneficial factor for data (dimensionalty) reduction, obtaining a new colorant vector space with reduced dimensionality that corresponds to the physical dimensionality of a given sample set, and the vector addition and scalar multiplication in new vector space should approximately describe the process of subtractive opaque coloration. Consider the subtractive opaque colorant mixing, the more colorants that are added for coloration, the darker the resultant mixture is. A vector space formed by adding reflectance factors is not realizable for opaque coloration. Thus, an empirical equation was derived based on these restrictions.
72
The new near-normal as well as reduced dimensionality space, Ψ, and its inverse transformation were determined and described by *
1
Ψλ = a − R λ2
(3-6)
and *
R λ = (a − Ψλ )2 ,
(3-7)
where Ψλ represents the new linear vector representation of an opaque colorant and *
*
a ≅ 1 which is empirically determined from a set of samples requiring reproduction. The *
process of optimizing a and the MATLAB programs are attached in Appendix B. *
Determination of a is to perform the transformation by Eq. (3-6) such that the resultant set of Ψλ is of the requested dimensionality upon color reproduction, i.e., the Ψλ for photographic material should be three dimensional. The use of square root transformation of the spectral reflectance factors improves normality (Johnson and Wichern, 1992) and the offset term is required to account for a subtractive opaque coloration.
C. DIMENSIONALITY REDUCTION: NORMAL VERSUS NONNORMAL POPULATIONS
The proof for multivariate normality used as a beneficial property for sample reproduction requiring data (dimension) reduction is to construct both multivariate normal and non-normal populations with known dimensionality and employ a fewer number of eigenvectors for spectral reconstruction. The metric for judging the reconstruction
73
accuracy should be the RMS error of reflectance factors, Rλ. If the Ψ or Φ space is used for linear modeling, then minimizing the RMS error of Ψλ or Φλ is not equivalent to minimizing the RMS of Rλ since the transformations is nonlinear for the Ψ or Φ spaces. In order to compare the reconstruction accuracy in terms of RMS error, the comparison for the superiority between a normal and a non-normal set linearly modeled by PCA is convenient using reflectance space. Hence, the simulation discussed next will use reflectance space for color mixing though it is not realizable by subtractive coloration, and the colorimetric error and metameric index can be shown. Colorimetric accuracy of eigenvector reconstruction is specified by ∆E*94 color difference equation (CIE Technical Report, 1995) under CIE standard illuminant D65 and 1931 standard observer. Spectral accuracy is indicated by metameric index based on parameric correction (Fairman, 1987) for D65 and the 1931 observer followed by a color difference calculation for illuminant A. The parameric correction is equivalent to modulating colorant concentration such that an exact colorimetric match is achieved for D65 and the color difference under A indicated the degree of metamerism. The higher the metamerism index, the larger the degree of spectral mismatch. The multivariate normal sample set is generated by six linearly independent colorant vectors modulated by coefficients which are randomly sampled from a beta distribution with parameters α = 5 and β = 5 (Dougherty, 1990). Hence, the resultant population, which is the linear combination of six bell-shape marginal distributions, is
74
approximately multivariate normally distributed. On the contrary, the same six colorant vectors were modulated by coefficients which come from six beta distributions with six different combinations of α and β to create a non-normal distribution. Sample sets of five hundred six-colorant mixtures, one multivariate normal and one non-normal, were created computationally. Normality is conformed by the inspection of its normal plot, shown in Fig. 3-2, revealing that the each marginal distribution is approximately normally distributed (straight lines).
Non-normality is conformed by the non-normal marginal
distributions plotted in Fig. 3-3.
0.999 0.997 0.99 0.98 0.95
Probability
0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001 0.1
0.2
0.3
0.4 0.5 Reflectance factor
0.6
0.7
Fig. 3-2: The normal plot of simulated reflectance factors at each sample wavelength obtained by linear combinations of six approximately normal distributions.
75
0.999 0.997 0.99 0.98 0.95
Probability
0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001 0.1
0.2
0.3
0.4 0.5 Reflectance factor
0.6
0.7
0.8
Fig. 3-3:The normal plot of simulated reflectance factors at each sample wavelength generated by linear combinations of six non-normal distributions. Since the two sample sets are distributed in six dimensional spaces, six eigenvectors should ideally span the entire six dimensional sample space. This simulation is to demonstrate that it is possible to approximate the entire six dimensional sample space by lower dimensional subspaces if they are multivariate normally distributed. Table 3-1 shows the colorimetric and spectral accuracy of the multivariate normal and the non-normal sample sets reconstructed with different dimensionalities at a random simulation. By comparing the two sets, the dimensionality can be reduced to three dimensional for the multivariate normal sample set indicated by low RMS error and satisfactory colorimatric
76
and spectral performance. Whereas, the non-normal sample set can not be effectively approximated below four dimensional reconstruction. Table 3-1: The colorimetric and spectral accuracy of the multivariate normal and nonnormal sample sets reconstructed with different number of dimensions, where Stdev stands for the standard deviation and RMS representing the total root mean square error of the reconstructed reflectance spectra. Multivariate Normal Sample set ∆E*94 Metamerism Index (∆E*94) Dimensionality six five four three six five four three Mean 0.0 0.0 0.8 0.9 0.0 0.0 0.1 0.3 Stdev 0.0 0.0 0.4 0.6 0.0 0.0 0.0 0.2 Maximum 0.0 0.1 2.0 2.9 0.0 0.0 0.2 1.3 Minimum 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 RMS 0.000 0.004 0.005 0.008
Dimensionality six Mean 0.0 SDV 0.0 Maximum 0.0 Minimum 0.0 RMS 0.000
Non-Normal Sample set five four three six 0.1 1.5 3.1 0.0 0.0 0.8 1.3 0.0 0.2 4.9 7.1 0.0 0.0 0.0 0.2 0.0 0.007 0.009 0.023
five 0.0 0.0 0.0 0.0
four 0.1 0.1 0.5 0.0
three 1.5 0.7 3.2 0.1
D. VERIFICATIONS
The first sample set used was the ANSI IT8.7/2 reflection target measured with a Gretag SPM 60 spectrophotometer. Since the IT8.7/2 reflection target is manufactured with three dyes, dimensionality should be theoretically three in a vector space via a suitable transformation. The Kubelka-Munk transformation for a transparent colorant layer in optical contact with an opaque support, Eqs. (3-4) and (3-5), was utilized. Table
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3-2 shows the percent variance (PV) of the significant eigenvectors calculated in both reflectance and absorption space of the IT8.7/2 reflection target. Table 3-2: The percent variance by eigenvector analysis in both reflectance and absorption space of IT8.7/2 reflection target.
The ith Eigenvector
Eigenvalue
1 2 3 4 5
0.9474 0.1845 0.0415 0.0033 0.0006
Reflectance Absorption Percent Cumulative Eigenvalue Percent Cumulative Variance PV % Variance PV % (PV) % (PV) % 80.43 80.43 7.5702 74.25 74.25 15.66 96.09 1.9038 18.67 92.92 0.35 99.62 0.7172 7.03 99.96 0.28 99.90 0.05 99.95
PCA requires five eigenvectors in reflectance space to explain the same variance of IT8.7/2 reflection target as that of absorption space with three eigenvectors. If Eq. (3-5) transforms the measured spectral reflectance factor of the IT8.7/2 reflection target into the representation of spectral absorption whose dimensionality is exactly three, then the multivariate normality of IT8.7/2 reflection target in absorption space is not crucial since three eigenvectors in absorption space already span the entire absorption space of IT8.7/2 reflection target, i.e., no data (dimensionality) reduction is required. Since the spectral measurements in absorption space for the IT8.7/2 reflection target is approximately three dimensional, properly modeled by Eqs. (3-4) and (3-5), the multivariate normality in absorption space is viewed as less crucial for three-eigenvector reconstruction. On the contrary, the dimensionality of the reflectance space of IT8.7/2 reflection target is
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obviously beyond three dimensions. In order to achieve high colorimatric and spectral accuracy by three-eigenvector reconstruction in reflectance space, the multivariate normality in reflectance space for the IT8.7/2 reflection target
is crucial for data
(dimension) reduction. The normality plot of reflectance factors at each sample wavelength (marginal distribution) is shown in Fig. 3-4.
0.997 0.99 0.98 0.95
P ro b a b ilit y
0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0
0.1
0.2
0.3
0.4 0.5 R e fle c t a nc e fa c to r
0.6
0.7
0.8
Fig. 3-4: The normal plot of marginal distributions of IT8.7/2 reflection target in reflectance space. Since the multivariate-normality is less crucial for IT8.7/2 reflection target in absorption space, the attention is focused on reflectance space. Recall the necessary condition of multivariate normal distribution is that the marginal distribution must be normal. By inspecting Fig. 3-4, the degree of normality is low as shown by the large amount of curvature. It is expected that three-eigenvector reconstruction in reflectance
79
will not yield satisfactory accuracy. The performance of three-eigenvector reconstruction and the statistical results of colorimetric and spectral metrics in both spaces are listed in Table 3-3. Table 3-3: The colorimetric and spectral performance of three eigenvector reconstruction in both spaces for an IT8.7/2 reflection target.
Mean Stdev Maximum Minimum RMS
∆E*94 1.8 1.5 8.5 0.0
Reflectance Metamerism Index 0.6 0.6 3.6 0.0 0.014
∆E*94 0.5 0.2 1.0 0.0
Absorption Metamerism Index 0.1 0.1 0.4 0.0 0.006
From Table 3-3, the colorimetric and spectral accuracy is far superior in absorption space compared with reflectance space. Tables 3-2 and 3-3 assure that Kubelka-Munk equations, Eqs. (3-4) and (3-5), are suitable transformations to model the IT8.7/2 reflection target. The accuracy by three-eigenvector reconstruction in absorption space agrees with the fact that the IT8.7/2 reflection target is comprised of three dyes. Modeling the IT8.7/2 reflection target should be a three dimensional problem instead of five dimensional. The ideal reproduction should be achieved using three eigenvectors since the IT8.7/2 reflection target is three dimensional. Whereas, the measurement noise and validity of Kubelka-Munk transparent transformation affect the accuracy. The analysis for IT8.7/2 is based on the Kubelka-Munk turbid media theory while color materials behave similarly to the underlying theoretical requirements. However,
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when the absorption and scattering behavior of the object violates the two flux assumption, spatial averaging occurs due to a large field of view, noise contributes to the measurements, and objects are not completely opaque, the Kubelka-Munk opaque transformations Eqs. (3-2) and (3-3) are no longer valid. The proposed transformation methods, shown as Eqs. (3-6) and (3-7), are employed. The second verification was performed on a paint target of 141 patches created by mixing six linearly independent Galeria acrylic colors which were measured using a Macbeth Color-Eye 7000 integrating sphere spectrophotometer with specular component included. By knowing that this set of samples is theoretically six dimensional, the spectral reproductions by the six significant eigenvectros in R (reflectance), Ψ, and Φ space were performed. The colorimetric and spectral accuracy, RMS, and percent variance (PV) explained by the six significant eigenvectros are tabulated in Table 3-4. Six-eigenvector reconstruction in Ψ space yielded the best performance in light of metameric and RMS error. Interestingly, the percent variance explained by six-eigenvectors in Φ space is the highest (99.92%) among the three, whereas, the spectral reconstruction yielded the highest RMS error, hence, the lowest colorimetric and spectral accuracy. Table 3-4: The statistical performance by six-eigenvector reconstruction for 141 acrylic colors in R, Ψ and Φ spaces.
Space R Ψ Φ
∆E*94 Metamerism Index (∆E*94) Mean SDV Max Min Mean SDV Max Min RMS 0.4 0.3 1.6 0.0 0.2 0.2 1.0 0.0 0.010 0.3 0.2 0.8 0.0 0.1 0.1 0.7 0.0 0.007 1.0 1.7 6.6 0.0 0.3 0.4 1.8 0.0 0.026
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PV (%) 99.84 99.85 99.92
Another verification was performed on a set of 105 color patches created by another set of six linearly independent colorants (two Sakura poster colors and four Pentel poster colors) using the same measuring instrument and geometry described above. The colorimetric and spectral accuracy, RMS, and percent variance explained by the six significant eigenvectros are shown in Table 3-5. Table 3-5: The statistical performance by six-eigenvector reconstruction for 105 poster colors in R, Ψ ,and Φ spaces.
Space R Ψ Φ
∆E*94 Metamerism Index (∆E*94) Mean SDV Max Min Mean SDV Max Min RMS 1.0 0.7 2.8 0.1 0.3 0.2 0.7 0.0 0.012 0.5 0.3 1.1 0.1 0.2 0.1 0.4 0.0 0.007 0.5 0.3 1.9 0.0 0.1 0.1 0.5 0.0 0.012
PV (%) 99.60 99.70 99.98
This results is again showing that even though the percent variance explained by six eigenvectors is the highest in Φ space (99.98%), the spectral reconstruction can still go wrong due to its nonlinear inverse transformation, Eq. (3-3). Figure 3-5 shows the nonlinear transformation is acting as an high gain amplifier at the low Φ value corresponding to high reflectance factor. Even a tiny mismatch around the low Φ region will be enhanced after inverse transformation causing high colorimetric error. Furthermore, the spectral regions with low Φ components are most likely reconstructed with error based on PCA if the number of eigenvectors is not sufficient for reconstruction. Kubelka-Munk transformation is, thus, sensitive to the real dimensionality of a sample set by this observation.
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1 0.9 0.8
Reflectance factor
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
5
10
15
K/S
Fig. 3-5: The Kubelka-Munk inverse transformation, Eq. (3-3), for opaque color. Figure 3-6 shows the spectral error is enhanced at the low Φ spectral region while the error is suppressed at the high Φ spectral region after Kubelka-Munk inverse transformation for opaque coloration. The proposed transformation was designed to overcome this type of problem by first ensuring that the dimensionality in Ψ space meets the demanded dimensionality; and second transforming from Ψ to R space, shown in Fig. 3-7, by Eq. (3-7) with less steepness does not over-amplify the spectral mismatch from Ψ space.
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2
1 0.9
1.5
0.8
Original Φ
0.7
Eq. (3-3)
Original R
K/S
0.5
Reconstructed Φ
0.5
0.6
Reflectancefactor
1
0.4
Reconstructed R
0.3
0
0.2 -0.5 400
450
500
550 Wavelength
600
650
0.1 400
700
450
500
550 Wavelength
600
650
700
Fig. 3-6: An example of enhanced spectral error after the transformation by Eq. (3-3).
1 0.9
R efle cta nce fa c to r
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
Fig. 3-7: The transformation from Ψ to R space by Eq. (3-7). Although the multivariate normality among three spaces is not easy to conclude by inspecting their normal plot (not shown) for these two examples, the example of IT8.7/2
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shown above as well as the observation on the percent variance indicate that multivariate normality allow a larger degree of data (dimensionality) reduction. The derivation for empirical transformation to Ψ space is based on the beneficial factor of multivariate normality.
E. CONCLUSIONS
Several linear colorant mixing spaces were discussed. Given a sample set, accurate spectral reconstruction can take advantage of the normality of a sample set represented in a linear space when reconstruction is limited to a lower number of dimensions. It was shown by the numerical simulation using linear combinations of basis colorant whose concentrations are generated from several approximately normal distributions and the example of IT8.7/2. Spectral error essentially lies in the higher dimensions beyond the reconstructibility with constrained dimensionality. These two examples imply that normality has lower magnitude of higher dimensional error such that PCA confined to a limited dimensionality can model a sample set with satisfactory accuracy. A new transformation was empirically defined when encountering a set of sample measurements modeled by PCA. The transformation to a reduced dimensionality and near normal space for PCA estimation can well approximate the sample measurements. It is also designed to account for the opaque coloration of physical materials. Examples shown for verification are highly accurate in terms of colorimetric and spectral performance.
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IV. COLORANT ESTIMATION OF ORIGINAL OBJECTS The goal of research for colorant estimation is to determine a set of six basis colorants which are the best representation of original objects such as paintings. That is, their spectral information can be accurately reconstructed by linear combinations of the six estimated colorants represented in a linear colorant mixing space. Since each painting is possibly created by different colorants, the six estimated colorants are image dependent. The clue leading to the six estimated colorants is the six eigenvectors determined from the corresponding spectral measurements. The relationship between the six eigenvectors and estimated colorants is merely the linear transformation (or geometrical rotation). Based on these observations, a constrained-rotation engine using MATLAB was devised to perform the transformation from the eigenvectors to a set of all-positive vectors as the estimated colorants. Once a set of reasonable colorants is uncovered, this set of colorants can be used to synthesize the original artwork with the least metameric effect between the reproductions and originals. This chapter will show the derivation of the relationship between eigenvectors and the statistical primaries as the desired colorant for synthesis of an original object. For simplicity of communication, all linear modeling techniques will utilize the Φ space for discussions. All the algorithms derived for this module were also applied in Ψ space.
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A. APPROXIMATELY LINEAR COLORANT MIXING SPACE
Kubelka-Munk turbid media theory is used as the first-order approximation transforming spectral reflectance factor, Rλ, into an approximately linear space, defined as Φ space. The mathematical description for linear color mixing, specified as Eq. (3-1), is re-expressed by Eq. (4-1), k
Φ λ , miture = ∑ c i φ λ , i ,
(4-1)
i =1
where c represents concentration and (k/s)λ is replaced with φλ. As an example, a still life painting of a floral arrangement was produced with six independent acrylic paints shown in Fig. 4-1.
Fig. 4-1: The still life painting of a floral arrangement creating with six opaque colorants.
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Each paint was applied on a paper stock at a thickness achieving opacity and measured spectrally using a Gretag SPM 60 spectrophotometer. Each reflectance vector had thirtyone components: 400 nm - 700 nm at 10 nm bandwidths and intervals. These reflectance vectors were transformed to Φλ. Ideally, one needs thirty-one “spectral colorants” with 10 nm bandwidth absorption and scattering properties at the sampled wavelengths in order to reconstruct the measured sample spectra. Realistically, colorants do not have such narrow band properties. Furthermore, reproducing a color by mixing thirty-one colorants is highly impractical for any real coloration process. Fortunately, chromatic stimuli are not originally created by such spectral colorants; hence, their Φλ do not span the entire thirtyone dimensional Φλ space.
Instead, they are distributed in a lower dimensional Φλ
subspace. If an original painting was only painted, for example, using six independent colorants, then, ideally, the measured set of Φλ should be distributed only in a sixdimensional subspace of Φλ space.
B. PRINCIPAL COMPONENT ANALYSIS
Principal component analysis (PCA) can provide a measure to statistically determine the dimensionality of the sample population. The linear combinations of the first p eigenvectors should describe the entire set of Φλ if the original was created by p colorants, i.e.,
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p
Φ λ , sample = ∑ b i e λ , i ,
(4-2)
i =1
where eλ, i is the ith eigenvector and bi is the corresponding coefficient to reconstruct a sample. Rewriting Eq. (4-2) in matrix form: Φ = EB.
(4-3)
E is the matrix of the first six eigenvectors and B is the coefficient matrix to reconstruct the sample population, Φ. Figure 4-2 shows the first six eigenvectors which were obtained from the still life painting explaining the most sample variations (99.98%) in Φλ space.
K /S
The 1 s t e ig e nve c tor.
The 2 nd e ig e nve c to r. 1
1
0 .5
0 .5
0 .5
0
0
0
-0 .5
-0 .5
-0 .5
-1 400
500
600
700
-1 400
The 4 th e ig e nve c tor.
K /S
The 3 rd e ig enve c to r.
1
500
600
700
-1 400
The 5 th e ig e nve c tor. 1
1
0 .5
0 .5
0 .5
0
0
0
-0 .5
-0 .5
-0 .5
500
600
W a ve le ng th
700
-1 400
500
600
W a ve le ng th
600
700
-1 400
500
600
W a ve le ng th
Fig. 4-2: The six eigenvectors obtained from the still life painting.
89
700
The 6 th e ig e nve c tor.
1
-1 400
500
700
C. COLORANT ESTIMATION
As shown in Fig. 4-2, the thirty-one components of the eigenvectors are often bipolar; consequently, they are not a set of physical colorants. Furthermore, their corresponding
coefficient
vectors
are
also
bipolar
not
representing
physical
concentrations. Real colorants should have all-positive Φλ as their vector components, and the corresponding concentrations should be all-positive. Since an original was created by mixing a set of existing physical colorants at different concentrations and the color mixing operation is mathematically described by Eq. (4-1), the sampled Φ are distributed in an allpositive space. Rewriting Eq. (4-1) in matrix form: Φ = φC,
(4-4)
where φ is the matrix of the basis colorants and C is the concentration matrix to reconstruct the sample population, Φ. Notice that Eq. (4-4) can be equated with Eq. (4-3) in order to obtain the relationship between the eigenvectors and the φλ of the basis colorants used for creating the original painting. Based on this observation, the relationship between the eigenvectors and the physical basis colorants is merely a linear transformation, or a geometric rotation. Since Φ = EB = φC,
(4-5)
φ = EBC- = EM,
(4-6)
this implies that
90
where C- stands for the pseudo-inverse of the concentration matrix and M is the representation of the matrix product of B and C-. The linear transformation from eigenvectors to physical basis colorants should result in two important properties. First, the rotated eigenvectors should be a set of all-positive vectors. Second, the concentration matrix should have all non-negative entries. These two constraints should result in colorant spectra that are very similar or linearly related to the actual colorants. This constrained rotation was previously performed by Ohta (1973). His research goal was to estimate the spectral density curves of an unknown dye set for photographic materials using only the spectra of color mixtures such as ANSI IT8 targets. A Monte Carlo method was also used to help identify the most likely dye set. It was a three dimensional vector transformation. This research extends the challenge to six dimensions. In the current analysis, a constrained-rotation engine using MATLAB as the calculation platform was devised to solve the problem. Since the ultimate goal of this research is to identify a set of printing inks that minimize metamerism between a set of objects and their printed reproduction, the dimensionality is limited to six, corresponding to six printing stations. If the dimensionality of the original Φ is greater than six, or if there is appreciable spectral measurement error, residual errors will result. Hence, goodness metrics are required. The spectral accuracy was quantified by an index of metamerism that consists of both a parameric correction for D50 and the use of CIE94 under illuminant A. The colorimetric accuracy is calculated using CIE94 under D50 for the 1931 observer.
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D. JUSTIFICATION OF EIGENVECTOR RECONSTRUCTION WITHOUT SAMPLE MEAN
In practice, the measured samples often reveal more than p dimensions due to measurement noise and limitations in the validity of the Kubelka-Munk transformations. Given the p limited dimensions for spectral reconstruction, one should employ Eq. (4-2) together with the sample mean for better accuracy. That is, p
Φ λ , sample = ∑ b i e λ , i + Φ λ , sample i =1
mean
.
(4-7)
The existence of a sample mean for spectral reconstruction poses several difficulties for this research. First, the sample mean is only a statistical result which specifies the average Φλ behavior for the set of samples. The sample mean does not represent any physical colorant. Second, in Eq. (4-7), the sample mean is acting as an offset vector which impedes the equality relationship in Eq. (4-5). Since the eigenvectors are the only clue leading to a set of possible colorants, the sample mean must be excluded for maintaining the rotation relationship between eigenvectors and the set of possible colorants which is specified by Eq. (4-6). Finally, the confidence for excluding the sample mean is that if the dimensionality of sample population is approximately the constrained number of dimensions, then the sample mean approximately resides in the reconstructed sample population. That is, the sample mean can be approximately expressed as a linear
92
combination by the limited number of eigenvectros. Henceforth, the sample mean in Eq. (4-7) can be excluded without significant error, i.e., Eq. (4-2) will be used.
E. VERIFICATIONS
Testing the Constrained-Rotation Engine by a Virtual Sample Population The constrained-rotation engine was first tested for a virtual sample population with three thousand random mixtures created by linear combinations of the six acrylicpaint spectra in Φλ space. These six spectra, plotted in Fig. 4-3, were carefully chosen and verified to be independent colorant vectors, i.e., no one colorant vector can be expressed as the linear combination of any other five colorant vectors. 1.2
Normalized K/S
1
yellow
0.8
magenta 0.6
cyan green
0.4
blue black
0.2
700
680
660
640
620
600
580
560
540
520
500
480
460
440
420
400
0 Wavelength
Fig. 4-3: The six acrylic paints used for the computer generated sample population.
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Hence, the virtual sample population is ensured to be six dimensional. The corresponding concentration vectors were randomly generated from uniform distributions. Thus, the resulted population of linear combinations of six uniform distributions is approximately multivariate normally distributed (convolution of six uniform distributions is approximately normal). Given that real sample populations can be confounded by processes and measurements, the idea of using the computer generated sample population to test the constrained-rotation engine is to provide a noise free sample population. This ensures that the rotated eigenvectors with all-positive vector components as the estimated colorant spectra should be identical or linearly related to the six acrylic-paint spectra if the proposed vector transformation theory expressed as Eq. (4-5) is valid. Six-eigenvector reconstruction without a sample mean vector, based on Eq. (4-2), yielded approximately zero spectral errors, hence, zero colorimetric errors since full dimensionalty was employed. Then, an arbitrary set of six colorant vectors were used as the initial values for the constrained-rotation engine. The resultant all-positive eigenvectors as the set of estimated colorant vectors are identical to the original six acrylic-paint spectra. In addition, another set of six block spectra evenly spaced within 400 nm to 700 nm representing an initial colorant vectors was utilized and the resulted estimated colorant vectors were also identical to the six acrylic-paint spectra. This is surprising since the vector transformation can not be unique; multiple solutions should
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exist. Whereas, those solution are linearly related with each other since they all are the linear transformations of the six eigenvectors. Thus, the first test shows that the constrained-rotation engine is able to converge to an all-positive representation of the eigenvectors. The MATLAB program for the implementation of constrained-rotation is attached in Appendix B. Colorant Estimation for a Kodak Q60C Target The second verification was performed on a Kodak Q60C, a photographic reflection target that was a precursor to the ANSI IT8 target. Three eigenvector reconstruction should yield low spectral and colorimetric errors corresponding to the fact that it is manufactured using three dyes. The Kulbeka-Munk transformation for transparent materials was used to transform reflectance factor to absorption. The spectral and colorimetric accuracy, based on the three-eigenvector reconstruction, is shown in Table 4-1. Ideally, this is a three dimensional problem. Whereas, the spectral and colorimetric accuracy is confounded by the manufacturing, processing and measuring noise, and the model accuracy limitations of Kulbeka-Munk theory. Table 4-1: The spectral and colorimetric accuracy of the three-eigenvector reconstruction for the Kodak Q60C.
Mean Stdev Max Min
∆E*94 Metamerism Index 0.48 0.19 0.20 0.17 1.12 1.00 0.00 0.00
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Uncovering the set of all-positive eigenvectors as the estimated dye spectra of the Q60C was preceded by using the first eigenvectors of cyan, magenta, and yellow ramps of the Q60C as the initial colorant vectors. The first eigenvector of each ramp, denoted as the "local eigenvector," is the first statistical estimation of the real dye spectrum (Berns and Shyu, 1995). By this approach, the advantage is to get a close solution and help expedite the rotation process. The estimated dye spectra (thick lines) and the local eigenvectors (dotted lines) are plotted in Fig. 4-4. Since the all-positive eigenvectors representing the estimated dyes are an exact linear transformation of the first three eigenvectors, denoted as global eigenvectors determined from the Kodak Q60C target, the spectral and colorimetric performance of estimated dyes is the same as that of global eigenvectors.
Normalized absorption coefficient
1.2
1
0.8
0.6
0.4
0.2
700
680
660
640
620
600
580
560
540
520
500
480
460
440
420
400
0 Wavelength
Fig. 4-4: The all-positive eigenvectors as the estimated dye spectra (thick line) and the local eigenvectors (dotted lines).
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It was found that the local eigenvectors were not the exact transformation of the global eigenvectors. The spectral reconstructibility by local eigenvectors was worse than that of the estimated dyes, as expected. Furthermore, the broader absorption bandwidths of the local eigenvectors symbolize the possible impurity contamination during the manufacturing, processing, and measuring. In comparison, the all-positive eigenvectors as the estimated dye spectra showing narrower absorption bandwidths may be close to the real dye spectra based on the support of low spectral and colorimetric errors. The testing for the proposed colorant-estimation engine favors the sense of reverse engineering, i.e., uncovering the spectral structures of real colorants. However, for the current research applications, it needs only one reasonable set of colorant spectra which can be used to search through the existing ink database or for a colorant chemist to synthesize the exact inks. Once one exact or similar set of inks is selected, spectral-based printing process can utilize the selected ink set to fulfill the least metameric reproduction. Hence, it is not critical for the proposed colorant estimation engine to converge to the exact colorant spectra which were used to manufacture the colored objects. Colorant Estimation for the Still Life Painting Another verification for the constrained-rotation engine was performed by spectral measurements of the still life painting mentioned previously. The painting was painted by six independent acrylic-paints whose Φλ spectra are plotted in Fig. 4-3. One hundred and twenty-six samples were obtained to represent the entire Φλ space of the painting whose
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six eigenvectors plotted in Fig. 4-2 explaining 99.98% of total variation indicated that this sample population is approximately six dimensional. The spectral and colorimetric accuracy of the six-eigenvector reconstruction is specified in Table 4-2. Table 4-2: The spectral and colorimetric accuracy of the six-eigenvector reconstruction for the still life painting.
Mean Stdev Max Min
∆E*94 Metamerism Index 0.21 0.18 0.14 0.16 0.75 0.95 0.02 0.01
Initially, the colorant estimation was intended to directly rotate the six eigenvectors to one set of all-positive representations. The resultant colorant spectra are plotted in Fig. 4-5 and show that there is a colorant (thin dotted line) with various absorption bands across the visible spectral region and the reasonable appearance of the rest of the five colorants. Several sets of colorant vectors were used as the initial estimation for the constrainedrotation engine. The resulting sets of estimated colorants all possessed the similar spectral properties. These initial attempts did not reveal the existence of a neutral colorant judged by the lack of a flat spectrum.
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1.2
Normalized K/S
1
0.8
0.6
0.4
0.2
700
680
660
640
620
600
580
560
540
520
500
480
460
440
420
400
0 Wavelength
Fig. 4-5: The six all-positive eigenvectors as the estimated colorants for the still life painting. The neutral colorant with an approximately flat spectrum can be approximated by the linear combination of the rest of the five estimated colorants. The lack of a neutral colorant indicated that the rest of the five estimated colorants did not explain sufficient spectral variation. Since the current research aims to uncover one neutral and five chromatic colorants for printing processes, the approach was to constrain the assumption of the existence of the neutral colorant. Hence, the colorant estimation for the still life painting was proceeded by: first, estimate the neutral colorant using linear regression to fit the perfect flat spectrum by the six eigenvectors. Second, rotate the most significant five eigenvectors to their all-positive representations. The resultant estimated colorants should explain a higher degree of spectral variation once the neutral dimension is constrained. The
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spectral and colorimetric accuracy of the resultant six estimated colorants is shown in Table 4-3 and their spectral curves (solid lines) are simultaneously plotted with the six original colorants (astroidal lines) used for the still life painting in Fig. 4-6. Table 4-3: The spectral and colorimetric accuracy of the six estimated colorants for the still life painting. ∆E*94 Metamerism Index 0.22 0.21 0.16 0.18 0.92 1.01 0.02 0.01
Mean Stdev Max Min
Normalized K/S
Y ellow
M agenta
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0 400
500
600
700
0 400
G reen Normalized K/S
Cyan
500
600
700
0 400
Blue 1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
500 600 W avelength
700
0 400
500 600 W avelength
600
700
Neutral
1
0 400
500
700
0 400
500 600 W avelength
700
Fig. 4-6: The estimated colorants (solid lines) and the original colorants (astroidal lines) used for the still life painting.
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The constrained-rotation of the six eigenvectors obtained from the still life painting yielded a reasonable set of estimated colorants. Judging from them, most colorants have similar spectral properties to the original which were utilized to create the still life painting. Whereas, the spectral property similar to green is absent in this set of estimated colorants. Instead, the constrained-rotation process gave out a spectrum equivalent to a yellow colorant. This can be attributed to the sampling error due to the usage of larger aperture size of spectrophotometer which violates the additive assumption in Φλ space. It was discussed in Chapter III that the possible field of view of a spectrophotometer may sample at a multi-color surface. Once the spectrophotometer samples at a spot where two or more contiguous colors are within its field of view, the reading is equivalent to the additive result of the spectral reflectance factors confined by the spot whose spectral reflectance is contributed by that of the two or more colors. Furthermore, the additive operation in reflectance space undergoing a nonlinear transformation such as Eqs. (3-2) and (3-3) results in the additive operation undefined in Φλ space. As a consequence, the behavior of samples in Φλ space are not predictable by the linear model of Eq. (4-2). This type of sampling error can be reduced when the spectral reflectance factor of a color object is estimated by a high resolution CCD camera with very narrow field of views for each pixel. The under-sampling of green and over-sampling of yellow-orange color are the other source of errors which cause the estimated colorants not to agree with the original colorant. Once the sample gamut is approximately uniform, i.e., each color has
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approximately equal probability of occurrence in the sample population, this type of error is minimized. Since the sample gamut of the still life was carefully controlled to be as uniform as possible, the lack of green colorant is mainly caused by the violation of the additive assumption in Φλ space due to the large field of view of the spectrophotometer. Colorant Estimation for 105 Mixtures Using Ψ Space Since the non-optimal aperture size of a spectrophotometer leads to sampling error and a spectral image captured by a multi-spectral acquisition system is still under development, the rational proof for the validity of the constrained-rotation mechanism is to avoid the type of sampling error leading to the additive failure in a linear color mixing space. Although the verification for Kodak Q60 target does not suffer this sampling error since the measurements were performed on color patches, this section repeats the verification with a set of six-color mixtures whose underlying primaries are known. If the rotation results can converge or nearly go to the original primary colorants, then it not only verifies the validity of the proposed rotation algorithms but also confirms the effectiveness of the linear colorant mixing space utilized for analysis. Accordingly, a set of 105 six-color mixtures, shown in Fig. 4-7, created by hand mixing six opaque poster paints (Sakura cerulean blue No. 25, Sakura rose violet No. 22, Pentel yellow No. 5, Pentel sap green No. 63, Pentel ultramarine N0. 25, and Pentel black No. 28), whose spectral reflectance factor are shown in Fig. 4-8, which are the exact set of colorants to paint the still life painting. Measurements were done by using a Macbeth
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Color Eye 7000 spectrophotometer with SPEX measuring geometry and an integrating sphere.
Fig. 4-7: The 105 mixtures created by hand mixing six opaque poster paints.
R e f e lc t a n c e
C e r u le a n b lu e
R o s e v i o le t 1
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1
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U lt r a m a r i n e
S a p g re e n
R e f e lc t a n c e
Y e llo w
1
700
0 400
500
W a v e le n g t h
Fig. 4-8: The spectral reflectance factors of the six poster colors used for creating the 105 opaque mixtures.
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Since the accuracy of Kubelka-Munk transformations for opaque colorant, Eqs (3-2) and (3-3), did not provide sufficient accuracy, the linear colorant mixing space obtained by the empirical transformation, Eq. (3-6), was utilized for analysis and process. The offset v
vector a , shown in Fig. 4-9 and in Eq. (3-6), was optimized according to the 105 mixtures in Ψ space such that the 105 mixtures in Ψ space are distributed in a near six dimensional vector space. The optimization procedures are shown in Appendix B. Six eigenvectors, shown in Fig. 4-10, explain 99.70% sample variation in Ψ space. The colorimetric and spectral accuracy of six eigenvector reconstruction is listed in Table 4-4. 1
0 .9
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Ψ
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0 400
450
500
550
600
650
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W a ve le n g th
v
Fig. 4-9: The offset vector, a , for transforming the 105 mixtures to Ψ space. From Table 4-4, the colorimetric and spectral errors are low for the six-eignvector reconstruction. This indicates that the 105 mixtures distributed in Ψ space are approximately six dimensional.
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Ψ
T he 1 s t e ig e nve c to r.
T he 2 nd e ig e nve c to r. 1
1
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T he 4 th e ig e nve c to r.
Ψ
T he 3 rd e ig e nve c to r.
1
5 00
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T he 5 th e ig e nve c to r. 1
1
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5 00 6 00 W a ve le ng th
7 00
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6 00
7 00
T he 6 th e ig e nve c to r.
1
-1 4 00
5 00
7 00
-1 4 00
5 00 6 00 W a ve le ng th
7 00
Fig. 4-10: The six eigenvectors of the 105 mixtures in Ψ space. Table 4-4: The colorimetric and spectral accuracy of the six eigenvector reconstruction for the 105 mixtures.
Mean Stdev Max Min RMS
∆E*94 Metamerism Index 0.22 0.21 0.16 0.18 0.92 1.01 0.02 0.01 0.007
Constrained rotation by the proposed algorithms were performed on the determined six eigenvectors derived from the 105 mixtures in Ψ space. Since the a priori knowledge about the primary colorants is known, the spectra of the six primaries were used as the initial vectors for rotation process. It was desired to conclude whether the
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constrained-rotation mechanism can perform "reverse engineering" if the initial colorant vectors were known for comparing to the estimated colorants, though it is not necessary for this research. The six rotated all-positive eigenvectors as the statistical primaries whose reflectance spectra are shown in Appendix G, shown in Fig. 4-11, are plotted with original six primary colorant in reflectance space. Five chromatic colorants were normalized to 0.9 units of reflectance factor and the black colorant was normalized to 0.1 and shifted up 0.8 to 0.9 units of reflectance factor for visual comparison.
R e fe lc ta n c e
C e ru le a n b lu e vs . P ri m a ry 1
R o s e vi o le t vs . P ri m a ry 2 1
1
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S a p g re e n vs . P ri m a ry 4
R e fe lc ta n c e
Ye llo w v s . P ri m a ry 3
1
1
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1
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W a ve le n g th
Fig. 4-11: The reflectance factors of the six original colorants (solid lines) and statistical primaries (dashed lines) derived by a constrained rotation from the six eigenvectors.
From Fig. 4-11, the six statistical primaries did not converge to the original colorants. They converged to a set of close solutions. The agreement between the six
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original and statistical primaries are high. Although the statistical primaries are not identical to the six originals, the probable causes leading to this discrepancy are unwanted contamination, measuring error, and the validity of the empirical transformation by Eq. (36). It was tested by a linear regression model to see if the six original primaries span the 105 mixtures in Ψ space, that is, for any vector of the 105 mixtures is a linear combination of the six original primaries. It was found that the six original primary do not span the 105 colorants in Ψ space judged by that negative concentrations were required to synthesis a spectrum. This implies that the six original primaries only span a partial colorant space that does not include all of the 105 mixtures. In another words, the six original primaries do not explain all the spectral variation of the 105 mixtures. This is strong evidence for unwanted contamination when hand mixing the 105 mixtures. Since the statistical primaries approximately span the entire 105 mixtures in Ψ space and the measurement was done by a highly accurate instrument. The unwanted contamination is attributed to be the main cause for the discrepancy. Nevertheless, the correlation between the six original and statistical primaries is high. The statistical primaries can be used as the basis information for synthesizing an object such as a painting created by unknown colorants.
I. CONCLUSIONS
An algorithm was developed for the colorant estimation of original objects through vector analysis and principal component analysis. The relationship between basis colorants
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and eigenvectors is elucidated by performing a constrained linear transformation. Since the basis colorants used for creating original objects can be statistically uncovered with sufficient accuracy, the color reproduction at the synthesis stage gains the maximum capability to spectrally reconstruct a sample from the original. Therefore, metamerism between the reproduction and original is minimized.
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V. OPTIMAL INK SELECTION Given a spectral image, the mission of spectral reproduction using a multiple-ink printing process can be a challenging task.
Even though the number of degrees of
freedom is increased, the spectrum corresponding to every pixel in a given spectral image may not well be inside the spectral gamut of the multiple-ink process. Theoretically, a given spectral image whose spectra of all pixels should be located inside the spectral gamut of the multiple-ink printing process. Then each spectrum requiring reproduction is described by a set mathematical functions of its printing primaries. The analytical descriptions of a multiple-ink printing process are discussed in chapter VII. Therefore, the set of printing primaries dominate the capability of spectral reproduction. Given a large set of inks, the decision for choosing an optimal ink set can be a tedious task. The combinations of candidate ink set can be a geometric figure. There are C(n, 6) combinations of possible ink sets for n inks in storage to be chosen for a six-color printing process. For example if n = 100, then there are 119,205,240 combinations of choices. For a practical example, even if n = 18 (Pantone 14 basic colors in addition to four process colors) then there still exist 18,564 combinations to choose from. To estimate the performance of each ink set in terms of colorimetric and spectral accuracy, a spectral printing model is needed to evaluate the spectral reconstruction. This requires construction of 18,564 six-color printing models for the 18 ink combinations and estimate
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the total performance model by model. The computation for trying 18,564 combinations is unreasonable, not to mention the model building effort. It can be safely said that is a "mission impossible." Clearly, testing each ink set is insufficient and a robust ink-selection algorithm is required.
A. FIRST ORDER INK SELECTION BY VECTOR CORRELATION
Although there are 18,564 combinations of ink sets by selecting six inks from an 18 ink database for a six-color printing process, a large portion of combinations are obviously impossible ink combinations for synthesis. The question boils down to this: on what basis can these impossible ink sets be excluded analytically? For a printing system which performs colorimetric other than spectral reproduction, it can be achieved visually by determining all the colorimetric values of a given image inside a colorimetric gamut spanned by a set of ink combinations. This is done by examining visualizations of a 2-D, 3D CIELAB, or chromaticity plots. Generally, the larger the colorimetric gamut a set of inks can provide, the more accurate colorimetric reproduction can be accomplished. The very reason for the current research project to derive a colorant estimation module for a multi-spectral output system is to eliminate the impossible combinations of ink sets analytically. The set of statistical primaries are utilized as a set of basis to remove the impossible ink combinations. Testing the performance of the irrational ink combinations can be avoided by well defined analytical printing models that describe the
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color mixing behavior. Once the statistical primaries are uncovered for an input spectral image, the process of the throughput for the proposed multiple-ink printing system minimizing metamerism is to correlate the statistical primaries to a set of physical inks which are the most capable of spectrally reproducing the spectral image during the synthesis stage. Theoretically, if an exact set of inks exist in a current manufacture line or in storage, then the use of the exact ink set will yield the ideal or closest spectral reproduction relative to the input spectral image. Since the statistical primaries are image dependent, the probability of an exact ink set is low. Hence, the exact spectral reproduction can not be achieved. A compromise has to be made to balance between colorimetric and spectral accuracy. Since the colorimetric match is the first priority for any color application, it is necessary to trade a slight decrease in spectral accuracy in exchange for higher colorimetric accuracy. This compromise will be discussed in a later section. Intuitively, the use of the statistical primaries is to search for the exact or similar inks in a given ink database. Vector correlation can be used to compare the similarity among them. A similarity measurement for ink1 and ink2, shown as Eq. (5-1), is quantified by the correlation coefficient, ρ, which is the cosine of the angle between a statistical primary and an ink from a large ink database, where Ψλ,ink1 is the linear colorant vector of ink1 and λ is a wavelength within the visible spectrum. Hence, the closer the correlation coefficient is to unity, the higher the similarity between a statistical primary and an ink in the database.
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700
ρ=
∑Ψ
λ = 400
λ ,ink1
(5-1)
700
700
∑Ψ
λ = 400
Ψλ , ink 2
2 λ , ink1
∑Ψ
λ = 400
2 λ , ink 2
Since the chances of selecting inks from a large ink database which are identical to the statistical primaries are low, candidate inks corresponding to each primary can be selected using a threshold of an acceptable correlation coefficient, say 0.90, or the highest twenty. Further filtration of the selected candidates is done by adopting the two candidates with the highest chroma for each statistical primary. A larger colorimetric gamut corresponding to a better possibility of colorimetric reproduction has been elucidated by various literature and experiential evidences (Ostromoukhov, 1993; Boll, 1994; Stollnitz, Ostromoukhov, and Salesin, 1998; Viggiano and Hoagland, 1998). The use of the highly chromatic primaries for colorant mixing yields a larger colorimetric gamut which is essentially desired when an exact spectral color reproduction cannot be accomplished. Compromises have to be made by trading decreased spectral accuracy in exchange for colorimetric accuracy. This is the reason for choosing candidates with the highest chroma for balancing between colorimetric and spectral accuracy. Based on this selection method, there are 64 (2 6) possible combinations of candidate ink sets. This is a significant reduction from 18,564. Nevertheless, it is still necessary to pinpoint the exact ink set for the application of the least metameric reproduction.
