OPTICAL REVIEW Vol. 16, No. 3 (2009) 296–303
Recovery of Spectral Data Using Weighted Canonical Correlation Regression Niloofar E SLAHI, Seyed Hossein A MIRSHAHI, and Farnaz A GAHIAN Department of Textile Engineering, Amirkabir University of Technology, Tehran 15914, Iran (Received December 24, 2008; Revised March 9, 2009; Accepted March 13, 2009) The weighted canonical correlation regression technique is employed for reconstruction of reflectance spectra of surface colors from the related XYZ tristimulus values of samples. Flexible input data based on applying certain weights to reflectance and colorimetric values of Munsell color chips has been implemented for each particular sample which belongs to Munsell or GretagMacbeth Colorchecker DC color samples. In fact, the colorimetric and spectrophotometric data of Munsell chips are selected as fundamental bases and the color difference values between the target and samples in Munsell dataset are chosen as a criterion for determination of weighting factors. The performance of the suggested method is evaluated in spectral reflectance reconstruction. The results show considerable improvements in terms of root mean square error (RMS) and goodness-of-fit coefficient (GFC) between the actual and reconstructed reflectance curves as well as CIELAB color difference values under illuminants A and TL84 for CIE1964 standard observer. # 2009 The Optical Society of Japan Keywords: canonical correlation analysis, canonical correlation regression, principal component analysis, colorimetric data, recovery of spectral reflectance
1.
investigated the modeling of spectral reflectance or spectral power distribution through dimensionality reduction techniques.6–10) The most common technique implements the well-known linear model based on the principal component analysis technique, abbreviated PCA.11–14) The method takes advantage of the fact that the reflectance spectra of natural and man-made surfaces are a generally smooth function of wavelength over the range in which the human visual system is sensitive. Such spectra are strongly correlated and may be represented as the weighted sum of a small number of orthogonal basis functions.7) These functions can be simply obtained from a suitable set of available spectra by applying the PCA technique. This technique was recently improved by Agahian et al.15) and they employed the weighted version of PCA to more efficiently recover the spectral reflectance of surface colors from their tristimulus color coordinates. Application of the more complicated version of the multivariate technique, namely canonical correlation analysis (CCA) in the reproduction of spectral behavior of different datasets from six channel camera data was also reported. Zhao and his colleagues16) employed the canonical correlation regression (CCR) technique in spectral color reproduction of samples. They compared the results of their method with those obtained by pseudo-inverse and matrix R routines and claimed that CCR led to the highest spectral accuracy among these methods. In this paper, weighted canonical correlation regression (wCCR) is introduced and employed for reconstruction of the reflectance data of 1269 Munsell color chips as well as 237 colored samples of GretagMacbeth Colorchecker DC from the corresponding XYZ tristimulus values of samples under a given set of illumination-viewing conditions. The performance of the proposed method in the spectral estimation of samples has been investigated by calculation of color difference values under different illuminants, root mean square errors and goodness-of-fit coefficients (GFC)
Introduction
The spectral reflectance of a colored surface provides the most useful information for specification of color under different viewing conditions as well as color reproduction efforts. For example, it is used as an input of computer color formulation of paints, textiles, plastics, and inks to calculate the concentrations of required colorants for matching of the XYZ tristimulus values of a target under a given set of conditions. Additionally, the reflectance data is critical for prediction of changes in the appearance of an object under different illuminants in computer aided design (CAD) and/ or computer aided manufacturing (CAM) applications, and also provides suitable input for many general computer graphic applications that require a wavelength-based approach to specify color.1) However, the spectral reflectance data of objects are not always available. For example, measurement of the reflectance behavior of paint arts requires special devices, i.e., a spectroradiometer with suitable resolution, which is not easily accessible. On the other hand, capturing of colorimetric data such as RGB by a non-contact device like digital cameras is becoming cheaper and more popular every day. In fact, the color spaces such as CIEYxy or RGB models, project semi infinite-dimensional spectral space to three-dimensional color space and allow the surface color information to be represented by a set of tristimulus values. Although the computation of colorimetric data from spectral information can be easily performed, the calculation of spectral reflectance from the colorimetric value is an unresolved problem and thus is not yet a routine procedure.2) In the past few years, there has been a tremendous interest in extracting the spectral data of samples as well as the spectral radiance distribution of light sources from their colorimetric specifications.2–5) Several researchers have also
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between the actual and the reconstructed spectra. Finally, the results of wCCR method are compared with those obtained from classical CCR, PCA, and wPCA techniques. 2.