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Now this has come to the point of which type of analytical description for printing process should be utilized to estimate the performance of these first order selections. The intuitive choice is the six-color printing model based on the Yule-Nielsen modified spectral Neugebauer equation. To build 64 six-color printing models, it is required to do the sample preparations for printing 64 sets of ramps of Neugebauer primaries and verification targets corresponding to the 64 sets of selected ink combinations. The efforts of model building are beyond economical consideration. Even if the sample preparations can be replaced by computer simulation under certain assumptions, the model building efforts are still computational costly and time inefficient. Hence, a more efficient mechanism for estimating the colorimetric and spectral performance of the 64 selected ink sets are desired.
B. CONTINUOUS TONE APPROXIMATION
The further removal of low proficient combinations among the 64 is judged by scrutinizing the ink sets which are incapable of spanning the vector space of a given set of color samples. For this task, an assumption is made that halftone reproduction can be approximated by a continuous-tone model for subtractive color mixing (Berns, Bose, and Tzeng, 1996; Van De Capelle and Meireson, 1997). The continuous-tone modeling techniques used by the research program (Berns, 1993; Berns and Shyu, 1995) at the Munsell Color Science Laboratory at Rochester Institute of Technology are mostly based
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on Kubelka-Munk turbid media theory. However, for this application, Kubelka-Munk theory has insufficient accuracy. Alternatively, an empirical transformation for approximating the color mixing behavior of a halftone printing process, shown as Eq. (52) whose inverse transformation is shown as Eq. (5-3), was derived as Ψλ = R
1 w λ , paper
1 w λ
−R
R λ = (R
1 w λ , paper
− Ψλ ) w ,
and
(5-2)
(5-3)
where the Rλ,paper is the spectral reflectance factor of the paper substrate being printed on by primary inks and 2 ≤ w ≤ ∞. The transformation of a reflectance factor to the empirically derived space is somewhat different from Eq. (3-6) since Eq. (3-6) is derived for opaque colorant. Whereas, they have basically the same structure, one offset vector accounting for subtractive color mixing and a higher order power to account for the nonlinearity. The use of R
1 w λ , paper
as the offset vector has a significant meaning. Consider
that transforming a spectrum, which is exactly Rλ,paper, to the linear color mixing space, the result is a zero vector. This corresponds to the fact that there is not any primary presented in the linear space. Furthermore, Eq. (5-3) transforms a zero in the linear space back to the exact reflectance spectrum of the paper, Rλ,paper. The justifications for the use of the proposed transformation for continuous tone approximation to be described in verification section. Equation (5-2) transforms spectral reflectance factor to the representation for a
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subtractive color mixing process. Hence, the synthesis, quantitatively described by Eq. (54), of color mixtures is the linear combinations of the primary colorants modulated by their corresponding concentrations k
Ψλ , mixture = ∑ c i ψ λ , i ,
(5-4)
i =1
where ψλ is the linear operand of a primary colorant normalized to its unit concentration, c is the corresponding concentration, and k is the number of the primary colorants. Based on the assumption that a multiple-ink halftone printing process can be approximated by continuous tone modeling using Eq. (5-2) and (5-3), a direct constrained regression model using Eq. (5-4) was employed to estimate the performance of each candidate ink set. The estimated concentration for each primary is constrained to be positive. Positivity symbolizes the capability of a candidate ink set to span the entire colorant vector space of the target. If a negative concentration is reported using Eq. (5-4) without this constraint, then the corresponding ink set is only spanning the partial colorant vector space of the target. As a consequence, the spectral reconstruction by the ink set yields spectral error when the constraint of positivity is enforced. The final decision should favor the ink set(s) whose spectral reconstruction for a given target achieves the higher colorimetric and spectral accuracy. For this research project, the colorimetric accuracy is specified by the CIE color difference equation ∆E*94 under standard illuminant D50 and the 1931 standard observer. The spectral accuracy is quantified by the metameric index
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which is calculated by ∆E*94 under standard illuminant A and the 1931 standard observer based on a parameric correction.
C. VERIFICATIONS AND RESULTS
Deriving a Linear Color Mixing Space for Continuous Tone Approximation Current research analysis is based on the validity for a linear colorant mixing space which approximates the color mixing behavior of a multiple-ink halftone printing process such that the vector correlation analysis and constrained regression can be performed. The verification of deriving an empirical transformation, Eq. (5-2), for a typical halftone printing process utilized the spectral reflectance factor of IT8/7.3 of 928 samples printed by SWOP standard at 133 LPI screen frequency (McDowell, 1995). The SWOP specified spectral reflectance factor of paper and process CMYK are plotted in Fig. 5-1. The parameter in Eq. (5-2) to be optimized is w. The optimization process is to use a nonnegative least square function, nnls( ), built in MATLAB, to set up a constrained regression model, based on Eq. (5-4), for synthesizing every spectrum of the IT8/7.3 target. The nnls( ) performs a least square match for each spectrum by constraining the corresponding concentration for each primary to be non-negative. The w corresponding to the highest colorimetric and spectral accuracy will be adopted. Figure 5-2 shows the colorimetric and spectral accuracy with respect to different w values.
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1
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Fig. 5-1: The SWOP specified process CMYK primaries and paper substrate. w vs . m a xim um D e lta E 9 4 10
0 .8
8 D e lta E 94
D e lta E 94
w vs . a vera g e D e lta E 9 4 1
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w vs . m a xim um M .I.
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Fig. 5-2: The colorimetric and spectral error vs. w values for the empirical transformation, Eq. (5-2), where M.I. represents metamerism index.
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It was found that colorimetric and spectral errors monotonically decease as w approached ∞. Since the slope of decreasing of average ∆E*94 and metamerism indices are small when w is higher than three, the change of w does not increase the average performance significantly. It only improved the maximum errors significantly as w increases. The adopted w was 3.5 for an empirical decision. Colorimetric and spectral accuracy for the four SWOP primaries synthesizing 928 samples of the IT8/7.3 in the proposed linear colorant mixing space is shown in Table 5-1. Low colorimetric and spectral error indicated by Table 5-1 reveals that the four SWOP primaries span the 928 samples of the IT8/7.3 target in the proposed linear color mixing space. That is, every sample is a linear combination of the CMYK primaries. The color formation for this halftone printing process is approximately described by mixing the CMYK in the linear colorant mixing space. Table 5-1: The colorimetric and spectral accuracy for the four SWOP primaries synthesizing the 928 samples of IT8/7.3 target in the proposed linear colorant mixing space.
Mean Stdev Max Min RMS
∆E*94 0.65 0.65 5.08 0.00 0.0047
Metamerism Index 0.14 0.16 1.41 0.00
Four reconstructed spectra corresponding to the highest four colorimetric and spectral error are shown in Fig. 5-3.
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R e fle c tanc e
1
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Fig. 5-3: The four spectra reconstructed with the highest colorimetric and spectral errors based on the linear colorant mixing space where solid line is the measured spectrum and the dashed line is the reconstructed spectrum .
Even though the four spectra are reconstructed with the highest colorimetric and spectral errors, the reconstructed curves are well tracing the originally measured spectra. This implies that the halftone printing process can be well described by a continuous-tone approximation based on the proposed transformation. Hence, the derived new Ψ space can be used for estimating the performance of the 64 ink sets without heavy halftone modeling efforts.
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Vector Correlation Analysis in Ψ Space The set of 105 color patches, Shown in Fig. 4-7, was employed as the presumed reproduction target of an arbitrary image. Its six statistical primaries, shown in Fig. 5-4, were estimated by the module of colorant estimation discussed in Chapter IV.
R e f le c t a n c e
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Fig. 5-4: The six statistical primaries derived from the 105 mixtures by the colorant estimation module.
Since the six statistical primaries are, theoretically, the exact rotation of the six eigenvectors of the reproduction target in the proposed linear color space, their colorimetric and spectral accuracy in reconstructing the 105 patches is, theoretically, identical to the accuracy, shown in Table 4-4 of the six-eigenvector reconstruction. Discrepancy between the accuracy reconstructed by two sets of basis vectors depends on the numerical precision of the constrained rotation. Nevertheless, in spite of this
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discrepancy, the six statistical primaries are treated equivalent to the six eigenvectors. Hence, the use of the six statistical primaries is capable of reproducing the 105 six-color mixtures with a desired accuracy. Their spectral information is the link to select a closest set of inks from a given ink database for six-color halftone reproduction. Pantone 14 basic and process CMYK printed coated paper, shown in Fig. 5-5, were utilized as an ink database to perform the analysis for ink selection algorithms. (Color names and their abbreviates are yellow (Y) , yellow 012 (Y 12), orange 021 (O 21), warm red (Wr), red 032 (R), rubine red (Rr), rhodamine red (Rh), purple (Pu), violet (V), blue 072 (B 72), reflex blue (Rb), process blue (Prs B), green (G), black (K), process yellow (Prs Y), process magenta (Prs M), process cyan (Prs C), and process black (Prs K).) 1
1
1
1
1
1
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0 400 500
600 700
0 400
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0 400
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600 700
Fig. 5-5: The Pantone 14 basic colors and the process CMYK as the ink database (Color name order corresponding to each spectrum is from left to right and top to bottom).
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All the spectral reflectance factors for the statistical primaries and the ink database were first transformed by Eq. (5-2) to a linear colorant mixing space. Then, correlation coefficients of all 18 inks in the database for the statistical primaries were calculated using Eq. (5-1). Up to two candidate inks for each statistical primary with the highest chroma among those highly correlated inks were chosen. Their correlation coefficients and chroma are tabulated in Table 5-2. Since the research development aims at using one black and five chromatic inks to perform spectral reproduction, the two black inks in the ink data base are the certain candidate inks for the continuous tone estimation. Hence, their chroma and correlation coefficients with the sixth primaries were not tabulated in Table 5-2. Table 5-2: The correation coefficeints and the chroma of the 18 inks with the five chromatic statistical primaries. Candidate Inks Primary 1 Primary 2 Primary 3 Primary 4 Primary 5 Chroma Process Blue 0.98 68.6 Process Cyan 0.98 64.1 Rhodamine Red 0.94 80.1 Purple 0.88 86.8 Yellow 0.98 112.6 Process Yellow 0.97 106.3 Green 0.36 82.3 Blue 072 0.66 88.8 Reflex Blue 0.65 75.4
There are two candidates for each statistical primary except for the primary 4 due to the lack of similar ink existing in the 18 ink database. It was forced to chose only one with the highest correlation coefficient with respective to primary 4 among the 18 inks. Another situation happens when an ink in the database is simultaneously chosen as the
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candidate for two or more primaries; that is, an ink is selected more than twice. Then this ink is a sure candidate. Thus, the candidacy of this ink should be removed for other primaries. This ensures that when forming ink combinations by all candidates, there are double or triple selected inks in an ink combination, consequently, leading to a combination of smaller gamut colorimetrically and spectrally. Colorimetric and Spectral Performance by Continuous Tone Approximation Thirty-two (2x2x2x1x2x2) ink sets were formed by 11 candidate inks. Their colorimetric and spectral accuracy were estimated based on the constrained regression model using Ψ space. Since the validity of continuous tone approximation has been verified for the IT8/7.3 printed by SWOP standard, it is generalized to any of the printing process meeting the SWOP specification. Three ink sets were designated as the optimal ink sets for reproducing the 105 mixtures based on their highest spectral accuracy specified by the metamerism index. Their spectral and colorimetric accuracy are listed in Table 5-3 and their ink combinations are described in Table 5-4. Table 5-3: The spectral and colorimetric accuracy of the three optimal ink sets. Metamerism Index Ink Set Mean Stdev Max Min 23 0.70 0.51 1.86 0.05 24 0.73 0.52 1.88 0.05 19 0.73 0.53 1.86 0.04
Mean 2.26 2.35 2.40
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∆E*94 Stdev Max 0.99 4.35 0.96 4.20 1.17 4.75
Min 0.37 0.37 0.20
RMS 0.028 0.028 0.028
Table 5-4: The Pantone color names of the three optimal ink sets. Ink Set Primary 1 23 Process Cyan 24 Process Cyan 19 Process Cyan
Primary 2 Primary 3 Primary 4 Primary 5 Primary 6 Rhodamine Yellow Green Blue 72 Process Red Black Rhodamine Yellow Green Blue 72 Black Red Rhodamine Process Green Blue 72 Process Red Yellow Black
The performance among the three ink sets are not significantly different. Set 23 and set 24 are only different in the use of the sixth primary. It is concluded that the use of process black and the black is approximately invariant with the resultant performance. In addition, set 23 and set 19 are different by the use of the third primary. It is concluded that the use of yellow and process yellow is also approximately invariant with performance. Three reconstructed spectra for a sample of 105 mixture corresponding to the maximum error predicted by the three sets are plotted in Fig. 5-6. The three reconstructed spectra are nearly identical. Based on this observation, the three optimal ink sets approximately span the same colorant mixing space. Three worst performed ink sets, whose colorimetric and spectral accuracy in predicting the 105 mixtures are shown in Table 5-5, are specified in Table 5-6. Performances of the three worst performing ink sets are about identical to each other. The same justification applied for the three optimal ink sets can also be applied for the three worst performers. Their reconstructed spectra for the sample, shown in Fig. 5-6, are depicted in Fig. 5-7.
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1
M e a s u re d Set 23 Set 24 Set 19
0 .9
0 .8
R e fle c ta nc e fa c to r
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 400
450
500
550 W a ve le n g th
600
650
700
Fig. 5-6: The three spectral reconstructions of a sample corresponding to the maximum prediction error by the three optimal ink sets. Table 5-5: The spectral and colorimetric accuracy of the three worst performing ink sets. Metamerism Index Ink Set Mean Stdev Max Min 26 0.93 0.74 2.68 0.03 14 0.93 0.73 2.68 0.05 9 0.96 0.73 2.67 0.05
Mean 3.73 3.69 3.73
∆E*94 Stdev Max 1.34 7.03 1.36 7.03 1.38 6.69
Min 0.48 0.43 0.43
RMS 0.037 0.037 0.037
Table 5-6: The ink combinations of the three worst performing ink sets. Ink Set Primary 1 Primary 2 Primary 3 Primary 4 Primary 5 Primary 6 26 Process Purple Process Green Reflex Process Cyan Yellow Blue Black 14 Process Purple Yellow Green Reflex Process Blue Blue Black 9 Process Purple Process Green Reflex Black Blue Yellow Blue
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1
M e a s ure d Set 26 Set 14 Set 9
0 .9
0 .8
Re fle ctance fa cto r
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 400
450
500
550 W a vele ng th
600
650
700
Fig. 5-7: The three spectral predictions of the sample, shown in Fig. 5-6, by the three worst performed ink sets.
Finally, two predictions for a sample by set 23 and set 26, plotted in Fig. 5-8, are shown. By the numerical results, the colorimetric and spectral accuracy of set 23 is higher than that of set 26. By Fig. 5-8, the shape of the spectrum reconstructed by set 23 whose RMS error is 0.041 is closer to the measured spectrum than that of the reconstruction by set 26 whose RMS error is 0.050. The implication is that the color space spanned by set 23 is closer to the color space of the 150 mixtures than the color space spanned by set 26. Since the decision of choosing optimal ink sets is based on the metamerism index, the above justification was made to correlate the effectiveness of metamerism index to a set of
126
ink combinations, which is the optimal selection. The implementation of ink selection subsystem is attached in C. 1
M e a s ure d Set 23 Set 26
0 .9
0 .8
R e fle c ta n c e fa c to r
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 4 00
4 50
5 00
5 50
6 00
6 50
7 00
W a v e le n g th
Fig. 5-8: The two reconstructed spectra by set 23 and set 26 for the sample used as the example in Fig. 5-6.
D. CONCLUSIONS
An optimal ink set selection algorithm was proposed and justified. Its primary goal is to bridge between multi-spectral acquisition systems and multiple-ink output systems for the least metameric color reproduction. It serves the purpose of removing the redundancy or the deficiency of large ink combinations of a given ink database. This research also proposed applying a linear color mixing space for a continuous tone approximation to a halftone printing process. It dramatically reduced large scale of modeling efforts in
127
estimating the validity of optimal ink selection. The ink selection algorithm was comprised of vector correlation analysis follow by a constrained regression analysis. The proposed approach was able to remove a large number of ink combinations from an ink database. It pinpointed the optimal ink combination for a spectral-based halftone color reproduction system minimizing metamerism.
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VI. SPECTRAL REFLECTANCE PREDICTION OF INK OVERPRINT USING KUBELKA-MUNK TURBID MEDIA THEORY Consider the microscopic structure of ink on paper as delivered by a typical halftone printing process, shown in Fig. 6-1; a three-color halftone print is shown for demonstration. The total spectral reflectance factor over a square area, which is assumed to be the area of interest, is a summation of the individual spectral reflectance factors of each color inside the area.
R λ, P1
R λ , P1P2
R λ, P1P3
R λ , P2
R λ , P3
R λ , P2 P3
R λ , white
R λ , p1 p2 p3
Fig. 6-1: The microscopic structure of color formation by a halftone printing process where Rλ,color represents the spectral reflectance factor of a color appearing in the square area.
As we can see, the colors appearing inside the area of interest for reproduction are not only the primary colors, white, primary one (P 1), primary two (P 2), and primary three (P3),
129
but also the overprints of the primaries, primary one on primary two (P1P2), primary one on primary three (P1P3), primary two on primary three (P2P3), and the three-primary overprint (P1P2P3). These colors are usually called the Neugebauer primaries. Intuitively, the total spectral reflectance factor, Rλ,mix, over the area of interest is the linear sum, known as the spectral Neugebauer equation, of each spectral reflectance factor of the Neugebauer primaries modulated by their corresponding probability of occurrences. Yule and Nielsen further introduced an empirical n-factor to modify the spectral Neugebauer equation in order to account for light scattering within the paper, usually referred to as "optical dot gain." The Yule-Nielsen modified spectral Neugebauer equation for a threecolor (P1, P2, and P3) printing process is defined as R λ , mix = [a P1 R 1λ/,nP1 + a P2 R 1λ/,nP2 + a P3 R 1λ/,nP3 + a P1P2 R 1λ/,nP1P2 + a P1P3 R 1λ/,nP1P3 + a P2 P3 R 1λ/,nP2 P3 + a P1P2 P3 R 1λ/,nP1P2 P3 + ,
(6-1)
(1 − a P1 − a P2 − a P3 − a P1P2 − a P1P3 − a P2 P3 − a P1P2 P3 )R 1/λ ,nwhite ]n
where a (indexed by P1, P2, P3, P1P2, P1P3, P2P3, and P1P2P3) represents the fractional dot area of a Neugebauer primary. In order to use the Yule-Nielsen modified spectral Neugebauer equation, the spectral reflectance factor of each ink overprinted at 100% dot area coverage (known as secondary, tertiary, quaternary primaries, and so forth) are required as a priori knowledge for predicting the fractional dot area coverage of given spectra from an input spectral image. There are 2j-j-1 overprints for a j-color halftone printing system. For example, a seven-color halftone printing process needs to print and measure 120 (27-7-1) overprints.
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As the number of colors used for halftone printing increases linearly, the number of overprints increases exponentially. Hence, an analytical method for predicting the spectral properties of overprints can avoid the necessity of exhaustively printing and measuring each overprint upon using different ink and paper materials.
A. PREVIOUS RESEARCH
Previous research for this task had been performed by Allen (1969). He proposed a three-layer model using Kubelka-Munk turbid media theory for translucent inks printed on top of a highly scattering support (Kubelka and Munk, 1931; Kubelka, 1948). Somehow, Allen abandoned the complex model proposed in 1969 in favor of a simpler approach (Allen and Hoffenberg, 1973). Basically, they applied a thin ink film on a Mylar film and backed the film with both black and white supports in optical contact in order to determine two surface reflectance measurements over the printed Mylar film. The two optical constants known as absorption, K, and scattering, S, coefficients of an arbitrary ink were numerically estimated by using the weight of the ink film to calibrate the thickness. According to Kubelka-Munk theory, the surface reflectance factor is a function of K, S, the thickness of the ink film alone, and the reflectance factor of the background. Van De Capelle and Meireson (1997) took a different approach in which the surface reflectance factor of an ink printed on an opaque support is a function of three parameters, conceptually similar to the absorption and scattering coefficients and an
131
additional interaction term. The determination of these three parameters requires printing an arbitrary ink onto white, gray, and black surfaces for setting up three simultaneous equations in order to solve for the three unknowns. Another type of approach was recently exercised by Stollnitz, et al. (1998). It was claimed that the surface reflectance factor of multiple ink layers sitting on top of an opaque support is a function of the transmitance factor of each ink layer, multiple internal reflections at each interface, and the reflectance factor of support. The scattering of each ink layer was not considered. Finally, Viggiano and Hoagland (1998) used the additivity of ink density to predict ink overprints. Unfortunately, the colorimetric and spectral accuracy for the Allen, Stollnitz, and Viggiano studies were not disclosed. The approach by Van De Capelle has been patented by Barco Co. Thus, it was of interest to explore whether Kubelka-Munk theory could predict these overprints with sufficient colorimetric and spectral accuracy for spectralbased color reproduction.
B. TECHNICAL APPROACH
The famous Kubelka-Munk basic equation (1931) is shown in Eq. (6-2): Rλ =
1 − R λ , g [ a λ − b λ coth( b λ S λ X )] a λ − R λ , g + b λ coth( b λ S λ X )
132
,
(6-2)
where λ is a wavelength within the visible spectrum, Rλ,g is the spectral reflectance factor of an opaque support, Kλ is the absorption coefficient, Sλ is the scattering coefficient, X is the thickness of the layer of colorant, aλ is equal to 1+(K/S)λ, bλ is equal to [(aλ)2 - 1]1/2, and coth( ) is the hyperbolic cotangent function. The determination of K and S is carried out by drawing down or printing, for example, a thin ink film on black and white contrast paper, depicted as Fig. 6-2. Contrast paper whose surface is covered by a transparent plastic layer or resin coating is normally used. This layer prevents the ink from submerging into the paper fiber. The process utilizing contrast paper with plastic or resin coating is similar to that of Allen and Hoffenberg’s preparation using Mylar film back-coated by black and white paints.
Rλ,Pk
Rλ,Pw
Rλ,k
Rλ,w
Fig. 6-2: An ink film applied on black and white contrast paper.
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Four spectra can be attained by this technique to set up the two nonlinear equations using Eq. (6-2) by assuming the ink thickness is unity and homogenous where Rλ,Pw and Rλ,Pk are the spectral reflectance factor of an primary ink (P) printed over white and black support, respectively, and R λ,w and Rλ,k are the spectral reflectance factor of the white and black areas of the contrast paper, respectively. First, the surface reflectance factor, Rλ,Pw, of the ink printed on top of the white background is described by R λ , Pw =
1 − R λ , w [a λ , P − b λ , P coth( b λ , P S λ , P )] a λ , P − R λ , w + b λ , P coth( b λ , P S λ , P )
,
(6-3)
where X is assumed to be unity. Second, the surface reflectance factor, Rλ,Pk, of the same primary ink printed on top of black background is described by R λ , Pk =
1 − R λ , k [a λ , P − b λ , P coth( b λ , P S λ , P )] a λ , P − R λ , k + b λ , P coth( b λ , P S λ , P )
.
(6-4)
Notice that the difference between Eqs. (6-3) and (6-4) is the term of Rλ,g in Eq. (6-2) which is substituted with the spectral reflectance factor of the white support in Eq. (6-3) and substituted with the spectral reflectance factor of the black support in Eq. (6-4). Thus, Eqs. (6-3) and (6-4) construct a nonlinear system with two equations and two unknowns, Kλ and Sλ. To solve this system of nonlinear equations, a numerical method based on the techniques of operational research can be employed to estimate these two optical constants. This is repeated for each ink of interest. Once all the optical constants related to the assumed thickness are numerically determined, the prediction of overprints depends on the thickness of inks actually printed
134
on a medium, such as the SWOP specified standard paper. The effective thickness for each ink can be estimated using Eq. (6-2) by the known optical constants for each ink, measured surface reflectance factors of each ink printed on a specific paper, and the measured reflectance factor, Rλ,paper, for the specific paper. The equation for estimating the thickness, typical of a printing or proofing process, of an primary ink (P) is set up by R λ ,P =
1 − R λ , paper [a λ , P − b λ , P coth( b λ , P S λ , P X P )] a λ , P − R λ , paper + b λ , P coth( b λ , P S λ , P X P )
,
(6-5)
where Rλ,P is the spectral reflectance factor of a primary ink printed on top of a specific paper with spectral reflectance factor, Rλ,paper, and XP is the effective thickness. XP again can be solved by a numerical method. Once Kλ, Sλ, and X for each ink are estimated, the prediction of ink overprints is simply a recursive calculation using Eq. (6-2). That is, taking a three-ink-layer overprint as an example, given that all sets of characteristic parameters for all ink layers were estimated, the prediction of the topmost spectral reflectance factor requires the knowledge about the spectral reflectance factor of the second layer which are predicted based on the a priori knowledge about the known spectral reflectance factor of the bottom layer.
135
R λ ,P1P2 P3
R λ,P2 P3 R λ,P3
R λ,paper
K λ ,P1 S λ ,P1 X P1 K λ ,P2 S λ ,P2 X P2 K λ ,P3 S λ ,P3 X P3
Paper
Fig. 6-3: The diagram of a three-ink-layer overprint. Figure 6-3 is shown to conceptualize this process where factor of primary three printed on paper with
Rλ,P3 is
the spectral reflectance
R λ,paper , R λ,P2P3 is
the spectral reflectance
factor of primary two printed on top of the primary three, and
R λ ,P1P2P3 is
the spectral
reflectance factor of primary one printed on the topmost layer. Analytically, the estimation of spectral reflectance factor for all three ink layers can be described by Eq. (6-6) for the bottom layer, Eq. (6-7) for the inner layer, and Eq. (6-8) for the topmost layer, respectively: R λ , P3 =
1 − R λ , paper [ a λ , P3 − b λ , P3 coth( b λ , P3 S λ , P3 X P3 )]
R λ , P2 P3 =
a λ , P3 − R λ , paper + b λ , P3 coth( b λ , P3 S λ , P3 X P3 ) 1 − R λ , P3 [a λ , P2 − b λ , P2 coth( b λ , P2 S λ , P2 X P2 )] a λ , P2 − R λ , P3 + b λ , P2 coth( b λ , P2 S λ , P2 X P2 )
136
,
(6-6)
,
(6-7)
R λ , P1 P2 P3 =
1 − R λ , P2 P3 [ a λ , P1 − b λ , P1 coth( b λ , P1 S λ , P1 X P1 )] a λ , P1 − R λ , P2 P3 + b λ , P1 coth( b λ , P1 S λ , P1 X P1 )
.
(6-8)
The accuracy of this process was defined using the CIE94 color difference equation calculated under standard illuminant D50 and the 1931 standard observer (CIE, 1995). The spectral accuracy is quantified both by root-mean-square (RMS) error in units of reflectance factor and the CIE94 color difference equation calculated under standard illuminant A and the 1931 standard observer as the metamerism index (M.I.) after parameric correction (Fairman, 1987).
C. EXPERIMENTAL
For the convenience of verifying the technical approach described above, the DuPont Water Proof system was used to print six primaries, which are cyan, magenta, yellow, red, green, and blue, on six pieces of contrast paper shown in Fig. 6-4. Twentyfive overprints, shown in Fig. 6-5, were generated using at most three-primary combinations. Among the 25 overprints, 14 are two-color overprints and 11 are threecolor overprints. The printing order corresponds to the color order putting cyan at the bottom-most layer, magenta on cyan, yellow on magenta, red on yellow, green on red, and blue on the top-most layer. These samples were measured using a Gretag Spectrolino by averaging five measurements for each color. Due to the different refractive indices among air, ink, and support, the Saunderson correction was employed to correct for refractiveindex discontinuity at each interface (Saunderson, 1942; Allen, 1987).
137
Fig. 6-4: The six primaries printed on contrast paper.
Fig. 6-5: Twenty-five overprints printed on coated paper.
138
D. RESULTS
Equations (6-3) and (6-4) were used to set up the system of nonlinear equations and solved for the two unknowns by assuming the thickness for each ink is unity. The estimated Kλ and Sλ, attached in Appendix H, of the six primaries are plotted in Fig. 6-6.
K or S
C yan
Magenta 3
3
2.5
2.5
2.5
2
2
2
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0 400
500
600
700
0 400
500
Red
K or S
Yellow
3
600
700
0 400
Green 3
3
2.5
2.5
2.5
2
2
2
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
500
600
Wavelength
700
0 400
500
600
700
600
700
Blue
3
0 400
500
600
Wavelength
700
0 400
500
Wavelength
Fig. 6-6: The spectral absorption (solid line) and scattering (dashed line multiplied by ten times) curves of the six primaries.
Thickness was then estimated for each primary printed on the coated paper shown in Fig. 6-5, using Eq. (6-2). Due to the high colorimetic and spectral accuracy for predicting
139
primaries, the difference spectra between measured and predicted primaries printed on the coated paper are plotted in Fig. 6-7 and their colorimetric and spectral accuracy as well as their statistical thicknesses are shown in Table 6-1. Thickness is a ratio related to the ink thickness of each primary printed on contrast paper.
Delta R
Cyan
Magenta 0.05
0.05
0
0
0
-0.05 400
500
600
700
-0.05 400
Red
Delta R
Yellow
0.05
500
600
700
-0.05 400
Green 0.05
0.05
0
0
0
500 600 Wavelength
700
-0.05 400
500 600 Wavelength
600
700
500 600 Wavelength
700
Blue
0.05
-0.05 400
500
700
-0.05 400
Fig. 6-7: The difference spectra between measured and predicted primaries. Table 6-1: The colorimetric accuracy, spectral accuracy, and the statistical thickness for the six primaries.
∆E Metamerism Index RMS Error Thickness * 94
Cyan Magenta Yellow Red 0.8 0.3 0.1 0.4 0.0 0.0 0.0 0.0 0.005 0.002 0.004 0.004 0.956 0.958 0.975 1.026
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Green 0.2 0.0 0.001 0.962
Blue 0.8 0.2 0.004 0.987
According to Table 6-1, the prediction of each primary is of high spectral accuracy as indicated by the near zero metamerism index and the low RMS error. Thus, the first verification ensures the success using of Kubelka-Munk theory to predict the translucent material backed by an opaque support. With the knowledge of optical constants, Kλ and Sλ, and the effective thickness of ink deposited by a typical printing process, the estimation of spectral reflectance factor can be accomplished whenever the paper support is changed under the assumption that there is no interaction between ink and paper (coated paper is preferred). Since the accuracy of the first prediction is high, the interaction between ink and paper is considered insignificant. Second, the prediction of overprints is based on the assumption that no chemical or physical interaction at interfaces of each ink layer. In our experiment, the statistical colorimetric and spectral performance of predictions for the 25 overprints, shown in Table 6-2, is considered high judged by the low average and standard deviation (Stdev) of metamerism indices whose histogram is shown in Fig. 6-8. It indicates that almost all the overprints are predicted with high accuracy since most of estimated metamerism indices of the 25 overprints are concentrated around 0.3 ∆E*94 units. Figures 6-9 and 6-10 are shown as examples of good predictions and predictions with relatively low accuracy in terms of their metamerism indices. However, the spectral predictions of these “relative low accuracy” samples are considered acceptable judged by their low colorimetric and spectral error. The predicted spectral curves correspond well to the
141
measured spectral curves indicated by the difference spectra. The implementation of ink overprint prediction is shown in Appendix D. Table 6-2: The colorimetric and spectral accuracy of the 25 overprints. ∆E*94 0.9 0.5 2.1 0.2
Mean Stdev Maximum Minimum RMS Error
Metamerism index 0.3 0.3 1.2 0.0 0.004
10 9 8
F re q u e n c y
7 6 5 4 3 2 1 0 0
0 .2
0 .4
0 .6
0 .8
1
1 .2
1 .4
M e ta m e ri s m In d e x
Fig. 6-8: Histogram of the metamerism indices for prediction of the 25 overprints.
142
Red on M agenta
Y ellow on M agenta
Delta R
0.05
0.05
∆E*94 0.2 M. I. 0.0
∆E*94 0.4 M. I. 0.1
0
0
-0.05 400
500
600
-0.05 400
700
Red on Y ellow
600
700
G reen on Y ellow
0.05
Delta R
500
0.05
0
0
∆E*94
∆E*94 0.4 M. I. 0.1
0.3 M. I. 0.1 -0.05 400
500 600 W avelength
-0.05 400
700
500 600 W avelength
700
Fig. 6-9: The difference spectra of four overprints best predicted with high accuracy.
G reen on Red
Blue on Y ellow
Delta R
0.05
0.05
∆E*94 2.1 M. I. 0.4
∆E*94 0.9 M. I. 0.5
0
0
-0.05 400
500
600
-0.05 400
700
Blue on Cyan
Delta R
600
700
Blue on Y ellow on Cyan
0.05
0.05
0
0
∆E*94
∆E*94 1.2 M. I. 1.1
1.4 M. I. 0.7 -0.05 400
500
500 600 W avelength
-0.05 400
700
500 600 W avelength
700
Fig. 6-10: The difference spectra of four overprints predicted with relatively low accuracy.
143
E. DISCUSSIONS
The errors contributed to the prediction using the technical approach described herein can be attributed to three types of error. The first is the uniformity of the paper support to be printed. The second is the homogeneity of the ink thickness delivered by the printing process. The last is the accuracy of the determination of the two optical constants, Kλ and Sλ. Recall that Kλ and Sλ are solved numerically. In the process of solving for Kλ and Sλ, the constraints of the positivity of these two optical constants should be superimposed in numerical estimation to account for the absorption and scattering properties of real material, i.e., the negativity of Kλ and Sλ is not realizable. Another realistic consideration for the numerical estimation is that the term aλ in Eq. (6-2) is a function of the ratio of absorption to scattering coefficient, (K/S)λ, at a sampled wavelength. When estimated Sλ is zero, the numerical error of division by zero happens. The process required setting the lower boundary of Sλ to be nonzero to prevent the numerical error of the division by zero in addition to that Eq. (6-2) can be confounded by the numerical result of zero divided by zero. This limitation causes the over-prediction for the reflectance factor when both the estimated Kλ and Sλ are low and close to zero. The strongest evidence is the prediction for cyan ink printed on the coated paper. Figure 6-11 indicates that the estimated low absorption (solid line) and scattering (dashed line) for cyan ink happens from 440 nm to 470 nm. Its corresponding prediction of spectral
144
reflectance factor is over-predicted (the difference spectrum is obtained by the predicted subtracted from the measured) at the same spectral region due to this numerical limitation. E s ti m a te d K & S
D i ffe re n c e S p e c tru m
2
0 .0 2
1 .8 0 .0 1 5 1 .6 0 .0 1 1 .4
0 .0 0 5 D e lta R
K or S
1 .2
1
0
0 .8 - 0 .0 0 5 0 .6 - 0 .0 1
0 .4
0 .2
- 0 .0 1 5
0 400
450
500
550
600
650
- 0 .0 2 400
700
450
W a v e le n g t h
500
550
600
650
700
W a v e le n g t h
Fig. 6-11: The estimated two optical constants for cyan ink and corresponding difference spectrum in units of reflectance factor.
F. CONCLUSIONS
An algorithm for predicting overprints by a proofing device was exercised and verified. The errors contributing to this verification process are the uniformity of a coated paper to be printed, homogeneity of the printed ink thickness, and numerical limitations when estimating the two optical constants. Nevertheless, high colorimetric and spectral accuracy was achieved by this approach using Kubelka-Munk turbid media theory.
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VII. SPECTRAL-BASED SIX-COLOR SEPARATION MINIMIZING METAMERISM Given an input image, the task in the synthesis stage of its color reproduction using halftone printing device is to determine the ink amount corresponding to each primary to be delivered onto a paper substrate. This task is so called "color separation". The determined ink amount in terms of fractional areas are stored as color separation records corresponding to each primary ink by digital storage or conventional high contrast lithographic film. The reproduction is accomplished by transferring the ink information in each color-separation record digitally by an inkjet printer or conventionally by a printing press. This chain completes the synthesis stage of a halftone printing process. Conventionally, color separation schemes often utilize empirical ink tables or multi-dimensional look up tables (CLUT). Such approaches require quite a few data samples and measurements. On the contrary, analytical models frequently require relatively small number of samples to build device profile. This is especially efficient whenever the printing process is subject to a change of ink and paper material. Most analytical models are based on the Murray-Davies and Neugebauer equations. The former is utilized as the analytical description for the color formation of a single ink printed on a substrate. Whereas, the latter is employed as the analytical description for the color formation of multi-color printing processes.
146
Four-color modeling techniques based on the Neugebauer equation for halftone printing process have long been disclosed and well established (Pobboravsky and Pearson, 1972; Balasubramanian, 1995; 1996; 1998; Viggiano, 1985). The quality of this type of color synthesis is based on the accuracy of colorimetric reproduction. It suffers from the problem of metamerism due to a lack of degrees of freedom. The use of more than four inks for the halftone printing process not only increases the colorimetric gamut but also increases the "spectral gamut" of the color mixtures produced by a halftone device. Spectral gamut is defined as the set of spectra which is spanned by a set of basis primaries used for halftone synthesis. Theoretically, the larger the spectral gamut, the higher the probability matching a sample from the input image spectrally. Based on the economy and production convenience as described in the introduction, this research extends the challenge to build an analytical model for a six-color halftone printing process.
A. SIX-COLOR YULE-NIELSEN MODIFIED SPECTRAL NEUGEBAUER EQUATION
An analytical estimation for the synthesis of a reflectance spectrum using six primary inks can employ the Yule-Nielsen modified spectral Neugebauer equation generalized to a six-primary expression, shown as Eq. (7-1), 64
R λ = [ ∑ a i R 1λ/,ni ]n , i =1
147
(7-1)
where terms are similarly defined as that of Eq. (2-36). Notice that Eq. (7-1) is extended to a linear sum of 64 Nuegebauer primaries. The corresponding 64 coefficients as the fractional dot areas can be expressed again by the Demichel probability model extended to a six-primary case shown as Eq. (7-2),
White
: a 1 = (1 - P1) (1 - P2) (1 - P3) (1 - P4) (1 - P5) (1 - P6)
Six Primaries P1 P2 P3 P4 P5 P6
: : : : : :
a2 a3 a4 a5 a6 a7
= P1 (1 - P2) (1 - P3) (1 - P4) (1 - P5) (1 - P6) = (1 - P1) P2 (1 - P3) (1 - P4) (1 - P5) (1 - P6) = (1 - P1) (1 - P2) P3 (1 - P4) (1 - P5) (1 - P6) = (1 - P1) (1 - P2) (1 - P3) P4 (1 - P5) (1 - P6) = (1 - P1) (1 - P3) (1 - P3) (1 - P4) P5 (1 - P6) = (1 - P1) (1 - P3) (1 - P3) (1 - P4) (1 - P5) P6
Secondaries (fifteen two-color overprints) P1 P2 P1 P3 P1 P4 P1 P5 P1 P6 P2 P3 P2 P4 P2 P5 P2 P6 P3 P4 P3 P5 P3 P6 P4 P5 P4 P6 P5 P6
: : : : : : : : : : : : : : :
a8 = P1 P2 (1 - P3) (1 - P4) (1 - P5) (1 - P6) a9 = P1 (1 - P2) P3 (1 - P4) (1 - P5) (1 - P6) a10 = P1 (1 - P2) (1 - P3) P4 (1 - P5) (1 - P6) a11 = P1 (1 - P2) (1 - P3) (1 - P4) P5 (1 - P6) a12 = P1 (1 - P2) (1 - P3) (1 - P4) (1 - P5) P6 a13 = (1 - P1) P2 P3 (1 - P4) (1 - P5) (1 - P6) a14 = (1 - P1) P2 (1 - P3) P4 (1 - P5) (1 - P6) a15 = (1 - P1) P2 (1 - P3) (1 - P4) P5 (1 - P6) a16 = (1 - P1) P2 (1 - P3) (1 - P4) (1 - P5) P6 a17 = (1 - P1) (1 - P2) P3 P4 (1 - P5) (1 - P6) a18 = (1 - P1) (1 - P2) P3 (1 - P4) P5 (1 - P6) a19 = (1 - P1) (1 - P2) P3 (1 - P4) (1 - P5) P6 a20 = (1 - P1) (1 - P2) (1 - P3) P4 P5 (1 - P6) a21 = (1 - P1) (1 - P2) (1 - P3) P4 (1 - P5) P6 a22 = (1 - P1) (1 - P2) (1 - P3) (1 - P4) P5 P6
Tertiaries (twenty three-color overprints)
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P1 P2 P3 P1 P2 P4 P1 P2 P5 P1 P2 P6 P1 P3 P4 P1 P3 P5 P1 P3 P6 P1 P4 P5 P1 P4 P6 P1 P5 P6 P2 P3 P4 P2 P3 P5 P2 P3 P6 P2 P4 P5 P2 P4 P6 P2 P5 P6 P3 P4 P5 P3 P4 P6 P3 P5 P6 P4 P5 P6
: a23 = P1 P2 P3 (1 - P4) (1 - P5) (1 - P6) : a24 = P1 P2 (1 - P3) P4 (1 - P5) (1 - P6) : a25 = P1 P2 (1 - P3) (1 - P4) P5 (1 - P6) : a26 = P1 P2 (1 - P3) (1 - P4) (1 - P5) P6 : a27 = P1 (1 - P2) P3 P4 (1 - P5) (1 - P6) : a28 = P1 (1 - P2) P3 (1 - P4) P5 (1 - P6) : a29 = P1 (1 - P2) P3 (1 - P4) (1 - P5) P6 : a30 = P1 (1 - P2) (1 - P3) P4 P5 (1 - P6) : a31 = P1 (1 - P2) (1 - P3) P4 (1 - P5) P6 : a32 = P1 (1 - P2) (1 - P3) (1 - P4) P5 P6, : a33 = (1 - P1) P2 P3 P4 (1 - P5) (1 - P6) : a34 = (1 - P1) P2 P3 (1 - P4) P5 (1 - P6) : a35 = (1 - P1) P2 P3 (1 - P4) (1 - P5) P6 : a36 = (1 - P1) P2 (1 - P3) P4 P5 (1 - P6) : a37 = (1 - P1) P2 (1 - P3) P4 (1 - P5) P6 : a38 = (1 - P1) P2 (1 - P3) (1 - P4) P5 P6 : a39 = (1 - P1) (1 - P2) P3 P4 P5 (1 - P6) : a40 = (1 - P1) (1 - P2) P3 P4 (1 - P5) P6 : a41 = (1 - P1) (1 - P2) P3 (1 - P4) P5 P6 : a42 = (1 - P1) (1 - P2) (1 - P3) P4 P5 P6
Quaternaries (fifteen four-color overprints) P1 P2 P3 P4 P1 P2 P3 P5 P1 P2 P3 P6 P1 P2 P4 P5 P1 P2 P4 P6 P1 P2 P5 P6 P1 P3 P4 P5 P1 P3 P4 P6 P1 P3 P5 P6 P1 P4 P5 P6 P2 P3 P4 P5 P2 P3 P4 P6 P2 P3 P5 P6 P2 P4 P5 P6 P3 P4 P5 P6
: : : : : : : : : : : : : : :
a43 = P1 P2 P3 P4 (1 - P5) (1 - P6) a44 = P1 P2 P3 (1 - P4) P5 (1 - P6) a45 = P1 P2 P3 (1 - P4) (1 - P5) P6 a46 = P1 P2 (1 - P3) P4 P5 (1 - P6) a47 = P1 P2 (1 - P3) P4 (1 - P5) P6 a48 = P1 P2 (1 - P3) (1 - P4) P5 P6 a49 = P1 (1 - P2) P3 P4 P5 (1 - P6) a50 = P1 (1 - P2) P3 P4 (1 - P5) P6 a51 = P1 (1 - P2) P3 (1 - P4) P5 P6 a52 = P1 (1 - P2) (1 - P3) P4 P5 P6 a53 = (1 - P1) P2 P3 P4 P5 (1 - P6) a54 = (1 - P1) P2 P3 P4 (1 - P5) P6 a55 = (1 - P1) P2 P3 (1 - P4) P5 P6 a56 = (1 - P1) P2 (1 - P3) P4 P5 P6 a57 = (1 - P1) (1 - P2) P3 P4 P5 P6
Quinaries (six five-color overprints)
149
(7-2)
P1 P2 P3 P4 P5 P1 P2 P3 P4 P6 P1 P2 P3 P5 P6 P1 P2 P4 P5 P6 P1 P3 P4 P5 P6 P2 P3 P4 P5 P6
: : : : : :
a58 = P1 P2 P3 P4 P5 (1 - P6) a59 = P1 P2 P3 P4 (1 - P5) P6 a60 = P1 P2 P3 (1 - P4) P5) P6 a61 = P1 P2 (1 - P3) P4 P5 P6 a62 = P1 (1 - P2) P3 P4 P5 P6 a63 = (1 - P1) P2 P3 P4 P5 P6
Hexary (one six-color overprint) P1 P2 P3 P4 P5 P6
: a64 = P1 P2 P3 P4 P5 P6
where P1, P2,…, P6 represent the six-primary inks.
Although Eq. (7-1) theoretically describes the synthesis of a desired color spectrum, whether or not an actual six-color halftone printing process which is capable of abiding by this analytical description is not known. The failure is due to the ink-trapping limitation which is defined as the capability in terms of percentage of ink amount for successive inks printed on top of the preceding ink. At the stage of ink-trapping, the preceding inks are still wet. Several practical observations have reported that ink-trapping lithography off-set printing is limited to 300~400% of total ink amount printed over a fixed area. Hence, it is very likely that the validity of Eq. (7-1) is confined by this physical limitation. From Eq. (7-1), a given reflectance spectrum requires fitting by the linear combination of 64 basis spectra† (Neugebauer primaries). The redundancy among those 64 basis spectra is worthwhile checking since the linear independency among them is not a
150
certainty. It can be reasoned by an example of six-color halftone printing process using cyan, magenta, yellow, green, orange, and black inks. Consider that the quinary and hexary as well as some quaternary whose spectra are flat are neutral colors. It is highly probable that those Neugebauer primaries can be approximated by linear combinations of others. If so, then the redundancy of those Neugebauer primaries is high. Their contribution to the synthesis of a spectrum is insignificant. Based on the physical limitation of a certain printing process, the use of six-color Yule-Nielsen modified spectral Neugebauer equation seems impractical. In addition, the speculation on the basis spectra indicated that the existence of five or six color overprints may not be significant for spectral reproduction. Thus, an alternative approach which takes advantage of more degrees of freedom as well as abiding by the physical limitation of a particular printing process is desired.
B. AN ALTERNATIVE APPROACH USING SIX-COLOR HALFTONE PRINTING PROCESS MINIMIZING METAMERISM
The current research project proposes a paradigmatic algorithm in balancing between the spectral gamut and physical limitation due to ink-trapping failure. These modules for the current research development are sequentially outlined in Fig. 7-1.
†
We don’t want to call the Neugebauer primaries as the basis vectors since they are not necessary linearly independent. Recall that basis vectors are defined as the set of vectors which are not only linearly independent but also span the entire vector space.
151
Subdivision of Six-Color Modeling
Forward Four-Color Halftone Spectral Printing Models
Backward Printing Models for Six-Color Separation Minimizing Metamerism
Proper Four-Color Sub-Model Selection
Fig. 7-1: The structure chart for the development of six-color separation minimizing metamerism. Subdivision of Six-Color Modeling The task of analytical modeling for a six-color halftone printing process is subdivided into k four-color modeling processes in order to comply with the ink-trapping limitation. The number, k, depends on the division algorithm and will be specified later in this section. The six-color printing model can be viewed as the super-model of the k fourcolor printing sub-models. The spectral gamut of the six-color printing model is assumed to be a good approximation to the spectral gamut of an input image by the ink-selection algorithm. If the ink-selection algorithm suggests an optimal inkset, for an arbitrary example, comprised by cyan (C), magenta (M), yellow (Y), green (G), orange (O), and
152
black (K) inks and a division algorithm recommends the use of one black and three chromatic inks to synthesize each pixel of a given spectral image, then there are ten fourcolor printing sub-models to be constructed. They are CMYK, CMGK, CMOK, CYGK, CYOK, CGOK, MYGK, MYOK, MGOK, and YGOK. Forward Four-Color Halftone Spectral Printing Models Since the modeling of six-color halftone printing process is subdivided into ten four-color modeling processes, a modeling process for CMYK halftone printing will be utilized for discussion. Then, it is generalized for any four-color printing process using different primaries. 1. The first-order forward model The mission for a forward model is to obtain an accurate estimation of a synthesized spectrum given a set of requested (or theoretical) dot areas "dialed in" by a printer. Owing to mechanical and optical dot gain, the effective dot size printed on a substrate is different from the theoretical dot size. Hence, the first task is to determine the Yule-Nielsen n-factor, the second is to relate the theoretical dot area to the effective dot area through a mathematical function or look up table (LUT), and finally, the estimated spectrum is estimated by the Yule-Nielsen modified spectral Neugebauer equation. The chain of first-order forward modeling process based on Yule-Nielsen modified spectral Neugebauer equation is outlined in Fig. 7-2.
153
cmyk
f(c, m, y, k)
(theoretical dot areas)
c’m’y’k’ (effective dot areas)
Rλ,synthesis
Nλ,n (c’, m’, y’, k’)
Fig. 7-2: The structure of a general forward halftone printing model where f( ) is a mathematical function or LUT describing the dot-gain effect and Nλ,n( ) is the function of Yule-Nielsen modified spectral Neugebauer equation.
In order to complete the forward printing model, the transfer function relating the theoretical to effective dot areas, f( ), and the Yule-Nielsen n-factor need to be uncovered for a particular halftone printing process. The n-factor is utilized to compensate for the non-linearity using the Neugebauer equation. Hence, the use of n-factor accounts for both the mechanical and optical dot-gain behavior. Solving for n-factor corresponding to a halftone printing system usually comes first followed by a determination of theoretical to effective dot area transfer function, f( ). This modeling procedure only requires a sample preparation of primary ramps of different fractional dot area and overprints of all combinations of them.
154
From the primary ramps, the color formation can be described by the Yule-Nielsen modified spectral Murray-Davies equation, specified as Eq. (2-33). By inverting Eq. (233), the effective dot area of each patch can be estimated. The inversion of Eq. (2-33) in terms of fractional dot area is shown as a=
R 1λ/,nk % − R 1λ/,npaper R 1λ/,n100% − R 1λ/,npaper
,
(7-3)
where a is the solved effective dot area corresponding to a primary printed at theoretical dot area to be k% (0 ≤ k ≤ 100 ) whose spectral reflectance factor is Rλ,k%, Rλ,100% is the spectral reflectance factor of the primary printed at 100% ink coverage, and Rλ,paper is the spectral reflectance factor of the paper substrate. Notice that a = k% if n=1, i.e, there is no dot-gain effect or the effective dot area is equal to the theoretical dot area. With the solved fractional dot area, a, the predicted spectral reflectance factor for each patch can be estimated by Eq. (2-33) given a known n-factor. Hence, the optimality of n-factor determines the predicting accuracy of the estimated reflectance spectra. The optimization of n-factor can be carried out by the procedure outlined in Fig. 7-3. Figure 73 suggests the use of all the patches of primary ramps to step through all values of n-factor with increment of ∆n and find an optimal n value corresponding to the smallest average ∆E*94 of the predictions of all patches.
155
n=1 n = n + ∆n
INV(YNMD)
R λ,k %
a
YNMD
R λ ,k %
no
Yes if
∆E*94
is minimum
n
Fig. 7-3: The algorithm structure of determining the Yule-Nielsen n-factor where INV(YNMD) stands for the inverse function, Eq. (7-3), of n-factor corrected spectral Murray-Davies equation.
Once the n-factor is optimized, the corresponding solved effective dot area, a, for each patch of each primary ramp is used to build the transfer function, f( ), by higher order polynomials or nonlinear interpolation by cubic spline functions. A set of transfer functions, depicted in Fig. 7-4, determined from a printing process is shown as an example.
156
1
0.9
0.8
0.7
Effective dot area
0.6
0.5
0.4
0.3
0.2
cyan magenta yellow black
0.1
0 0
0.1
0.2
0.3
0.4 0.5 0.6 Theoretical dot area
0.7
0.8
0.9
1
Fig. 7-4: A set of theoretical to effective dot area transfer functions determined from a CMYK halftone printing process. 2. Second order improvement (modeling for ink- and optical-trapping) The first-order forward printing model does not take ink trapping and optical interaction into account. Its model accuracy is confined by these two physical effects. It assumes that 100% ink-trapping capability and the optical dot gain of a multi-layer ink overprint is the same as that of single primary layer. Since the percentage of ink trapping varies for each printing process, the 100% assumption is not optimal. Furthermore, the optical dot gain is obviously different among multi-layer overprints. This concept, termed as optical-trapping, can be found in the literature published by Iino and Berns (1998). Both ink trapping and optical trapping caused the first-order prediction to be too dark for a sample; that is, the effective dot areas by printing the successive ink on top of the preceding ink are smaller than expected. Hence, by overestimating the effective dot areas
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for synthesizing a sample requiring reproduction, the resultant reflectance spectrum by the first-order forward printing model is under predicted. Iino and Berns proposed the use of a correction factor, q, to rectify the effective dot area estimated by the theoretical to effective dot area transfer function, f( ), for the first-order printing model. This correction factor originated from an observation on the dot gain effect of a primary ramp at the existence of another primary. Dot-gain, g, can be quantitatively defined as the difference between effective dot area, aeff, and theoretical dot area, atheo, shown as Eq. (7-4), g = aeff - atheo .
(7-4)
Figure 7-5 shows the dot gain curves (g vs. theoretical dot area) of the CMYK ramps given in Fig. 7-4. 0 .4
c ya n ma ge nta ye llo w bla c k
D o t-g a in in units of fra cti o n al d o t a r e a
0 .3 5
0 .3
0 .2 5
0 .2
0 .1 5
0 .1
0 .0 5
0 0
0 .1
0 .2
0 .3
0 .4 0 .5 0 .6 The oretica l d o t a re a
0 .7
0 .8
0 .9
1
Fig. 7-5: The dot-gain functions of the CMYK ramps given in Fig. 7-4.
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Iino and Berns hypothesized each primary ink has its inherent dot-gain function which can be obtained from its corresponding primary ramp. They further hypothesized that the dot-gain functions with respect to a primary varies their extent but not shape given the existence of other primaries. For example, the family of dot-gain curves of a cyan ramp given that the magenta ink is present at 0%, 25% 50%, and 75% fractional dot areas is plotted in Fig. 7-6. 0.35
mag=0% mag=25% mag=50% mag=75%
0.3
Dot-gain in units of fractional dot area
0.25
0.2
0.15
0.1
0.05
0 0
0.1
0.2
0.3
0.4 0.5 0.6 Theoretical dot area
0.7
0.8
0.9
1
Fig. 7-6: The family of dot-gain curves of a cyan ramp when magenta ink presents at 0%, 25%, 50%, and 75% fractional dot areas.
Thus, when magenta ink is present at 0%, the corresponding curve whose extent is the highest describes the dot-gain behavior of the cyan primary alone, denoted as gc. When
159
magenta ink presents at nonzero dot area, the dot-gain of the cyan ramp is reduced. The percentage of reduction is represented by a scalar, q, which depends on the existence of other primary inks, where 0 ≤ q ≤ 1. Hence, quantitative description of global dot-gain int the presence of another primary, g, is the product of q and the dot-gain, gp, of a primary ramp, i.e., g = qgp = q(aeff, priamry - atheo,primary), where aeff,
priamry
(7-5)
and atheo,primary are the effective and theoretical dot areas of a primary,
respectively. Intuitively, the correction scalar, q, for a primary ink is a function of the theoretical dot area of the second ink. It was assumed that the dot-gain variance for an arbitrary primary printed at 50% theoretical dot area is maximized although variance of dot-gain curves shown in Fig. 7-6 seems to peak around 45%. (The location of peak variance is related to the dot shape formed by a specific halftone screening process. Usually, the elliptical, diamond dot shape, and the dot shape formed by FM screening peaks around 45% theoretical dot area, whereas, the round and square dot shape peak around at 50% theoretical dot area.) Hence, to model this correction scalar, it is required to print a primary at 50% theoretical dot area and varying the second ink from 0% to 100% by assuming the dot-gain variance peaks at 50% theoretical area for a particular halftone printing process. Let the gi=50%,j be the dot gain of a primary i printed at 50% theoretical dot area given that the second primary j is present at a known theoretical dot area,
160
gi=50%,j = aeff,i=50%,j - 0.5 ,
(7-6)
where aeff,i=50%,j is the effective dot area estimated using the inverse Yule-Nielsen modified spectral Neugebauer equation of the primary i at 50% theoretical dot area given that the second primary j is present at a known theoretical dot area. Further, the dot gain, g i=50%, of the primary i alone printed on a paper substrate at 50% theoretical dot area is the difference of the corresponding effective dot area, aeff,i=50% and atheo,i, i.e., gi=50% = aeff,i=50% - atheo,i ,
(7-7)
where atheo = 0.5. Thereby, the correction scalar, q, is the ratio of gi=50%,j to gi=50%, i.e., q = gi=50%,j / gi=50% = (aeff,i=50%,j - atheo,i) / (aeff,i=50% - atheo,i) .
(7-8)
Since q varies with the theoretical dot area of the second primary j, atheo,j, q is defined as a function of atheo,j, that is, q = fi_j(atheo,j). Table 7-1 is the example published by Iino and Berns (1998). They fixed cyan ink at 50% theoretical dot area and overlapping the magenta ink at 0%, 25%, 50%, 75% and 100%. Table 7-1: The effective dot areas and the correction scalar, q, of cyan fixed at 50% theoretical dot area by overlapping the secondary magenta ink at various theoretical dot areas (Iino and Berns, 1998).
Theoretical Dot area 1 2 Cyan Magenta 0.500 0.000 0.500 0.250 0.500 0.500 0.500 0.750 0.500 1.000
Effective Dot area 3 4 Cyan Magenta 0.629 0.000 0.613 0.320 0.597 0.605 0.592 0.809 0.594 1.000
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Correction scalar 5 q 1.000 0.876 0.752 0.713 0.729
Using Eq. (7-6) the dot-gain of the cyan ink when the secondary magenta ink exists, gc=50%,m, is the difference between the column 3 and column 1. Since the dot-gain of cyan ink at 50% printed at paper alone (magenta ink is not present) by Eq. (7-7) is 0.129 (0.629 - 0.5). Hence, by Eq. (7-8), the correction scalar, q, listed in column 5, of the cyan ink at 50% when the magenta ink is present is gc=50%,m / 0.129. Iino and Berns further generalized the dot-gain correction scalar, q, for tertiary, and quaternary overprint of a four-color halftone process. They hypothesized that the dotgain correction scalar is the product of correction scalar of each combination of secondary overprint. For example if i, j and s primaries coexist, the correction scalar for primary i, q i, is equal to the product of fi_j(atheo,j) and fi_s(atheo,s). For the CMYK halftone printing process, the global correction scalar for each primary is defined as qc = fc_m(atheo,m) fc_y(atheo,y) fc_k(atheo,k) qm = fm_c(atheo,c) fm_y(atheo,y) fm_k(atheo,k) . qy = fy_c(atheo,c) fy_m(atheo,m) fy_k(atheo,k) qk = fk_c(atheo,c) fk_m(atheo,m) fk_y(atheo,y)
(7-9)
Hence, final effective dot area is the sum of the theoretical dot area and dot-gain, i.e., aeff = atheo + g = atheo + qgp = atheo + q(aeff,primary - atheo,primary).
(7-10)
3. Alternative second order improvement The algorithms proposed by Iino and Berns resulted in significant and impressive improvement for CMYK processed. Whereas, these algorithms implemented for processes
162
using primaries other than CMYK (e.g., CMGK, CMOK, MYOK,…, etc.) do not significantly improve or sometimes worsen the accuracy relative to the first-order printing model. Whether or not the dot-gain characteristic of the process using other than CMYK does not agree with the Iino and Berns' assumptions, it is difficult to conclude. Since our goal is to use various four-color processes to enhance the capability of spectral reproduction, it is necessary to uncover other algorithms which are capable of modeling the ink- and optical-trapping of any primary combination for the second order improvement. Conceptually, the proposed modification is not too different from the algorithms proposed by Iino and Berns. The difference is in the determination of the correction scalar, q. The alternative approach (using the Iino and Berns' notation) is to look at the dot-gain effect of a primary ramp given that the secondary is present at 50% theoretical dot area. The use of secondary at 50% theoretical dot area is hypothesized to have the average influence on the dot-gain characteristic of the primary ramp. Hence, dot gain of a primary j at theoretical dot area p% given that the existence of the secondary i at 50% theoretical dot area† is gi=50%,j=p% = aeff,i=50%,j=p% - p/100 .
(7-11)
By Eq.(7-7), the dot gain of the primary j printed alone (secondary i is not presented) at p% theoretical dot area is †
The intention of assigning j to be the primary and i to be the secondary is to be consistent with the Iino and Benrs' notation.
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gj=p% = aeff,j=p% - atheo,j ,
(7-12)
where atheo,j = p/100. Therefore, the correction scalar, q, is the ratio of gi=50%,j=p% to gj=p%, i.e, q = gi=50%,j=p% / gj=p% ,
(7-13)
Similarly, q varies with the theoretical dot area of the primary j, a theo,j, q is again defined as a function of atheo,j, that is, q = g i_j(atheo,j). Table 7-2 shows an example of the determined q scalar and dot gains of primary (magenta) given that the secondary (cyan) is present where the estimation of effective dot area is performed by inverting the Yule-Nielsen modified spectral Neugebauer equation. Table 7-2: The determined q scalars by proposed modification and the dot-gain of the primary (magenta) given that the secondary (cyan) is present. Theoretical 1 Cyan
Dot area 2 Magenta
0.500 0.500 0.500 0.500 0.500
0.000 0.250 0.500 0.700 0.900
Effective 3 Magenta (with Cyan) 0 0.503 0.808 0.919 0.979
Dot area 4 Magenta (without Cyan) 0 0.546 0.813 0.931 0.990
Correction scalar 5 q
1.000 0.856 0.984 0.946 0.872
By Eq. (7-13), the correction scalar, q, column 5, is obtained by (column 3 - column 2) / (column 4 - column 2).
164
To generalize the dot gain correction scalar, q, for tertiary and quaternary overprints of a four-color halftone process, it uses the similar hypothesis to the Iino and Berns that the dot-gain correction scalar is the product of the correction scalar of each combination of secondary (two-color) overprint. For example if i, j and s primaries coexist, the correction scalar for primary j, qj , is equal to the product of gi_j(atheo,j) and gs_j(atheo,j). For the CMYK halftone printing process, the global correction scalar for each primary is defined as qc = gm_c(atheo,c) gy_c(atheo,c) gk_c(atheo,c) qm = gc_m(atheo,m) gy_m(atheo,m) gk_m(atheo,m) . qy = gc_y(atheo,y) gm_y(atheo,y) gk_y(atheo,y) qk = gc_k(atheo,k) gm_k(atheo,k) gy_k(atheo,k)
(7-14)
Iino and Berns' algorithms are different from the proposed alternative algorithms. Conceptually, Iino and Berns' algorithms suggested that, for example, for cyan and magenta ink mixtures, whose theoretical dot areas are m% and k%, respectively, the dot gain of the cyan is normalized to the dot-gain locus of the cyan when the magenta is present at k%. Similarly, the dot gain of the magenta is normalized to the dot gain locus of the magenta when the cyan is present at m%. Whereas, the proposed algorithms recommend that the optimal dot gain of the cyan is normalized to the dot-gain locus of the magenta presenting at 50% theoretical dot area whenever the magenta is present at any theoretical dot area other than zero.
165
To be more specific, when cyan and magenta are present at 50% and 25% theoretical dot areas, respectively. The dot gain of the cyan estimated by the first-order model is the point a3 in Fig. 7-7. After Iino and Berns' modification the estimated dot gain of the cyan subjected to the optical trapping is at point b3. Whereas, the suggested dot gain of the cyan subjected to the optical trapping by the proposed algorithms is at point c3.
0.35
a2
0.3
mag=0% mag=25% mag=50% mag=75%
a3
Dot-gain in units of fractional dot area
0.25
a4 0.2
b3 a1
0.15
c2
c3
0.1
c4
c1
0.05
d3
0 0
0.1
0.2
0.3
0.4 0.5 0.6 Theoretical dot area
0.7
0.8
0.9
1
Fig. 7-7: The dot-gain loci of a cyan ramp when a magenta ink is present at different theoretical dot areas where the locus goes through a3, b3, and c3 are the dot gains esimated by the first-order model, Iino and Berns' algoritms, and the proposed algorithms, respectively. Iino and Berns modeled the optical trapping by defining the dot gain correction scalar, q = fc_m(atheo,m), exemplified by Fig. 7-7, to be the ratio of the dot gain value of the cyan at
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50% theoretical dot area when the magenta is present at atheo,m to the dot gain value of the cyan at 50% theoretical dot area when the magenta is not present. Whereas, the proposed algorithm modeled the optical trapping by defining the dot gain correction scalar, q = gc_m(atheo,c), to be the ratio of the dot gain value of the cyan at atheo,c theoretical dot area when the magenta is present at any theoretical area other than zero to the dot gain value of the cyan at atheo,c theoretical dot area when the magenta is not present. The difference between the two second-order improvements is shown by Fig. 7- 8.
T h e c o rr e c ti o n s c a la r b y Ii n o a n d B e r n s
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Fig. 7- 8: The functions of dot gain corrrection scalar by Iino and Berns (left) and the proposed (right) algorithms.
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The examples in Fig. 7- 8 pictorially show how the correction scalars associated with the two second-order improvements were obtained. Notice that the fc_m, shown as an monotonic function, is not necessary an monotonic function since the dot gain loci of real situation is not necessary as regularly shaped and spaced as that shown in Fig. 7-7. 4. Modeling by matrix transformation One straightforward approach to model a halftone printing process is to directly relate the theoretical dot area to the effective dot area of the printed sample through a matrix transformation. This requires extensive sampling of the colorimetric or spectral gamut to achieve high accuracy. Ideally, the set of effective dot areas of the representative sampling on a particular printing process is equal to the product of a transformation and the set of corresponding theoretical dot areas, i.e., aeff = M atheo ,
(7-15)
where aeff and atheo are the vector-matrix representations of effective and theoretical dot areas, respectively and M is a transformation matrix. Realistically, the effective dot area obtained by inverting Yule-Nielsen modified spectral Neugebauer equation is highly nonlinearly related to the theoretical dot area. Thus, the theoretical dot area needs to undergo a transformation to a representation which is linearly related to the effective dot area. Thereafter, the transformation matrix, M, can be derived by the pseudo-inverse matrix operation.
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The most frequently used process in finding a transformation is through polynomial regression. Its process is to represent the theoretical dot area by a higher order polynomial. For example, assuming the set of theoretical dot areas for a CMYK printing process is represented as c, m, y, and k, in vector-matrix form, the corresponding matrix of a second order polynomial representation is [c m y k c2 m2 y 2 k2 cm cy ck my mk yk]. Hence, the transformation matrix, M, can be obtained by M = pinv([c m y k c2 m2 y 2 k2 cm cy ck my mk yk]) aeff ,
(7-16)
where pinv( ) stands for the pseudo-inverse function. Proper Four-Color Sub-Model Selection At the color separation stage, it is necessary to decide a set of four inks (one black and three chromatic inks) associated with a four-color sub-model for synthesizing an input spectrum abiding by the ink limiting scheme. Conventionally, the selection can be done by locating a colorimetric value requiring reproduction inside a colorimetric gamut spanned by a set of candidate inks. The decision is based on whether the set of candidate inks whose colorimetric gamut includes the colorimetric value requiring reproduction. This requires a printing model to populate the colorimetric gamut boundary of the set of candidate inks and an inclusion algorithm to determine whether the colorimetric value is interior to the colorimetric gamut. This conventional method is not suitable for our challenge to accomplish spectral reproduction, since the higher dimensional spectral gamut is impossible to visualize. The
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use of inclusion algorithms is computationally intensive for determining an interior point inside a spectral gamut spanned by a set of candidate inks. If there exists a linear color mixing space that approximates the color formation of the employed halftone printing process, then it is possible to select the most significant set of four inks through regression or select the most significant set of four inks with the minimum reconstructing error in the linear color mixing space. It is of great interest to derive a linear color mixing space for a halftone printing process utilized by the experiment. 1. Deriving a linear color mixing transformation for a halftone printing process Although the derivation of the linear color mixing space for halftone printing process has been shown in Chapter V, it is described here again for the completeness of this chapter. First, it should agree with subtractive color mixing. Second, the dimensionality should coincide with the number of primaries to span the spectral gamut in the resultant mixing space. The transformation and its inverse transformation for halftone color are empirically derived as 1
1
Ψλ = R λw, paper − R λw and
(7-17)
1
R λ = ( R λw, paper − Ψλ ) w ,
(7-18)
respectively, where the Rλ,paper is the spectral reflectance factor of the paper substrate being printed on by primary inks and 2 ≤ w ≤ ∞. The transformation of reflectance factor to the empirically derived space is somewhat different from Eq. (3-6) since Eq. (3-6) is
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derived for opaque colorants. Whereas, they have basically the same structure, one offset vector accounting for subtractive color mixing and a higher order power to account for 1
the nonlinearity. The use of R λw, paper as the offset vector has a significant meaning. Consider that transforming a spectrum, which is exactly Rλ,paper, to the linear color mixing space, the result is a zero vector. This corresponds to the fact that there is not any primary presented in the linear space. Furthermore, Eq. (7-18) transforms a zero in the linear space back to the exact reflectance spectrum of the paper, Rλ,paper. 2. Selecting the most significant four primaries from the defined six-colors inkset The next step is to set up a regression model using the linear color mixing space to determined the suitable set of three chromatic primaries by constraining on the absolute existence of black ink for a given input spectral image pixel by pixel. One can utilize any statistical software with a stepwise option to remove the two least significant chromatic inks out of five. Whereas, for a robust process, it is suggested to try all the ten combinations of a set of four primaries, which are partitioned by the proposed method described in the section of sub-division of six color modeling, for a given pixel. The set with the least spectral error reconstruction is the candidate for the particular pixel. Backward Printing Models for Six-Color Separation Minimizing Metamerism The last component for the proposed six-color separation algorithm, which minimizes the metamerism between a given input spectral image and its spectral reproduction, is to estimate the set of theoretical dot areas corresponding to each pixel.
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The estimated theoretical dot areas will be stored as six separation records corresponding to each primary ink. To accomplish this, a backward printer model needs to be derived. Since it is not possible to analytically invert the forward printing model, the most feasible approach is to numerically invert the forward printing model. Thus, the backward sixcolor printing model is comprised of the proposed six-color forward spectral halftone printing model and a optimization module using the Simplex or Newton-Raphson iterative method to estimated a set of optimal theoretical dot areas which minimizes metamerism pixel by pixel between an input spectral image and it reproduction. For this prototypical research, the constrained optimization function in the optimization toolbox of MATLAB is utilized for the numerical engine. The structure of the six-color backward spectral printing model is depicted in Fig. 7-9. In Fig. 7-9, those modules inside the dashed box construct the proposed six-color forward spectral halftone printing model. Outside the dashed box, the process is handled by a numerical optimization engine. The process of spectral-based six-color separation starts from a given reflectance spectrum depicted as an input module located at the lower left corner. The proper four inks for reproducing an input spectrum are determined by the four-ink selector. Then, the optimization initializes a set of theoretical dot areas, which are the concentrations estimated in the linear color mixing space, corresponding to the selected four inks in the throughput processes for estimating the reconstructed spectrum, that is the output of the forward model and located at the lower right corner. The decision
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module in the diamond box located in the lower center then compares the input and output spectra at this iteration stage. If the spectral reconstruction satisfies the error criteria then output the set of theoretical dot areas at current iteration stage and terminate the iteration. If not, the following module located near the input module decides and feeds back the modification of the set of theoretical areas for the next iteration.
Six-Color Forward Spectral Halftone Printing Model cmyk
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No
Minimum Spectral Error or Minimum Four Tristimulus Error
Rλ,estimated
Yes
Fig. 7-9: The structure of the six-color backward spectral printing model using cyan, magenta, yellow, green, orange and black ink as printing primaries.
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Initially, the optimization criteria is to minimize the spectral error. If the spectral minimization yields unsatisfactory accuracy, then the optimization pursues to balance between spectral and colorimetric accuracy. Because the six-color forward spectral printing model is composed by ten forward four-color sub-models, the optimization criteria for balancing between spectral and colorimetric accuracy is to minimize the total error of four tristimulus values. The four tristimulus values are obtain by the tristimulus values under standard viewing illumant such as D50 for printing industry. Then, the fourth value is the X value calculated under second standard viewing illumant such as A, based on the research by Allen (1980).
C. EXPERIMENTAL AND VERIFICATION
In order to verify the proposed algorithm in building a six-color spectral halftone printing model, the current research utilized the DuPont Waterproof® system to represent a halftone printing process. The standard cyan (C), magenta (M), yellow (Y), black (K), green (G), and orange (O) designed for DuPont Waterproof® system were advised to be used as the six printing primaries for a stable proofing process. The choice, which was an arbitrary decision, of these colors was not defined by the optimal inkset selection algorithm since a spectral image was not given. Although the six primaries were not defined by the optimal inkset selection algorithm, the development of the six-color spectral printing model is independent from any of previous colorant estimation, optimal inkset
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selection, and overprint prediction modules. Thus, this research assumes that the usage of the six primaries is independent from the proposed algorithms developed for a general spectral-based six-color printing system. The spectral reflectance factors of the printed six primaries and the paper substrate (100# Vintage Gloss Text) are plotted in Fig. 7-10. 1
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Fig. 7-10: The reflectance spectra of the printed six primaries and substrate. Sample Preparation Usually, modeling of halftone printing process requires a set of ramps including primary secondary, tertiary, quaternary,…, and so forth and a verification target representing the device gamut. Since this research aims at developing a six-color printing system, the preparation for each printed sample patch utilized FM screening to avoid moiré patterns. The screen frequency was chosen at 175 LPI resolution or equivalent (the
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resolution of stochastic screening can not be represented in units of LPI) to account for the most common condition for practical use. 1. Preparation for ramps Each ramp was printed at 5%, 10%, 15%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, and 100% theoretical dot areas. For a four-color halftone printing process, there are secondary (two-ink overprints), tertiary (three-ink overprints), and quaternary (four-ink overprints). Although this research used six primary inks for halftone reproduction, the approach to reproduce an input color (spectrally) is limited to use three chromatic and one black inks for synthesis. Therefore, five-ink and six-ink overprints were not generated in the six-color printing process. The overprints are categorized by their number of inks and listed as follows. There were fifteen two-ink overprint combinations (C(6,2)) selecting two inks out of six. They were CM, CY, CG, CO, CK, MY, MG, MO, MK, YG, YO, YK, GO, GK, and OK. There were twenty three-ink overprint combinations (C(6,3)) selecting three inks out of six. They were CMY, CMG, CMO, CMK, CYG, CYO, CYK, CGO, CGK, COK, MYG, MYO, MYK, MGO, MGK, MOK, YGO, YGK, YOK, and GOK. There were ten four-ink overprint combinations (C(5,3)) selecting three chromatic inks out of five plus one black due to the constraint mentioned previously. They are CMYK, CMGK, CMOK, CYGK, CYOK, CGOK, MYGK, MYOK, MGOK, and YGOK. 2. Preparation of the verification target (5x5x5x5 combinatorial design for mixtures)
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Again, the constraint is to use black and three chromatic inks to reproduce each pixel from a spectral image. The six-color halftone printing model can be viewed as a super set of several sets of four-color halftone models. In this case, there were ten subsets of four-color halftone models. They are CMYK, CMGK, CMOK, CYGK, CYOK, CGOK, MYGK, MYOK, MGOK, and YGOK. There were 625 combinations for each subset, the 5x5x5x5 combinatorial design of mixtures modulated by five different fractional dot areas. To printing the 625 mixtures for CMYK sub-model, twenty-five samples were printed on paper by varying the theoretical dot areas of cyan and magenta inks at the 0% of yellow and 0% of black ink. This printing procedure was repeated 25 times by varying the yellow and black inks at five different fractional dot areas ( 0%, 25%, 50%, 70%, and 90%). Since there are ten sets of four-ink combinations to compose the proposed six-color model, the total number of samples was 6,250 for the 5x5x5x5 combinatorial design. 3. Sample measurements Samples were measured using a Gretag Spectrolino, whose sampling geometry is 0/45, with automatic station to obtain reflectance spectra. The adopted spectral range was from 400 nm to 700 nm at 10 nm intervals. The spectral data of ramps (total 612 patches) were obtained by the average of four measurements on each patch. Due to the large number of patches of the verification target (total 6,250), the spectral data of each patch were only measured once.
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4. Accuracy metric The colorimetric accuracy was specified by the CIE94 color difference equation for standard illuminant D50 and 1931 standard observer. The spectral accuracy was specified by a metamerism index which is quantified by the CIE94 color difference equation for standard illuminant A and 1931 standard observer after a parameric correction. Determining the Yule-Nielsen n-Factor Performing the estimation using the algorithm outlined in Fig. 7-3, the n-factor was found to be 2.2 for this set of six-primary ramps. The mean colorimetric accuracy, specified in units of ∆E*94, in predicting all 72 samples of the six primary ramps against different n values is plotted in Fig. 7-11. With n = 2.2, the colorimetric and spectral accuracy is listed in Table 7-3 and the reconstructed spectra for each primary ramp are plotted in Fig. 7-12. 1.6
1.4
Mean Delta E94 under D50
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Fig. 7-11: The mean prediction accuracy of all 72 samples of the six primary ramps as function of n-factor.
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Table 7-3: The colorimetric and spectral accuracy of n = 2.2 in predicting all 72 samples of the six primary ramps where Stdev stands for the standard deviation and RMS represents the root-mean-square error in unit of reflectance factor. ∆E*94 0.27 0.28 1.42 0.00 0.005
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Metamerism Index 0.04 0.05 0.20 0.00
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Fig. 7-12: The measured and the predicted reflectance spectra by Eqs. (7-3) and (2-33) using n = 2.2 where the solid lines are measured spectra and the dashed lines are the predicted spectra.
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First, Fig. 7-12 provides a visual confirmation for the spectral prediction using n = 2.2. Second, from Table 7-3 the mean and maximum colorimetric errors in predicting the primary ramp are 0.27 and 1.4 unit of ∆E*94, respectively. Their low mean and maximum metamerism index as well as low RMS error indicate that the determined Yule-Neilsen nfactor was optimal for the primary ramp prediction. This n-factor (n = 2.2) will be used as the optimal parameter to the construction of the six-color spectral halftone printing model for this particular printing process. Accuracy of the First-Order Six-Color Forward Printing Model By adopting n = 2.2, the theoretical to effective dot area transfer function, f( ), for each primary ramp was attained by discretely sampling at the effective dot area forming continuous transfer functions using cubic spline interpolation. The theoretical to effective dot area transfer functions for the six primaries are plotted in Fig. 7-13. 1
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Fig. 7-13: The theoretical to effective transfer functions of the six primaries.
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The performance of the first order forward halftone printing model in predicting the 6,250 samples of the verification target is specified in Table 7-4 and the histogram of the colorimetric accuracy is plotted in Fig. 7-14. Table 7-4: The colorimetric and spectral accuracy of the first order forward model in predicting the 6,250 samples of the verification target. ∆E*94 1.95 1.51 9.7 0.00 0.008
Mean Stdev Max Min RMS
Metamerism Index 0.22 0.22 1.58 0.00
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Fig. 7-14: Histogram of the colorimetric performance using the first-order forward printing model in predicting the 6,250 samples of the verification target.
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Observing the accuracy with an average ∆E*94 =1.95 and the histogram of color differences for the verification target, the first impression of this model is that it is quite capable to describe the color formation of the Waterproof® system since the most of samples (approximately 4,000 samples) can be matched under two units of ∆E*94. Nevertheless, there are still about 824 samples which were predicted with colorimetric error higher than four units of ∆E*94. As Iino and Berns suggested, these prediction errors are mainly due to the overestimating effective dot areas, thereby, under predicting the spectral reflectance factors. On account of the non ideal ink and optical trapping, the effective dot areas were actually smaller than that estimated by the first-order approximation since it does not these the trapping factors into account. Since the printing device used for this verification generates samples similar to Matchprint, the ink-trapping ability is assumed to be ideal; thus the influence is mainly referred to as optical-trapping. The visual evidence of overestimating the effective dot areas leading to the under prediction of reflectance factors can be provided by the vector plot of the L* vs. a*, shown in Fig. 7-15. Due to the large number of data (6,250), it only shows the vector plots of the 100 predicted samples with the highest colorimetric errors. As pictured by Fig. 7-15, the trend of the predicted L* indicates the over estimation of effective dot areas. Another visual aid, Fig. 7-16, shows that the four estimated samples with large prediction errors have been reproduced with lower reflectance factors.
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Fig. 7-15: The vector plot of the 100 predicted samples with the highest colorimeric errors by the first-order forward model. The vector tail represents the measured coordinate and the vector head represents the prediction. 0 .1 4
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Fig. 7-16: Four example spectra showing under prediction by the first order model.
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Second-Order Modification (By Iino and Berns' Suggestion) Iino and Berns' algorithms were implemented to estimate the colorimetric and spectral performance in predicting the verification target of 6,250 mixtures. This secondorder printer model required determining the dot-gain correction scalar, q, as a function of theoretical dot area, mentioned previously. The database for modeling q was obtained from some of the samples in the verification target with the exact theoretical dot area combinations. For example, to determine the fc_m described in Eq. (7-9), the five samples with the theoretical dot area combinations for cyan = 50% and magenta = 0%, 25%, 50%, 70%, and 90% were selected and cubic spline was employed to interpolate the data in between. The sample of cyan = 50% and magenta = 100% was not included at the beginning of designing the verification target. Hence, the corresponding fc_m(am=100%) was extrapolated. Since the designed verification target does not include this sample, the effectiveness or the validity of the extrapolation will not affect the verification accuracy. There are 12 fi_j functions determined for each four-color sub-model. Three of the example fi_j functions, fc_m, fc_y, and fc_k, determined for the CMYK sub-model is shown in Fig. 7-17. In Iino and Berns' article, the fi_j functions are second-order polynomials since those function were found to be monotonically decreasing by their experiment. Whereas, the fi_j functions, shown in Fig. 7-17, were not monotonic. This indicates that the two-dimensional dot gain surface of two-ink mixtures were not smooth and monotonically decreasing for the Waterproof® samples. It was suspected that this
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discrepancy is not caused by different printing systems but the halftone screening shcemes. That is, the regular dot pattern generated by the conventional rotated-screen (utilized by Iino and Berns) has smoother dot gain characteristic, whereas, the stochastic dot pattern (utilized by this research) has relatively irregular dot gain behavior.
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Fig. 7-17: The three example functions of dot-gain correction scalar determined for CMYK sub-model based on Iino and Berns' algorithms.
After the determination of all q functions the correction scalar for each primary, qc, qm, qy, qg, qo, and qk, were calculated by Eq. (7-9) for each theoretical dot area combination. Colorimetric and spectral accuracy, shown by Table 7-5, was estimated and the histogram of colorimetric errors is shown in Fig. 7-18.
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Table 7-5: The colorimetric and spectral accuracy in predicting the verification of 6,250 sample by the algorithms suggested by Iino and Berns. ∆E*94 Metamerism Index 1.93 0.32 1.41 0.32 9.79 2.41 0.00 0.00 0.008
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Fig. 7-18: Histogram of the colorimetric error in units of ∆E*94 for the 6,250 samples predicted by Iino and Berns' algorithms.
At first glance, the performance of this second-order modification seems not significantly different from that of the first-order printing model due to the similar average and maximum values. However, in predicting the large number of samples, the standard
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deviation and histogram of ∆E*94 should be examined and compared. The Iino and Berns model performed with a smaller standard deviation of colorimetric errors. Furthermore, it improves the first-order printing model by lowering the number of high colorimetric errors above 4 units of ∆E*94 from 824 to 507 samples. This can be visually confirmed by comparing the histograms in Figs. 7-14 and 7-18. The vector plot , shown in Fig. 7-19, of L* vs. a* for the exact same 100 samples, plotted in Fig. 7-15, indicates that this secondorder modification significantly reduced the lightness errors relative to the first- order model. 70 60 50
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Fig. 7-19: The vector plot of L* vs. a* for the 100 samples used as examples in Fig. 7-15 predicted by Iino and Berns' algorithms.
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The spectral predictions of the four samples used as examples are Fig. 7-16 is depicted in Fig. 7-20. Compared with Fig. 7-16, the spectral predictions are significantly improved. Based on this analysis, it is concluded that the Iino and Bern algorithms significantly corrects the under prediction of mixtures due to optical-trapping. However, given that the average predicted colorimetric error is similar to that of the first- order prediction, the other alternative second-order improvements need to evaluated. Hopefully, one of these will yield high colorimetric and spectral accuracy.
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Fig. 7-20: The spectral predictions of the four example samples used as examples in Fig. 7-16 by the Iino and Berns' algorithms.
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Alternative Second-Order Modification (Proposed Algorithm) This proposed alternative second-order modification attacks the (ink and) optical trapping from the stand point of modeling the optical-trapping of primary-secondary interactions under the assumption of average influence of a secondary ink presenting at 50% theoretical dot area. To determine the gm_c described in Eq. (7-14), the five samples with the theoretical dot area combinations for magenta = 50% and cyan = 0%, 25%, 50%, 70%, and 90% were selected and cubic spline was employed to interpolate the data in between. The sample of magenta = 50% and cyan = 100% was not included at the beginning of designing the verification. Hence, the corresponding gm_c(ac=100%) was set to one since there will be no dot-gain for cyan at 100% theoretical dot area. There are 12 q functions determined for each four-color sub-model. Three of the example q functions, gm_c, gy_c, and gk_c, determined for the CMYK sub-model are shown in Fig. 7-21. One logical consideration using Eq. (7-14) is worth mentioning. Consider a set of the theoretical dot areas with ac = t%, am = 0%, ay = 0%, ak = 0%, the corresponding correction scalar for each primary should be 0 ≤ qc ≤ 1, qm = 1, qy = 1, and qk = 1 where 0 ≤ t ≤ 100. That is, when a sample is exactly a primary color, there is no correction scalar needed for its dot gain determined by the first-order theoretical to effective dot area transfer function, f( ). Whereas, if Eq. (7-14) is utilized without modification then the q c in Eq. (7-14), being a product of g m_c, gy_c, and gk_c, will yield correction when cyan appears alone. This error also is encountered for two or three primary mixtures. To be more
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specific, Table 7-6 shows some samples' theoretical dot areas and their correction scalars for each ramp using Eq. (7-14) without logical operator correction.
1
0 .9 5
q
0 .9
0 .8 5
0 .8
g mc g
0 .7 5
yc
gk c 0 .7 0
0 .1
0 .2
0 .3
0 .4 0 .5 T he o re ti c a l d o t a re a
0 .6
0 .7
0 .8
0 .9
Fig. 7-21: The three example functions of dot-gain correction scalar determined for CMYK sub-model based on proposed algorithms. Table 7-6: The theoretical dot areas and their correction scalar by Eq. (7-14) without logical correction. ac 0 0.25 0.5 0.7 0.9 0 0.25 0.5 0.7 0.9
am 0 0 0 0 0 0.25 0.25 0.25 0.25 0.25
ay 0 0 0 0 0 0 0 0 0 0
ak 0 0 0 0 0 0 0 0 0 0
qc 1 0.260 0.470 0.645 0.477 1 0.260 0.470 0.645 0.477
qm 1 1 1 1 1 0.346 0.346 0.346 0.346 0.346
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qy 1 1 1 1 1 1 1 1 1 1
qk 1 1 1 1 1 1 1 1 1 1
It can be seen that the first five samples are pure cyan at different dot areas. Their correction scalars, qcs are not unity as expected, i. e., no correction is needed for its dot gain. Hence, a logical modification is required to rectify this error. The logical modification was derived by g'i_j = gi_j.*{[(1./gi_j).*(~atheo,i)]+(~~atheo,i)} ,
(7-19)
where .* and ./ are the vector-matrix componentwise multiplication and division, respectively, and ~ is a logical "not" operator (MATLAB, 1996). It will be shown by example when applying the logical modification, Eq. (7-19), to the correction scalar, q, for each ramp. It turns out that the logical modification is needed only when the secondary is present. Let i be m and j be c, thus c is assumed as primary and m is assumed as secondary. Case 1: if atheo,m is zero then ~atheo,m = 1 and ~~atheo,c = 0. This dot area combination indicates the samples are cyan ramp along. Using Eq. (7-19) results in g'm_c(atheo,c) = gm_c(atheo,c).*{[(1./gm_c(atheo,c)).*(~atheo,m)]+(~~atheo,m)} = gm_c(atheo,c).*{[(1./gm_c(atheo,c)).*(1)]+(0)} = gm_c(atheo,c).*[1./gm_c(atheo,c)] = 1. Case 2: If atheo,m is nonzero then ~atheo,m = 0 and ~~atheo,c = 1. Using Eq. (7-19) results in g'm_c(atheo,c) = gm_c(atheo,c).*{[(1./gm_c(atheo,c)).*(~atheo,m)]+(~~atheo,m)} = gm_c(atheo,c).*{[(1./gm_c(atheo,c)).*(0)]+(1)} = gm_c(atheo,c).*(0 + 1) = gm_c(atheo,c). That is, the dot-gain correction scalars are not changed when a secondary is present.