Mathematical Background
2.1 Definition Canonical correlation analysis was developed by Hotelling17) as a method of measuring the linear relationship between two multidimensional variables. This technique seeks to identify and quantify two bases, one for each variable, that are optimal with respect to correlation while at the same time finding the corresponding correlations.18) More explicitly, it focuses on the correlation between linear combinations of two sets of variables. The idea is first to determine the pair of linear combinations having the largest correlation, and next, to identify the pair of linear combinations having the largest correlation among all pairs uncorrelated with the initially selected pair. This process is repeated until all possible correlations are extracted. The pairs of linear combinations are called the canonical variables and their correlations are called canonical correlations. The maximization aspect of the technique represents an attempt to concentrate a high-dimensional relationship between two sets of variables into pairs of canonical variables.19) The canonical model selects linear functions that have maximum covariances between domains, subject to restrictions of orthogonality. Geometrically, the canonical model can be considered an exploration of the extent to which individuals occupy the same relative positions in one measured space as they do in the other.20) An important property of canonical correlations is the fact that they are invariant with respect to affine transformations of the variables. This is the most important difference between CCA and ordinary correlation analysis which highly depends on the basis on which the variables are described.18) In comparison to PCA, it could be stated that the PCA is a simple version of canonical correlation that only uses one set of variables. However, the relation is complicated and some authors have suggested first to extract the principal components of each variable separately and then perform the canonical correlation analysis on them.11) 2.2 Application of CCR in reflectance reconstruction In the spectral reflectance recovery, two sets of variables X and Y which respectively represent XYZ tristimulus values and reflectance data of a set of colored samples could be considered.16) X and Y are, respectively, n p and n q matrices where n refers to the number of specimens and p and q show the number of color coordinates and wavelengths, respectively. The covariance matrix of both datasets could be simply determined by X ; CovðXÞ ¼ 11
CovðYÞ ¼ CovðX; YÞ ¼
X 22 X 12
;
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As emphasized earlier, the CCA finds two sets of basis vectors, one for X and the other for Y, such that the correlations between the projections of the variables onto these basis vectors are mutually maximized. Obviously, in the case of spectral and colorimetric data, the number of wavelengths is greater than the number of colorimetric data (p q), so the matrices share the same p largest eigenvalues, 21 22 2p . Linear combinations provide simple summary measures of a set of variables. Canonical correlation analysis seeks matrices A and B such that the variables U and V maximize the correlation CovðU; VÞ ¼ corrðU; VÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : VarðUÞ VarðVÞ So, for coefficient matrices A and B, with the dimensions of p d and q d, where d ¼ minðrankðXÞ; rankðYÞÞ ¼ p, two sets of canonical variables could be created as U¼XA¼X
1=2 X
E;
ð2Þ
F;
ð3Þ
11
V¼YB¼Y
1=2 X 22
while EPis a pP pP matrix which is the eigenvector of the 1=2 1 P P1=2 and, similarly, F is a matrix 22 12 21 11 11 qP p P matrix which is the eigenvector of the matrix P1 P P1=2 1=2 . These two eigenvectors are 22 22 11 21 12 the normalized canonical correlation basis vectors. It should be noted that the two sets of variables X and Y must be full rank and non-singular, otherwise canonical correlation does not work properly because the rows of A or B corresponding to dependent columns of X or Y return zeros. As emphasized earlier, the first pair of canonical variables which could be shown by U1 and V1 benefits from the largest correlation and is equal to the square root of the first eigenvalues. These properties are valid for the other pairs and generally could be shown by corrðUk ; Vk Þ ¼ k ; VarðUk Þ ¼ VarðVk Þ ¼ 1;