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Table 7-7 shows the theoretical dot areas and the corresponding correction scalar for each ramp of the same samples shown in Table 7-6 after logical correction. After logical modification the first five samples, which are cyan primaries printed at different areas, have the correct corresponding correction scalars, qc, of unity. The rest of correction scalars for none-appearing secondaries are all one as expected. In addition, the sixth through tenth samples, which are cyan and magenta mixtures, whose correction scalars for the cyan ramp are different from the correction scalars for the cyan ramp in Table 7-6, neither for the magenta ramp. Hence, the logical error is dramatically affecting the accuracy of the dot-gain correction scalar, q, for the proposed algorithms. (It was investigated to see if Iino and Berns' algorithms also required logical modification for dotgain correction scalar. It was found that the dot-gain correction scalar for each primary by Eq. (7-9) was invariant with this logical modification.) Table 7-7: The theoretical dot areas and their correction scalar by Eq. (7-14) with logical correction. ac 0 0.25 0.5 0.7 0.9 0 0.25 0.5 0.7 0.9
am 0 0 0 0 0 0.25 0.25 0.25 0.25 0.25
ay 0 0 0 0 0 0 0 0 0 0
ak 0 0 0 0 0 0 0 0 0 0
qc 1 1 1 1 1 1 0.848 0.960 0.992 0.977
qm 1 1 1 1 1 1 0.856 0.856 0.856 0.856
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qy 1 1 1 1 1 1 1 1 1 1
qk 1 1 1 1 1 1 1 1 1 1
The colorimetric and spectral accuracy of the proposed algorithms in predicting the 6,250 samples is listed in Table 7-8 and the histogram is plotted in Fig. 7-22. Table 7-8: The colorimeric and spectral accuracy of the proposed algorithms in predicting the verification target of 6,250 sample mixtures. ∆E*94 1.44 1.16 7.59 0.00 0.006
Mean Stdev Max Min RMS
Metamerism Index 0.22 0.25 1.75 0.00
2500
F re q u e n c y
2000
1500
1000
500
0 0
1
2
3
4
5
6
7
8
D e lta E 9 4
Fig. 7-22: Histogram of the colorimeric accuracy of the proposed algorithms in predicting the verification target of 6,250 sample mixtures.
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Table 7-8 reveals that the average and maximum colorimetric errors are 1.44 and 7.59 ∆E*94, respectively. In addition, it also indicates that the majority of sample mixtures can be reproduced with lower colorimeric errors by the proposed algorithms given the tight standard deviation. There are only 251 as opposed to 507 (by Iino and Berns' modification) or 824 (by the first order prediction) samples predicted with higher than four units of ∆E*94. Colorimetrically, this is a significant improvement. The spectral accuracy specified by the low average metamerism index indicates that the verification target is spectrally well predicted for all three models discussed so far. Performance of spectral prediction for all three models is not significantly different judging by the statistical results of the metamerism index. The L* vs. a* and the four predicted sample spectra used for the previous example are plotted in Figs. 7-23 and 7-24, respectively. 70 60 50
L*
40 30 20 10 0 -40
-30
-20
-10
0 a*
10
20
30
40
Fig. 7-23: The vector plot of L* vs. a* for the 100 samples used as examples in Fig. 7-15 predicted by the proposed algorithms.
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At first glance of Figs 7-23 and 7-24, the proposed algorithms tend to over predict the spectral reflectance factors or under estimate the effective dot areas for low lightness mixtures. Although the colorimetric and the spectral performance of the
proposed
algorithm is considered as superior, the appearance of the spectral matches shown by Fig. 7-24 is disappointing.
Reflectance fac tor
0.2
0.15 0.1 0.1 0.05 0.05
0 400
Reflectance fac tor
0.15
450
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W avelength
Fig. 7-24: The spectral prediction of the four samples used as examples in Fig. 7-16 by the proposed algorithms.
By comparing at the L * vs. a* vector plots of Figs 7-21 and 7-13, it was found that the prediction trend by the proposed algorithms is approximately opposite to that of the first-order prediction. Thus, it was hypothesized that the accurate estimation of effective
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dot areas linearly lies between the estimation by the proposed algorithms and by the first order prediction. With this observation, the final estimation of effective dot area, aeff,est is modified by a linear combination of aeff,proposed and aeff,1st, i. e., aeff,est = taeff,proposed + (1-t)aeff,1st ,
(7-20)
where t is a linear scalar and 0 ≤ t ≤ 1, aeff,proposed is the effective dot area estimated by the proposed algorithm, and aeff,1st is the effective area estimated by the first order printing model. Equation (7-20) can be expressed as Eq. (7-21) based on Eq. (7-10), aeff,est = t(atheo + q(aeff,primary - atheo,primary)) + (1-t)aeff,1st .
(7-21)
It was found that best spectral prediction by Eq. (7-20) occurred when t = 0.62. The colorimetric and spectral performance of the proposed algorithms modified by Eq. (7-20) in predicting the verification target is shown in Table 7-9 and the histogram of the colorimetric error is plotted in Fig. 7-25. The implementation for the six-color forward printing model based on the proposed algorithm is shown in Appendix E. Table 7-9: The colorimeric and spectral accuracy of the proposed algorithms modified by Eq. (7-20) in predicting the verification target of 6,250 sample mixtures.
Mean Stdev Max Min RMS
∆E*94 0.90 0.75 6.06 0.00 0.004
Metamerism Index 0.10 0.10 1.19 0.00
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30 00
25 00
F reque nc y
20 00
15 00
10 00
50 0
0 0
1
2
3
4
5
6
7
D elta E 94
Fig. 7-25: Histogram of the colorimeric accuracy of the proposed algorithms modified by Eq. (7-20) in predicting the verification target of 6,250 sample mixtures. The modification by Eq (7-20) yielded excellent accuracy improvement in predicting the 6,250 samples judged by the statistical results of colorimetric and spectral error. It also rectifies the problem of over prediction of spectral reflectance factors for low L* samples. Visual evidence is provided by the vector plot, shown in Fig. 7-26, of L* vs. a*, L* vs. b*, and b* vs. a* for the 300 samples whose colorimetric error is predicted higher than 2.41 units of ∆E*94.
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100
90
90
80
80
70
70
60
60
L*
L*
100
50
50
40
40
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30
20
20
10
10
0 -60
-40
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0 a*
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20
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-20
0 b*
20
40
60
60
40
b*
20
0
-20
-40
-60 -60
20
40
60
Fig. 7-26: The vector plot of L* vs. a*, L* vs. b*, and b* vs a* for the 300 samples whose colorimetric error is predicted higher than 2.41 units of ∆E*94.
Observing Fig. 7-26, lightness errors of the reproduction are dramatically reduced. There are only three to four samples with large lightness errors and two to three with large
198
chromatic errors. The rest of sample are predicted with low colorimetric error. Recall that Fig. 7-26 represents the 300 samples predicted with higher colorimetric error. Thus, the proposed algorithm together with the modification Eq. (7-20) provided the best performance second-order printing model for this research project. The spectral prediction for the four samples used as examples in Fig. 7-16, shown in Fig. 7-27, reflects this superiority.
0 .1 5
0 .1
Re fle c ta nce fa cto r
0 .0 8 0 .1 0 .0 6
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M e a s ure d P re dic te d
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Fig. 7-27: The spectral prediction by the proposed algorithms modified by Eq. (7-20) for the four samples used as examples in Fig. 7-16. Modeling by Matrix Transformation A modeling method was evaluated by directly relating a set of theoretical dot areas of a multi-ink mixture to the corresponding set of effective dot areas estimated by
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inverting the Yule-Nielsen modified spectral Neugebauer equation. This requires a large sample database to represent major sample behavior. Hence, its accuracy relies on the size of database. Since the theoretical dot area is nonlinearly related to the effective dot area due to the mechanical dot-gain, optical dot-gain, and ink and optical trapping, a direct linear transformation using Eq. (7-15) is not possible. Hence, this research project implemented a higher-order polynomial regression using Eq. (7-16) seeking the matrix relationship between the theoretical and effective dot areas. It was found that the order of the polynomial matrix should be higher than four to achieve the similar accuracy predicted by the proposed algorithms. The process of the six-color modeling by matrix method is similar to the structure of the six-color forward printing model depicted in Fig. 7-9. The only difference is that the ten theoretical to effective dot area transfer functions, f( )s, and the second order improvements are replaced with ten matrices for each four-color submodel. The fourth order polynomial matrix, as the input to the matrix transformation, for each four-color sub-model was implemented with 27 terms to account for any of nonlinearity between the theoretical and effective dot area. This research project investigated the matrix method in predicting the verification of 6,250 samples. Its colorimetric and spectral accuracy is shown Table 7-10 and the histogram of the predicted colorimetric errors is plotted in Fig. 7-28.
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Table 7-10: The colorimetric and spectral performance of the verification target predicted by the matrix method. ∆E*94 1.66 1.39 14.08 0.02 0.012
Mean Stdev Max Min RMS
Metamerism Index 0.20 0.21 3.04 0.00
4000
3500
3000
F re q ue ncy
2500
2000
1500
1000
500
0 0
2
4
6
8
10
12
14
D e lta E 9 4
Fig. 7-28: Histogram of the colorimeric accuracy of the matrix method in predicting the verification target of 6,250 sample mixtures. From Table 7-10 and Fig. 7-28, the performance of the matrix method is not a favorable approach relative to the proposed algorithms in predicting the verification target. Furthermore, it requires a relatively large sample database to explain the color mixing of a
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halftone printing process. Furthermore, the estimated effective dot area obtained by the matrix method sometime is a negative value leading to a non-rational synthesis by YuleNielsen modified spectral Neugebauer equation. Hence, the matrix method is not adopted by the research development of six-color separation minimizing metamerism. Six-Color Separation Minimizing Metamerism It was desired to verify the validity of the backward printing model which is comprised by a four-ink selector, a forward printing model, and a numerical engine. The proposed algorithms modified by Eq, (7-20) were utilized as the forward printing model since it has the best accuracy for the Waterproof® system. This research project chose the Gretag Macbeth Color Checker as a separation target and tried to reproduce the spectrum for each color with minimal metamerism. Owing to the original spectral image (the Gretag Macbeth Color Checker) was unavailable, the use of CMYGOK as printing primaries was not a decision from the optimal inkset selection module. Hence, some of color spectra may be outside the spectral gamut of the developed six-color spectral printing model using the Waterproof® system. Nevertheless, the objective of this section is to verify the spectral reproducibility simultaneously balancing with the colorimetric accuracy of the proposed six-color modeling process. As mentioned previously, given a reflectance factor of a color sample, the task of color separation is to determine a set of theoretical dot areas (or sometime called request dot areas), corresponding to the printing primaries, for a printer to synthesize the desired
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spectrum. The estimated 24 sets of theoretical dot areas with respect to the 24 colors in the Color Checker, the corresponding 24 colorimetric errors in units of ∆E*94, the corresponding 24 metamerism indices, and the corresponding RMS error in units of reflectance factor are shown in Table 7-11 and the statistical results in predicting the 24 colors using the proposed six-color printing model is listed in Table 7-12. Table 7-11: The predicted theoretical dot areas, colorimetric and spectral errors of the 24 colors in Macbeth color checker where M.I. represents the metamerism index. Color Name Dark Skin Light Skin Blue Sky Foliage Blue Flower Blue Green Orange Purplish Blue Moderate Red Purple Yellow Green Orange Yellow Blue Green Red Yellow Magenta Cyan White N8 N6.5 N5 N3.5 Black
ac 0.000 0.000 0.305 0.026 0.278 0.144 0.069 0.555 0.000 0.461 0.000 0.000 0.921 0.000 0.059 0.000 0.092 0.562 0.000 0.005 0.012 0.021 0.034 0.207
am 0.040 0.140 0.106 0.000 0.209 0.000 0.000 0.323 0.479 0.499 0.000 0.000 0.418 0.000 0.763 0.000 0.448 0.047 0.000 0.000 0.000 0.000 0.005 0.110
ay 0.121 0.227 0.000 0.308 0.012 0.000 0.460 0.000 0.262 0.171 0.621 0.648 0.000 0.297 0.515 0.820 0.004 0.000 0.000 0.000 0.000 0.000 0.000 0.155
ag 0.000 0.000 0.000 0.099 0.000 0.103 0.000 0.000 0.020 0.000 0.120 0.031 0.000 0.250 0.000 0.015 0.000 0.039 0.000 0.001 0.001 0.000 0.000 0.000
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ao 0.200 0.000 0.000 0.000 0.000 0.000 0.364 0.000 0.000 0.000 0.032 0.240 0.000 0.025 0.000 0.078 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
ak ∆E*94 0.346 0.01 0.093 0.02 0.133 0.41 0.310 0.11 0.022 0.24 0.000 0.94 0.021 0.20 0.000 1.48 0.031 0.03 0.050 0.07 0.000 0.32 0.001 0.36 0.000 5.57 0.074 0.02 0.040 0.35 0.008 0.17 0.000 0.36 0.086 0.33 0.000 1.60 0.088 0.15 0.189 0.29 0.317 0.24 0.466 0.28 0.601 0.46
M.I. 0.09 0.11 0.86 0.34 0.69 0.17 0.56 1.63 0.16 0.73 0.65 1.01 1.55 0.17 0.39 0.39 0.49 0.89 0.35 0.30 0.57 0.45 0.52 0.90
RMS 0.007 0.031 0.041 0.013 0.051 0.044 0.021 0.037 0.024 0.079 0.026 0.041 0.028 0.021 0.071 0.021 0.115 0.032 0.062 0.054 0.043 0.027 0.013 0.005
Table 7-12: The statistical results in predicting the 24 colors using the proposed six-color printing model.
Mean Stdev Max Min RMS
∆E*94 0.58 1.14 5.57 0.02 0.045
Metamerism Index 0.58 0.41 1.63 0.11
First, the overall colorimetric accuracy in predicting the spectral reflectance of the 24 colors with average 0.57 and maximum 5.57 units of ∆E*94 is satisfactory. There are 20 out of 24 colors predicted with colorimetric errors below 0.5 units of ∆E*94. Furthermore, the Blue and White are the out of gamut colors. Excluding these out of colorimetric gamut colors, the statistical results of the colorimetric and spectral performance for 22 in gamut colors is listed in Table 7- 13. Table 7- 13: The colorimetric and spectral performance of the 22 in gamut colors.
Mean Stdev Max Min RMS
∆E*94 0.31 0.33 1.48 0.01 0.045
Metamerism Index 0.55 0.36 1.63 0.11
Since the primary goal for this research development is to minimize the metamerism between the original and its color reproduction, special attention is drawn to metamerism. The metamerism index represents the color difference under the second reference
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illuminant (A) while the color difference between the original and reproduction is zero under the first standard illuminant (D50) after parameric correction (Fairman, 1987). Thus, the average, 0.55, and maximum, 1.63, in units ∆E*94 indicates that the metamerism is not severe at all for this reproduction. There are 21 out of 24 colors which were predicted under one unit of metamerism index. This implies that the spectral mismatch between the original and reproduction is small such that their color matches are approximately invariant and constant across illuminants. The four best and four worst predicted spectra in terms of metamerism index are plotted in Figs. 7-29 and 7-30, respectively.
1
1
Re fle c ta nce
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Fig. 7-29: The four best predicted spectra in terms of metamerism index of the 24 colors.
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1
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Purplish Blue
Blue
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Fig. 7-30: The four worst predicted spectra in terms of metamerism index of the 24 colors. The implementation of the spectral-based six-color separation algorithm is shown in Appendix F.
D. SPECTRAL PERFORMANCE COMPARISONS FOR THREE-, FOUR-, AND SIX-COLOR PRINTING PROCESSES In order to substantiate the superiority of the spectral performance by adopting more number of degrees of freedom, two three-color (CMY) continuous tone printing devices, Fujix Pictrography 3000 and Kodak Professional 8670 PS thermal printer, and one four-color (CMYK) printing processes, DuPont Waterproof®, were utilized to formulate the color reproductions of the Macbeth Color Checker computationally.
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Kubelka-Munk transformations, Eqs. (2-9) and (2-10), for transparent material in optical contact with opaque support were used to describe the color synthesis for the two continuous tone printers. Two sets of one thousand samples of 10x10x10 combinatorial mixtures in units of monitor RGB were printed by the two three-color printers. The three eigenvectors derived in the absorption space for each printer were used as the pseudoprimaries for color synthesis (recall that the eigenvectors are theoretically linearly related to that actual primary colorants, described in Chapter IV). The proposed algorithm for modeling a four-color halftone printing process was employed to describe the color synthesis of DuPont Waterproof® system using CMYK. Exact colorimetric matches not including the out of colorimetric gamut colors under illuminant D50 and 1931 observer for the Macbeth Color Checker were obtained by all the three- and four-color printing processes. The color difference in unit of ∆E*94 under illuminant A and F2, shown in Table 7-14, were calculated for the prediction by Fujix, Kodak, Waterproof® CMYK, and Waterproof® CMYKGO printing processes whose predicted spectral reflectance factors are attached in Appendices L, M, and N, respectively. It can be seen that the average color difference under illuminant A for the Waterproof® CMYK process was relatively large, whereas, that of rest of the three printing processes were small. In addition, the average color differences under illuminant F2 for the Fujix, Kodak, Waterproof® CMYK processes were relatively large, whereas, the color difference predicted by the Waterproof® CMYKGO was the smallest.
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Table 7-14: The color difference of the predicted Macbeth Color Checker under illuminant A and F2 by four different printing processes, where the color names corresponding to the bold faced entries are the out of colorimetric gamut colors of each device. Fujix ∆E*94 under A Dark Skin 0.78 Light Skin 0.05 Blue Sky 0.64 Foliage 0.36 Blue Flower 1.00 Blue Green 0.98 Orange 1.59 Purplish Blue 0.23 Moderate Red 0.05 Purple 0.84 Yellow Green 0.23 Orange Yellow 1.69 Blue 0.15 Green 0.31 Red 0.63 Yellow 0.46 Magenta 0.63 Cyan 2.70 White 0.99 N8 0.51 N6.5 0.87 N5 0.82 N3.5 0.95 Black 0.90 Mean Stdev Max Min
0.77 0.59 2.70 0.05
∆E*94 under F2 1.39 1.24 0.77 1.80 1.65 1.07 1.85 1.13 0.36 3.33 0.37 2.14 1.87 0.59 1.32 0.60 2.17 1.54 0.41 1.00 1.35 1.60 1.80 1.82 1.38 0.69 3.33 0.36
Kodak RMS ∆E*94 under A 0.057 0.23 0.041 0.21 0.060 1.41 0.050 0.59 0.058 1.72 0.060 1.16 0.080 1.70 0.039 1.08 0.033 0.24 0.050 1.42 0.052 0.23 0.070 1.90 0.050 0.99 0.048 0.33 0.037 0.77 0.031 0.73 0.087 0.72 0.052 2.92 0.106 0.04 0.068 0.16 0.070 0.26 0.064 0.16 0.051 0.20 0.034 0.27
∆E*94 under F2 2.72 2.30 1.78 2.94 2.30 1.68 2.16 1.64 1.34 3.68 0.95 2.50 2.05 0.93 1.23 1.06 1.82 2.03 0.25 1.69 2.70 3.40 3.76 3.61
RMS
0.043 0.066 0.041 0.048 0.082 0.066 0.089 0.044 0.059 0.095 0.063 0.090 0.046 0.039 0.062 0.071 0.106 0.051 0.028 0.037 0.046 0.045 0.035 0.022
0.81 2.10 0.73 0.94 2.92 3.76 0.04 0.25
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Waterproof CMYK * ∆E 94 ∆E*94 RMS under under A F2 2.47 2.26 0.029 0.76 1.39 0.034 1.77 1.23 0.047 2.67 1.79 0.034 0.81 1.27 0.05 0.65 1.70 0.055 2.23 2.49 0.079 1.60 1.53 0.038 0.28 0.56 0.023 0.79 2.90 0.078 1.79 1.56 0.064 2.23 2.64 0.081 1.25 2.25 0.031 1.81 2.00 0.049 0.44 1.53 0.074 0.91 1.02 0.042 0.53 2.09 0.118 0.50 0.92 0.034 0.35 0.18 0.062 1.64 1.27 0.067 2.98 2.16 0.067 3.61 2.55 0.052 3.92 2.51 0.032 3.18 1.68 0.014
Waterproof CMYKGO * ∆E 94 ∆E*94 RMS under under A F2 0.09 0.04 0.007 0.11 0.89 0.031 0.86 0.63 0.041 0.34 0.90 0.013 0.69 1.21 0.051 0.17 0.98 0.044 0.56 0.45 0.021 1.63 1.57 0.037 0.16 0.45 0.024 0.73 2.85 0.079 0.65 0.67 0.026 1.01 1.39 0.041 1.55 2.66 0.028 0.17 0.93 0.021 0.39 1.41 0.071 0.39 0.53 0.021 0.49 1.95 0.115 0.89 0.67 0.032 0.35 0.18 0.062 0.30 0.39 0.054 0.57 0.55 0.043 0.45 0.43 0.027 0.52 0.30 0.013 0.90 0.53 0.005
1.63 1.09 3.92 0.28
0.58 0.41 1.63 0.09
1.73 0.69 2.90 0.18
0.94 0.73 2.85 0.04
Observing the individual color match predicted by these three- and four printing processes, most colors such as blue, foliage, purple, neutral colors, and so forth were highly metameric. Whereas, the predictions by the proposed six-color spectral printing process was the least metameric results. Hence, the color reproductions by the Fujix, Kodak, Waterproof® CMYK are of high degree of metamerism and the degree of metamerism synthesized by the proposed six-color spectral printing process using Waterproof® CMYKGO can be minimized.
E. CONCLUSIONS
Although the six-color Yule-Nielsen modified spectral Neugebauer equation should be employed to construct a six-color printing model, the physical printing limitation such as ink-trapping failure confines the existence or stability of real five-color and sixcolor overprints which are the Neugauer primaries. Since a six-color Yule-Nielsen modified spectral Neugebauer uses up to six-color overprints as its basis spectra for spectral reconstruction, stable five-color and six-color overprints are necessary for accurate reconstruction of a given spectral reflectance factor. This especially is a problem when attempting to derive a more than six-color printing model. There are 97 (27 -31) five-color, six-color, and seven-color overprints for a seven-color printing process. Even the sample preparation of these Neugebauer primaries will be a critical issue. Although the analytical prediction of those overprints were presented in Chapter VI, the physical
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feasibility is still questionable due to the ink-trapping limitation. Hence, the proposed approach by this research development seems a reasonable paradigm. The modeling for a k-color printing process is suggested to subdivide into several four-color sub-models. Their union of their spectral gamut can be a good approximation of the spectral gamut spanned by the all combinations of k primaries since the color of the more than four overprints tend to be neutral and low lightness. Spectral-based six-color separation algorithms were derived in building a six-color forward and backward spectral printing model. Four six-color forward spectral printing models were compared for their performance in terms of colorimetric and spectral accuracy using Waterproof®, a halftone proofing system. They are the first-order forward printing model comprised of a theoretical to effective dot area transfer function, f( ), and the Yule-Nielsen modified spectral Neugebauer equation, the second order modification suggested by Iino and Berns, an alternative second order improvement by the proposed algorithms, and the printing process modeled by matrix transformation method. The proposed algorithms together with the modification by Eq. (7-20) were the favorite forward printing model for its highest performance both in terms of high colorimetric and spectral accuracy. A backward spectral printing model was constructed by a four-ink selector, the adopted six-color forward spectral printing model based on the proposed algorithms, and a numerical engine. The performance of this backward printing model was tested in
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estimating the theoretical dot areas of 24 colors of the Gretag Macbeth Color Checker. The colorimetric and spectral accuracy, shown in Table 7-12, of the predictions corresponding to the estimated theoretical dot areas by the proposed backward model is high. One thing worth mentioning is that the four-selector is able to select a proper set of four inks from the defined six inks for reproducing the Gretag Macbeth Color Checker. The validity for the hypothesis of continuous tone approximation and the use of Eq. (717), derived empirically, are reassured by the accuracy of the color separation performed on the Color Checker. An analysis was made by comparing the spectral performance of three- and four printing processed to the proposed six-color technology. It was shown that the degree of metamerism for the prediction by the three- (Fujix Pictrography 3000 and Kodak Professional 8670 PS thermal printer) and four-color (Dupont Waterproof® using CMYK) printing processes were relatively high. Based upon all of these analyses, it is concluded that the proposed color separation algorithms are capable of minimizing metamerism between the original and its reproduction while maintaining high colorimetric accuracy.
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VIII. MULTIPLE-INK DIRECTING PRINTING The last stage of this dissertation is to utilize the implemented spectral-based color separation algorithm in building a multi-spectral output system. This requires predefining a multiple-ink halftone printing process to be used as an output device since forward and backward halftone printing models are device dependent. The Waterproof® system was utilized to verify the research development of the six-color spectral halftone printing models and a set of six color separation records for the Gretag Macbeth Color Checker was calculated corresponding to the six primaries for this printing process. The final verification for the research development in building a multi-spectral output system is to output the six color separation records, shown in Table 7-11, by employed printing process and observe the colorimetric and spectral accuracy of the reproduction.
A. VERIFICATIONS
The six color separation records of the Gretag Macbeth Color Checker were output by the DuPont Waterproof® system using the specified six primaries. Four six-color printed Checker were measured five times for each by the Gretag Spectrolino (since it was utilized for the samples to model the six-color printing process) and averaged to account for the print to print variation and spatial uniformity. The colorimetric and spectral accuracy for each individual color is shown in Table 8-1. The statistical colorimetric and
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spectral performance for original vs. reproduction and prediction vs. reproduction are listed Table 8-2.
Table 8-1: The colorimetric and spectral accuracy of original vs. reproduction and prediction vs. reproduction for the Gretag Macbeth Color Checker.
Color Name Dark Skin Light Skin Blue Sky Foliage Blue Flower Blue Green Orange Purplish Blue Moderate Red Purple Yellow Green Orange Yellow Blue Green Red Yellow Magenta Cyan White N8 N6.5 N5 N3.5 Black
Origianl vs. Reproduction M.I. RMS ∆E*94 0.013 1.64 0.51 0.038 1.57 0.14 0.038 2.09 0.68 0.016 1.70 0.49 0.056 2.21 0.20 0.041 1.26 0.21 0.019 1.76 0.33 0.037 2.53 1.51 0.028 1.54 0.12 0.083 1.87 1.08 0.022 1.50 0.52 0.046 2.81 1.06 0.028 5.80 1.39 0.020 1.05 0.29 0.078 0.91 0.56 0.027 1.52 0.32 0.123 0.96 0.80 0.034 1.55 1.74 0.059 1.45 0.40 0.054 1.44 0.16 0.050 2.89 0.25 0.033 3.08 0.48 0.014 2.00 0.11 0.005 1.55 0.63
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Prediction vs. Reproduction M.I. RMS ∆E*94 1.64 0.60 0.010 1.57 0.27 0.017 1.73 0.49 0.010 1.71 0.34 0.009 2.07 0.79 0.020 0.98 0.08 0.011 1.80 0.39 0.020 1.49 0.66 0.012 1.54 0.12 0.011 1.81 0.50 0.008 1.27 0.32 0.016 3.02 0.29 0.028 0.74 0.37 0.007 1.04 0.23 0.008 0.86 0.17 0.008 1.60 0.19 0.020 0.83 0.18 0.012 1.72 0.38 0.011 0.26 0.05 0.004 1.42 0.47 0.022 2.90 0.95 0.033 3.09 1.01 0.024 2.01 0.67 0.009 1.45 0.58 0.004
Table 8-2: The statistical colorimetric and spectral accuracy corresponding to the predicted and reproduced Gretag Macbeth Color Checker.
Mean Stdev Max Min RMS
Origianl vs. Reproduction Prediction vs. Reproduction * ∆E 94 Metamerism Index ∆E*94 Metamerism Index 1.94 0.58 1.61 0.42 1.01 0.46 0.69 0.26 5.80 1.74 3.09 1.01 0.91 0.11 0.26 0.05 0.048 0.016
Ideally, the colorimetric and spectral error between the prediction and the actual reproduction should be zero. Actual average colorimateric and spectral error between the prediction and the reproduction were 1.61 ∆E*94 and 0.42 unit of metamerism index, respectively. This discrepancy is due to the print to print variation and calibration status of the printing process. (The proposed six-color spectral printing model was established one month before the Gretag Macbeth Color Checker was printed.) Nevertheless, this accuracy is still satisfactory. Figure 8-1 shows the spectral reflectance factors of six colors arbitrarily chosen for demonstration. It can been seen that the spectral reflectance factors of reproduction was systematically lower than the prediction. This was attributed to the print to print variation and calibration status of the printing process. The reproduced Gratag Macbeth Color Checker is bounded at the end of this chapter and the original, predicted, and reproduced spectral reflectance factors are shown in Appendices I, J, and K, respectively.
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Fig. 8-1: The original, predicted, and reproduced spectral reflectance factors of the six Gretag Macbeth Colors.
B. CONCLUSIONS
The six color separation records in terms of the theoretical dot areas, shown in Tabel 7-11, for the Gretag Macbeth Color Checker were output by the DuPont Waterproof® system using the cyan, magenta, yellow, black, green, and orange, the standard DuPont Waterproof® primaries, which were utilized to established the six-color spectral printing model. Curve shapes of the reproduced reflectance spectra were approximately parallel to the spectra predicted by the proposed six-color printing model.
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The discrepancy is due to the print to print variation and calibration status of the physical printing device. Nevertheless, the similarity of the spectral curves between the reproduced and predicted spectra indicates that the proposed six-color printing model is capable of predicting the color formulated by the actual Dupont Waterproof ® printing process.
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IX. CONCLUSIONS DISCUSSIONS AND SUGGESTIONS FOR FUTURE RESEARCH
Research for bridging a multi-spectral image acquisition system and a spectralbased output system has been described. It includes colorant estimation, optimal inkselection, spectral reflectance factor prediction for ink overprints, spectral-based six-color separation minimizing metamerism, and verification from direct printing. Results show a promising future for the feasibility of the proposed algorithms in terms of colorimetric and spectral accuracy. Since linear modeling techniques are heavily employed for the first-order analysis for this research project, a linear colorant mixing space was desired. Kubelka-Munk turbid media theory has frequently been used to represent opaque colorant mixtures in a linear vector space. It was observed for the colorant mixtures used in this research that the accuracy of spectral prediction based on Kubelka-Munk theory was not satisfactory. Furthermore, excessively large fields of view for conventional spectrophotometry result in additivity failure when spectral data are transformed to Kubelka-Munk representations. As a consequence, an empirical transformation was derived, as described in chapter III, resulting in improved prediction accuracy for a set of opaque colorant mixtures requiring reproduction. The proposed transformation was designed according to the coloration process of subtractive colorant mixing and the exact or reduced dimensionality of a set of mixtures in the linear colorant mixing space after the proposed
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transformation. The former was concerned with the observation that the more colorants used for coloration, the darker the resultant colorant mixture. The latter accounted for the underlying number of colorants used for creating the set of colorant mixtures. The accuracy of the proposed transformation was high. In the analysis stage, a colorant estimation algorithm, described in chapter IV, statistically decomposed an input spectral image according to the six directions explaining major variation of the spectral image in a linear colorant mixing space. These directions are known as the directions of six significant eigenvectors. Hence, each pixel of the spectral image represented in a linear colorant mixing space is a linear combination of the six eigenvectors corresponding to the input spectral image. Since the six eigenvectors did not represent physical colorant spectra judged by their bipolar appearance across the visible spectrum, a constrained rotation algorithm was applied to transform them to a set of all-positive vector representations leading to non-negative coordinates for describing every pixel in the input spectral image. The six all-positive eigenvectors, as the statistical primaries, symbolized a set of six primary colorants utilized for creating the original object corresponding to the captured spectral image. The non-negative coordinates associated with each pixel account for a typical coloration process, i.e. no negative concentrations, can be used for color synthesis. Based on these two physical constraints, the proposed colorant estimation algorithm was able to converge to a set of reasonable basis colorants
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whose prediction accuracy were identical to that of the six eigenvectors derived for the set of input spectra requiring synthesis. Once the set of statistical primaries corresponding to a given image were uncovered, the optimal ink-selection module correlated the statistical primaries to an existing ink database in order to attain the most similar ink set for a halftone printing process using multiple inks. Since the number of six-ink combinations generated from a large ink database is enormous, the effort of inspecting the performance of every combination was prohibitive. The obviously infeasible ink combinations, such as those sets formed by the inks of same hue, should be removed analytically. The proposed inkselection algorithm, described in chapter V, was capable of removing the impossible ink combinations resulting in significant reduction, i.e., from 18,564 down to 32 for a database of 18 inks. Small numbers of candidate ink sets were much more feasible to evaluate for spectral reconstruction. The Yule-Nielsen modified spectral Neugebauer equation was an obvious choice in building halftone printing models for performance evaluation. Whereas, parameters such as dot gain factor and reflectance spectra of ink overprints of a printing process were frequently unknown. The halftone model building effort was not quite feasible at this phase. Practically, this research suggested a continuous-tone approximation to a halftone printing process based on the proposed transformation described in Chapter III. The transformation was revised in describing the colorant mixing for a halftone printing
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process. The verification of the IT8/7.3 target printed by SWOP standard conditions showed that the continuous-tone approximation was reasonable. Since the optimal ink set for minimizing metamerism is image dependent, the optimal ink set varies from image to image. In order to construct an analytical printing model using the Yule-Nielsen modified Neugebauer equation, the ink overprints, known as the Neugebauer primaries, were the basis information required to use this equation. Due to the number of overprints increasing exponentially with each additional ink, the number of overprints was enormous when the number of primary inks was large. This would lead to an unreasonable number of samples to be prepared and measured, in addition to necessitating resampling and remeasurement upon changes in consumable (e.g., ink and paper). This research, therefore, included the analytical prediction of ink overprints using the Kubelka-Munk theory for translucent materials as opposed to exhaustively printing and measuring. Two optical constants, absorption and scattering coefficients, of each ink were estimated numerically using the reflectance measurements of an ink printed on a white and black surfaces and the reflectance factors of the white and black surfaces, respectively. Hence, the surface spectral reflectance factor of an ink overprint was a function of the absorption and scattering coefficients, thickness of the top-most ink layer, and the spectral reflectance factor of the layer underneath the top-most layer. This research tested the algorithm discussed in Chapter VI using the DuPont Waterproof® system, a halftone proofing process. Twenty-five overprints, which were at
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most three-color overprints, were generated from a combination of six primaries. The prediction results were excellent. Relatively poorer predictions resulted for some of the three-color overprints. There was a systematic trend in which the samples were too dark. It was reasoned that the prediction using this algorithm for dark overprints was sensitive to noise. Noise was greatly amplified when it underwent the transformation using the translucent equation which is highly non-linear. Nevertheless, the colorimetric and spectral accuracy was still high. The development for the synthesis stage of this research was to construct a highly accurate six-color printing model. DuPont Waterproof® was also utilized to print six-color ramps and a verification target of 6,250 samples. The six-color printing model was comprised of ten four-color printing sub-models. The modeling of the six-color printing process was a collection of modeling ten four-color printing processes. Four different models were evaluated: the first-order Yule-Nielsen modified spectral Nuegebauer equation together with the theoretical to effective dot area transfer function, the secondorder improvement published by Iino and Berns, an alternative second-order modification, and a higher-order matrix transformation. These models were described and evaluated in Chapter VII. By inspecting two-dimensional vector plots of CIELAB L* versus a*, it was revealed that the first-order estimation tended to under predict the spectral reflectance factors, due to an overestimation of the effective dot areas of multiple-ink mixtures for the dark samples. The error of overestimating effective dot areas was mainly caused by an
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“optical-trapping” effect. A second-order modification proposed by Iino and Berns, accounting for optical-trapping, was tested to improve the model accuracy. Although it yielded approximately the same accuracy as that of the first-order approximation for this type of printing process, it was capable of correcting the prediction of some of the dark colors. It was found that the prediction errors were mostly chromatic errors. The third algorithm, developed during this dissertation as an alternate model of optical trapping, was more effective in reducing the errors for these dark samples. The difference between the newly developed algorithm and that described by Iino and Berns was that, conceptually, Iino and Berns' algorithms suggested that, for example, for cyan and magenta ink mixtures, whose theoretical dot areas are m% and k%, respectively, the dot gain of the cyan is normalized to the dot-gain locus of the cyan when the magenta is present at k%. Similarly, the dot gain of the magenta is normalized to the dot gain locus of the magenta when the cyan is present at m%. Whereas, the proposed algorithms recommend that the more optimal dot gain of the cyan is normalized to the dot-gain locus of the magenta presenting at 50% theoretical dot area whenever the magenta is present at any theoretical dot area other than zero. It was found that the new approach tended to over predict spectral reflectance factor (or under estimate the effective dot areas) for the 6,250 mixtures. It is not a surprising nature of the newly developed algorithms since the approximation by the dot gain of a primary given a secondary presenting at 50% theoretical will overestimate the effective dot area of the primary when the secondary is
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actually present at less than 50% theoretical dot area and vice versa. Nevertheless, this drawback is corrected by Eq. (7-21), as a consequence, yielding the best accuracy. By inspecting the vector plots of CIELAB values, it was also found that the reproduction trend was approximately opposite to that of the first-order approximation. Thus, it was concluded that the real effective dot areas for predicting the 6,250 samples was linearly located between the effective dot areas estimated by the first-order approximation and that estimated by the new algorithm. A weighted sum was statistically estimated and found to yield the highest colorimetric and spectral accuracy. The matrix method was undesirable for this research since it required a large sample database for modeling process. In addition, it occasionally produced negative effective dot areas. This research tested the six-color spectral printing model by estimating the dot areas required to reproduce a Gretag Macbeth Color Checker. The six color separations, defined in terms of theoretical dot areas, would lead to color-reproduction accuracy of an average of 0.57 and a maximum of 5.57 ∆E*94 and an average of 0.78 and a maximum of 2.09 units of metamerism index. Thus, the colorimetric and spectral accuracy was satisfactory. Since the six primaries were not optimized for the Gretag Macbeth Color Checker (having been arbitrarily defined by DuPont), the maximum errors were due to spectral and colorimetric gamut limitations. Comparisons among the spectral gamut limitation for two three-color printing processes (Fujix Pictrography 3000 and Kodak Professional 8670 PS thermal printer), one four-color printing process (DuPont
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Waterproof® system using CMYK), and the proposed six-color technology were made computationally. The color formulation by the three- and four-color printing processes were highly metameric relative to the proposed six-color technology. It was concluded that the spectral-based color separation algorithm was capable of minimizing metamerism while maintaining high colorimetric accuracy. The six separation records for the 24 color patches of the Gretag Macbeth Color Checker were printed using DuPont Waterproof® using the defined six primaries. Accuracy comparisons among the original, prediction by the six-color model, and reproduction based on the six-color separations in terms of the theoretical dot areas predicted by the proposed six-color printing model were shown. It was found that the colorimetric and spectral error between the prediction and the reproduction was systematic owing to the print to print variation and the calibration status of the employed printing process. However, the average colorimetric and spectral error with 1.61 ∆E*94 and 0.42 units of metamerism index, respectively was satisfactory. The stability of the DuPont Waterproof® is considered high. Hence, the validity of the proposed six-color printing model is substantiated by this verification. This research has accomplished a theoretical development and several prototypical implementations. There is still a need for improvement in order to implement this research in practice. The colorant-estimation model was implemented by a constrained rotation process using MATLAB. It takes minutes to hours converging to a set of statistical
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primaries depending on the initial estimation and the requested numerical precision. It is possible to statistically guess the initial set of colorants by directly selecting the spectra located at the vertices of the colorimetric gamut of an input spectral image. The Monte Carlo simulation suggested by Ohta (1973) may be an alternative to expedite the converging speed. The Pantone 14 basic and four process colors were utilized as the ink database for ink selection. There is only one green in the ink database. From the development of optimal ink-selection, the performance is partially limited by the lack of green inks to choose from though it is not the absolute dominant factor. This implies that a wide variety of candidate inks are desirable, in general. The development for ink overprint prediction was verified by a proofing process. Color patches were printed by holding colorant on a laminate, then applying on a substrate. Unlike a conventional printing process, the phenomenon of colorant penetration into the substrate was negligible. The penetration of ink causes the estimation of the scattering coefficient to be difficult. Further research could test the ink overprint predictions using process-printed samples. During the development of the six-color separation algorithm, the spectral image acquisition system was still under development. Hence, it was not possible to test each component developed during this dissertation within a system. Clearly, the next step is to
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combine these results with an spectral image acquisition system and develop and test the complete spectral color reproduction system. This research has demonstrated the capability of spectral reproduction six-color spectral printing process. Colorimetric and spectral accuracy for reproducting the Gretag Macbeth Color Checker was very high even though the six primaries (CMYKGO) were not defined from the ink selection algorithm. However, the derived six-color process was not capable of reproducing the blue and purple colors with high accuracy. It is hypothesized that by adding one degree of freedom to derive a seven-color printing process, the capability of spectral reproduction can be greatly improved. It is also hypothesized that by a properly defined ink set of seven colors can approximately bypass the process of ink selection based on the predicting accuracy of the Gretag Macbeth Color Checker estimated by the derived six-color printing model. Since the proposed color separation algorithm is constrained to use one black and three chromatic inks to reproduce a pixel from an original, the modeling process for a seven-color printing model is at the expense of establishing ten more four-color submodels. There are 20 (C(6, 3)) four-color sub-models for a seven-color printing process based on the proposed ink-limiting scheme. It is worth mentioning that the file size will be the same for storing the k-color separation records, where k ≥ 5. To be more specific, consider the file structure of the separation records diagram as below:
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Tag of an ink set 1 5 3 . . 8
Theoretical dot area of the 1st ink 0.05 0.66 . . . 0.83
Theoretical dot area of the 2nd ink 0.81 0.32 . . . 0.90
Theoretical dot area of the 3rd ink 0.21 0.12 . . . 0.11
Theoretical dot area of the black ink 0.43 0.02. . . . 0.11
The tag of an ink set specifies which ink combination should be used for the particular pixel and the corresponding four theoretical dot areas, shown in the same row, required for spectral reproduction. For a six-color printing process the integer tag is ranged from one to ten, from 1 to 20 (C(6, 3)) for a seven-color printing process, and from 1 to 35 (C(7,3)) for a eight-color printing process. Only 6 bits are required to store the information of the four-ink combinations up to eight-color printing process. There are only five channels for a multiple-ink printing process abiding by the proposed ink-limiting scheme. Hence, it is suggested by this research to speculate the performance of a seven-color printing process, which is an extension of the proposed six-color technology, without large additional modeling efforts and storage issues.
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X. APPENDICES This research utilized MATLAB V5.2 as the algorithm developing tool. A collection of program scripts developed for the calculation of fundamental colorimetry, such as XYZ tristimulus values, L*a*b*, L*ch, ∆E*ab, ∆E*94, metamerism index,…, and so forth, Kubelka-Munk translucent equation, functions for Saunderson correction, and database for colorimetry, such as color matching functions, ASTM weights for standard illuminants (D65, D50 , A,…, etc.) is stored as the MATLAB library or toolbox named under "munsell". When running the attached program scripts in each appendix, the "munsell" library is indispensable. User should include the "munsell" in the search path when working with the MATLAB environment. The program scripts developed for each module of this research are shown in the following appendices. This research adopted the spectral range from 400 nm to 700 nm at 10 nm interval. Hence, all the spectral data should be of 31 components. Most sets of spectral data were stored as column vectors. Whereas, a few sets of spectral data from large measurements are stored as row vectors. User should be aware of the dimensionality of these matrices before running the program.
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APPENDIX A : MATLAB PROGRAMS FOR THE "MUNSELL" LIBRARY asaunder.m %This is the fuchtion of inverse Saunderson correction. The argument R_inf %is the corrected reflectance factor by Saunderson correction. function R_meas=asaunder(R_inf) k1=0.04; k2=0.6; R_meas=((1-k1)*(1-k2)*R_inf)./(1-k2*R_inf);
cost_eig.m %This the objective function of pos_tran.m performing all-positive %vector rotation. the first argument is the initially estimated coefficient %matrix which might rotate the set of input eigenvector (second argument) %matrix to an all positive basis vector matrix close to the third argument %(target-Spectra). function [f, g]=cost_eig(coefficient, eigenvec, target_spectra) new_basis = eigenvec*coefficient'; f=rms(target_spectra, new_basis); % rms: function for root-mean-square error g(1,:)= [-new_basis]'; %the constraints for the positivity of the resultant vector
del_e94.m %This program calculates the color difference in units of CIE94 color difference equation for two samples with ref1 and ref2, respectively, %where the white_point is a 31x1 reflectance vector of the reference white. %The illum_option is 1x2 vector of specifying the illuminant and observer function De94=del_e94(ref1, ref2, white_point, illum_option) if (nargin (Lab2(i,2)*Lab1(i,3)) del_H(i,1)= -sqrt(2*(lch1(i,2)*lch2(i,2)-Lab1(i,2)*Lab2(i,2)-Lab1(i,3)*Lab2(i,3))); else del_H(i,1)= sqrt(2*(lch1(i,2)*lch2(i,2)-Lab1(i,2)*Lab2(i,2)-Lab1(i,3)*Lab2(i,3))); end end Cab=(lch1(:,2).*lch2(:,2)).^(0.5); %Cab=lch1(:,2); SL=1; SC=1+0.045*Cab; SH=1+0.015*Cab; De94=sqrt((del_L./SL).^2 + (del_C./SC).^2 + (del_H./SH).^2);
etok.m %This program is the objective function of the constrained optimization %process for the colorant estimation subsystem, where k_ink is the variable %of a set of estimated colorants represented in a linear color mixing space %such as K/S space. The pseudo is the coefficient of a set of eigenvectors, %named as eigenvec. The k_black is a black colorant in linear color mixing %space to be constrained for its pre-existence in the set of mixtures. function [f,g]=etok(k_ink, pseudo, eigenvec, k_black) k1=eigenvec*pseudo; k1=(abs(k1)+k1)/2; %conc=pinv([k_ink k_black])*k1; %if need to constrain the pre-existence of black %k2=[k_ink k_black]*conc; %then use these two line conc=pinv(k_ink)*k1; k2=k_ink*conc; f=rms(k1,k2); %calculate the RMS error between two reconstructed set of samples g=-conc;
% concentration is constrained to be all-positive
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k_pca.m %This program calculates a required numbers (num_of_eig) of eigenvectors %(K_eig) for a set of sample (kovs) in linear color mixing space, where %the prin_fig is the option for displaying the plot of eigenvectors %if that is required. function [K_eig, report]=k_pca(kovs, num_of_eig, prin_fig) if nargin == 0 fprintf('The first argument is a matrix of K/S,\n'); fprintf('The second argument is the number of required eigenvectors.\n'); elseif nargin == 1 fprintf('The default number of required eigenvectors is six.\n'); num_of_eig=6; prin_fig=0; elseif nargin == 2 prin_fig=0; end lambda=[400:10:700]; sigma=cov(kovs'); [vk,ek]=eig(sigma); ek=flipud(diag(ek)); tot_per_var=sum(ek(1:num_of_eig))/sum(ek); per_var= ek/sum(ek); cum_var=zeros(31,1); cum_var(1)=per_var(1); for i=2:31 cum_var(i)=cum_var(i-1)+per_var(i); end if prin_fig==1 figure subplot(2,1,1) plot(ek, 'r+') title('The 31 eigenvalues based on K/S.') xlabel('The ith eigenvectors.') ylabel('Eigenvalue') subplot(2,1,2) plot(cum_var*100, 'r+') title('The cumulative variance% of i eigenvectors based on K/S.') xlabel('The ith eigenvectors.') ylabel('Variance%')
fprintf('The total percent variance described by %g\n', num_of_eig) fprintf('eigenvectors in K/S space is %f\n', tot_per_var*100); if num_of_eig >help meta_idx for details. \n\n'); elseif nargin == 2
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fprintf('The default white point is prd and.\n'); fprintf('The default defult option = [1 2 1 2], Type >>help meta_idx for details.\n'); white_point=ones(31,1); option = [1 2 1 2]; elseif nargin == 3 fprintf('The default defult option = [1 2 1 2], Type >>help meta_idx for details.\n'); option = [1 2 1 2]; end if option(1)==option(2) fprintf('ERROR :The standard and referencial viewing illuminant are the same.\n'); fprintf(' The results of metameric index are not reliable.\n\n'); end corrected_spectra=pcorrect(standard, trial, [option(1) option(3)]); if option(4)==1 meta_index=del_e(standard, corrected_spectra, white_point, [option(2) option(3)]); elseif option(4)==2 meta_index=del_e94(standard, corrected_spectra, white_point, [option(2) option(3)]); end
off_fn.m %This is the objective function for the optimization process of estimating %the offset vector of the empirical transformation derived for opaque %colorant represented in the resultant linear color mixing space, where %b is the offset vector, sample is a reflectance matrix of a set of sample %requiring reproduction, and po is the power (default=0.5). function [f,g]=off_fn(b, sample, po, numofeig) [n,m]=size(sample); tmp=b; for i=2:m tmp=[tmp b]; end k_paint=tmp-(sample.^po); [eig_k, repk]=k_pca(k_paint,numofeig); psu_con=pinv(eig_k)*k_paint; ks_recon=eig_k*psu_con; ks_recon=(abs(ks_recon)+ks_recon)/2; %R_pred=(tmp-ks_recon).^(1/po); f=rms(k_paint, ks_recon); %f=rms(R_pred, sample); g=-k_paint;
pcorrect.m %This function calculate the corrected spectrum for two sample which are paramour %based on the article of Fairman (1987). Refer to the program meta_idx.m for the %definitions of the input arguments. function corrected_spectra=pcorrect(standard, trial, illum_option)
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if nargin < 2 fprintf('The first argument is a matrix of standard reflectance,\n'); fprintf('The second argument is a matrix of trial reflectance.\n'); fprintf('The third argument is character string for a viewing illuminant.\n'); fprintf('It could be illumD65, illumA, or illumd50.\n\n'); elseif nargin == 2 fprintf('The default illuminant is D65 of 2 degree ASTM weight.\n'); illum_option=[1 1]; end
if (illum_option(1)==1)&(illum_option(2)==1) load wd65_2.dat; light_source=wd65_2; elseif (illum_option(1)==1)&(illum_option(2)==2) load wd65_10.dat; light_source=wd65_10; elseif (illum_option(1)==2)&(illum_option(2)==1) load wa_2.dat; light_source=wa_2; elseif (illum_option(1)==2)&(illum_option(2)==2) load wa_10.dat; light_source=wa_10; elseif (illum_option(1)==3)&(illum_option(2)==1) load wd50_2.dat; light_source=wd50_2; elseif (illum_option(1)==3)&(illum_option(2)==2) load wd50_10.dat; light_source=wd50_10; elseif (illum_option(1)==4)&(illum_option(2)==1) load wf2_2.dat; light_source=wf2_2; elseif (illum_option(1)==4)&(illum_option(2)==2) load wf2_10.dat; light_source=wf2_10; elseif (illum_option(1)==5) load wf7_2.dat; light_source=wf7_2; end
R=light_source*inv(light_source'*light_source)*light_source'; [m,n]=size(R); identity=diag(ones(m,1)); corrected_spectra=R*standard + (identity-R)*trial;
saunder.m %This fuction performs the Saunderson's correction to the discontinuity %of the refractive index for two material in optical contact with each other function R_inf=saunder(R_meas) k1=0.04; k2=0.6;
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R_inf=R_meas./(1-k1-k2+k1*k2+k2*R_meas);
xyz.m %This program calculates the XYZ tristimulus. Refer to the lab.m for the definition of arguments. function [XYZ, xy]=xyz(reflect1, illum_option) if nargin == 0 fprintf('The first argument is a matrix of reflectance,\n'); fprintf('The second argument is character string for a viewing illuminant.\n'); fprintf('It could be illumD65_2, illumD65_10, illumA_2, or illumA_10.\n\n'); return; elseif nargin == 1 fprintf('The default illuminant is D65 of 2 degree ASTM weight.\n'); illum_option=[1 1]; end if (illum_option(1)==1)&(illum_option(2)==1) load wd65_2.dat; XYZ=[wd65_2'* reflect1]'; elseif (illum_option(1)==1)&(illum_option(2)==2) load wd65_10.dat; XYZ=[wd65_10'* reflect1]'; elseif (illum_option(1)==2)&(illum_option(2)==1) load wa_2.dat; XYZ=[wa_2'* reflect1]'; elseif (illum_option(1)==2)&(illum_option(2)==2) load wa_10.dat; XYZ=[wa_10'* reflect1]'; elseif (illum_option(1)==3)&(illum_option(2)==1) load wd50_2.dat; XYZ=[wd50_2'* reflect1]'; elseif (illum_option(1)==3)&(illum_option(2)==2) load wd50_10.dat; XYZ=[wd50_10'* reflect1]'; elseif (illum_option(1)==4)&(illum_option(2)==1) load wf2_2.dat; XYZ=[wf2_2'* reflect1]'; elseif (illum_option(1)==4)&(illum_option(2)==2) load wf2_10.dat; XYZ=[wf2_10'* reflect1]'; elseif (illum_option(1)==5)&(illum_option(2)==1) load wf7_2.dat; XYZ=[wf7_2'* reflect1]'; end [m,n]=size(XYZ); for i=1:m xy(i,1)=XYZ(i,1)/sum(XYZ(i,:)); xy(i,2)=XYZ(i,2)/sum(XYZ(i,:)); end
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APPENDIX B : MATLAB PROGRAMS FOR THE COLORANT ESTIMATION SUBSYSTEM The following diagram depicts the program structure of the process for the colorant estimation subsystem.
Main Program (colorant_est.m)
Optimization for the offset vector of the empirical transformation
Principal component analysis
Colorant estimation
colorant_est.m %This program was implemented based on the algorithm of the colorant estimation research. %It estimates the underlying colorant of a set of 105 opaque mixtures obtained by hand mixing %six opaque Poster Colorants close all; clear all; load target105.txt %the 105 patches of the hand mixed poster colors paint=target105; %now paint is the matrix of the reflectance spectra %of the 105 patches lambda=[400:10:700]'; load white.txt; R_paper=white;
%white is the reflectance spectrum of the reference white %now R_paper is the reference white
%************************************************************************************ %This section of scripts performance the optimization to obtain the offset vector of the %empirical transformation for opaque color represented in a linear colorant mixing space %named psi space %************************************************************************************ b=ones(31,1);
%initializing the offset vector of the empirical transformation
numeig=6;
%number of eigenvector is six due to the 105 samples were mixed with six poster colors
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vlb=zeros(31,1); vub=ones(31,1); options(1)=1; options(2)=1e-3; options(3)=1e-3; options(4)=1e-3; options(14)=10000000; po=1/2;
%power of the empirical transformation
b=constr('off_fn', b, options, vlb, vub, '', paint, po, numeig); [n,m]=size(paint); tmp=b; for i=2:m tmp=[tmp b]; end b=tmp; clear tmp;
%now b is a 31x105 matrix
k_meas=b-paint.^(po); %the 105 mixtures were transformed to a linear colorant mixing representation [eig6_k, report]=k_pca(k_meas, numeig, 1); %deriving the six eigenvectors from the psi space fprintf('The total percent variance described by 6 eigenvectors in psi space is %f\n', report(numeig,3)*100); %************************************************************************************ % Spectral reconstruction by PCA analysis % Model : K_reconstruction = eig6_k * C %************************************************************************************ pseudo_con100=pinv(eig6_k(:,1:6))*k_meas; k_recon=eig6_k(:,1:6)*pseudo_con100; R_predicted=(b-k_recon).^(1/po); delta_e94=del_e94(R_predicted, paint, R_paper); fprintf('The statistical results of the delta E94') fprintf( 'Mean \t\t%f\n', mean(delta_e94) ); fprintf( 'Standard Deviation \t\t%f\n', std(delta_e94) ); fprintf( 'Maximum \t\t%f\n', max(delta_e94) ); fprintf( 'Minimum \t\t%f\n\n', min(delta_e94) ); spectral_RMS = RMS( R_predicted, paint ); midx=meta_idx(R_predicted, paint, R_paper); fprintf('The statistical results of the metamerism index') fprintf( 'Mean \t\t%f\n', mean(midx) ); fprintf( 'Standard Deviation \t\t%f\n', std(midx) ); fprintf( 'Maximum \t\t%f\n', max(midx) ); fprintf( 'Minimum \t\t%f\n\n', min(midx) );
fprintf( 'The root mean square error by 6 eigenvectors in reflectance space is %f\n\n', spectral_RMS ); figure [yy,xx]=hist(delta_e94); bar(xx,yy,'r') title('The histogram color difference between measured and predicted by PCA.') xlabel('Delta E94')
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ylabel('Frequency') %************************************************************************************ %This section is performing the colorant estimation for the 105 mixtures based on the %proposed constrained vector rotation theory % pos_est.txt is the stored text file of the best estimation in psi space %************************************************************************************ k_ink=b(:,1:6)-primary.^(1/2); %k_ink is the initial estimated six colorant for the 105 mixtures vlb=ones(31,6)*0.05; vub=b(:,1:6); options(1)=1; options(2)=1e-3; options(3)=1e-3; options(4)=1e-3; options(14)=100000000000000000; Black=ones(31,1)*0.95; %if the black is constrained to be the pre-existing colorant k_ink=constr('etok', k_ink, options, vlb, vub, '',pseudo_con100, eig6_k(:,1:6), black); conc=pinv([black k_ink])*eig6_k(:,1:6)*pseudo_con100; k_recon=[black k_ink]*conc; %conc=pinv(k_ink)*eig6_k(:,1:6)*pseudo_con100; %if the black is not constrained, use these two lines %k_recon=k_ink*conc; R_predicted=(b-k_recon).^(1/po); delta_e94=del_e94(R_predicted,paint, R_paper); fprintf('The average color difference between measured and predicted by choosed ink-set %f\n', mean(delta_e94)); fprintf( 'Mean \t\t%f\n', mean(delta_e94) ); fprintf( 'Standard Deviation \t\t%f\n', std(delta_e94) ); fprintf( 'Maximum \t\t%f\n', max(delta_e94) ); fprintf( 'Minimum \t\t%f\n\n', min(delta_e94) ); spectral_RMS = RMS( R_predicted, paint ); fprintf( 'The root mean square error of the 100 measurements is %f\n\n', spectral_RMS ); figure [yy,xx]=hist(delta_e94); bar(xx,yy,'r') title('The histogram of color difference between measured and predicted by 6C K-M.') xlabel('Delta E94') ylabel('Frequency') midx=meta_idx(R_predicted, paint, R_paper); fprintf('The average color difference between measured and predicted undek illum A by choosed ink-set %f\n', mean(midx)); fprintf( 'Mean \t\t%f\n', mean(midx) ); fprintf( 'Standard Deviation \t\t%f\n', std(midx) ); fprintf( 'Maximum \t\t%f\n', max(midx) ); fprintf( 'Minimum \t\t%f\n\n', min(midx) ); figure [yy,xx]=hist(midx); bar(xx,yy,'r') title('The histogram of color difference between measured and predicted by parametric method.') xlabel('Delta E under illum A for corrected paramers') ylabel('Frequency')
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APPENDIX C : MATLAB PROGRAMS FOR THE INK SELECTION SUBSYSTEM The following diagram depicts the program structure of the process for the ink selection subsystem.
Main Program (inkselection.m)
Vector correlation analysis
Highest chroma selection
inkselection.m %This program performs the analysis based on the ink selection algorithm close all; clear all; load pan18.txt; %the ink database of 14 Pantone basic color and 4 process CMYK load pos_rest.txt; %the statistical primaries of the 105 mixtures lambda=[400:10:700]'; figure; for i=1:18 subplot(3,6,i); plot(lambda, pan18(:,i)); axis([400 700 0 1]); end
load panwhite.txt R_paper=panwhite; %the reference white is the coated paper printed with Pantone inks pan16=[pan18(:,1:13) pan18(:,15:17)]; % the two black inks are the sure candidates % only 16 colors need to be correlated [r,c]=size(pan16); po=1/3.5; %the power of the empirical transformation is 1/3.5 for transparent/translucent %ink in optical contact with an opaque substrate (R_paper) tmp=R_paper.^po; %the offest vector in this case is R_paper.^(1/3.5)
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Performance evaluation for 32 candidate ink sets by continuous tone approximation
for i=2:c tmp=[tmp R_paper.^po]; end b=tmp; clear tmp; %************************************************************************************ % Vector correlation analysis %************************************************************************************ coef_tab=zeros(5,c); for i=1:5 for j=1:c tmpcorr=corrcoef(b(:,1)-pos_rest(:,i).^po, b(:,1)-pan16(:,j).^po); coef_tab(i,j)=tmpcorr(1,2); end end coef_tab=coef_tab'; %correlation coefficients are stored in coef_tab %************************************************************************************ % Selecting the two highest chroma values %************************************************************************************ lab_pan16=lab(pan16,R_paper,[3 1]); chroma=sqrt(lab_pan16(:,2).^2+lab_pan16(:,3).^2); [coef1,idx1]=sortrows([coef_tab(:,1) chroma]); coef1=flipud(coef1); idx1=flipud(idx1); [coef2,idx2]=sortrows([coef_tab(:,2) chroma]); coef2=flipud(coef2); idx2=flipud(idx2); [coef3,idx3]=sortrows([coef_tab(:,3) chroma]); coef3=flipud(coef3); idx3=flipud(idx3); [coef4,idx4]=sortrows([coef_tab(:,4) chroma]); coef4=flipud(coef4); idx4=flipud(idx4); [coef5,idx5]=sortrows([coef_tab(:,5) chroma]); coef5=flipud(coef5); idx5=flipud(idx5);
tmpcoef=[coef1(:,1) coef2(:,1) coef3(:,1) coef4(:,1) coef5(:,1)]; tmpchroma=[coef1(:,2) coef2(:,2) coef3(:,2) coef4(:,2) coef5(:,2)]; tmpidx=[idx1 idx2 idx3 idx4 idx5]; %************************************************************************************ % Performance analysis for each candidate ink set by Continuous tone approximation %************************************************************************************ load target105.txt %the reflectance spectra of the 105 mixtures [m,n]=size(target105); load inkset.txt %32 sets of ink combinations formed by the 11 selected inks tmp=R_paper.^po; for i=2:n
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tmp=[tmp R_paper.^po]; end b=tmp; clear tmp; k_tar=b-target105.^po;
%the 105 mixtures represented in psi space
pan18=[pan16 pan18(:,14) pan18(:,18)]; performance=zeros(32,9); %accuracy of the 32 sets are stored in performance for i=1:32 conc=zeros(6,n); primary=[pan18(:,inkset(i,1)) pan18(:,inkset(i,2)) pan18(:,inkset(i,3))]; primary=[primary pan18(:,inkset(i,4)) pan18(:,inkset(i,5)) pan18(:,inkset(i,6))]; k_prmy=b(:,1:6)-primary.^po; for j=1:n conc(:,j)=nnls(k_prmy, k_tar(:,j)); end k_recon=k_prmy*conc; R_predicted=(b-k_recon).^(1/po); delta_e94=del_e94(target105, R_predicted,R_paper,[3 1]); midx=meta_idx(target105, R_predicted, R_paper, [3 1 2 1]); performance(i,1)=mean(midx); performance(i,2)=std(midx); performance(i,3)=max(midx); performance(i,4)=min(midx); performance(i,5)=mean(delta_e94); performance(i,6)=std(delta_e94); performance(i,7)=max(delta_e94); performance(i,8)=min(delta_e94); performance(i,9)=rms(target105,R_predicted); end [mdx,imdx]=sortrows(performance); load sam94set.txt temp=sam94set; figure; plot(lambda,temp(:,1),'b', lambda,temp(:,2),'m:',lambda,temp(:,3),'g-.', lambda,temp(:,4),'k--') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('Measured', 'Set 23','Set 24','Set 19') figure; plot(lambda,temp(:,1),'b', lambda,temp(:,5),'m:',lambda,temp(:,6),'g-.', lambda,temp(:,7),'k--') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('Measured', 'Set 26','Set 14','Set 9') figure; plot(lambda,temp(:,1),'b', lambda,temp(:,2),'m:',lambda,temp(:,5),'k-.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('Measured', 'Set 23','Set 26')
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APPENDIX D : MATLAB PROGRAMS FOR THE INK OVERPRINT PREDICTION SUBSYSTEM There are two parts of this subsystem. The first estimates the absorption (K) and scattering (S) coefficients of translucent ink. The second uses the estimated K and S to predicted an overprint by using the K, S, and thickness data of the topmost layer ink and the reflectance data the layer underneath the topmost layer as the parameters of the Kubelka-Munk basic equation. The first process is the program named ksfind.m and the second is named overprint.m The following diagram is the structure chart of the first component for K and S estimation for translucent inks.
Main Program (ksfind.m)
K&S estimation for Yellow ink
K&S estimation for magenta ink
K&S estimation for cyan ink
K&S estimation for red ink
K&S estimation for green ink
K&S estimation for blue ink
Objective function (ksest.m)
Objective function (ksest.m)
Objective function (ksest.m)
Objective function (ksest.m)
Objective function (ksest.m)
Objective function (ksest.m)
ksfind.m %This program estimates the absorption (K) and scattering (S) coefficients for translucent %inks clear all; close all; load black_b.txt; %reflectance spectrum of the black surface of contrast paper load white_b.txt; %reflectance spectrum of the white surface of contrast paper load r_on_ba.txt; %reflectance spectra of primary inks printed on contrast paper %data order is ink printed on white followed by ink printed on black r_on_ba2=saunder(r_on_ba); %reflectance spectra of primary inks printed on contrast % paper after saunderson's correction lambda=[400:10:700]'; %************************************************************************************ % Determining the K and S for yellow ink %************************************************************************************ ks_yel(:,1)=white_b-r_on_ba2(:,1); %initialing K for yellow ink ks_yel(:,2)=r_on_ba2(:,2)-black_b; %initialing S for yellow ink %K&S data are stored as K followed by S ks_yel=abs(ks_yel);
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vlb=ones(31,2)*0.000000000001; vub=ones(31,2)*100; options(1)=1; options(2)=1e-3; options(3)=1e-3; options(4)=1e-3; options(14)=1000000; ks_yel=constr('ksest', ks_yel, options, vlb, vub, '',r_on_ba(:,1), r_on_ba(:,2), white_b, black_b); figure plot(lambda, ks_yel(:,1),'b', lambda, ks_yel(:,2), 'm') title('The estimated spectral absorption and scattering coefficient of yellow ink.') xlabel('Wavelength') ylabel('Absorption and Scattering Coefficient') legend('b', 'Absorbtion', 'm','Scattering') ay=1+(ks_yel(:,1)./ks_yel(:,2)); by= sqrt(ay.^2-1); %a and b parameters of the Kubelka-Munk basic equation white_b=saunder(white_b); Ry_est= (1-white_b.*(ay-by.*coth(by.*ks_yel(:,2))))./(ay-white_b+by.*coth(by.*ks_yel(:,2))); %Reflectance estimation by the Kubelka-Munk basic equation Ry_est=asaunder(Ry_est); deltaE94=del_e94(r_on_ba(:,1), Ry_est, white_b) T=num2str( deltaE94, 4 ); figure plot(lambda, r_on_ba(:,1),'b', lambda, Ry_est,'m') title('The measured and predicted reflectance of yellow ink.') xlabel('Wavelength') ylabel('Reflectance factor') legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)
%************************************************************************************ % Determining the K and S for magenta ink %************************************************************************************ load white_b.txt; %reflectance spectrum of the white surface of contrast paper ks_mag(:,1)=white_b-r_on_ba2(:,3); %initialing K for magenta ink ks_mag(:,2)=r_on_ba2(:,4)-black_b; %initialing S for magenta ink ks_mag=abs(ks_mag); vlb=ones(31,2)*0.000000000001; vub=ones(31,2)*100; options(1)=1; options(2)=1e-3; options(3)=1e-3; options(4)=1e-3; options(14)=1000000; ks_mag=constr('ksest', ks_mag, options, vlb, vub, '',r_on_ba(:,3), r_on_ba(:,4), white_b, black_b); figure plot(lambda, ks_mag(:,1),'b', lambda, ks_mag(:,2), 'm') title('The estimated spectral absorption and scattering coefficient of magenta ink.') xlabel('Wavelength') ylabel('Absorption and Scattering Coefficient') legend('b', 'Absorbtion', 'm','Scattering') am=1+(ks_mag(:,1)./ks_mag(:,2)); bm= sqrt(am.^2-1); %a and b parameters of the Kubelka-Munk basic equation white_b=saunder(white_b);
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Rm_est= (1-white_b.*(am-bm.*coth(bm.*ks_mag(:,2))))./(am-white_b+bm.*coth(bm.*ks_mag(:,2))); %Reflectance estimation by the Kubelka-Munk basic equation Rm_est=asaunder(Rm_est); deltaE94=del_e94(r_on_ba(:,3), Rm_est, white_b) T=num2str( deltaE94, 4 ); figure plot(lambda, r_on_ba(:,3),'b', lambda, Rm_est,'m') title('The measured and predicted reflectance of magenta ink.') xlabel('Wavelength') ylabel('Reflectance factor') legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) %************************************************************************************ % Determining the K and S for cyan ink %************************************************************************************ load white_b.txt; %reflectance spectrum of the white surface of contrast paper ks_cyan(:,1)=white_b-r_on_ba2(:,5); %initialing K for cyan ink ks_cyan(:,2)=r_on_ba2(:,6)-black_b; %initialing S for cyan ink ks_cyan=abs(ks_cyan); vlb=ones(31,2)*0.000000000001; vub=ones(31,2)*10; options(1)=1; options(2)=1e-3; options(3)=1e-3; options(4)=1e-3; options(14)=1000000; ks_cyan=constr('ksest', ks_cyan, options, vlb, vub, '',r_on_ba(:,5), r_on_ba(:,6), white_b, black_b); figure plot(lambda, ks_cyan(:,1),'b', lambda, ks_cyan(:,2), 'm') title('The estimated spectral absorption and scattering coefficient of cyan ink.') xlabel('Wavelength') ylabel('Absorption and Scattering Coefficient') legend('b', 'Absorbtion', 'm','Scattering') ac=1+(ks_cyan(:,1)./ks_cyan(:,2)); bc= sqrt(ac.^2-1); %a and b parameters of the Kubelka-Munk basic equation white_b=saunder(white_b); Rc_est= (1-white_b.*(ac-bc.*coth(bc.*ks_cyan(:,2))))./(ac-white_b+bc.*coth(bc.*ks_cyan(:,2))); %Reflectance estimation by the Kubelka-Munk basic equation Rc_est=asaunder(Rc_est); deltaE94=del_e94(r_on_ba(:,5), Rc_est, white_b) T=num2str( deltaE94, 4 ); figure plot(lambda, r_on_ba(:,5),'b', lambda, Rc_est,'m') title('The measured and predicted reflectance of cyan ink.') xlabel('Wavelength') ylabel('Reflectance factor') legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) %************************************************************************************ % Determining the K and S for red ink %************************************************************************************ load white_b.txt; %reflectance spectrum of the white surface of contrast paper ks_red(:,1)=white_b-r_on_ba2(:,7); %initialing K for red ink ks_red(:,2)=r_on_ba2(:,8)-black_b; ks_red=abs(ks_red);
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vlb=ones(31,2)*0.000000000001; vub=ones(31,2)*10; options(1)=1; options(2)=1e-3; options(3)=1e-3; options(4)=1e-3; options(14)=1000000; ks_red=constr('ksest', ks_red, options, vlb, vub, '',r_on_ba(:,7), r_on_ba(:,8), white_b, black_b); figure plot(lambda, ks_red(:,1),'b', lambda, ks_red(:,2), 'm') title('The estimated spectral absorption and scattering coefficient of red ink.') xlabel('Wavelength') ylabel('Absorption and Scattering Coefficient') legend('b', 'Absorbtion', 'm','Scattering') ar=1+(ks_red(:,1)./ks_red(:,2)); br= sqrt(ar.^2-1); %a and b parameters of the Kubelka-Munk basic equation white_b=saunder(white_b); Rr_est= (1-white_b.*(ar-br.*coth(br.*ks_red(:,2))))./(ar-white_b+br.*coth(br.*ks_red(:,2))); %Reflectance estimation by the Kubelka-Munk basic equation Rr_est=asaunder(Rr_est); deltaE94=del_e94(r_on_ba(:,7), Rr_est, white_b) T=num2str( deltaE94, 4 ); figure plot(lambda, r_on_ba(:,7),'b', lambda, Rr_est,'m') title('The measured and predicted reflectance of red ink.') xlabel('Wavelength') ylabel('Reflectance factor') legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) %************************************************************************************ % Determining the K and S for green ink %************************************************************************************ load white_b.txt; %reflectance spectrum of the white surface of contrast paper ks_grn(:,1)=white_b-r_on_ba2(:,9); %initialing K for green ink ks_grn(:,2)=r_on_ba2(:,10)-black_b; %initialing S for green ink ks_grn=abs(ks_grn); vlb=ones(31,2)*0.000000000001; vub=ones(31,2)*10; options(1)=1; options(2)=1e-3; options(3)=1e-3; options(4)=1e-3; options(14)=1000000; ks_grn=constr('ksest', ks_grn, options, vlb, vub, '',r_on_ba(:,9), r_on_ba(:,10), white_b, black_b); figure plot(lambda, ks_grn(:,1),'b', lambda, ks_grn(:,2), 'm') title('The estimated spectral absorption and scattering coefficient of green ink.') xlabel('Wavelength') ylabel('Absorption and Scattering Coefficient') legend('b', 'Absorbtion', 'm','Scattering') ag=1+(ks_grn(:,1)./ks_grn(:,2)); bg= sqrt(ag.^2-1); %a and b parameters of the Kubelka-Munk basic equation white_b=saunder(white_b);
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Rg_est= (1-white_b.*(ag-bg.*coth(bg.*ks_grn(:,2))))./(ag-white_b+bg.*coth(bg.*ks_grn(:,2))); %Reflectance estimation by the Kubelka-Munk basic equation Rg_est=asaunder(Rg_est); deltaE94=del_e94(r_on_ba(:,9), Rg_est, white_b) T=num2str( deltaE94, 4 ); figure plot(lambda, r_on_ba(:,9),'b', lambda, Rg_est,'m') title('The measured and predicted reflectance of green ink.') xlabel('Wavelength') ylabel('Reflectance factor') legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) %************************************************************************************ % Determining the K and S for blue ink %************************************************************************************ load white_b.txt; %reflectance spectrum of the white surface of contrast paper ks_blu(:,1)=white_b-r_on_ba2(:,11); %initialing K for blue ink ks_blu(:,2)=r_on_ba2(:,12)-black_b; %initialing S for blue ink ks_blu=abs(ks_blu); vlb=ones(31,2)*0.000000000001; vub=ones(31,2)*10; options(1)=1; options(2)=1e-3; options(3)=1e-3; options(4)=1e-3; options(14)=1000000; ks_blu=constr('ksest', ks_blu, options, vlb, vub, '',r_on_ba(:,11), r_on_ba(:,12), white_b, black_b); figure plot(lambda, ks_blu(:,1),'b', lambda, ks_blu(:,2), 'm') title('The estimated spectral absorption and scattering coefficient of blue ink.') xlabel('Wavelength') ylabel('Absorption and Scattering Coefficient') legend('b', 'Absorbtion', 'm','Scattering') ab=1+(ks_blu(:,1)./ks_blu(:,2)); bb= sqrt(ab.^2-1); %a and b parameters of the Kubelka-Munk basic equation white_b=saunder(white_b); Rb_est= (1-white_b.*(ab-bb.*coth(bb.*ks_blu(:,2))))./(ab-white_b+bb.*coth(bb.*ks_blu(:,2))); %Reflectance estimation by the Kubelka-Munk basic equation Rb_est=asaunder(Rb_est); deltaE94=del_e94(r_on_ba(:,11), Rb_est, white_b) T=num2str( deltaE94, 4 ); figure plot(lambda, r_on_ba(:,11),'b', lambda, Rb_est,'m') title('The measured and predicted reflectance of blue ink.') xlabel('Wavelength') ylabel('Reflectance factor') legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) ks=[ks_yel ks_mag ks_cyan ks_red ks_grn ks_blu]; save 'du_ks.txt' ks -ASCII -DOUBLE -TABS
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ksest.m %This is program of the objective function for estimating the absorption and scattering %coefficients for translucent inks function [f,g]=ksest(ks, Ronw, Ronbk, white, black) aw=1+(ks(:,1)./ks(:,2)); bw= sqrt((aw.^2)-1); %a and b parameters of the Kubelka-Munk basic equation white=saunder(white); black=saunder(black); Rw_est= (1-white.*(aw-bw.*coth(bw.*ks(:,2))))./(aw-white+bw.*coth(bw.*ks(:,2))); %Reflectance estimation by the Kubelka-Munk basic equation Rw_est=asaunder(Rw_est); ak=1+(ks(:,1)./ks(:,2)); bk= sqrt(ak.^2-1); %a and b parameters of the Kubelka-Munk basic equation Rk_est= (1-black.*(ak-bk.*coth(bk.*ks(:,2))))./(ak-black+bk.*coth(bk.*ks(:,2))); %Reflectance estimation by the Kubelka-Munk basic equation Rk_est=asaunder(Rk_est); f=rms([Ronw Ronbk], [Rw_est Rk_est]); g=-[Rw_est Rk_est];
The following diagram is the structure chart of the ink overprint prediction.
Main Program (overprint.m)
Overpr1.m
Overpr2.m
Overpr3.m
Overpr4.m
Thickness estimation
Ink overprint estimation
Thickness estimation
Ink overprint estimation
Thickness estimation
Ink overprint estimation
Thickness estimation
Ink overprint estimation
thickest.m
km_trans.m
thickest.m
km_trans.m
thickest.m
km_trans.m
thickest.m
km_trans.m
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overprint.m %This program performs the spectral reflectance factor prediction of ink overprint load ks_dup.sn1; %the absorption and scattering information of the CMYRGB primaries %are stored in a text file named ks_dup.sn1 figure subplot(2,3,1) plot(lambda, ks_dup(:,1),'b',lambda, ks_dup(:,2)*10,'k:') title('Cyan') ylabel('K or S') axis([400 700 0 3]) subplot(2,3,2) plot(lambda, ks_dup(:,3),'b',lambda, ks_dup(:,4)*10,'k:') title('Magenta') axis([400 700 0 3]) subplot(2,3,3) plot(lambda, ks_dup(:,5),'b',lambda, ks_dup(:,6)*10,'k:') title('Yellow') axis([400 700 0 3]) subplot(2,3,4) plot(lambda, ks_dup(:,7),'b',lambda, ks_dup(:,8)*10,'k:') title('Red') xlabel('Wavelength') ylabel('K or S') axis([400 700 0 3]) subplot(2,3,5) plot(lambda, ks_dup(:,9),'b',lambda, ks_dup(:,10)*10,'k:') title('Green') xlabel('Wavelength') axis([400 700 0 3]) subplot(2,3,6) plot(lambda, ks_dup(:,11),'b',lambda, ks_dup(:,12)*10,'k:') title('Blue') xlabel('Wavelength') axis([400 700 0 3]) overpr1; %prediction of the 1st set of ink overprints proceduced by Waterproof o_est=over_est; overpr2; %prediction of the 2nd set of ink overprints proceduced by Waterproof o_est=[o_est over_est]; overpr3; %prediction of the 3rd set of ink overprints proceduced by Waterproof o_est=[o_est over_est]; overpr4; %prediction of the 4th set of ink overprints proceduced by Waterproof o_est=[o_est over_est]; load duover.txt; %25 ink overprints produced by Waterproof deltaE94=del_e94(duover, o_est, ones(31,1), [3 1]); fprintf('The average color difference between measured and predicted DuPont overprints is \n'); fprintf( 'Mean \t\t%f\n', mean(deltaE94) ); fprintf( 'Standard Deviation \t\t%f\n', std(deltaE94) ); fprintf( 'Maximum \t\t%f\n', max(deltaE94) ); fprintf( 'Minimum \t\t%f\n\n', min(deltaE94) ); figure [yy,xx]=hist(deltaE94); bar(xx,yy,'r')
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title('The histogram color difference between measured and predicted DuPont overprints.') xlabel('Delta E94') ylabel('Frequency')
spectral_RMS = RMS(duover, o_est); midx=meta_idx(duover, o_est, ones(31,1)); fprintf('The average color difference between measured and predicted DuPont overprints under illum A is %f\n', mean(midx)); fprintf( 'Mean \t\t%f\n', mean(midx) ); fprintf( 'Standard Deviation \t\t%f\n', std(midx) ); fprintf( 'Maximum \t\t%f\n', max(midx) ); fprintf( 'Minimum \t\t%f\n\n', min(midx) );
fprintf( 'The root mean square error of predicted DuPont overprints is %f\n\n', spectral_RMS ); figure [yy,xx]=hist(midx); bar(xx,yy,'r') title('The histogram of the index of metamerism between measured and predicted DuPont overprints.') xlabel('Index of Metamerism') ylabel('Frequency') deltaE94=[deltaE94(1:11);deltaE94(15:21); deltaE94(28:30); deltaE94(34:37)]; midx=[midx(1:11);midx(15:21); midx(28:30); midx(34:37)]; fprintf('The average color difference between measured and predicted DuPont overprints is \n'); fprintf( 'Mean \t\t%f\n', mean(deltaE94) ); fprintf( 'Standard Deviation \t\t%f\n', std(deltaE94) ); fprintf( 'Maximum \t\t%f\n', max(deltaE94) ); fprintf( 'Minimum \t\t%f\n\n', min(deltaE94) ); figure [yy,xx]=hist(deltaE94); bar(xx,yy,'r') title('The histogram color difference between measured and predicted DuPont overprints.') xlabel('Delta E94') ylabel('Frequency')
fprintf('The average color difference between measured and predicted DuPont overprints under illum A is %f\n', mean(midx)); fprintf( 'Mean \t\t%f\n', mean(midx) ); fprintf( 'Standard Deviation \t\t%f\n', std(midx) ); fprintf( 'Maximum \t\t%f\n', max(midx) ); fprintf( 'Minimum \t\t%f\n\n', min(midx) );
figure [yy,xx]=hist(midx); bar(xx,yy,'r') %title('The histogram of the index of metamerism between measured and predicted DuPont overprints.') xlabel('Metameric Index') ylabel('Frequency') figure subplot(2,2,1) plot(lambda,duover(:,5),'b',lambda,o_est(:,5),'m:') %xlabel('Wavelength') ylabel('Reflectance') title('Red on Magenta') %legend('measured','predicted','DE94 0.2', 'MI 0.0') axis([400 700 0 1])
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subplot(2,2,2) plot(lambda,duover(:,21),'b',lambda,o_est(:,21),'m:') title('Yellow on Magenta') %xlabel('Wavelength') %ylabel('Reflectance') %legend('measured','predicted','DE94 0.4', 'MI 0.1') axis([400 700 0 1]) subplot(2,2,3) plot(lambda,duover(:,15),'b',lambda,o_est(:,15),'m:') title('Red on Yellow') xlabel('Wavelength') ylabel('Reflectance') %legend('measured','predicted','DE94 0.3', 'MI 0.1') axis([400 700 0 1]) subplot(2,2,4) plot(lambda,duover(:,16),'b',lambda,o_est(:,16),'m:') title('Green on Yellow') xlabel('Wavelength') %ylabel('Reflectance') %legend('measured','predicted','DE94 0.4', 'MI 0.1') axis([400 700 0 1]) figure subplot(2,2,1) plot(lambda,duover(:,5)-o_est(:,5),'b') %xlabel('Wavelength') ylabel('Delta R') title('Red on Magenta') %legend('measured','predicted','DE94 0.2', 'MI 0.0') axis([400 700 -0.05 0.05]) subplot(2,2,2) plot(lambda,duover(:,21)-o_est(:,21),'b') title('Yellow on Magenta') %xlabel('Wavelength') %ylabel('Reflectance') %legend('measured','predicted','DE94 0.4', 'MI 0.1') axis([400 700 -0.05 0.05]) subplot(2,2,3) plot(lambda,duover(:,15)-o_est(:,15),'b') title('Red on Yellow') xlabel('Wavelength') ylabel('Delta R') %legend('measured','predicted','DE94 0.3', 'MI 0.1') axis([400 700 -0.05 0.05]) subplot(2,2,4) plot(lambda,duover(:,16)-o_est(:,16),'b') title('Green on Yellow') xlabel('Wavelength') %ylabel('Reflectance') %legend('measured','predicted','DE94 0.4', 'MI 0.1') axis([400 700 -0.05 0.05]) figure subplot(2,2,1) plot(lambda,duover(:,34),'b',lambda,o_est(:,34),'m:') %xlabel('Wavelength') ylabel('Reflectance')
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title('Green on Red') %legend('measured','predicted','DE94 2.1', 'MI 0.4') axis([400 700 0 1]) subplot(2,2,2) plot(lambda,duover(:,17),'b',lambda,o_est(:,17),'m:') title('Blue on Yellow') %xlabel('Wavelength') %ylabel('Reflectance') %legend('measured','predicted','DE94 0.9', 'MI 0.5') axis([400 700 0 1]) subplot(2,2,3) plot(lambda,duover(:,4),'b',lambda,o_est(:,4),'m:') title('Blue on Cyan') xlabel('Wavelength') ylabel('Reflectance') %legend('measured','predicted','DE94 1.4', 'MI 0.7') axis([400 700 0 1]) subplot(2,2,4) plot(lambda,duover(:,20),'b',lambda,o_est(:,20),'m:') title('Blue on Yellow on Cyan') xlabel('Wavelength') %ylabel('Reflectance') %legend('measured','predicted','DE94 1.2', 'MI 1.1') axis([400 700 0 1]) figure subplot(2,2,1) plot(lambda,duover(:,34)-o_est(:,34),'b') %xlabel('Wavelength') ylabel('Delta R') title('Green on Red') %legend('measured','predicted','DE94 2.1', 'MI 0.4') axis([400 700 -0.05 0.05]) subplot(2,2,2) plot(lambda,duover(:,17)-o_est(:,17),'b') title('Blue on Yellow') %xlabel('Wavelength') %ylabel('Reflectance') %legend('measured','predicted','DE94 0.9', 'MI 0.5') axis([400 700 -0.05 0.05]) subplot(2,2,3) plot(lambda,duover(:,4)-o_est(:,4),'b') title('Blue on Cyan') xlabel('Wavelength') ylabel('Delta R') %legend('measured','predicted','DE94 1.4', 'MI 0.7') axis([400 700 -0.05 0.05]) subplot(2,2,4) plot(lambda,duover(:,20)-o_est(:,20),'b') title('Blue on Yellow on Cyan') xlabel('Wavelength') %ylabel('Reflectance') %legend('measured','predicted','DE94 1.2', 'MI 1.1') axis([400 700 -0.05 0.05]) load prmy_est.txt load primary.txt
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figure subplot(1,2,1) plot(lambda,ks_dup(:,1),'b',lambda,ks_dup(:,2),'k:') %grid on title('Estimated K & S') xlabel('Wavelength') ylabel('K or S') axis([400 700 -0.1 2]) subplot(1,2,2) plot(lambda,primary(:,3)-prmy_est(:,1),'b') grid on title('Difference Spectrum') xlabel('Wavelength') ylabel('Delta R') axis([400 700 -0.02 0.02])
overpr1.m %This program performs ink overprints of the first set of Waterproof samples clear all; close all; lambda=[400:10:700]'; load set1.txt; %measured reflectance of overprint [r,c]=size(set1); inkidx=set1(1,3:c); Rg=set1(2:32,2); %spectral reflectance factor of the coated paper prints=set1(2:r,3:c); clear set1; i=1; while inkidx(i) < 7 i=i+1; end num_of_prmy=i-1; pidx=inkidx(:,num_of_prmy+1:c-2); %index of overprint primary=prints(:,1:num_of_prmy); overprint=prints(:,num_of_prmy+1:c-2); clear prints; load ks_dup.sn1; %The determined k and S data are stored in ks_dup.sn1 %************************************************************************************ % Thickness estimation %************************************************************************************ thickness=ones(1,6)*0.001; prmy_est=zeros(31,6); for i=1:num_of_prmy ks=ks_dup(:,(inkidx(i)-1)*2+1:(inkidx(i)-1)*2+2); vlb=0; vub=10;
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options(1)=1; options(2)=1e-5; options(3)=1e-5; options(4)=1e-5; options(14)=1000000; thickness(inkidx(i))=constr('thickest', thickness(inkidx(i)), options, vlb, vub, '',ks, saunder(Rg), saunder(primary(:,i))); prmy_est(:,inkidx(i))=asaunder(km_trans(thickness(inkidx(i)), ks, saunder(Rg))); deltaE94=del_e94(primary(:,i), prmy_est(:,inkidx(i)), Rg, [3 1]) T=num2str( deltaE94, 4 ); figure plot(lambda, primary(:,i),'b', lambda, prmy_est(:,inkidx(i)),'m') %title('The measured and predicted reflectance of yellow ink.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) end [r,c]=size(pidx); over_est=zeros(31,c); %thickness=[0.9754 0.9584 0.95605 1.02565 0.961775 0.987025]; %************************************************************************************ % Ink overprint prediction %************************************************************************************ for i=1:c %c tmpstr=num2str(pidx(i)); num_of_color=length(tmpstr); if num_of_color==2 substrate=prmy_est(:,str2num(tmpstr(2))); ith_prmy=str2num(tmpstr(1)); else substrate=over_est(:,1); ith_prmy=str2num(tmpstr(1)); end ks=ks_dup(:,(ith_prmy-1)*2+1:(ith_prmy-1)*2+2); over_est(:,i)=asaunder(km_trans(thickness(ith_prmy), ks, saunder(substrate))); deltaE94=del_e94(overprint(:,i), over_est(:,i), Rg, [3 1]); T=num2str( deltaE94, 4 ); figure plot(lambda, overprint(:,i),'b', lambda, over_est(:,i),'m') title('The measured and predicted reflectance of overprint.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) end
overpr2.m %This program performs ink overprints of the second set of Waterproof samples lambda=[400:10:700]'; load set2.txt; %measured reflectance of overprints [r,c]=size(set2); inkidx=set2(1,3:c);
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Rg=set2(2:32,2);
%spectral reflectance factor of the coated paper
prints=set2(2:r,3:c); clear set2; i=1; while inkidx(i) < 7 i=i+1; end num_of_prmy=i-1; pidx=inkidx(:,num_of_prmy+1:c-2); %index of overprint primary=prints(:,1:num_of_prmy); overprint=prints(:,num_of_prmy+1:c-2); clear prints; load ks_dup.sn1; %************************************************************************************ % Thickness estimation %************************************************************************************ thickness=ones(1,6)*0.001; prmy_est=zeros(31,6); for i=1:num_of_prmy ks=ks_dup(:,(inkidx(i)-1)*2+1:(inkidx(i)-1)*2+2); vlb=0; vub=10; options(1)=1; options(2)=1e-5; options(3)=1e-5; options(4)=1e-5; options(14)=1000000; thickness(inkidx(i))=constr('thickest', thickness(inkidx(i)), options, vlb, vub, '',ks, saunder(Rg), saunder(primary(:,i))); prmy_est(:,inkidx(i))=asaunder(km_trans(thickness(inkidx(i)), ks, saunder(Rg))); deltaE94=del_e94(primary(:,i), prmy_est(:,inkidx(i)), Rg, [3 1]) T=num2str( deltaE94, 4 ); figure plot(lambda, primary(:,i),'b', lambda, prmy_est(:,inkidx(i)),'m') title('The measured and predicted reflectance of yellow ink.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) end [r,c]=size(pidx); over_est=zeros(31,c); %thickness=[0.9754 0.9584 0.95605 1.02565 0.961775 0.987025]; %************************************************************************************ % Ink overprint prediction %************************************************************************************ for i=1:c %c
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tmpstr=num2str(pidx(i)); num_of_color=length(tmpstr); if num_of_color==2 substrate=prmy_est(:,str2num(tmpstr(2))); ith_prmy=str2num(tmpstr(1)); else substrate=over_est(:,1); ith_prmy=str2num(tmpstr(1)); end ks=ks_dup(:,(ith_prmy-1)*2+1:(ith_prmy-1)*2+2); over_est(:,i)=asaunder(km_trans(thickness(ith_prmy), ks, saunder(substrate))); deltaE94=del_e94(overprint(:,i), over_est(:,i), Rg, [3 1]); T=num2str( deltaE94, 4 ); figure plot(lambda, overprint(:,i),'b', lambda, over_est(:,i),'m') title('The measured and predicted reflectance of overprint.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) end
overpr3.m %This program performs ink overprints of the second set of Waterproof samples lambda=[400:10:700]'; load set3.txt; %measured reflectance of overprints [r,c]=size(set3); inkidx=set3(1,3:c); Rg=set3(2:32,2); %spectral reflectance factor of the coated paper prints=set3(2:r,3:c); clear set3; i=1; while inkidx(i) < 7 i=i+1; end num_of_prmy=i-1; pidx=inkidx(:,num_of_prmy+1:c-2); %index of overprint primary=prints(:,1:num_of_prmy); overprint=prints(:,num_of_prmy+1:c-2); clear prints; load ks_dup.sn1; %************************************************************************************ % Thickness estimation %************************************************************************************ thickness=ones(1,6)*0.001; prmy_est=zeros(31,6); for i=1:num_of_prmy ks=ks_dup(:,(inkidx(i)-1)*2+1:(inkidx(i)-1)*2+2); vlb=0;
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vub=10; options(1)=1; options(2)=1e-5; options(3)=1e-5; options(4)=1e-5; options(14)=1000000; thickness(inkidx(i))=constr('thickest', thickness(inkidx(i)), options, vlb, vub, '',ks, saunder(Rg), saunder(primary(:,i))); prmy_est(:,inkidx(i))=asaunder(km_trans(thickness(inkidx(i)), ks, saunder(Rg))); deltaE94=del_e94(primary(:,i), prmy_est(:,inkidx(i)), Rg, [3 1]) T=num2str( deltaE94, 4 ); figure plot(lambda, primary(:,i),'b', lambda, prmy_est(:,inkidx(i)),'m') title('The measured and predicted reflectance of yellow ink.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) end [r,c]=size(pidx); over_est=zeros(31,c); %thickness=[0.9754 0.9584 0.95605 1.02565 0.961775 0.987025]; %************************************************************************************ % Ink overprint prediction %************************************************************************************ for i=1:c %c tmpstr=num2str(pidx(i)); num_of_color=length(tmpstr); if num_of_color==2 substrate=prmy_est(:,str2num(tmpstr(2))); ith_prmy=str2num(tmpstr(1)); else substrate=over_est(:,1); ith_prmy=str2num(tmpstr(1)); end ks=ks_dup(:,(ith_prmy-1)*2+1:(ith_prmy-1)*2+2); over_est(:,i)=asaunder(km_trans(thickness(ith_prmy), ks, saunder(substrate))); deltaE94=del_e94(overprint(:,i), over_est(:,i), Rg, [3 1]); T=num2str( deltaE94, 4 ); figure plot(lambda, overprint(:,i),'b', lambda, over_est(:,i),'m') title('The measured and predicted reflectance of overprint.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) end
overpr4.m %This program performs ink overprints of the second set of Waterproof samples lambda=[400:10:700]'; load set4.txt; %measured reflectance of overprints [r,c]=size(set4); inkidx=set4(1,3:c); Rg=set4(2:32,2); %spectral reflectance factor of the coated paper
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prints=set4(2:r,3:c); clear set4; i=1; while inkidx(i) < 7 i=i+1; end num_of_prmy=i-1; pidx=inkidx(:,num_of_prmy+1:c-2); %index of overprint primary=prints(:,1:num_of_prmy); overprint=prints(:,num_of_prmy+1:c-2); clear prints; load ks_dup.sn1; %************************************************************************************ % Thickness estimation %************************************************************************************ thickness=ones(1,6)*0.001; prmy_est=zeros(31,6); for i=1:num_of_prmy ks=ks_dup(:,(inkidx(i)-1)*2+1:(inkidx(i)-1)*2+2); vlb=0; vub=10; options(1)=1; options(2)=1e-5; options(3)=1e-5; options(4)=1e-5; options(14)=1000000; thickness(inkidx(i))=constr('thickest', thickness(inkidx(i)), options, vlb, vub, '',ks, saunder(Rg), saunder(primary(:,i))); prmy_est(:,inkidx(i))=asaunder(km_trans(thickness(inkidx(i)), ks, saunder(Rg))); deltaE94=del_e94(primary(:,i), prmy_est(:,inkidx(i)), Rg, [3 1]) T=num2str( deltaE94, 4 ); figure plot(lambda, primary(:,i),'b', lambda, prmy_est(:,inkidx(i)),'m') title('The measured and predicted reflectance of yellow ink.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) end [r,c]=size(pidx); over_est=zeros(31,c); %thickness=[0.9754 0.9584 0.95605 1.02565 0.961775 0.987025]; %************************************************************************************ % Ink overprint prediction %************************************************************************************ for i=1:c %c tmpstr=num2str(pidx(i)); num_of_color=length(tmpstr);
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if num_of_color==2 substrate=prmy_est(:,str2num(tmpstr(2))); ith_prmy=str2num(tmpstr(1)); else substrate=over_est(:,1); ith_prmy=str2num(tmpstr(1)); end ks=ks_dup(:,(ith_prmy-1)*2+1:(ith_prmy-1)*2+2); over_est(:,i)=asaunder(km_trans(thickness(ith_prmy), ks, saunder(substrate))); deltaE94=del_e94(overprint(:,i), over_est(:,i), Rg, [3 1]); T=num2str( deltaE94, 4 ); figure plot(lambda, overprint(:,i),'b', lambda, over_est(:,i),'m') title('The measured and predicted reflectance of overprint.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T) end
thickest.m %This is the objective fuction of estimating the actual ink thickness of an printed on %an opaque substrate function [f,g]=thickest(thick, ks, Rg, R_sample) R_est=km_trans(thick, ks, Rg); % Kubelka-Munk equation for translucent material f=rms(R_est, R_sample); g=-R_est;
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APPENDIX E : MATLAB PROGRAMS FOR THE PROPOSED SIXCOLOR FORWARD PRINTING MODEL Since a six-color printing model is the union of 10 four-color sub-models, the following structure chart only shows the program structure for a CMYK modeling process.
Main Program (secondorder.m)
Determining n-factor (nfind.m)
Finding data for modeling optical-trapping (optrap1.m)
...
CMYK modeling
Determining the dot gain correction function (interink.m)
Reflectance estimation by Neugebauer equation (neug4c.m)
Determining the effective dot area
The first order estimation (theo2eff.m)
YGOK modeling
The second order dot gain correction (iino_q.m)
Demichel equation (dmi4c.m)
secondorder.m %This program is a forward second order six-color printing model proposed by this research %modeling optical trapping effect to modified the accuracy of the first order printing model %which is comprised of theoretical to effective dot area transfer function and the Yule-Nielsen modified %spectral Neugebauer equation clear all; close all; load ave_ramp.txt %the spectral reflectance factors of 13-step ramps c_ramp=(ave_ramp(1:13, 7:37))'; m_ramp=(ave_ramp(14:26, 7:37))'; y_ramp=(ave_ramp(27:39, 7:37))';
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.. .
g_ramp=(ave_ramp(40:52, 7:37))'; o_ramp=(ave_ramp(53:65, 7:37))'; k_ramp=(ave_ramp(66:78, 7:37))'; clear ave_ramp; lambda=[400:10:700]'; R_paper=c_ramp(:,13);
dc=[100;90;80;70;60;50;40;30;20;15;10;5;0]/100; %the theoretical dot areas of each ramp n=nfind(c_ramp,m_ramp,y_ramp,g_ramp,o_ramp,k_ramp,R_paper,20,0.1); %determining the Yule-Nielsen n-factor (n=2.2) %the effective dot area of primary ramps solved by n-modified Murray-Davies equation a_c=inv_murr(c_ramp,R_paper,n); a_m=inv_murr(m_ramp,R_paper,n); a_y=inv_murr(y_ramp,R_paper,n); a_g=inv_murr(g_ramp,R_paper,n); a_o=inv_murr(o_ramp,R_paper,n); a_k=inv_murr(k_ramp,R_paper,n); %prediction of primary ramps by n-modified Murray-Davies equation reconc=murray(c_ramp(:,1), a_c, R_paper, n); reconm=murray(m_ramp(:,1), a_m, R_paper, n); recony=murray(y_ramp(:,1), a_y, R_paper, n); recong=murray(g_ramp(:,1), a_g, R_paper, n); recono=murray(o_ramp(:,1), a_o, R_paper, n); reconk=murray(k_ramp(:,1), a_k, R_paper, n); figure; plot(dc, [a_c a_m a_y a_g a_o a_k]); axis([0 1 0 1]) xlabel('Theoretical dot area') ylabel('Effective dot area') title('Mechanical dot gain of six-color ramps') figure; plot(dc, a_c,'k-',dc,a_m,'k:',dc, a_y,'k-.',dc, a_k,'k--'); axis([0 1 0 1]) xlabel('Theoretical dot area') ylabel('Effective dot area') %title('Mechanical dot gain of six-color ramps') legend('cyan','magenta', 'yellow','black') figure; plot(dc, a_c-dc,'k-',dc,a_m-dc,'k:',dc, a_y-dc,'k-.',dc, a_k-dc,'k--'); axis([0 1 0 0.4]) xlabel('Theoretical dot area') ylabel('Dot-gain in units of fractional dot area') %title('Mechanical dot gain of six-color ramps') legend('cyan','magenta', 'yellow','black') tmp1=a_c-dc; tmp2=(a_c-dc)*0.75; tmp3=(a_c-dc)*0.50; tmp4=(a_c-dc)*0.25;
figure; plot(dc, tmp1,'k-',dc,tmp2,'k:',dc, tmp3,'k-.',dc, tmp4,'k--');
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axis([0 1 0 0.35]) xlabel('Theoretical dot area') ylabel('Dot-gain in units of fractional dot area') %title('Mechanical dot gain of six-color ramps') legend('mag=0%','mag=25%', 'mag=50%','mag=75%') hold on plot([0.1; 0.1],[0; tmp1(11)],'k:'); plot([0.3; 0.3],[0; tmp1(8)],'k:'); plot([0.5; 0.5],[0; tmp1(6)],'k:'); plot([0.7; 0.7],[0; tmp1(4)],'k:'); figure subplot(1,2,1) plot([0;0.25;0.5;0.75], [1; tmp2(6)/tmp1(6); tmp3(6)/tmp1(6); tmp4(6)/tmp1(6)].^(2)); xlabel('a_t_h_e_o_,_m') ylabel('f_c_m') axis([0 0.8 0 1]) title('The correction scalar by Iino and Berns') subplot(1,2,2) plot([0;0.1;0.3;0.5;0.7], [1; 0.85; 0.87; 0.86; 0.9]); xlabel('a_t_h_e_o_,_c') ylabel('g_m_c') axis([0 0.8 0.5 1]) title('The correction scalar by the proposed algorithm')
c=c_ramp(:,1); m=m_ramp(:,1); y=y_ramp(:,1); g=g_ramp(:,1); o=o_ramp(:,1); k=k_ramp(:,1); w=R_paper; figure; plot(lambda,c,'c',lambda,m,'m',lambda,y,'y',lambda,g,'g',lambda,o,'r',lambda,k,'k',lambda,w,'k:') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('Cyan','Magenta','Yellow','Green','Orange','Black','Substrate')
figure; plot(lambda, c_ramp,'b', lambda, reconc,'m:') xlabel('Wavelength') ylabel('Reflectance factor') title('Measured and predicted c_ramp') axis([400 700 0 1]) legend('b','Measured', 'm:', 'Predicted') figure; plot(lambda, m_ramp,'b', lambda, reconm,'m:') xlabel('Wavelength') ylabel('Reflectance factor') title('Measured and predicted m_ramp') axis([400 700 0 1]) legend('b','Measured', 'm:', 'Predicted') figure; plot(lambda, y_ramp,'b', lambda, recony,'m:') xlabel('Wavelength') ylabel('Reflectance factor') title('Measured and predicted y_ramp')
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axis([400 700 0 1]) legend('b','Measured', 'm:', 'Predicted') figure; plot(lambda, g_ramp,'b', lambda, recong,'m:') xlabel('Wavelength') ylabel('Reflectance factor') title('Measured and predicted g_ramp') axis([400 700 0 1]) legend('b','Measured', 'm:', 'Predicted') figure; plot(lambda, o_ramp,'b', lambda, recono,'m:') xlabel('Wavelength') ylabel('Reflectance factor') title('Measured and predicted o_ramp') axis([400 700 0 1]) legend('b','Measured', 'm:', 'Predicted') figure; plot(lambda, k_ramp,'b', lambda, reconk,'m:') xlabel('Wavelength') ylabel('Reflectance factor') title('Measured and predicted k_ramp') axis([400 700 0 1]) legend('b','Measured', 'm:', 'Predicted')
dac=[0 25 50 70 90]/100; %the theoretical dot areas for modeling ink trapping dac=dac'; sc=0.62; %scalar for adjusting the actual effective dot areas %************************************************************************************ % The forward CMYK submodel %************************************************************************************ load cmyk.txt dc_1=cmyk(:,2:5)/100; %the theoretical dot areas of the CMYK target cmyk5555=cmyk(:,6:36)'; %the 625 CMYK mixtures load neprmy1.txt
%the 16 Neugebauer primaries of CMYK printing process
%optrap1=trapfind(cmyk); %the procedure searching for data in the verification target %for modeling optical-trapping effect %fij=interink(optrap1, neprmy1, n, a_c, a_m, a_y, a_k, dc, dac); %save 'fij_1.txt' fij -ASCII -DOUBLE -TABS %the fij is the dot gain correction scalar function discussed in the thesis %here, fij is the gij described in the thesis load fij_1.txt; %the dot gain correction scalar function for CMYK printing process fij=fij_1; qscalar=iino_q(fij, dc_1, dac); %the dot gain correction scalars for the CMYK verification target eff=theo2eff(dc_1, dc, c_ramp, m_ramp, y_ramp, k_ramp, R_paper, n); %the first order estimated theoretical dot areas for CMYK verification target eff1=dc_1+qscalar.*(eff-dc_1); %the effective dot areas for CMYK verification target after the second order %improvement by dot gain correction
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eff1=(dc_1+qscalar.*(eff-dc_1))*sc + eff*(1-sc); %the final effective dot areas for CMYK verification target lies between the first %order estimation and the second order estimation pred1=neug4c(eff1, neprmy1, n); %spectral reconstruction by n-modified Neugebauer equation del1=del_e94(cmyk5555, pred1, R_paper, [3 1]); %************************************************************************************ % The forward CMGK submodel %************************************************************************************ load cmgk.txt dc_2=cmgk(:,2:5)/100; cmgk5555=cmgk(:,6:36)'; load neprmy2.txt %optrap2=trapfind(cmgk); %fij=interink(optrap2, neprmy2, n, a_c, a_m, a_g, a_k, dc, dac); %save 'fij_2.txt' fij -ASCII -DOUBLE -TABS load fij_2.txt; fij=fij_2; qscalar=iino_q(fij, dc_2, dac); eff=theo2eff(dc_2, dc, c_ramp, m_ramp, g_ramp, k_ramp, R_paper, n); eff2=dc_2+qscalar.*(eff-dc_2); eff2=(dc_2+qscalar.*(eff-dc_2))*sc + eff*(1-sc); pred2=neug4c(eff2, neprmy2, n); del2=del_e94(cmgk5555, pred2, R_paper, [3 1]); %************************************************************************************ % The forward CMOK submodel %************************************************************************************ load cmok.txt dc_3=cmok(:,2:5)/100; cmok5555=cmok(:,6:36)'; load neprmy3.txt %optrap3=trapfind(cmok); %fij=interink(optrap3, neprmy3, n, a_c, a_m, a_o, a_k, dc, dac); %save 'fij_3.txt' fij -ASCII -DOUBLE -TABS load fij_3.txt; fij=fij_3; qscalar=iino_q(fij, dc_3, dac); eff=theo2eff(dc_3, dc, c_ramp, m_ramp, o_ramp, k_ramp, R_paper, n); eff3=dc_3+qscalar.*(eff-dc_3); eff3=(dc_3+qscalar.*(eff-dc_3))*sc + eff*(1-sc); pred3=neug4c(eff3, neprmy3, n);
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del3=del_e94(cmok5555, pred3, R_paper, [3 1]); %************************************************************************************ % The forward CYGK submodel %************************************************************************************ load cygk.txt dc_4=cygk(:,2:5)/100; cygk5555=cygk(:,6:36)'; load neprmy4.txt %optrap4=trapfind(cygk); %fij=interink(optrap4, neprmy4, n, a_c, a_y, a_g, a_k, dc, dac); %save 'fij_4.txt' fij -ASCII -DOUBLE -TABS load fij_4.txt; fij=fij_4; qscalar=iino_q(fij, dc_4, dac); eff=theo2eff(dc_4, dc, c_ramp, y_ramp, g_ramp, k_ramp, R_paper, n); eff4=dc_4+qscalar.*(eff-dc_4); eff4=(dc_4+qscalar.*(eff-dc_4))*sc + eff*(1-sc); pred4=neug4c(eff4, neprmy4, n); del4=del_e94(cygk5555, pred4, R_paper, [3 1]); %************************************************************************************ % The forward CYOK submodel %************************************************************************************ load cyok.txt dc_5=cyok(:,2:5)/100; cyok5555=cyok(:,6:36)'; load neprmy5.txt %optrap5=trapfind(cyok); %fij=interink(optrap5, neprmy5, n, a_c, a_y, a_o, a_k, dc, dac); %save 'fij_5.txt' fij -ASCII -DOUBLE -TABS load fij_5.txt; fij=fij_5; qscalar=iino_q(fij, dc_5, dac); eff=theo2eff(dc_5, dc, c_ramp, y_ramp, o_ramp, k_ramp, R_paper, n); eff5=dc_5+qscalar.*(eff-dc_5); eff5=(dc_5+qscalar.*(eff-dc_5))*sc + eff*(1-sc); pred5=neug4c(eff5, neprmy5, n); del5=del_e94(cyok5555, pred5, R_paper, [3 1]); %************************************************************************************ % The forward CGOK submodel %************************************************************************************ load cgok.txt
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dc_6=cgok(:,2:5)/100; cgok5555=cgok(:,6:36)'; load neprmy6.txt %optrap6=trapfind(cgok); %fij=interink(optrap6, neprmy6, n, a_c, a_g, a_o, a_k, dc, dac); %save 'fij_6.txt' fij -ASCII -DOUBLE -TABS load fij_6.txt; fij=fij_6; qscalar=iino_q(fij, dc_6, dac); eff=theo2eff(dc_6, dc, c_ramp, g_ramp, o_ramp, k_ramp, R_paper, n); eff6=dc_6+qscalar.*(eff-dc_6); eff6=(dc_6+qscalar.*(eff-dc_6))*sc + eff*(1-sc); pred6=neug4c(eff6, neprmy6, n); del6=del_e94(cgok5555, pred6, R_paper, [3 1]); %************************************************************************************ % The forward MYGK submodel %************************************************************************************ load mygk.txt dc_7=mygk(:,2:5)/100; mygk5555=mygk(:,6:36)'; load neprmy7.txt %optrap7=trapfind(mygk); %fij=interink(optrap7, neprmy7, n, a_m, a_y, a_g, a_k, dc, dac); %save 'fij_7.txt' fij -ASCII -DOUBLE -TABS load fij_7.txt; fij=fij_7; qscalar=iino_q(fij, dc_7, dac); eff=theo2eff(dc_7, dc, m_ramp, y_ramp, g_ramp, k_ramp, R_paper, n); eff7=dc_7+qscalar.*(eff-dc_7); eff7=(dc_7+qscalar.*(eff-dc_7))*sc + eff*(1-sc); pred7=neug4c(eff7, neprmy7, n); del7=del_e94(mygk5555, pred7, R_paper, [3 1]); %************************************************************************************ % The forward MYOK submodel %************************************************************************************ load myok.txt dc_8=myok(:,2:5)/100; myok5555=myok(:,6:36)'; load neprmy8.txt %optrap8=trapfind(myok);
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%fij=interink(optrap8, neprmy8, n, a_m, a_y, a_o, a_k, dc, dac); %save 'fij_8.txt' fij -ASCII -DOUBLE -TABS load fij_8.txt; fij=fij_8; qscalar=iino_q(fij, dc_8, dac); eff=theo2eff(dc_8, dc, m_ramp, y_ramp, o_ramp, k_ramp, R_paper, n); eff8=dc_8+qscalar.*(eff-dc_8); eff8=(dc_8+qscalar.*(eff-dc_8))*sc + eff*(1-sc); pred8=neug4c(eff8, neprmy8, n); del8=del_e94(myok5555, pred8, R_paper, [3 1]); %************************************************************************************ % The forward MGOK submodel %************************************************************************************ load mgok.txt dc_9=mgok(:,2:5)/100; mgok5555=mgok(:,6:36)'; load neprmy9.txt %optrap9=trapfind(mgok); %fij=interink(optrap9, neprmy9, n, a_m, a_g, a_o, a_k, dc, dac); %save 'fij_9.txt' fij -ASCII -DOUBLE -TABS load fij_9.txt; fij=fij_9; qscalar=iino_q(fij, dc_9, dac); eff=theo2eff(dc_9, dc, m_ramp, g_ramp, o_ramp, k_ramp, R_paper, n); eff9=dc_9+qscalar.*(eff-dc_9); eff9=(dc_9+qscalar.*(eff-dc_9))*sc + eff*(1-sc); pred9=neug4c(eff9, neprmy9, n); del9=del_e94(mgok5555, pred9, R_paper, [3 1]);
%************************************************************************************ % The forward YGOK submodel %************************************************************************************ load ygok.txt dc_10=ygok(:,2:5)/100; ygok5555=ygok(:,6:36)'; load neprmy10.txt %optrap10=trapfind(ygok); %fij=interink(optrap10, neprmy10, n, a_y, a_g, a_o, a_k, dc, dac); %save 'fij_10.txt' fij -ASCII -DOUBLE -TABS load fij_10.txt; fij=fij_10;
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qscalar=iino_q(fij, dc_10, dac); eff=theo2eff(dc_10, dc, y_ramp, g_ramp, o_ramp, k_ramp, R_paper, n); eff10=dc_10+qscalar.*(eff-dc_10); eff10=(dc_10+qscalar.*(eff-dc_10))*sc + eff*(1-sc); pred10=neug4c(eff10, neprmy10, n); del10=del_e94(ygok5555, pred10, R_paper, [3 1]); tmp1=[mean(del1);mean(del2);mean(del3);mean(del4);mean(del4);mean(del6)]; meande=[tmp1;mean(del7);mean(del8);mean(del9);mean(del10)] tmp1=[max(del1);max(del2);max(del3);max(del4);max(del4);max(del6)]; maxde=[tmp1;max(del7);max(del8);max(del9);max(del10)] tmp1=[std(del1);std(del2);std(del3);std(del4);std(del4);std(del6)]; stdde=[tmp1;std(del7);std(del8);std(del9);std(del10)] measured=[cmyk5555 cmgk5555 cmok5555 cygk5555 cyok5555 cgok5555]; measured=[measured mygk5555 myok5555 mgok5555 ygok5555]; predicted=[pred1 pred2 pred3 pred4 pred5 pred6 pred7 pred8 pred9 pred10]; delta_e94=del_e94(measured, predicted, R_paper, [3 1]); fprintf('The statistical color difference between measured and predicted ramps is \n'); fprintf( 'Mean \t\t%f\n', mean(delta_e94) ); fprintf( 'Standard Deviation \t\t%f\n', std(delta_e94) ); fprintf( 'Maximum \t\t%f\n', max(delta_e94) ); fprintf( 'Minimum \t\t%f\n\n', min(delta_e94) ); spectral_RMS = RMS( measured, predicted ); midx=meta_idx(measured, predicted, R_paper, [3 2 1 2]); fprintf('The statistical color difference between measured and predicted under illum A %f\n', mean(midx)); fprintf( 'Mean \t\t%f\n', mean(midx) ); fprintf( 'Standard Deviation \t\t%f\n', std(midx) ); fprintf( 'Maximum \t\t%f\n', max(midx) ); fprintf( 'Minimum \t\t%f\n\n', min(midx) ); figure [yy,xx]=hist(delta_e94); bar(xx,yy,'r') %title('The histogram color difference between measured and predicted') xlabel('Delta E94') ylabel('Frequency')
nfind.m %This program searches and optimizes the best n-factor for reconstructing the ramps with %minimum error function n_factor=nfind(ramp1, ramp2, ramp3, ramp4, ramp5, ramp6, R_paper, num_of_step, step) n=1; idx_deltaE=zeros(num_of_step,2); for i=1:num_of_step dotarea1=inv_murr(ramp1,R_paper,n);
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dotarea2=inv_murr(ramp2,R_paper,n); dotarea3=inv_murr(ramp3,R_paper,n); dotarea4=inv_murr(ramp4,R_paper,n); dotarea5=inv_murr(ramp5,R_paper,n); dotarea6=inv_murr(ramp6,R_paper,n); recon1=murray(ramp1(:,1), dotarea1, R_paper, n); recon2=murray(ramp2(:,1), dotarea2, R_paper, n); recon3=murray(ramp3(:,1), dotarea3, R_paper, n); recon4=murray(ramp4(:,1), dotarea4, R_paper, n); recon5=murray(ramp5(:,1), dotarea5, R_paper, n); recon6=murray(ramp6(:,1), dotarea6, R_paper, n); measured=[ramp1 ramp2 ramp3 ramp4 ramp5 ramp6]; predicted=[recon1 recon2 recon3 recon4 recon5 recon6]; idx_deltaE(i,1)=mean(real(del_e94(measured, predicted, R_paper, [3 1]))); idx_deltaE(i,2)=n; n=n+step; end figure; plot(idx_deltaE(:,2), idx_deltaE(:,1)); xlabel('n-factor') ylabel('Delta E94 for D50') [sorted, idx]=sortrows(idx_deltaE); n_factor=sorted(1,2);
trapfind.m %This fuction search the ink mixture in the verification target for modeling optical trapping function trapdata=trapfind(cmyk) load trapdac.txt % the theoretical dot areas of ink mixtures which are required for modeling optical trapping dc_1=cmyk(:,2:5)/100; cmyk5555=cmyk(:,6:36)'; [p,q]=size(trapdac); [s,t]=size(dc_1); trap_idx=zeros(p,1); for i=1:p for j=1:s if trapdac(i,:)-dc_1(j,:)*100==0 trap_idx(i)=j; end end end optrap1=cmyk(trap_idx(1),:); for i=2:p optrap1=[optrap1 ;cmyk(trap_idx(i),:)]; end
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trapdata=optrap1(:,6:36)'; interink.m %This program models the optical-trapping and builds dot gain correction functions for each %verification target function fij=interink(optrap1, neprmy1, n, a_c, a_m, a_y, a_k, dc, dac) rcm=optrap1(:,1:5); rcy=optrap1(:,6:10); rck=optrap1(:,11:15); rmc=optrap1(:,16:20); rmy=optrap1(:,21:25); rmk=optrap1(:,26:30); ryc=optrap1(:,31:35); rym=optrap1(:,36:40); ryk=optrap1(:,41:45); rkc=optrap1(:,46:50); rkm=optrap1(:,51:55); rky=optrap1(:,56:60); fcm=r_neug2c(rcm, [neprmy1(:,1:3) neprmy1(:,7)], n); fcy=r_neug2c(rcy, [neprmy1(:,1:2) neprmy1(:,4) neprmy1(:,6)], n); fck=r_neug2c(rck, [neprmy1(:,1:2) neprmy1(:,9:10)], n); fmc=r_neug2c(rmc, [neprmy1(:,1:3) neprmy1(:,7)], n); fmy=r_neug2c(rmy, [neprmy1(:,1) neprmy1(:,3:5)], n); fmk=r_neug2c(rmk, [neprmy1(:,1) neprmy1(:,3) neprmy1(:,9) neprmy1(:,11)], n); fyc=r_neug2c(ryc, [neprmy1(:,1) neprmy1(:,2) neprmy1(:,4) neprmy1(:,6)], n); fym=r_neug2c(rym, [neprmy1(:,1) neprmy1(:,3) neprmy1(:,4) neprmy1(:,5)], n); fyk=r_neug2c(ryk, [neprmy1(:,1) neprmy1(:,4) neprmy1(:,9) neprmy1(:,12)], n); fkc=r_neug2c(rkc, [neprmy1(:,1) neprmy1(:,2) neprmy1(:,9) neprmy1(:,10)], n); fkm=r_neug2c(rkm, [neprmy1(:,1) neprmy1(:,3) neprmy1(:,9) neprmy1(:,11)], n); fky=r_neug2c(rky, [neprmy1(:,1) neprmy1(:,4) neprmy1(:,9) neprmy1(:,12)], n); %r_neug2c() is a inverse Neugebauer equation for a two color case tmp=[fcm(:,2) fcy(:,2) fck(:,2) fmc(:,1) fmy(:,2) fmk(:,2)]; tmp=[tmp fyc(:,1) fym(:,1) fyk(:,2) fkc(:,1) fkm(:,1) fky(:,1)]; c=interp1(dc, a_c, dac, 'cubic'); m=interp1(dc, a_m, dac, 'cubic'); y=interp1(dc, a_y, dac, 'cubic'); k=interp1(dc, a_k, dac, 'cubic'); tmp2=ones(5,12); tmp2(2:5,1)=(tmp(2:5,1)-dac(2:5))./(m(2:5,1)-dac(2:5)); tmp2(2:5,2)=(tmp(2:5,2)-dac(2:5))./(y(2:5,1)-dac(2:5)); tmp2(2:5,3)=(tmp(2:5,3)-dac(2:5))./(k(2:5,1)-dac(2:5)); tmp2(2:5,4)=(tmp(2:5,4)-dac(2:5))./(c(2:5,1)-dac(2:5)); tmp2(2:5,5)=(tmp(2:5,5)-dac(2:5))./(y(2:5,1)-dac(2:5)); tmp2(2:5,6)=(tmp(2:5,6)-dac(2:5))./(k(2:5,1)-dac(2:5)); tmp2(2:5,7)=(tmp(2:5,7)-dac(2:5))./(c(2:5,1)-dac(2:5)); tmp2(2:5,8)=(tmp(2:5,8)-dac(2:5))./(m(2:5,1)-dac(2:5)); tmp2(2:5,9)=(tmp(2:5,9)-dac(2:5))./(k(2:5,1)-dac(2:5)); tmp2(2:5,10)=(tmp(2:5,10)-dac(2:5))./(c(2:5,1)-dac(2:5)); tmp2(2:5,11)=(tmp(2:5,11)-dac(2:5))./(m(2:5,1)-dac(2:5)); tmp2(2:5,12)=(tmp(2:5,12)-dac(2:5))./(y(2:5,1)-dac(2:5)); fij=tmp2;
%This program determines the global correction scalar for the dot gain of each primary %ramp due to the ink and optical trapping function qscalar=iino_q(fij, dc_1, dac) fmc=interp1(dac, fij(:,4), dc_1(:,1), 'cubic'); fyc=interp1(dac, fij(:,7), dc_1(:,1), 'cubic'); fkc=interp1(dac, fij(:,10), dc_1(:,1), 'cubic');
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fcm=interp1(dac, fij(:,1), dc_1(:,2), 'cubic'); fym=interp1(dac, fij(:,8), dc_1(:,2), 'cubic'); fkm=interp1(dac, fij(:,11), dc_1(:,2), 'cubic'); fcy=interp1(dac, fij(:,2), dc_1(:,3), 'cubic'); fmy=interp1(dac, fij(:,5), dc_1(:,3), 'cubic'); fky=interp1(dac, fij(:,12), dc_1(:,3), 'cubic'); fck=interp1(dac, fij(:,3), dc_1(:,4), 'cubic'); fmk=interp1(dac, fij(:,6), dc_1(:,4), 'cubic'); fyk=interp1(dac, fij(:,9), dc_1(:,4), 'cubic'); %the following check for the existence of secondary %fmc should be corrected by the existence of magenta in as the secondary ink fmc_c=fmc.*(((1./fmc).*(~dc_1(:,2)))+(~~dc_1(:,2))); %corrected fmc fyc_c=fyc.*(((1./fyc).*(~dc_1(:,3)))+(~~dc_1(:,3))); fkc_c=fkc.*(((1./fkc).*(~dc_1(:,4)))+(~~dc_1(:,4))); fcm_c=fcm.*(((1./fcm).*(~dc_1(:,1)))+(~~dc_1(:,1))); fym_c=fym.*(((1./fym).*(~dc_1(:,3)))+(~~dc_1(:,3))); fkm_c=fkm.*(((1./fkm).*(~dc_1(:,4)))+(~~dc_1(:,4))); fcy_c=fcy.*(((1./fcy).*(~dc_1(:,1)))+(~~dc_1(:,1))); fmy_c=fmy.*(((1./fmy).*(~dc_1(:,2)))+(~~dc_1(:,2))); fky_c=fky.*(((1./fky).*(~dc_1(:,4)))+(~~dc_1(:,4))); fck_c=fck.*(((1./fck).*(~dc_1(:,1)))+(~~dc_1(:,1))); fmk_c=fmk.*(((1./fmk).*(~dc_1(:,2)))+(~~dc_1(:,2))); fyk_c=fyk.*(((1./fyk).*(~dc_1(:,3)))+(~~dc_1(:,3)));
qscalar=[fmc_c.*fyc_c.*fkc_c fcm_c.*fym_c.*fkm_c fcy_c.*fmy_c.*fky_c fck_c.*fmk_c.*fyk_c];
neug2c.m %This program performs the spectral reflectance estimation for a two-color printing process. %The estimation is based the n-modified spectral Neugebauer equation function R_predicted=neug2c(cm, neuprimary, nf) a=demi2c(cm); a1=a(:,1); %notice a1 is the area of paper a2=a(:,2); a3=a(:,3); a4=a(:,4); clear a; R1=neuprimary(:,1); %notice R1 is the R_paper R2=neuprimary(:,2); R3=neuprimary(:,3); R4=neuprimary(:,4); [n,m]=size(cm);
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[p,q]=size(neuprimary); tmp=zeros(p,n); for i=1:n tmp(:,i)=a1(i)*(R1.^(1/nf))+a2(i)*(R2.^(1/nf))+a3(i)*(R3.^(1/nf))+a4(i)*(R4.^(1/nf)); tmp(:,i)=tmp(:,i).^nf; end R_predicted=tmp;
neug4c.m %This program performs the spectral reflectance estimation for a four-color printing process. %The estimation is based the n-modified spectral Neugebauer equation function R_predicted=neug4c(cmyk, neuprimary, nf) a=demi4c(cmyk); a1=a(:,1); %notice a1 is the area of paper a2=a(:,2); a3=a(:,3); a4=a(:,4); a5=a(:,5); a6=a(:,6); a7=a(:,7); a8=a(:,8); a9=a(:,9); a10=a(:,10); a11=a(:,11); a12=a(:,12); a13=a(:,13); a14=a(:,14); a15=a(:,15); a16=a(:,16); clear a; R1=neuprimary(:,1); %notice R1 is the R_paper R2=neuprimary(:,2); R3=neuprimary(:,3); R4=neuprimary(:,4); R5=neuprimary(:,5); R6=neuprimary(:,6); R7=neuprimary(:,7); R8=neuprimary(:,8); R9=neuprimary(:,9); R10=neuprimary(:,10); R11=neuprimary(:,11); R12=neuprimary(:,12); R13=neuprimary(:,13); R14=neuprimary(:,14); R15=neuprimary(:,15); R16=neuprimary(:,16); [n,m]=size(cmyk); [p,q]=size(neuprimary); tmp=zeros(p,n);
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for i=1:n tmp(:,i)=a1(i)*(R1.^(1/nf))+a2(i)*(R2.^(1/nf))+a3(i)*(R3.^(1/nf))+a4(i)*(R4.^(1/nf)); tmp(:,i)=tmp(:,i)+a5(i)*(R5.^(1/nf))+a6(i)*(R6.^(1/nf))+a7(i)*(R7.^(1/nf))+a8(i)*(R8.^(1/nf)); tmp(:,i)=tmp(:,i)+a9(i)*(R9.^(1/nf))+a10(i)*(R10.^(1/nf))+a11(i)*(R11.^(1/nf))+a12(i)*(R12.^(1/nf)); tmp(:,i)=tmp(:,i)+a13(i)*(R13.^(1/nf))+a14(i)*(R14.^(1/nf))+a15(i)*(R15.^(1/nf))+a16(i)*(R16.^(1/nf)); tmp(:,i)=tmp(:,i).^nf; end R_predicted=tmp;
demi4c.m %This function calculated the dot area coverage based on the Demichel's probability model function area=demi4c(cmyk) f1 =(1-cmyk(:,1)).*(1-cmyk(:,2)).*(1-cmyk(:,3)).*(1-cmyk(:,4)); %white f2 f3 f4 f5 f6 f7 f8
= cmyk(:,1).*(1-cmyk(:,2)).*(1-cmyk(:,3)).*(1-cmyk(:,4)); %cyan =(1-cmyk(:,1)).* cmyk(:,2).*(1-cmyk(:,3)).*(1-cmyk(:,4)); %magenta =(1-cmyk(:,1)).*(1-cmyk(:,2)).* cmyk(:,3).*(1-cmyk(:,4)); %yellow =(1-cmyk(:,1)).* cmyk(:,2).* cmyk(:,3).*(1-cmyk(:,4)); %red = cmyk(:,1).*(1-cmyk(:,2)).* cmyk(:,3).*(1-cmyk(:,4)); %green = cmyk(:,1).* cmyk(:,2).*(1-cmyk(:,3)).*(1-cmyk(:,4)); %blue = cmyk(:,1).* cmyk(:,2).* cmyk(:,3).*(1-cmyk(:,4)); %3cblack
f9 =(1-cmyk(:,1)).*(1-cmyk(:,2)).*(1-cmyk(:,3)).* cmyk(:,4); %1cblack f10 = cmyk(:,1).*(1-cmyk(:,2)).*(1-cmyk(:,3)).* cmyk(:,4); %kcyan f11 =(1-cmyk(:,1)).* cmyk(:,2).*(1-cmyk(:,3)).* cmyk(:,4); %kmagenta f12 =(1-cmyk(:,1)).*(1-cmyk(:,2)).* cmyk(:,3).* cmyk(:,4); %kyellow f13 =(1-cmyk(:,1)).* cmyk(:,2).* cmyk(:,3).* cmyk(:,4); %kred f14 = cmyk(:,1).*(1-cmyk(:,2)).* cmyk(:,3).* cmyk(:,4); %kgreen f15 = cmyk(:,1).* cmyk(:,2).*(1-cmyk(:,3)).* cmyk(:,4); %kblue f16 = cmyk(:,1).* cmyk(:,2).* cmyk(:,3).* cmyk(:,4); %4cblack area=[f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16];
theo2eff.m %The module builds an transfer function for theoretical to effective dot area %using one-dimensional look up table and cubic spline for interpolation function eff=theo2eff(cmyk, dc, c_ramp, m_ramp, y_ramp, k_ramp, R_paper, nf); efflut_c=inv_murr(c_ramp, R_paper, nf); efflut_m=inv_murr(m_ramp, R_paper, nf); efflut_y=inv_murr(y_ramp, R_paper, nf); efflut_k=inv_murr(k_ramp, R_paper, nf); [row, col]=size(cmyk); eff=zeros(row,col); eff(:,1)=interp1(dc, efflut_c, cmyk(:,1),'cubic'); eff(:,2)=interp1(dc, efflut_m, cmyk(:,2),'cubic'); eff(:,3)=interp1(dc, efflut_y, cmyk(:,3),'cubic'); eff(:,4)=interp1(dc, efflut_k, cmyk(:,4),'cubic');
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APPENDIX F : MATLAB PROGRAMS FOR THE SPECTRALBASED SIX-COLOR SEPARATION MINIMIZING METAMERISM The following diagram depicts the structure of the MLTLAB program which performs the spectral-based six-color separation to determine the ink amount corresponding to each printing primary.
Main Program (checksep.m)
Four-ink selector (selector.m)
Six color backward printing model (six_sep.m)
Nonnegative least square fitting (nnls.m)
CMYK backward printing model (cmyk_sep.m)
Objective function minimizing metamerism (sep1_obj.m)
...
YGOK backward printing model (ygok_sep.m)
Six color forward printing model (neuge6.m)
CMYK forward printing model (cmyk_sub.m)
Spectral Objective function reconstruction by minimizing the proposed printing metamerism model (sep1_obj.m) (forward2nd.m)
checksep.m %This program performs spectral-based six-color separation of the Gretag Macbeth Color Checker %minimizing the metamerism between the original and its reproduction clear all; close all; load ave_ramp.txt
%the six-color ramps for the forward modeling
c_ramp=(ave_ramp(1:13, 7:37))'; m_ramp=(ave_ramp(14:26, 7:37))'; y_ramp=(ave_ramp(27:39, 7:37))'; g_ramp=(ave_ramp(40:52, 7:37))'; o_ramp=(ave_ramp(53:65, 7:37))'; k_ramp=(ave_ramp(66:78, 7:37))'; clear ave_ramp; lambda=[400:10:700]';
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...
YGOK forward printing model (ygok_sub.m)
Spectral reconstruction by the proposed printing model (forward2nd.m)
R_paper=c_ramp(:,13); dc=[100;90;80;70;60;50;40;30;20;15;10;5;0]/100; %theoretical dot area for ramps n=2.2; %the determined n-factor is 2.2 prmy_ramp=[c_ramp m_ramp y_ramp g_ramp o_ramp k_ramp]; primary=[c_ramp(:,1) m_ramp(:,1) y_ramp(:,1) g_ramp(:,1) o_ramp(:,1) k_ramp(:,1)]; load checker.txt %spectral reflectance of the Gretag Macbeth Color Checker po=1/3.5; %po can be form 1/4 to 1/3 checktag=selector(checker, R_paper, primary, po); %the selector determines best four ink for a spectrum requiring six-color reproduction dc_est=six_sep(checktag, checker, prmy_ramp,R_paper, n); %the six-color backward printing model check_dc =dc_est(:,2:5); R_predicted=neuge6(check_dc, checktag, prmy_ramp, R_paper, n); %the proposed six-color forward printing model delta_e94=del_e94(R_predicted, checker, R_paper, [3 1]); fprintf('The average color difference between measured and predicted is \n'); fprintf( 'Mean \t\t%f\n', mean(delta_e94) ); fprintf( 'Standard Deviation \t\t%f\n', std(delta_e94) ); fprintf( 'Maximum \t\t%f\n', max(delta_e94) ); fprintf( 'Minimum \t\t%f\n\n', min(delta_e94) ); spectral_RMS = RMS( R_predicted, checker); midx=meta_idx(R_predicted, checker, R_paper, [3 2 1 2]); fprintf('The average color difference between measured and predicted under illum A %f\n', mean(midx)); fprintf( 'Mean \t\t%f\n', mean(midx) ); fprintf( 'Standard Deviation \t\t%f\n', std(midx) ); fprintf( 'Maximum \t\t%f\n', max(midx) ); fprintf( 'Minimum \t\t%f\n\n', min(midx) ); fprintf( 'The root mean square error of reflectance factor is %f\n\n', spectral_RMS ); figure [yy,xx]=hist(delta_e94); bar(xx,yy,'r') title('The histogram color difference between measured and predicted') xlabel('Delta E94') ylabel('Frequency') lab_meas=lab(checker,R_paper,[3 1]); lab_pred=lab(R_predicted,R_paper, [3 1]); primary=[c_ramp(:,1) m_ramp(:,1) y_ramp(:,1) g_ramp(:,1) o_ramp(:,1) k_ramp(:,1)]; lab_prmy=lab(primary,R_paper,[3 1]); figure; plot(lab_meas(:,2),lab_meas(:,3),'*') hold on plot(lab_pred(:,2),lab_pred(:,3),'mo') hold on plot(lab_prmy(:,2), lab_prmy(:,3),'g+') figure;
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plot(lab_meas(:,2),lab_meas(:,1),'*') hold on plot(lab_pred(:,2),lab_pred(:,1),'mo') figure; plot(lab_meas(:,3),lab_meas(:,1),'*') hold on plot(lab_pred(:,3),lab_pred(:,1),'mo') delta_e94=delta_e94(1:24); midx=midx(1:24); a=[midx delta]; [y,i]=sort(a); figure; subplot(2,2,1) plot(lambda,checker(:,1),'k', lambda,R_predicted(:,1),'k:') axis([400 700 0 1]) %xlabel('Wavelength') ylabel('Reflectance') legend('Dark Skin') subplot(2,2,2) plot(lambda,checker(:,2),'k', lambda,R_predicted(:,2),'k:') axis([400 700 0 1]) %xlabel('Wavelength') %ylabel('Reflectance') legend('Light Skin') subplot(2,2,3) plot(lambda,checker(:,14),'k', lambda,R_predicted(:,14),'k:') axis([400 700 0 1]) xlabel('Wavelength') ylabel('Reflectance') legend('Green') subplot(2,2,4) plot(lambda,checker(:,6),'k', lambda,R_predicted(:,6),'k:') axis([400 700 0 1]) xlabel('Wavelength') %ylabel('Reflectance') legend('Blue Green')
figure; subplot(2,2,1) plot(lambda,checker(:,8),'k', lambda,R_predicted(:,8),'k:') axis([400 700 0 1]) %xlabel('Wavelength') ylabel('Reflectance') legend('Purplish Blue') subplot(2,2,2) plot(lambda,checker(:,13),'k', lambda,R_predicted(:,13),'k:') axis([400 700 0 1]) %xlabel('Wavelength') %ylabel('Reflectance') legend('Blue') subplot(2,2,3) plot(lambda,checker(:,18),'k', lambda,R_predicted(:,18),'k:') axis([400 700 0 1])
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xlabel('Wavelength') ylabel('Reflectance') legend('Cyan') subplot(2,2,4) plot(lambda,checker(:,12),'k', lambda,R_predicted(:,12),'k:') axis([400 700 0 1]) xlabel('Wavelength') %ylabel('Reflectance') legend('Orange Yellow') load check_dc.txt; checktag=check_dc(:,1); check_dc=check_dc(:,2:5); predicted=neuge6(check_dc, checktag, prmy_ramp, R_paper, n); load check_6c_print.txt printed=check_6c_print; figure; subplot(2,3,1) plot(lambda, checker(:,5),'b', lambda, predicted(:,5),'m',lambda, printed(:,5),'g') ylabel('Reflectance') axis([400 700 0 1]) title('Blue flower') subplot(2,3,2) plot(lambda, checker(:,7),'b', lambda, predicted(:,7),'m',lambda, printed(:,7),'g') %ylabel('Reflectance') axis([400 700 0 1]) title('Orange') subplot(2,3,3) plot(lambda, checker(:,8),'b', lambda, predicted(:,8),'m',lambda, printed(:,8),'g') %ylabel('Reflectance') legend('Original','Predicted','Reproduction') axis([400 700 0 1]) title('Purplish blue') subplot(2,3,4) plot(lambda, checker(:,11),'b', lambda, predicted(:,11),'m',lambda, printed(:,11),'g') ylabel('Reflectance') xlabel('Wavelength') axis([400 700 0 1]) title('Yellow green') subplot(2,3,5) plot(lambda, checker(:,20),'b', lambda, predicted(:,20),'m',lambda, printed(:,20),'g') xlabel('Wavelength') axis([400 700 0 1]) title('Neutral 8') subplot(2,3,6) plot(lambda, checker(:,21),'b', lambda, predicted(:,21),'m',lambda, printed(:,21),'g') xlabel('Wavelength') axis([400 700 0 1]) title('Neutral 6.5')
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selector.m %This program performs the proper four inks for a given input spectrum requiring %six color reproduction function inktag=selector(sample, R_paper, primary, po) if nargin==3 po=1/3.5; end [n,m]=size(sample); tmp=R_paper.^po; for i=2:m tmp=[tmp R_paper.^po]; end k_prmy=tmp(:,1:6)-primary.^po; k_c=k_prmy(:,1); k_m=k_prmy(:,2); k_y=k_prmy(:,3); k_g=k_prmy(:,4); k_o=k_prmy(:,5); k_k=k_prmy(:,6); k_cmyk=[k_prmy(:,1) k_prmy(:,2) k_prmy(:,3) k_prmy(:,6)]; k_cmgk=[k_prmy(:,1) k_prmy(:,2) k_prmy(:,4) k_prmy(:,6)]; k_cmok=[k_prmy(:,1) k_prmy(:,2) k_prmy(:,5) k_prmy(:,6)]; k_cygk=[k_prmy(:,1) k_prmy(:,3) k_prmy(:,4) k_prmy(:,6)]; k_cyok=[k_prmy(:,1) k_prmy(:,3) k_prmy(:,5) k_prmy(:,6)]; k_cgok=[k_prmy(:,1) k_prmy(:,4) k_prmy(:,5) k_prmy(:,6)]; k_mygk=[k_prmy(:,2) k_prmy(:,3) k_prmy(:,4) k_prmy(:,6)]; k_myok=[k_prmy(:,2) k_prmy(:,3) k_prmy(:,5) k_prmy(:,6)]; k_mgok=[k_prmy(:,2) k_prmy(:,4) k_prmy(:,5) k_prmy(:,6)]; k_ygok=[k_prmy(:,3) k_prmy(:,4) k_prmy(:,5) k_prmy(:,6)]; k_sample=tmp-(sample.^po); %tranform ink samples to psi space tmp=tmp(:,1:10); tmp_tag=zeros(m,1); for i=1:m R_spectra=sample(:,i); for j=1:9 R_spectra=[R_spectra sample(:,i)]; end k_spectra=zeros(31,10); conc1=nnls(k_cmyk, k_sample(:,i)); %nonnegative least square fittting by CMYK inks conc2=nnls(k_cmgk, k_sample(:,i)); conc3=nnls(k_cmok, k_sample(:,i)); conc4=nnls(k_cygk, k_sample(:,i)); conc5=nnls(k_cyok, k_sample(:,i)); conc6=nnls(k_cgok, k_sample(:,i)); conc7=nnls(k_mygk, k_sample(:,i)); conc8=nnls(k_myok, k_sample(:,i)); conc9=nnls(k_mgok, k_sample(:,i)); conc10=nnls(k_ygok, k_sample(:,i));
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k_spectra(:,1)=k_cmyk*conc1; k_spectra(:,2)=k_cmgk*conc2; k_spectra(:,3)=k_cmok*conc3; k_spectra(:,4)=k_cygk*conc4; k_spectra(:,5)=k_cyok*conc5; k_spectra(:,6)=k_cgok*conc6; k_spectra(:,7)=k_mygk*conc7; k_spectra(:,8)=k_myok*conc8; k_spectra(:,9)=k_mgok*conc9; k_spectra(:,10)=k_ygok*conc10; R_pred10=(tmp-k_spectra).^(1/po); criteria = sqrt(mean((R_pred10-R_spectra).^2)); %the root-mean-squrae error %xyz_sample=[xyz(R_spectra, [1 1]) xyz(R_spectra, [2 1]) xyz(R_spectra, [3 1])]'; %xyz_pred=[xyz(R_pred10, [1 1]) xyz(R_pred10, [2 1]) xyz(R_pred10, [3 1])]'; %criteria=sqrt(mean((xyz_pred-xyz_sample).^2)); [y,ink_idx]=sort(criteria); tmp_tag(i)=ink_idx(1); end inktag=tmp_tag;
six_sep.m %This program performs the proposed spectral-based six-color separation algorithm where %the parameter "inktag" is the vector of enumerated ink combination formed by CMYKGO function dc_est=six_sep(inktag, sample, prmy_ramp,R_paper, n, dc) [p,q]=size(sample); prmy_cmyk=[prmy_ramp(:,1:39) prmy_ramp(:,66:78)]; prmy_cmgk=[prmy_ramp(:,1:26) prmy_ramp(:,40:52) prmy_ramp(:,66:78)]; prmy_cmok=[prmy_ramp(:,1:26) prmy_ramp(:,53:65) prmy_ramp(:,66:78)]; prmy_cygk=[prmy_ramp(:,1:13) prmy_ramp(:,27:39) prmy_ramp(:,40:52) prmy_ramp(:,66:78)]; prmy_cyok=[prmy_ramp(:,1:13) prmy_ramp(:,27:39) prmy_ramp(:,53:65) prmy_ramp(:,66:78)]; prmy_cgok=[prmy_ramp(:,1:13) prmy_ramp(:,40:78)]; prmy_mygk=[prmy_ramp(:,14:52) prmy_ramp(:,66:78)]; prmy_myok=[prmy_ramp(:,14:39) prmy_ramp(:,53:65) prmy_ramp(:,66:78)]; prmy_mgok=[prmy_ramp(:,14:26) prmy_ramp(:,40:78)]; prmy_ygok=[prmy_ramp(:,27:78)]; dc_tmp=zeros(q,5); for i=1:q i switch inktag(i) case 1, dc_tmp(i,:)= cmyk_sep(sample(:,i), prmy_cmyk, R_paper, n); case 2, dc_tmp(i,:)= cmgk_sep(sample(:,i), prmy_cmgk, R_paper, n); case 3, dc_tmp(i,:)= cmok_sep(sample(:,i), prmy_cmok, R_paper, n); case 4, dc_tmp(i,:)= cygk_sep(sample(:,i), prmy_cygk, R_paper, n); case 5, dc_tmp(i,:)= cyok_sep(sample(:,i), prmy_cyok, R_paper, n); case 6, dc_tmp(i,:)= cgok_sep(sample(:,i), prmy_cgok, R_paper, n); case 7, dc_tmp(i,:)= mygk_sep(sample(:,i), prmy_mygk, R_paper, n); case 8, dc_tmp(i,:)= myok_sep(sample(:,i), prmy_myok, R_paper, n); case 9, dc_tmp(i,:)= mgok_sep(sample(:,i), prmy_mgok, R_paper, n); case 10, dc_tmp(i,:)= ygok_sep(sample(:,i), prmy_ygok, R_paper, n); otherwise fprintf('error'); end
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end dc_est=dc_tmp;
cmyk_sep.m %This program is the backward CMYK printing model function dot_est=cmyk_sep(sample, ramp, R_paper, n); load fij_1.txt; fij_1=[fij_1; ones(1,12)]; load neprmy1.txt
cmyk_est=ones(1,4)*0.5; vlb=ones(1,4)*0.0001; vub=ones(1,4)*0.99; options(1)=1; options(2)=1e-3; options(3)=1e-3; options(4)=1e-3; options(14)=10000;
cmyk_est=constr('sep1_obj',cmyk_est,options,vlb,vub,'',sample,ramp,fij_1,neprmy1, R_paper, n); dot_est=[1 cmyk_est];
sep1_obj.m %This is the objective function for a four-color forward printing model function [f,g]=sep1_obj(dot_est, sample, ramp, fij, neprmy, R_paper, n); sc=0.62; dac=[0 25 50 70 90 100]/100; dac=dac'; dc=[100;90;80;70;60;50;40;30;20;15;10;5;0]/100; qscalar=iino_q(fij, dot_est, dac); eff=theo2eff(dot_est, dc, ramp(:,1:13), ramp(:,14:26), ramp(:,27:39), ramp(:,40:52), R_paper, n); eff1=(dot_est+qscalar.*(eff-dot_est))*sc + eff*(1-sc); pred=neug4c(eff1, neprmy, n); %xyz_sample=[xyz(sample, [3 1]) xyz(sample, [2 1])]'; %xyz_pred=[xyz(pred, [3 1]) xyz(pred, [2 1])]'; %f=rms(xyz_sample(1:4), xyz_pred(1:4)); f=del_e94(sample, pred, R_paper, [3 1]); tmpcmyk=sum(dot_est); g(1)=-tmpcmyk; g(2)=tmpcmyk-3; g(3)=dot_est(4)-0.001; g(4)=-dot_est(4);
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neuge6.m %This program performs the proposed spectral-based six-color synthesis based on the union %of 10 four-color forward printing models where the parameter "inktag" is the vector %of enumerated ink combination formed by CMYKGO function R_predicted=neuge6(check_dc, inktag,prmy_ramp,R_paper, n); [q,p]=size(check_dc); prmy_cmyk=[prmy_ramp(:,1:39) prmy_ramp(:,66:78)]; prmy_cmgk=[prmy_ramp(:,1:26) prmy_ramp(:,40:52) prmy_ramp(:,66:78)]; prmy_cmok=[prmy_ramp(:,1:26) prmy_ramp(:,53:65) prmy_ramp(:,66:78)]; prmy_cygk=[prmy_ramp(:,1:13) prmy_ramp(:,27:39) prmy_ramp(:,40:52) prmy_ramp(:,66:78)]; prmy_cyok=[prmy_ramp(:,1:13) prmy_ramp(:,27:39) prmy_ramp(:,53:65) prmy_ramp(:,66:78)]; prmy_cgok=[prmy_ramp(:,1:13) prmy_ramp(:,40:78)]; prmy_mygk=[prmy_ramp(:,14:52) prmy_ramp(:,66:78)]; prmy_myok=[prmy_ramp(:,14:39) prmy_ramp(:,53:65) prmy_ramp(:,66:78)]; prmy_mgok=[prmy_ramp(:,14:26) prmy_ramp(:,40:78)]; prmy_ygok=[prmy_ramp(:,27:78)]; tmp=zeros(31,q); for i=1:q switch inktag(i) case 1, tmp(:,i)= cmyk_sub(check_dc(i,:), prmy_cmyk, R_paper, n); case 2, tmp(:,i)= cmgk_sub(check_dc(i,:), prmy_cmgk, R_paper, n); case 3, tmp(:,i)= cmok_sub(check_dc(i,:), prmy_cmok, R_paper, n); case 4, tmp(:,i)= cygk_sub(check_dc(i,:), prmy_cygk, R_paper, n); case 5, tmp(:,i)= cyok_sub(check_dc(i,:), prmy_cyok, R_paper, n); case 6, tmp(:,i)= cgok_sub(check_dc(i,:), prmy_cgok, R_paper, n); case 7, tmp(:,i)= mygk_sub(check_dc(i,:), prmy_mygk, R_paper, n); case 8, tmp(:,i)= myok_sub(check_dc(i,:), prmy_myok, R_paper, n); case 9, tmp(:,i)= mgok_sub(check_dc(i,:), prmy_mgok, R_paper, n); case 10, tmp(:,i)= ygok_sub(check_dc(i,:), prmy_ygok, R_paper, n); otherwise fprintf('error'); end end R_predicted=tmp;
cmyk_sub.m %This program performs the spectral reconstruction using CMYK inks function pred= cmyk_sub(dot_est, ramp, R_paper, n); load fij_1.txt; fij_1=[fij_1; ones(1,12)]; load neprmy1.txt pred=forward2nd(dot_est, ramp, fij_1, neprmy1, R_paper, n);
forward2nd.m %This program is the proposed four-color forward printing model
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function pred=forward2nd(dottiest, ramp, fij, neprmy, R_paper, n); sc=0.62; dac=[0 25 50 70 90 100]/100; dac=dac'; dc=[100;90;80;70;60;50;40;30;20;15;10;5;0]/100; qscalar=iino_q(fij, dot_est, dac); eff=theo2eff(dot_est, dc, ramp(:,1:13), ramp(:,14:26), ramp(:,27:39), ramp(:,40:52), R_paper, n); eff1=(dot_est+qscalar.*(eff-dot_est))*sc + eff*(1-sc); pred=neug4c(eff1, neprmy, n);
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APPENDIX G : THE SIX ESTIMATED COLORANTS FOR THE 105 MIXTURES OF POSTER COLORS
Wavelength Primary 1 Primary 2 Primary 3 Primary 4 Primary 5 Primary 6 400 0.278 0.457 0.004 0.591 0.556 0.003 410 0.319 0.480 0.001 0.559 0.651 0.003 420 0.349 0.500 0.000 0.537 0.728 0.002 430 0.387 0.513 0.000 0.487 0.789 0.001 440 0.454 0.496 0.000 0.359 0.791 0.001 450 0.538 0.443 0.002 0.204 0.690 0.001 460 0.631 0.363 0.010 0.075 0.522 0.001 470 0.708 0.281 0.027 0.012 0.355 0.001 480 0.738 0.208 0.052 0.000 0.223 0.001 490 0.720 0.149 0.104 0.000 0.125 0.003 500 0.704 0.090 0.260 0.020 0.039 0.007 510 0.641 0.033 0.542 0.224 0.004 0.008 520 0.512 0.004 0.757 0.619 0.002 0.007 530 0.373 0.000 0.835 0.887 0.010 0.006 540 0.262 0.000 0.841 0.881 0.034 0.005 550 0.173 0.005 0.832 0.747 0.080 0.005 560 0.107 0.025 0.830 0.552 0.143 0.006 570 0.062 0.072 0.845 0.371 0.211 0.008 580 0.036 0.157 0.875 0.214 0.251 0.010 590 0.027 0.270 0.892 0.108 0.232 0.010 600 0.027 0.406 0.891 0.041 0.160 0.009 610 0.032 0.531 0.883 0.010 0.094 0.008 620 0.037 0.610 0.872 0.002 0.060 0.008 630 0.042 0.656 0.865 0.000 0.047 0.009 640 0.045 0.685 0.867 0.000 0.043 0.010 650 0.049 0.708 0.871 0.001 0.038 0.010 660 0.053 0.738 0.877 0.005 0.028 0.009 670 0.061 0.776 0.877 0.013 0.013 0.007 680 0.075 0.823 0.866 0.034 0.003 0.005 690 0.096 0.876 0.849 0.074 0.000 0.004 700 0.123 0.901 0.836 0.143 0.000 0.006
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APPENDIX H : THE ESTIMATED SPECTRAL ABSORPTION AND SCATTERING COEFFICIENTS FOR THE WATERPROOF® CMYRGB PRIMARIES
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
Kcyan 0.282 0.158 0.118 0.087 0.055 0.034 0.030 0.031 0.036 0.046 0.062 0.086 0.125 0.181 0.258 0.369 0.540 0.755 0.956 1.100 1.240 1.360 1.380 1.350 1.280 1.160 1.020 0.963 0.991 1.060 1.180
Scyan 0.027 0.021 0.015 0.012 0.009 0.007 0.006 0.005 0.004 0.003 0.002 0.002 0.002 0.003 0.004 0.007 0.011 0.015 0.019 0.022 0.025 0.028 0.029 0.028 0.026 0.023 0.020 0.018 0.018 0.020 0.023
Kmagenta 0.514 0.460 0.455 0.449 0.452 0.485 0.544 0.621 0.734 0.872 0.990 1.160 1.450 1.650 1.530 1.530 1.790 2.100 1.150 0.417 0.169 0.086 0.057 0.048 0.044 0.042 0.042 0.041 0.040 0.039 0.038
295
Smagenta 0.022 0.016 0.016 0.015 0.015 0.017 0.019 0.021 0.023 0.025 0.028 0.034 0.044 0.048 0.041 0.041 0.054 0.087 0.088 0.066 0.046 0.033 0.025 0.020 0.016 0.013 0.012 0.010 0.009 0.009 0.007
Kyellow 0.959 1.140 1.310 1.370 1.350 1.200 1.000 0.877 0.749 0.516 0.279 0.121 0.044 0.015 0.006 0.003 0.002 0.002 0.002 0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Syellow 0.036 0.043 0.050 0.051 0.054 0.051 0.044 0.038 0.041 0.041 0.035 0.026 0.020 0.015 0.012 0.010 0.008 0.008 0.006 0.006 0.005 0.005 0.005 0.004 0.003 0.003 0.003 0.003 0.003 0.002 0.002
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
Kred
Sred
Kgreen
Sgreen
Kblue
Sblue
1.200 1.100 1.140 1.130 1.150 1.220 1.380 1.740 2.020 2.170 2.250 2.290 2.310 2.320 2.320 2.300 2.250 2.210 1.910 0.690 0.240 0.084 0.032 0.014 0.007 0.004 0.003 0.002 0.001 0.001 0.001
0.100 0.065 0.075 0.069 0.070 0.075 0.086 0.113 0.121 0.114 0.111 0.111 0.109 0.111 0.127 0.160 0.204 0.232 0.247 0.132 0.101 0.080 0.066 0.056 0.049 0.044 0.039 0.036 0.033 0.031 0.029
3.130 2.270 1.960 1.670 1.460 1.220 0.961 0.763 0.604 0.416 0.245 0.165 0.159 0.191 0.252 0.346 0.483 0.671 0.941 1.350 1.940 2.460 2.560 2.580 2.600 2.600 2.560 2.410 2.080 1.740 1.620
0.219 0.156 0.136 0.113 0.103 0.093 0.081 0.074 0.070 0.067 0.062 0.054 0.045 0.039 0.036 0.035 0.036 0.040 0.045 0.058 0.076 0.094 0.095 0.100 0.103 0.107 0.116 0.119 0.111 0.087 0.087
0.800 0.614 0.489 0.393 0.297 0.221 0.200 0.212 0.256 0.342 0.464 0.629 0.877 1.220 2.240 2.560 2.650 2.660 2.660 2.650 2.640 2.630 2.610 2.590 2.570 2.560 2.540 2.540 2.550 2.560 2.550
0.100 0.050 0.041 0.035 0.029 0.025 0.022 0.020 0.019 0.019 0.021 0.022 0.027 0.029 0.102 0.060 0.041 0.042 0.043 0.046 0.047 0.049 0.051 0.052 0.054 0.054 0.053 0.051 0.047 0.049 0.052
296
APPENDIX I : THE REFLECTANCE SPECTRA OF THE ORIGINAL GRETAG MACBETH COLOR CHECKER
Wavelength Dark skin 400 0.059 410 0.059 420 0.059 430 0.059 440 0.059 450 0.059 460 0.059 470 0.059 480 0.059 490 0.060 500 0.062 510 0.067 520 0.073 530 0.075 540 0.077 550 0.080 560 0.087 570 0.100 580 0.118 590 0.134 600 0.144 610 0.148 620 0.152 630 0.158 640 0.166 650 0.174 660 0.179 670 0.179 680 0.175 690 0.171 700 0.171
Light skin
0.195 0.214 0.216 0.216 0.217 0.220 0.225 0.233 0.243 0.250 0.256 0.267 0.282 0.288 0.293 0.304 0.311 0.321 0.353 0.396 0.438 0.478 0.515 0.540 0.555 0.563 0.568 0.571 0.574 0.577 0.579
Blue sky
Foliage
0.249 0.294 0.305 0.311 0.321 0.326 0.323 0.313 0.302 0.290 0.273 0.250 0.227 0.211 0.196 0.175 0.154 0.143 0.151 0.158 0.152 0.145 0.141 0.140 0.141 0.144 0.149 0.151 0.147 0.140 0.133
0.057 0.058 0.060 0.062 0.065 0.067 0.068 0.068 0.070 0.073 0.083 0.110 0.155 0.187 0.188 0.167 0.142 0.123 0.114 0.109 0.104 0.100 0.100 0.103 0.108 0.112 0.115 0.113 0.108 0.104 0.104
297
Blue flower Blue green
0.301 0.377 0.405 0.417 0.423 0.425 0.417 0.401 0.376 0.341 0.304 0.275 0.249 0.225 0.211 0.206 0.200 0.194 0.191 0.198 0.214 0.229 0.239 0.251 0.276 0.308 0.338 0.353 0.350 0.343 0.343
0.262 0.311 0.329 0.343 0.359 0.383 0.421 0.467 0.516 0.563 0.592 0.598 0.584 0.557 0.522 0.479 0.433 0.388 0.344 0.300 0.259 0.232 0.219 0.211 0.206 0.202 0.204 0.210 0.220 0.230 0.238
Wavelength Orange 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
0.055 0.055 0.055 0.055 0.055 0.056 0.056 0.057 0.058 0.061 0.069 0.094 0.134 0.165 0.182 0.204 0.250 0.328 0.423 0.502 0.543 0.556 0.563 0.572 0.582 0.589 0.592 0.588 0.585 0.587 0.595
Purplish Moderate blue red 0.232 0.134 0.283 0.139 0.312 0.138 0.338 0.137 0.365 0.136 0.377 0.135 0.364 0.132 0.332 0.127 0.286 0.121 0.235 0.116 0.190 0.111 0.157 0.105 0.131 0.100 0.114 0.097 0.103 0.097 0.096 0.097 0.091 0.099 0.087 0.109 0.086 0.159 0.085 0.271 0.085 0.410 0.084 0.503 0.084 0.550 0.084 0.568 0.087 0.572 0.091 0.573 0.098 0.580 0.107 0.596 0.120 0.617 0.139 0.636 0.166 0.649
298
Purple 0.173 0.206 0.214 0.197 0.168 0.139 0.114 0.094 0.081 0.072 0.064 0.060 0.058 0.055 0.054 0.054 0.055 0.054 0.053 0.054 0.059 0.071 0.092 0.119 0.145 0.170 0.198 0.232 0.275 0.332 0.399
Yellow green 0.060 0.061 0.062 0.064 0.067 0.073 0.083 0.100 0.128 0.174 0.249 0.356 0.457 0.514 0.530 0.521 0.500 0.476 0.450 0.411 0.361 0.324 0.302 0.289 0.280 0.272 0.273 0.282 0.296 0.313 0.327
Orange yellow 0.066 0.067 0.066 0.066 0.066 0.067 0.069 0.073 0.080 0.090 0.103 0.126 0.186 0.289 0.379 0.432 0.479 0.530 0.573 0.602 0.614 0.615 0.620 0.638 0.667 0.695 0.715 0.713 0.695 0.691 0.703
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
Blue 0.114 0.151 0.185 0.223 0.274 0.325 0.332 0.294 0.225 0.155 0.104 0.074 0.059 0.051 0.047 0.045 0.044 0.043 0.043 0.043 0.042 0.043 0.043 0.043 0.043 0.044 0.044 0.044 0.044 0.045 0.046
Green 0.053 0.054 0.055 0.056 0.059 0.064 0.074 0.091 0.118 0.162 0.226 0.299 0.333 0.325 0.307 0.284 0.252 0.221 0.195 0.167 0.133 0.108 0.093 0.084 0.078 0.074 0.071 0.071 0.072 0.075 0.078
Red 0.059 0.058 0.057 0.057 0.057 0.057 0.056 0.055 0.053 0.052 0.051 0.050 0.049 0.049 0.049 0.050 0.052 0.057 0.072 0.116 0.230 0.374 0.487 0.551 0.583 0.600 0.618 0.636 0.657 0.678 0.695
Yellow 0.057 0.058 0.058 0.060 0.063 0.068 0.078 0.093 0.118 0.159 0.225 0.328 0.456 0.556 0.611 0.641 0.662 0.681 0.696 0.708 0.717 0.724 0.731 0.737 0.743 0.749 0.753 0.755 0.760 0.767 0.771
299
Magenta 0.280 0.339 0.351 0.341 0.320 0.293 0.263 0.236 0.207 0.180 0.160 0.143 0.123 0.105 0.100 0.104 0.104 0.109 0.138 0.200 0.291 0.391 0.491 0.575 0.647 0.705 0.748 0.778 0.799 0.814 0.826
Cyan 0.185 0.224 0.245 0.269 0.302 0.338 0.378 0.415 0.432 0.428 0.403 0.364 0.314 0.262 0.214 0.171 0.135 0.111 0.097 0.088 0.080 0.076 0.074 0.073 0.073 0.074 0.076 0.077 0.076 0.075 0.073
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
White 0.437 0.684 0.838 0.883 0.891 0.894 0.895 0.895 0.895 0.894 0.892 0.891 0.888 0.885 0.885 0.885 0.883 0.883 0.885 0.888 0.890 0.889 0.888 0.888 0.890 0.892 0.894 0.894 0.895 0.896 0.896
Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5 0.372 0.283 0.178 0.087 0.509 0.340 0.196 0.089 0.557 0.353 0.199 0.090 0.566 0.356 0.200 0.091 0.569 0.357 0.201 0.092 0.570 0.358 0.201 0.092 0.569 0.357 0.202 0.091 0.567 0.356 0.202 0.090 0.565 0.354 0.202 0.090 0.564 0.353 0.201 0.089 0.564 0.352 0.200 0.089 0.564 0.351 0.200 0.089 0.564 0.350 0.198 0.089 0.564 0.349 0.197 0.089 0.564 0.349 0.197 0.089 0.564 0.349 0.197 0.089 0.564 0.352 0.197 0.089 0.565 0.356 0.197 0.089 0.567 0.360 0.199 0.089 0.567 0.360 0.199 0.089 0.566 0.355 0.198 0.089 0.564 0.349 0.196 0.088 0.563 0.344 0.194 0.088 0.561 0.340 0.192 0.087 0.559 0.337 0.190 0.086 0.557 0.335 0.188 0.086 0.555 0.334 0.187 0.085 0.553 0.335 0.186 0.085 0.551 0.337 0.186 0.084 0.549 0.338 0.185 0.084 0.547 0.338 0.185 0.083
300
Black 0.033 0.034 0.034 0.034 0.034 0.034 0.034 0.034 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032
APPENDIX J : THE REFLECTANCE SPECTRA OF THE PREDICTED GRETAG MACBETH COLOR CHECKER BY THE PROPOSED SIX-COLOR SEPARATION ALGORITHM
Wavelength Dark skin 400 0.028 410 0.041 420 0.055 430 0.058 440 0.059 450 0.060 460 0.059 470 0.059 480 0.060 490 0.062 500 0.065 510 0.068 520 0.070 530 0.073 540 0.078 550 0.085 560 0.089 570 0.096 580 0.111 590 0.132 600 0.146 610 0.154 620 0.158 630 0.160 640 0.163 650 0.166 660 0.169 670 0.172 680 0.173 690 0.174 700 0.175
Light skin
0.079 0.138 0.200 0.221 0.226 0.228 0.228 0.228 0.229 0.240 0.262 0.283 0.292 0.295 0.302 0.303 0.295 0.293 0.329 0.409 0.472 0.506 0.522 0.528 0.532 0.536 0.540 0.543 0.546 0.547 0.548
Blue sky
Foliage
0.086 0.171 0.263 0.305 0.327 0.333 0.326 0.319 0.308 0.294 0.280 0.262 0.239 0.217 0.200 0.178 0.151 0.132 0.129 0.141 0.147 0.148 0.150 0.152 0.156 0.165 0.176 0.181 0.179 0.174 0.166
0.027 0.039 0.051 0.055 0.058 0.061 0.066 0.073 0.080 0.092 0.112 0.138 0.159 0.167 0.163 0.155 0.144 0.134 0.123 0.113 0.106 0.102 0.101 0.101 0.102 0.103 0.105 0.107 0.110 0.112 0.113
301
Blue flower Blue green
0.113 0.230 0.354 0.408 0.434 0.436 0.419 0.401 0.376 0.350 0.329 0.302 0.267 0.239 0.225 0.202 0.169 0.148 0.163 0.210 0.241 0.253 0.260 0.265 0.272 0.286 0.302 0.310 0.307 0.298 0.285
0.101 0.199 0.311 0.360 0.386 0.406 0.426 0.463 0.492 0.506 0.527 0.564 0.590 0.579 0.539 0.484 0.421 0.366 0.319 0.281 0.252 0.237 0.233 0.233 0.234 0.239 0.246 0.252 0.256 0.259 0.259
Wavelength Orange 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
0.028 0.038 0.046 0.048 0.050 0.052 0.056 0.059 0.065 0.079 0.101 0.124 0.142 0.159 0.189 0.227 0.252 0.295 0.381 0.477 0.540 0.570 0.586 0.593 0.599 0.608 0.618 0.623 0.623 0.620 0.615
Purplish Moderate blue red 0.089 0.055 0.191 0.092 0.291 0.125 0.342 0.136 0.371 0.139 0.374 0.137 0.352 0.133 0.328 0.127 0.294 0.119 0.259 0.116 0.230 0.120 0.197 0.118 0.157 0.105 0.129 0.098 0.115 0.104 0.094 0.103 0.068 0.090 0.053 0.084 0.060 0.138 0.082 0.288 0.091 0.431 0.091 0.513 0.093 0.548 0.095 0.562 0.100 0.568 0.113 0.573 0.130 0.577 0.137 0.582 0.133 0.586 0.123 0.590 0.110 0.593
302
Purple 0.047 0.087 0.122 0.138 0.147 0.146 0.138 0.126 0.111 0.100 0.093 0.081 0.062 0.051 0.049 0.043 0.032 0.027 0.041 0.076 0.099 0.107 0.111 0.114 0.118 0.129 0.143 0.150 0.146 0.139 0.128
Yellow green 0.032 0.044 0.056 0.060 0.063 0.071 0.085 0.103 0.127 0.174 0.260 0.381 0.490 0.541 0.542 0.520 0.484 0.450 0.414 0.376 0.344 0.328 0.323 0.321 0.320 0.320 0.323 0.328 0.337 0.346 0.353
Orange yellow 0.031 0.041 0.049 0.051 0.054 0.059 0.067 0.075 0.088 0.117 0.168 0.230 0.284 0.320 0.355 0.394 0.418 0.457 0.528 0.599 0.641 0.661 0.671 0.675 0.676 0.678 0.680 0.684 0.688 0.693 0.696
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
Blue 0.072 0.156 0.235 0.277 0.302 0.302 0.280 0.254 0.220 0.185 0.159 0.129 0.095 0.074 0.066 0.052 0.034 0.026 0.030 0.042 0.043 0.039 0.039 0.040 0.043 0.053 0.066 0.072 0.068 0.060 0.049
Green 0.025 0.037 0.050 0.056 0.060 0.067 0.080 0.099 0.121 0.146 0.190 0.258 0.321 0.345 0.332 0.301 0.261 0.222 0.182 0.144 0.115 0.101 0.096 0.095 0.094 0.094 0.096 0.100 0.106 0.113 0.118
Red 0.029 0.042 0.051 0.054 0.055 0.057 0.058 0.056 0.053 0.054 0.058 0.055 0.043 0.036 0.040 0.039 0.030 0.027 0.064 0.196 0.341 0.427 0.465 0.481 0.490 0.499 0.508 0.514 0.516 0.515 0.512
Yellow 0.034 0.044 0.052 0.053 0.056 0.065 0.080 0.093 0.114 0.167 0.266 0.395 0.504 0.564 0.596 0.619 0.629 0.648 0.681 0.713 0.731 0.741 0.746 0.748 0.749 0.751 0.753 0.756 0.760 0.764 0.766
303
Magenta 0.099 0.191 0.280 0.314 0.321 0.306 0.278 0.250 0.216 0.184 0.162 0.138 0.109 0.095 0.099 0.095 0.080 0.073 0.125 0.268 0.402 0.477 0.510 0.525 0.533 0.545 0.556 0.563 0.563 0.559 0.553
Cyan 0.082 0.170 0.264 0.313 0.347 0.367 0.371 0.378 0.376 0.366 0.355 0.340 0.314 0.277 0.238 0.192 0.144 0.108 0.089 0.081 0.074 0.069 0.069 0.070 0.073 0.081 0.091 0.095 0.093 0.088 0.081
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
White 0.235 0.471 0.756 0.858 0.879 0.873 0.860 0.858 0.855 0.853 0.851 0.850 0.850 0.849 0.848 0.850 0.848 0.853 0.854 0.856 0.857 0.859 0.862 0.863 0.863 0.866 0.868 0.869 0.871 0.873 0.873
Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5 0.162 0.108 0.066 0.035 0.316 0.203 0.118 0.057 0.501 0.315 0.178 0.080 0.567 0.355 0.199 0.089 0.582 0.365 0.205 0.092 0.580 0.364 0.206 0.093 0.572 0.360 0.204 0.092 0.572 0.360 0.204 0.093 0.571 0.360 0.204 0.093 0.570 0.359 0.204 0.093 0.569 0.359 0.204 0.093 0.568 0.358 0.203 0.092 0.568 0.357 0.202 0.092 0.566 0.355 0.201 0.091 0.565 0.353 0.199 0.090 0.564 0.352 0.198 0.089 0.561 0.349 0.195 0.087 0.562 0.348 0.194 0.086 0.560 0.347 0.193 0.086 0.561 0.347 0.193 0.087 0.561 0.347 0.193 0.087 0.562 0.348 0.194 0.087 0.564 0.349 0.194 0.088 0.565 0.350 0.196 0.089 0.567 0.353 0.198 0.091 0.570 0.357 0.202 0.093 0.574 0.360 0.205 0.096 0.576 0.363 0.208 0.098 0.578 0.365 0.209 0.099 0.579 0.365 0.209 0.099 0.579 0.365 0.209 0.099
304
Black 0.017 0.024 0.030 0.032 0.033 0.034 0.034 0.034 0.035 0.035 0.036 0.037 0.037 0.035 0.034 0.032 0.030 0.028 0.028 0.030 0.032 0.032 0.033 0.033 0.035 0.036 0.039 0.040 0.040 0.040 0.039
APPENDIX K : THE REFLECTANCE SPECTRA OF THE REPRODUCED GRETAG MACBETH COLOR CHECKER USING DUPONT WATERPROOF® SYSTEM
Wavelength Dark skin 400 0.029 410 0.040 420 0.050 430 0.052 440 0.053 450 0.053 460 0.053 470 0.053 480 0.053 490 0.055 500 0.058 510 0.060 520 0.062 530 0.065 540 0.071 550 0.078 560 0.083 570 0.090 580 0.105 590 0.123 600 0.135 610 0.141 620 0.144 630 0.146 640 0.148 650 0.151 660 0.155 670 0.157 680 0.159 690 0.159 700 0.160
Light skin
0.084 0.136 0.186 0.200 0.204 0.206 0.208 0.209 0.210 0.224 0.250 0.273 0.279 0.281 0.287 0.288 0.279 0.275 0.313 0.397 0.460 0.490 0.503 0.509 0.512 0.516 0.519 0.522 0.525 0.526 0.527
Blue sky
Foliage
0.094 0.177 0.261 0.297 0.316 0.321 0.315 0.308 0.298 0.285 0.273 0.257 0.236 0.216 0.199 0.176 0.147 0.126 0.122 0.131 0.135 0.134 0.136 0.138 0.142 0.151 0.161 0.166 0.164 0.159 0.151
0.028 0.038 0.047 0.049 0.051 0.055 0.059 0.066 0.073 0.085 0.105 0.128 0.146 0.152 0.149 0.143 0.133 0.124 0.114 0.105 0.098 0.094 0.092 0.092 0.093 0.094 0.096 0.099 0.102 0.104 0.106
305
Blue flower Blue green
0.121 0.234 0.347 0.396 0.418 0.418 0.401 0.382 0.358 0.333 0.313 0.288 0.254 0.229 0.215 0.192 0.159 0.136 0.150 0.194 0.219 0.227 0.232 0.237 0.243 0.256 0.272 0.280 0.276 0.267 0.255
0.114 0.210 0.311 0.354 0.379 0.398 0.420 0.458 0.488 0.503 0.527 0.566 0.592 0.578 0.532 0.473 0.407 0.352 0.306 0.269 0.240 0.224 0.219 0.217 0.218 0.223 0.230 0.238 0.243 0.246 0.246
Wavelength Orange 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
0.029 0.036 0.041 0.041 0.041 0.043 0.046 0.049 0.054 0.066 0.086 0.105 0.119 0.135 0.167 0.207 0.235 0.281 0.371 0.465 0.521 0.546 0.559 0.565 0.571 0.580 0.589 0.595 0.594 0.590 0.584
Purplish Moderate blue red 0.095 0.057 0.191 0.090 0.284 0.117 0.329 0.125 0.355 0.126 0.355 0.125 0.333 0.121 0.309 0.116 0.276 0.107 0.243 0.104 0.217 0.108 0.186 0.105 0.150 0.091 0.125 0.084 0.113 0.090 0.093 0.089 0.065 0.077 0.049 0.071 0.055 0.125 0.075 0.281 0.082 0.427 0.080 0.506 0.081 0.539 0.083 0.551 0.088 0.557 0.101 0.561 0.116 0.565 0.124 0.570 0.119 0.575 0.109 0.579 0.097 0.582
306
Purple 0.051 0.091 0.123 0.137 0.145 0.143 0.136 0.124 0.110 0.099 0.093 0.081 0.063 0.052 0.051 0.044 0.032 0.026 0.040 0.072 0.091 0.096 0.098 0.101 0.106 0.116 0.130 0.136 0.133 0.125 0.114
Yellow green 0.032 0.041 0.048 0.050 0.053 0.060 0.074 0.090 0.112 0.160 0.251 0.378 0.490 0.541 0.541 0.517 0.479 0.442 0.404 0.363 0.327 0.306 0.298 0.294 0.292 0.292 0.296 0.304 0.315 0.327 0.335
Orange yellow 0.030 0.036 0.040 0.041 0.042 0.046 0.053 0.060 0.070 0.096 0.141 0.193 0.234 0.265 0.303 0.347 0.376 0.423 0.508 0.586 0.626 0.641 0.648 0.650 0.650 0.652 0.654 0.659 0.666 0.672 0.676
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
Blue 0.076 0.156 0.230 0.268 0.291 0.288 0.266 0.240 0.207 0.175 0.151 0.123 0.092 0.073 0.066 0.052 0.034 0.024 0.029 0.041 0.041 0.037 0.036 0.037 0.040 0.050 0.063 0.068 0.065 0.056 0.046
Green 0.026 0.036 0.045 0.049 0.053 0.060 0.072 0.091 0.112 0.138 0.185 0.254 0.317 0.340 0.327 0.296 0.256 0.216 0.176 0.137 0.107 0.091 0.085 0.083 0.082 0.082 0.084 0.089 0.096 0.104 0.110
Red 0.030 0.041 0.048 0.049 0.050 0.051 0.053 0.051 0.048 0.049 0.054 0.051 0.039 0.032 0.036 0.035 0.026 0.023 0.062 0.199 0.341 0.421 0.455 0.469 0.477 0.486 0.495 0.500 0.501 0.500 0.497
Yellow 0.035 0.041 0.045 0.044 0.046 0.053 0.067 0.080 0.098 0.150 0.249 0.372 0.471 0.526 0.560 0.587 0.601 0.626 0.666 0.701 0.717 0.724 0.727 0.728 0.728 0.730 0.732 0.736 0.740 0.744 0.747
307
Magenta 0.105 0.192 0.273 0.303 0.308 0.293 0.266 0.238 0.205 0.174 0.154 0.130 0.103 0.091 0.095 0.092 0.076 0.069 0.120 0.267 0.398 0.465 0.494 0.507 0.515 0.526 0.537 0.544 0.543 0.539 0.533
Cyan 0.088 0.174 0.260 0.303 0.334 0.352 0.356 0.363 0.362 0.354 0.345 0.332 0.307 0.271 0.231 0.184 0.133 0.097 0.078 0.070 0.062 0.057 0.056 0.058 0.061 0.068 0.078 0.082 0.080 0.075 0.068
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
White 0.247 0.481 0.759 0.857 0.879 0.874 0.862 0.860 0.858 0.856 0.854 0.853 0.854 0.853 0.852 0.854 0.852 0.858 0.858 0.860 0.861 0.863 0.866 0.867 0.867 0.869 0.870 0.872 0.874 0.875 0.876
Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5 0.173 0.112 0.067 0.036 0.323 0.198 0.112 0.054 0.489 0.289 0.158 0.073 0.547 0.320 0.174 0.079 0.560 0.328 0.178 0.081 0.558 0.328 0.179 0.082 0.551 0.324 0.177 0.082 0.550 0.324 0.177 0.082 0.549 0.324 0.178 0.082 0.548 0.324 0.178 0.083 0.547 0.323 0.178 0.082 0.546 0.322 0.177 0.082 0.546 0.322 0.177 0.082 0.544 0.320 0.175 0.081 0.542 0.319 0.174 0.080 0.541 0.318 0.174 0.080 0.537 0.315 0.171 0.078 0.538 0.314 0.170 0.077 0.537 0.313 0.169 0.077 0.537 0.314 0.170 0.078 0.537 0.314 0.170 0.078 0.538 0.314 0.170 0.078 0.540 0.315 0.171 0.079 0.541 0.317 0.172 0.080 0.543 0.319 0.174 0.082 0.546 0.323 0.178 0.084 0.550 0.327 0.181 0.087 0.552 0.329 0.184 0.089 0.554 0.331 0.185 0.090 0.554 0.331 0.185 0.090 0.554 0.331 0.185 0.090
308
Black 0.017 0.022 0.026 0.028 0.029 0.030 0.030 0.030 0.031 0.031 0.033 0.033 0.033 0.032 0.031 0.029 0.027 0.025 0.025 0.027 0.028 0.029 0.029 0.030 0.031 0.033 0.035 0.036 0.036 0.036 0.035
APPENDIX L : THE REFLECTANCE SPECTRA OF THE PREDICTED GRETAG MACBETH COLOR CHECKER USING FUJIX PICTROGRAPH 3000
Wavelength Dark skin 400 0.037 410 0.053 420 0.066 430 0.063 440 0.059 450 0.054 460 0.052 470 0.055 480 0.062 490 0.073 500 0.084 510 0.089 520 0.085 530 0.078 540 0.072 550 0.070 560 0.073 570 0.086 580 0.109 590 0.138 600 0.159 610 0.162 620 0.156 630 0.150 640 0.154 650 0.163 660 0.173 670 0.195 680 0.245 690 0.327 700 0.428
Light skin
0.093 0.160 0.230 0.237 0.232 0.218 0.213 0.219 0.233 0.256 0.281 0.296 0.296 0.287 0.279 0.280 0.289 0.318 0.364 0.421 0.470 0.497 0.505 0.507 0.515 0.529 0.541 0.563 0.605 0.661 0.717
Blue sky
Foliage
0.110 0.207 0.318 0.341 0.338 0.321 0.311 0.306 0.299 0.290 0.275 0.252 0.224 0.199 0.179 0.167 0.161 0.163 0.168 0.171 0.161 0.142 0.124 0.114 0.114 0.121 0.128 0.147 0.194 0.274 0.374
0.036 0.052 0.064 0.063 0.061 0.058 0.058 0.065 0.078 0.101 0.131 0.156 0.165 0.160 0.150 0.141 0.134 0.134 0.136 0.135 0.123 0.104 0.088 0.080 0.080 0.086 0.092 0.108 0.149 0.222 0.322
309
Blue flower Blue green
0.136 0.265 0.418 0.451 0.447 0.421 0.403 0.389 0.368 0.343 0.313 0.278 0.244 0.216 0.198 0.189 0.189 0.200 0.221 0.243 0.250 0.241 0.225 0.215 0.218 0.228 0.238 0.262 0.316 0.400 0.495
0.127 0.238 0.374 0.410 0.417 0.407 0.409 0.425 0.450 0.489 0.531 0.562 0.571 0.559 0.533 0.499 0.457 0.415 0.366 0.315 0.261 0.215 0.184 0.167 0.167 0.174 0.182 0.204 0.256 0.339 0.438
Wavelength Orange 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
0.035 0.048 0.056 0.053 0.050 0.046 0.046 0.052 0.065 0.090 0.127 0.164 0.188 0.196 0.199 0.206 0.224 0.263 0.328 0.420 0.516 0.589 0.633 0.654 0.670 0.688 0.702 0.718 0.741 0.772 0.805
Purplish Moderate blue red 0.117 0.070 0.228 0.115 0.360 0.156 0.388 0.153 0.382 0.144 0.358 0.130 0.338 0.122 0.318 0.120 0.287 0.120 0.249 0.122 0.208 0.121 0.167 0.113 0.134 0.101 0.109 0.090 0.094 0.084 0.086 0.086 0.083 0.098 0.087 0.127 0.096 0.185 0.103 0.279 0.100 0.390 0.087 0.481 0.074 0.535 0.067 0.560 0.068 0.580 0.073 0.600 0.079 0.616 0.093 0.636 0.132 0.670 0.202 0.714 0.300 0.759
310
Purple 0.066 0.114 0.159 0.160 0.151 0.137 0.127 0.120 0.111 0.101 0.087 0.071 0.056 0.044 0.038 0.036 0.038 0.047 0.063 0.086 0.102 0.104 0.098 0.093 0.096 0.103 0.111 0.128 0.172 0.249 0.349
Yellow green 0.043 0.061 0.077 0.076 0.075 0.072 0.075 0.089 0.118 0.174 0.265 0.375 0.467 0.516 0.527 0.517 0.495 0.471 0.440 0.406 0.365 0.326 0.297 0.281 0.282 0.292 0.302 0.327 0.381 0.461 0.551
Orange yellow 0.039 0.053 0.063 0.060 0.058 0.054 0.055 0.064 0.085 0.124 0.186 0.259 0.318 0.349 0.363 0.375 0.393 0.430 0.486 0.557 0.625 0.677 0.708 0.724 0.736 0.751 0.762 0.775 0.792 0.813 0.836
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
Blue 0.101 0.197 0.306 0.326 0.318 0.293 0.271 0.246 0.210 0.169 0.128 0.093 0.067 0.051 0.041 0.037 0.036 0.039 0.046 0.052 0.052 0.044 0.036 0.032 0.032 0.035 0.039 0.048 0.076 0.131 0.218
Green 0.039 0.056 0.071 0.071 0.070 0.068 0.071 0.082 0.106 0.148 0.210 0.275 0.319 0.329 0.315 0.292 0.262 0.233 0.202 0.167 0.130 0.098 0.078 0.068 0.068 0.072 0.078 0.092 0.130 0.200 0.298
Red 0.041 0.060 0.074 0.068 0.062 0.054 0.050 0.049 0.051 0.052 0.053 0.049 0.042 0.036 0.033 0.035 0.041 0.060 0.102 0.181 0.290 0.389 0.451 0.480 0.504 0.527 0.546 0.568 0.607 0.662 0.717
Yellow 0.043 0.059 0.072 0.070 0.068 0.065 0.067 0.080 0.109 0.165 0.262 0.386 0.499 0.570 0.605 0.623 0.632 0.649 0.669 0.695 0.719 0.740 0.754 0.761 0.769 0.780 0.789 0.800 0.814 0.831 0.850
311
Magenta 0.117 0.218 0.329 0.338 0.322 0.291 0.267 0.246 0.221 0.192 0.162 0.132 0.106 0.089 0.081 0.082 0.092 0.120 0.174 0.261 0.363 0.444 0.489 0.509 0.526 0.545 0.561 0.582 0.622 0.675 0.727
Cyan 0.108 0.207 0.327 0.361 0.367 0.358 0.355 0.357 0.356 0.353 0.343 0.320 0.289 0.256 0.225 0.197 0.171 0.149 0.126 0.102 0.075 0.052 0.039 0.033 0.032 0.035 0.038 0.047 0.074 0.130 0.216
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
White 0.213 0.437 0.734 0.819 0.833 0.805 0.791 0.789 0.785 0.790 0.798 0.804 0.805 0.801 0.795 0.791 0.784 0.783 0.779 0.780 0.784 0.789 0.792 0.793 0.796 0.802 0.809 0.819 0.834 0.848 0.861
Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5 0.169 0.122 0.082 0.048 0.332 0.227 0.142 0.074 0.537 0.348 0.205 0.098 0.588 0.372 0.212 0.097 0.592 0.370 0.208 0.093 0.569 0.353 0.196 0.086 0.558 0.345 0.191 0.084 0.560 0.349 0.195 0.087 0.563 0.357 0.204 0.093 0.575 0.371 0.217 0.102 0.587 0.383 0.228 0.110 0.590 0.385 0.228 0.110 0.584 0.373 0.217 0.101 0.570 0.356 0.200 0.090 0.557 0.340 0.186 0.080 0.550 0.331 0.178 0.075 0.545 0.328 0.177 0.075 0.551 0.337 0.185 0.080 0.561 0.353 0.200 0.092 0.572 0.369 0.215 0.103 0.576 0.371 0.216 0.103 0.570 0.358 0.203 0.093 0.561 0.342 0.186 0.081 0.554 0.331 0.176 0.074 0.557 0.334 0.178 0.076 0.568 0.345 0.187 0.081 0.578 0.356 0.196 0.087 0.599 0.381 0.219 0.103 0.639 0.435 0.271 0.143 0.689 0.511 0.354 0.215 0.739 0.593 0.453 0.314
312
Black 0.024 0.033 0.039 0.036 0.034 0.031 0.030 0.031 0.035 0.040 0.044 0.044 0.039 0.033 0.028 0.025 0.025 0.028 0.034 0.040 0.040 0.034 0.028 0.024 0.025 0.028 0.031 0.039 0.062 0.113 0.195
APPENDIX M : THE REFLECTANCE SPECTRA OF THE PREDICTED GRETAG MACBETH COLOR CHECKER USING KODAK PROFESSIONAL 8670 PS THERMAL PRINTER
Wavelength Dark skin 400 0.174 410 0.180 420 0.139 430 0.092 440 0.058 450 0.040 460 0.032 470 0.036 480 0.057 490 0.085 500 0.098 510 0.093 520 0.082 530 0.074 540 0.068 550 0.067 560 0.072 570 0.087 580 0.113 590 0.141 600 0.163 610 0.174 620 0.171 630 0.159 640 0.146 650 0.134 660 0.125 670 0.121 680 0.121 690 0.125 700 0.134
Light skin
0.340 0.399 0.363 0.295 0.231 0.189 0.170 0.180 0.230 0.286 0.311 0.307 0.293 0.280 0.272 0.274 0.286 0.319 0.369 0.424 0.471 0.505 0.520 0.519 0.512 0.505 0.498 0.494 0.495 0.500 0.509
Blue sky
Foliage
0.254 0.348 0.382 0.373 0.343 0.311 0.286 0.279 0.296 0.304 0.284 0.251 0.219 0.193 0.175 0.164 0.160 0.167 0.177 0.180 0.172 0.155 0.133 0.112 0.095 0.083 0.075 0.070 0.070 0.074 0.081
0.150 0.163 0.132 0.091 0.060 0.043 0.036 0.043 0.074 0.125 0.161 0.170 0.162 0.151 0.140 0.133 0.130 0.134 0.141 0.143 0.134 0.118 0.099 0.081 0.067 0.057 0.051 0.047 0.047 0.050 0.056
313
Blue flower Blue green
0.321 0.439 0.483 0.478 0.448 0.413 0.383 0.364 0.362 0.349 0.314 0.273 0.237 0.212 0.195 0.188 0.190 0.206 0.231 0.252 0.260 0.254 0.238 0.216 0.196 0.181 0.170 0.164 0.164 0.169 0.180
0.299 0.410 0.450 0.443 0.414 0.388 0.374 0.393 0.460 0.534 0.576 0.583 0.570 0.543 0.511 0.481 0.449 0.417 0.378 0.335 0.285 0.237 0.196 0.162 0.138 0.121 0.110 0.104 0.104 0.108 0.117
Wavelength Orange 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
0.237 0.215 0.141 0.079 0.044 0.028 0.023 0.029 0.058 0.112 0.162 0.185 0.190 0.191 0.194 0.202 0.220 0.261 0.328 0.412 0.500 0.581 0.641 0.678 0.697 0.710 0.716 0.718 0.720 0.720 0.720
Purplish Moderate blue red 0.226 0.302 0.327 0.331 0.383 0.275 0.401 0.202 0.388 0.143 0.361 0.107 0.329 0.089 0.299 0.090 0.279 0.113 0.248 0.133 0.204 0.131 0.161 0.115 0.129 0.099 0.108 0.089 0.094 0.086 0.086 0.088 0.085 0.100 0.092 0.131 0.103 0.190 0.109 0.272 0.106 0.368 0.095 0.463 0.080 0.538 0.065 0.583 0.053 0.608 0.045 0.623 0.040 0.630 0.037 0.634 0.037 0.637 0.039 0.640 0.044 0.642
314
Purple 0.195 0.242 0.234 0.198 0.156 0.124 0.104 0.097 0.104 0.104 0.089 0.069 0.053 0.043 0.038 0.036 0.039 0.049 0.067 0.089 0.105 0.112 0.108 0.098 0.087 0.077 0.071 0.068 0.067 0.071 0.078
Yellow green 0.219 0.222 0.167 0.108 0.069 0.049 0.043 0.056 0.112 0.223 0.345 0.432 0.478 0.495 0.496 0.490 0.476 0.463 0.446 0.424 0.393 0.357 0.320 0.287 0.260 0.240 0.226 0.219 0.218 0.223 0.235
Orange yellow 0.255 0.234 0.156 0.089 0.051 0.033 0.028 0.036 0.076 0.156 0.242 0.298 0.326 0.340 0.350 0.364 0.384 0.423 0.483 0.553 0.619 0.678 0.720 0.744 0.757 0.766 0.769 0.771 0.772 0.771 0.770
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
Blue 0.175 0.261 0.314 0.335 0.326 0.299 0.265 0.229 0.201 0.164 0.123 0.088 0.065 0.051 0.042 0.038 0.038 0.042 0.050 0.056 0.055 0.048 0.039 0.030 0.023 0.019 0.016 0.015 0.015 0.016 0.018
Green 0.151 0.168 0.140 0.100 0.068 0.051 0.044 0.055 0.103 0.188 0.268 0.311 0.322 0.312 0.293 0.273 0.252 0.232 0.209 0.181 0.147 0.115 0.088 0.067 0.053 0.044 0.038 0.035 0.035 0.037 0.042
Red 0.229 0.222 0.159 0.098 0.059 0.038 0.030 0.031 0.044 0.058 0.060 0.051 0.042 0.037 0.035 0.036 0.043 0.064 0.106 0.175 0.266 0.366 0.452 0.508 0.540 0.558 0.567 0.572 0.576 0.579 0.581
Yellow 0.271 0.253 0.172 0.101 0.059 0.040 0.035 0.046 0.099 0.212 0.346 0.451 0.517 0.556 0.580 0.599 0.613 0.635 0.664 0.698 0.728 0.754 0.770 0.777 0.779 0.782 0.782 0.781 0.781 0.780 0.781
315
Magenta 0.362 0.449 0.439 0.386 0.324 0.272 0.236 0.217 0.210 0.193 0.161 0.128 0.104 0.090 0.084 0.085 0.096 0.126 0.181 0.259 0.346 0.430 0.493 0.528 0.544 0.552 0.555 0.557 0.560 0.565 0.570
Cyan 0.186 0.280 0.342 0.371 0.371 0.358 0.341 0.337 0.360 0.374 0.358 0.325 0.286 0.249 0.216 0.190 0.169 0.152 0.133 0.110 0.083 0.059 0.041 0.029 0.021 0.016 0.013 0.012 0.012 0.013 0.015
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
White 0.559 0.770 0.859 0.888 0.886 0.879 0.874 0.876 0.881 0.888 0.891 0.890 0.888 0.881 0.874 0.874 0.870 0.872 0.873 0.878 0.881 0.885 0.886 0.885 0.885 0.888 0.890 0.892 0.893 0.892 0.891
Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5 0.443 0.345 0.253 0.164 0.591 0.445 0.313 0.192 0.632 0.456 0.304 0.171 0.618 0.420 0.260 0.131 0.581 0.370 0.211 0.095 0.547 0.330 0.176 0.072 0.528 0.308 0.157 0.061 0.534 0.314 0.163 0.064 0.570 0.357 0.201 0.089 0.603 0.398 0.239 0.117 0.610 0.407 0.248 0.124 0.598 0.391 0.233 0.113 0.580 0.368 0.212 0.098 0.561 0.347 0.193 0.085 0.547 0.331 0.180 0.077 0.540 0.323 0.173 0.072 0.540 0.324 0.174 0.073 0.553 0.340 0.187 0.082 0.571 0.362 0.208 0.096 0.586 0.380 0.224 0.108 0.591 0.385 0.227 0.110 0.587 0.376 0.218 0.103 0.572 0.357 0.200 0.091 0.553 0.333 0.179 0.076 0.535 0.311 0.159 0.064 0.522 0.294 0.145 0.055 0.511 0.281 0.135 0.049 0.506 0.274 0.129 0.046 0.506 0.274 0.129 0.046 0.511 0.280 0.133 0.049 0.521 0.292 0.143 0.054
316
Black 0.094 0.102 0.082 0.055 0.034 0.023 0.018 0.019 0.031 0.047 0.052 0.045 0.037 0.030 0.026 0.024 0.025 0.029 0.036 0.042 0.043 0.039 0.032 0.025 0.019 0.016 0.013 0.012 0.012 0.013 0.015
APPENDIX N : THE REFLECTANCE SPECTRA OF THE PREDICTED GRETAG MACBETH COLOR CHECKER USING DUPONT WATERPROOF® WITH CMYK PRIMARIES
Wavelength Dark skin 400 0.025 410 0.036 420 0.043 430 0.045 440 0.049 450 0.054 460 0.061 470 0.064 480 0.067 490 0.079 500 0.100 510 0.110 520 0.101 530 0.090 540 0.086 550 0.076 560 0.059 570 0.051 580 0.070 590 0.120 600 0.157 610 0.172 620 0.179 630 0.184 640 0.190 650 0.203 660 0.220 670 0.228 680 0.224 690 0.215 700 0.202
Light skin
0.076 0.133 0.191 0.211 0.220 0.226 0.230 0.231 0.233 0.249 0.282 0.309 0.313 0.308 0.309 0.299 0.275 0.263 0.302 0.400 0.478 0.518 0.536 0.545 0.551 0.561 0.571 0.577 0.577 0.573 0.567
Blue sky
Foliage
0.085 0.171 0.257 0.298 0.323 0.333 0.328 0.320 0.306 0.297 0.295 0.284 0.253 0.223 0.202 0.172 0.134 0.110 0.114 0.141 0.154 0.157 0.160 0.163 0.170 0.184 0.202 0.210 0.206 0.195 0.181
0.025 0.035 0.041 0.044 0.048 0.056 0.067 0.074 0.084 0.110 0.156 0.193 0.197 0.182 0.165 0.139 0.107 0.086 0.088 0.109 0.119 0.120 0.121 0.124 0.129 0.142 0.158 0.165 0.161 0.152 0.139
317
Blue flower Blue green
0.112 0.230 0.353 0.407 0.433 0.436 0.419 0.401 0.376 0.351 0.331 0.305 0.268 0.240 0.224 0.201 0.166 0.145 0.161 0.210 0.243 0.256 0.263 0.269 0.276 0.290 0.308 0.316 0.313 0.303 0.289
0.102 0.199 0.301 0.348 0.379 0.405 0.424 0.439 0.456 0.497 0.558 0.610 0.620 0.587 0.532 0.466 0.392 0.333 0.295 0.274 0.258 0.248 0.247 0.250 0.256 0.272 0.291 0.300 0.296 0.285 0.269
Wavelength Orange 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
0.017 0.019 0.019 0.018 0.019 0.023 0.027 0.029 0.031 0.037 0.045 0.044 0.031 0.025 0.028 0.027 0.020 0.017 0.058 0.240 0.464 0.607 0.672 0.696 0.708 0.716 0.721 0.726 0.731 0.734 0.736
Purplish Moderate blue red 0.088 0.054 0.188 0.091 0.286 0.123 0.335 0.133 0.363 0.137 0.364 0.137 0.342 0.134 0.317 0.128 0.283 0.120 0.248 0.118 0.220 0.125 0.187 0.123 0.149 0.108 0.122 0.099 0.110 0.104 0.091 0.101 0.065 0.087 0.051 0.081 0.060 0.134 0.084 0.285 0.096 0.430 0.096 0.513 0.098 0.550 0.101 0.565 0.106 0.574 0.119 0.584 0.136 0.593 0.143 0.599 0.139 0.600 0.129 0.598 0.116 0.595
318
Purple 0.046 0.087 0.121 0.137 0.146 0.146 0.138 0.127 0.112 0.101 0.096 0.083 0.063 0.050 0.049 0.042 0.030 0.025 0.040 0.077 0.100 0.107 0.111 0.114 0.119 0.131 0.147 0.155 0.151 0.141 0.128
Yellow green 0.031 0.041 0.047 0.049 0.053 0.065 0.085 0.101 0.126 0.191 0.313 0.456 0.546 0.562 0.537 0.493 0.436 0.391 0.368 0.365 0.360 0.355 0.356 0.359 0.365 0.379 0.397 0.406 0.402 0.393 0.380
Orange yellow 0.030 0.037 0.042 0.042 0.045 0.052 0.065 0.075 0.089 0.129 0.201 0.285 0.341 0.366 0.383 0.386 0.373 0.369 0.423 0.551 0.655 0.713 0.738 0.747 0.751 0.756 0.759 0.763 0.766 0.768 0.769
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
Blue 0.068 0.144 0.215 0.252 0.272 0.268 0.244 0.217 0.183 0.151 0.127 0.100 0.071 0.055 0.050 0.041 0.027 0.022 0.030 0.047 0.052 0.049 0.049 0.050 0.054 0.064 0.078 0.085 0.081 0.072 0.061
Green 0.027 0.038 0.043 0.046 0.051 0.062 0.079 0.093 0.113 0.165 0.258 0.351 0.387 0.367 0.325 0.269 0.206 0.160 0.140 0.137 0.131 0.126 0.126 0.128 0.134 0.148 0.166 0.174 0.170 0.159 0.145
Red 0.029 0.042 0.051 0.053 0.055 0.057 0.058 0.057 0.054 0.056 0.061 0.058 0.045 0.038 0.042 0.040 0.031 0.028 0.066 0.197 0.338 0.421 0.458 0.474 0.483 0.493 0.503 0.509 0.510 0.508 0.504
Yellow 0.034 0.042 0.048 0.048 0.052 0.062 0.079 0.093 0.115 0.173 0.285 0.425 0.535 0.588 0.609 0.613 0.603 0.601 0.629 0.692 0.741 0.767 0.779 0.784 0.786 0.790 0.794 0.797 0.800 0.801 0.801
319
Magenta 0.099 0.190 0.280 0.313 0.321 0.306 0.279 0.251 0.218 0.186 0.165 0.140 0.111 0.098 0.101 0.098 0.082 0.076 0.127 0.267 0.397 0.470 0.502 0.517 0.525 0.536 0.547 0.554 0.554 0.550 0.545
Cyan 0.083 0.173 0.264 0.312 0.348 0.369 0.374 0.372 0.366 0.365 0.370 0.362 0.327 0.282 0.239 0.188 0.134 0.097 0.082 0.081 0.076 0.070 0.070 0.072 0.077 0.089 0.104 0.112 0.107 0.098 0.085
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
White 0.235 0.471 0.756 0.858 0.879 0.873 0.860 0.858 0.855 0.853 0.851 0.850 0.850 0.849 0.848 0.850 0.848 0.853 0.854 0.856 0.857 0.859 0.862 0.863 0.863 0.866 0.868 0.869 0.871 0.873 0.873
Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5 0.157 0.102 0.061 0.032 0.309 0.195 0.112 0.053 0.483 0.292 0.159 0.068 0.549 0.332 0.179 0.075 0.573 0.352 0.193 0.082 0.580 0.362 0.203 0.089 0.577 0.365 0.208 0.095 0.576 0.365 0.209 0.096 0.573 0.363 0.208 0.097 0.580 0.374 0.220 0.107 0.596 0.398 0.245 0.126 0.608 0.414 0.259 0.133 0.605 0.406 0.247 0.119 0.591 0.386 0.226 0.102 0.578 0.369 0.211 0.094 0.558 0.341 0.186 0.079 0.526 0.301 0.151 0.058 0.505 0.274 0.129 0.047 0.513 0.286 0.141 0.057 0.549 0.332 0.183 0.085 0.573 0.364 0.209 0.099 0.585 0.377 0.218 0.102 0.592 0.385 0.224 0.104 0.596 0.390 0.229 0.106 0.601 0.397 0.236 0.111 0.611 0.411 0.250 0.124 0.622 0.427 0.268 0.141 0.628 0.435 0.277 0.148 0.627 0.432 0.273 0.144 0.622 0.424 0.263 0.134 0.615 0.412 0.249 0.121
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Black 0.015 0.021 0.023 0.024 0.027 0.031 0.036 0.038 0.039 0.046 0.056 0.056 0.043 0.033 0.032 0.026 0.018 0.016 0.023 0.037 0.039 0.035 0.034 0.035 0.037 0.046 0.058 0.064 0.060 0.052 0.042
APPENDIX O : THE ACCURACY OF THE GRETAG SPECTROLINO SPECTROPHOTOMETER The accuracy of spectral reproduction is sensitive to the accuracy of measuring instruments. Since a Gretag Spectrolino with an automatic station was used to measure the 6,250 printed samples, an evaluation of the accuracy of this spectrophotometer is required. It is hypothesized that a Gretag SPM 60 is highly accurate, hence, designated as a standard spectrophotometer. Accuracy of the Gretag Spectrolino was evaluated by the colorimetric and spectral accuracy of the reflectance spectra of the Gretag Macbeth Color Checker measured by the two instruments. The colorimectric and spectral accuracy is shown below. ∆E*94 0.44 0.17 0.72 0.12 0.004
Mean Stdev Max Min RMS
Metamerism Index 0.04 0.03 0.14 0.00
The following figure is the histogram of the colorimetric error. 6
5
F re q u e nc y
4
3
2
1
0 0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
D e lt a E 9 4
According to this accuracy comparison, the agreement between the two instruments were very high revealed by the very low ∆E*94, metamerism index, and the RMS error. Hence,
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this ensures the spectral reproduction quality and prediction accuracy of the proposed sixcolor separation mechanism such that the resultant spectral reflectance spectra of the actual reproduction is consistent with the theoretical prediction. Spectral reflectance factors of the Gretag Macbeth Color Checker measured by the Gretag SPM 60 spectrophotometer
Wavelength Dark skin 400 0.060 410 0.061 420 0.060 430 0.060 440 0.060 450 0.060 460 0.060 470 0.060 480 0.061 490 0.061 500 0.063 510 0.068 520 0.074 530 0.077 540 0.078 550 0.080 560 0.086 570 0.100 580 0.119 590 0.138 600 0.148 610 0.151 620 0.155 630 0.163 640 0.174 650 0.185 660 0.193 670 0.195 680 0.189 690 0.183 700 0.183
Light skin
0.206 0.231 0.230 0.228 0.228 0.229 0.230 0.233 0.244 0.268 0.295 0.306 0.302 0.297 0.296 0.291 0.289 0.297 0.363 0.455 0.495 0.503 0.513 0.530 0.555 0.578 0.593 0.590 0.572 0.555 0.552
Blue sky
Foliage
0.260 0.319 0.325 0.332 0.343 0.349 0.347 0.335 0.322 0.309 0.292 0.266 0.239 0.221 0.207 0.183 0.158 0.144 0.156 0.166 0.159 0.150 0.146 0.144 0.144 0.148 0.155 0.158 0.153 0.146 0.136
0.053 0.054 0.055 0.057 0.060 0.062 0.063 0.064 0.065 0.068 0.074 0.099 0.148 0.184 0.188 0.170 0.144 0.125 0.115 0.110 0.105 0.101 0.100 0.103 0.108 0.112 0.115 0.113 0.109 0.105 0.104
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Blue flower Blue green
0.301 0.400 0.425 0.434 0.441 0.444 0.441 0.424 0.401 0.363 0.321 0.289 0.259 0.230 0.217 0.214 0.205 0.198 0.199 0.208 0.225 0.238 0.244 0.251 0.275 0.309 0.343 0.360 0.357 0.347 0.346
0.254 0.314 0.331 0.346 0.362 0.386 0.421 0.469 0.514 0.553 0.579 0.590 0.584 0.564 0.537 0.498 0.453 0.406 0.358 0.307 0.259 0.228 0.212 0.202 0.196 0.192 0.192 0.200 0.212 0.224 0.232
Wavelength Orange 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
0.048 0.048 0.048 0.049 0.050 0.051 0.051 0.052 0.053 0.056 0.061 0.084 0.130 0.162 0.176 0.195 0.240 0.327 0.438 0.542 0.594 0.599 0.595 0.588 0.582 0.576 0.568 0.559 0.562 0.571 0.586
Purplish Moderate blue red 0.246 0.136 0.309 0.139 0.340 0.137 0.365 0.136 0.391 0.135 0.403 0.134 0.393 0.132 0.361 0.127 0.312 0.121 0.255 0.116 0.204 0.111 0.165 0.105 0.135 0.100 0.114 0.097 0.102 0.098 0.094 0.098 0.087 0.100 0.083 0.107 0.081 0.151 0.080 0.276 0.080 0.423 0.079 0.516 0.079 0.557 0.079 0.567 0.082 0.568 0.087 0.567 0.094 0.570 0.104 0.583 0.118 0.603 0.139 0.623 0.167 0.637
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Purple 0.164 0.201 0.216 0.204 0.179 0.150 0.122 0.098 0.081 0.068 0.060 0.055 0.052 0.049 0.048 0.049 0.050 0.051 0.051 0.051 0.056 0.069 0.093 0.120 0.144 0.167 0.193 0.225 0.263 0.312 0.360
Yellow green 0.058 0.059 0.059 0.061 0.064 0.069 0.079 0.095 0.123 0.167 0.245 0.371 0.493 0.553 0.570 0.564 0.549 0.527 0.493 0.445 0.388 0.347 0.326 0.315 0.308 0.304 0.309 0.325 0.349 0.373 0.393
Orange yellow 0.068 0.068 0.067 0.068 0.068 0.069 0.071 0.072 0.075 0.079 0.089 0.128 0.223 0.323 0.379 0.415 0.464 0.532 0.586 0.615 0.622 0.620 0.630 0.649 0.677 0.701 0.714 0.705 0.691 0.683 0.690
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
Blue 0.115 0.156 0.193 0.232 0.283 0.349 0.359 0.312 0.235 0.160 0.106 0.075 0.058 0.050 0.046 0.043 0.040 0.040 0.039 0.039 0.039 0.039 0.039 0.039 0.040 0.040 0.041 0.041 0.042 0.043 0.044
Green 0.052 0.052 0.053 0.055 0.057 0.061 0.069 0.085 0.111 0.153 0.222 0.314 0.358 0.343 0.322 0.297 0.257 0.221 0.192 0.162 0.129 0.104 0.089 0.081 0.076 0.073 0.070 0.069 0.070 0.072 0.075
Red 0.049 0.048 0.048 0.048 0.048 0.047 0.047 0.046 0.045 0.044 0.044 0.043 0.043 0.043 0.044 0.045 0.048 0.053 0.065 0.099 0.180 0.315 0.474 0.595 0.665 0.701 0.721 0.733 0.741 0.749 0.755
Yellow 0.056 0.056 0.057 0.058 0.061 0.065 0.074 0.088 0.111 0.148 0.210 0.318 0.466 0.573 0.627 0.656 0.677 0.698 0.714 0.725 0.730 0.734 0.738 0.738 0.741 0.744 0.747 0.750 0.757 0.763 0.769
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Magenta 0.291 0.363 0.372 0.362 0.340 0.311 0.281 0.250 0.219 0.188 0.165 0.148 0.125 0.104 0.098 0.103 0.103 0.105 0.132 0.197 0.289 0.394 0.509 0.611 0.686 0.736 0.768 0.786 0.798 0.806 0.811
Cyan 0.180 0.221 0.236 0.260 0.293 0.329 0.371 0.422 0.448 0.445 0.417 0.374 0.321 0.268 0.218 0.173 0.134 0.108 0.094 0.084 0.077 0.072 0.070 0.069 0.069 0.070 0.072 0.073 0.073 0.072 0.069
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
White 0.371 0.657 0.823 0.860 0.870 0.880 0.888 0.889 0.890 0.894 0.893 0.894 0.894 0.891 0.891 0.889 0.884 0.882 0.882 0.884 0.884 0.883 0.887 0.886 0.888 0.894 0.897 0.897 0.899 0.899 0.897
Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5 0.332 0.265 0.167 0.079 0.509 0.335 0.184 0.082 0.558 0.347 0.187 0.083 0.566 0.349 0.191 0.085 0.571 0.353 0.195 0.087 0.575 0.354 0.197 0.087 0.577 0.353 0.197 0.087 0.573 0.350 0.195 0.086 0.571 0.348 0.194 0.086 0.572 0.347 0.194 0.085 0.573 0.346 0.194 0.086 0.574 0.346 0.195 0.086 0.575 0.346 0.196 0.086 0.575 0.346 0.197 0.087 0.576 0.346 0.197 0.087 0.575 0.345 0.196 0.086 0.574 0.345 0.196 0.086 0.575 0.348 0.196 0.086 0.574 0.350 0.195 0.085 0.574 0.351 0.195 0.085 0.572 0.350 0.194 0.084 0.570 0.349 0.192 0.084 0.570 0.347 0.191 0.083 0.566 0.344 0.189 0.082 0.565 0.342 0.187 0.082 0.564 0.340 0.186 0.081 0.562 0.338 0.184 0.080 0.560 0.336 0.183 0.080 0.559 0.335 0.182 0.079 0.557 0.332 0.180 0.079 0.555 0.330 0.178 0.078
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Black 0.033 0.033 0.032 0.032 0.033 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.033 0.033 0.033
Spectral reflectance factors of the Gretag Macbeth Color Checker measured by the Gretag Spectrolino spectrophotometer
Wavelength Dark skin 400 0.059 410 0.060 420 0.059 430 0.059 440 0.059 450 0.059 460 0.059 470 0.059 480 0.059 490 0.059 500 0.061 510 0.067 520 0.073 530 0.075 540 0.076 550 0.079 560 0.086 570 0.100 580 0.119 590 0.137 600 0.147 610 0.151 620 0.156 630 0.164 640 0.175 650 0.187 660 0.195 670 0.196 680 0.191 690 0.187 700 0.188
Light skin
0.205 0.231 0.232 0.230 0.229 0.229 0.230 0.234 0.246 0.271 0.296 0.306 0.303 0.299 0.297 0.294 0.291 0.307 0.375 0.461 0.500 0.511 0.522 0.542 0.566 0.590 0.604 0.602 0.586 0.572 0.571
Blue sky
Foliage
0.254 0.313 0.322 0.329 0.338 0.345 0.341 0.331 0.317 0.304 0.286 0.261 0.235 0.218 0.202 0.180 0.156 0.145 0.155 0.163 0.157 0.149 0.145 0.143 0.144 0.148 0.154 0.156 0.152 0.145 0.136
0.051 0.053 0.054 0.056 0.058 0.060 0.061 0.062 0.063 0.066 0.074 0.100 0.145 0.179 0.182 0.165 0.140 0.122 0.112 0.107 0.102 0.098 0.098 0.101 0.105 0.109 0.112 0.110 0.106 0.103 0.103
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Blue flower Blue green
0.297 0.392 0.421 0.433 0.438 0.441 0.435 0.420 0.395 0.358 0.317 0.285 0.255 0.228 0.215 0.212 0.203 0.197 0.199 0.209 0.225 0.238 0.244 0.254 0.278 0.312 0.344 0.360 0.359 0.352 0.352
0.249 0.309 0.328 0.343 0.360 0.384 0.420 0.466 0.510 0.549 0.575 0.585 0.580 0.561 0.532 0.495 0.449 0.404 0.354 0.304 0.258 0.228 0.212 0.203 0.197 0.192 0.194 0.202 0.214 0.225 0.234
Wavelength Orange 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
0.047 0.047 0.047 0.047 0.048 0.048 0.049 0.050 0.051 0.053 0.060 0.085 0.128 0.159 0.175 0.197 0.246 0.335 0.447 0.547 0.597 0.605 0.601 0.595 0.589 0.583 0.576 0.571 0.574 0.586 0.603
Purplish Moderate blue red 0.242 0.131 0.306 0.137 0.338 0.135 0.364 0.134 0.387 0.133 0.398 0.132 0.387 0.129 0.356 0.124 0.307 0.119 0.251 0.113 0.201 0.108 0.163 0.103 0.134 0.098 0.113 0.096 0.101 0.096 0.093 0.096 0.087 0.098 0.083 0.109 0.081 0.162 0.080 0.285 0.080 0.425 0.079 0.518 0.079 0.558 0.080 0.571 0.083 0.572 0.088 0.571 0.095 0.576 0.106 0.591 0.120 0.612 0.141 0.633 0.169 0.648
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Purple 0.161 0.199 0.213 0.202 0.178 0.149 0.120 0.096 0.080 0.067 0.058 0.053 0.050 0.048 0.047 0.048 0.049 0.050 0.050 0.050 0.056 0.070 0.095 0.122 0.146 0.169 0.196 0.229 0.269 0.316 0.365
Yellow green 0.058 0.058 0.059 0.060 0.063 0.069 0.078 0.095 0.123 0.169 0.249 0.369 0.485 0.546 0.563 0.559 0.543 0.522 0.487 0.438 0.385 0.346 0.325 0.315 0.308 0.304 0.309 0.326 0.350 0.374 0.393
Orange yellow 0.066 0.066 0.065 0.065 0.065 0.067 0.068 0.069 0.072 0.077 0.088 0.130 0.221 0.316 0.372 0.411 0.461 0.527 0.579 0.608 0.616 0.618 0.628 0.649 0.674 0.697 0.708 0.702 0.690 0.684 0.690
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
Blue 0.113 0.152 0.189 0.228 0.279 0.336 0.345 0.301 0.226 0.154 0.102 0.072 0.056 0.048 0.044 0.041 0.039 0.038 0.037 0.037 0.037 0.037 0.038 0.038 0.038 0.039 0.040 0.040 0.041 0.041 0.043
Green 0.051 0.051 0.052 0.053 0.056 0.060 0.068 0.084 0.111 0.154 0.223 0.307 0.349 0.338 0.317 0.291 0.253 0.218 0.188 0.158 0.126 0.102 0.088 0.080 0.075 0.072 0.069 0.068 0.069 0.071 0.074
Red 0.048 0.048 0.047 0.046 0.046 0.046 0.045 0.044 0.043 0.042 0.042 0.041 0.041 0.041 0.042 0.044 0.046 0.052 0.066 0.104 0.189 0.327 0.482 0.600 0.669 0.706 0.726 0.739 0.749 0.758 0.765
Yellow 0.056 0.056 0.056 0.057 0.060 0.065 0.073 0.087 0.111 0.149 0.213 0.322 0.463 0.568 0.623 0.655 0.677 0.700 0.714 0.725 0.732 0.737 0.740 0.742 0.744 0.747 0.750 0.755 0.762 0.770 0.778
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Magenta 0.281 0.353 0.364 0.357 0.335 0.307 0.275 0.245 0.214 0.184 0.161 0.143 0.121 0.101 0.096 0.101 0.101 0.106 0.136 0.202 0.293 0.402 0.516 0.616 0.688 0.737 0.769 0.789 0.802 0.811 0.819
Cyan 0.177 0.219 0.238 0.262 0.293 0.331 0.374 0.422 0.446 0.444 0.416 0.372 0.321 0.267 0.218 0.172 0.134 0.109 0.094 0.084 0.077 0.072 0.070 0.070 0.070 0.071 0.072 0.074 0.074 0.073 0.071
Wavelength 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
White 0.374 0.634 0.803 0.856 0.869 0.878 0.883 0.887 0.888 0.892 0.893 0.894 0.896 0.895 0.895 0.895 0.890 0.891 0.889 0.891 0.892 0.893 0.895 0.897 0.898 0.903 0.907 0.909 0.911 0.912 0.913
Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5 0.329 0.256 0.159 0.076 0.489 0.323 0.179 0.081 0.544 0.338 0.184 0.082 0.558 0.343 0.188 0.084 0.563 0.345 0.192 0.085 0.567 0.347 0.194 0.086 0.567 0.345 0.194 0.085 0.566 0.343 0.192 0.085 0.564 0.341 0.191 0.084 0.565 0.341 0.191 0.084 0.566 0.340 0.192 0.084 0.567 0.340 0.193 0.085 0.570 0.341 0.194 0.085 0.571 0.341 0.195 0.085 0.571 0.341 0.195 0.085 0.573 0.342 0.195 0.085 0.572 0.342 0.195 0.085 0.574 0.346 0.196 0.085 0.573 0.348 0.195 0.085 0.573 0.349 0.194 0.084 0.572 0.348 0.193 0.084 0.572 0.348 0.192 0.083 0.571 0.346 0.191 0.082 0.569 0.344 0.189 0.082 0.567 0.342 0.188 0.081 0.566 0.340 0.186 0.081 0.564 0.338 0.185 0.080 0.564 0.337 0.184 0.079 0.563 0.336 0.183 0.079 0.562 0.334 0.182 0.078 0.562 0.332 0.180 0.078
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Black 0.032 0.032 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.032 0.032