ð4Þ
corrðUk ; Vi Þ ¼ 0; corrðUk ; Ui Þ ¼ 0; corrðVk ; Vi Þ ¼ 0; if k 6¼ i;
k; i ¼ 1; 2; . . . ; p:
ð5Þ
In the reconstruction of reflectance data from the XYZ tristimulus values, matrix X (colorimetric data) is known and the coefficient matrices A and B as well as the canonical correlations between X and Y, i.e., , could be calculated. Hence, the reflectance data of sample (Ysamp ) of which the XYZ tristimulus values (Xsamp ) are known could be easily calculated from estimated canonical variables ^ ¼ Usamp cc ¼ Xsamp A cc ; V
ð6Þ
where cc is
ð1Þ
cc ¼ diagð1 ; 2 ; . . . ; p Þ;
ð7Þ
^ Bþ : ^ samp ¼ V Y
ð8Þ
finally :
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2
The ‘‘+’’ sign indicates the pseudo-inverse of the proposed matrix. The mathematics of classical canonical correlation analysis has been fully described by Johnson and Wichern19) and its application in spectral reconstruction was discussed by Zhao et al.16) 2.3 Weighted canonical correlation regression (wCCR) The increase in similarity between the reflectance spectra of target and samples in the dataset could lead to more efficient reflectance reproduction. Garcia-Beltran et al.10) used the clustering method on the Munsell chips, based on the hues to improve the performance of the recovery process. In addition, Skocaj et al.21) presented a batch algorithm for weighted PCA which considered individual pixels and images selectively depending on the corresponding weights. A method for estimating high-dimensional spectra signals from low-dimensional sensor responses was also introduced by DiCarlo and Wandell22) which found certain nonlinear structure in the spectral data. They applied a kernel function to weight each training data response. In the standard CCR method all spectral and colorimetric data have equal influence on the reconstruction. So, in order to selectively control the influence of the data in the recovery process, both reflectance and XYZ tristimulus values of samples in the database have been weighted by their color difference values from the proposed sample prior to extraction of canonical terms. The main goal of the wCCR algorithm is to minimize the weighted squared reconstruction error q X wðRa ðÞ Rr ðÞÞ2 ! min; ð9Þ "¼ ¼1
where Ra ðÞ and Rr ðÞ are respectively the actual and the reconstructed reflectance. Also, w is determined according to the importance of each sample in the dataset.15) Hence, a diagonal matrix called W has been introduced as 2 3 w1 0 . . . 0 6 .. 7 6 7 6 0 w2 0 . 7 7 : ð10Þ W¼6 6 . 7 .. 6 . 7 0 . 0 5 4 . 0 . . . 0 wn nn It is obvious that if the weight of a sample is large, the reconstruction error of such sample will decrease. In this case, the input data, i.e. X and Y, change to new variables, i.e. Xw and Yw by Xw ¼ W X; Yw ¼ W Y:
ð11Þ ð12Þ
Then, the classical canonical correlation regression is employed for two sets of weighted variables, i.e., Xw and Yw . Since the purpose of this work was to provide canonical variables which prepare the highest correlation, the weighting matrix was chosen as
1
6 E1 þ s 6 6 6 0 6 W¼6 6 .. 6 6 . 6 4 0
0 1 E2 þ s
...
0
0
.. .
0
..
...
0
.
0 1
3 7 7 7 7 7 7 7 7 7 7 5
;
ð13Þ
En þ s nn where Ei refers to color difference values between the samples which build the original dataset (i ¼ 1; . . . ; nÞ and the specimen of which the spectrum reconstruction is desirable. s is a very small constant, e.g., s ¼ 0:01, which prevents leading to infinity if any sample with the Ei ¼ 0 exists in the dataset. Other forms of weighted matrices such as Euclidian distance in CIEXYZ color space were also examined but did not lead to better results. In fact, implementation of the weighted version of CCR enables us to logically vary the influence of reflectance spectra and the colorimetric data of the original dataset in estimating coefficient matrices A and B as well as canonical variables. As eq. (13) shows, by increasing the color difference value between a given colored sample and a sample in the original dataset, the corresponding weight in matrix W and consequently the influence of that sample on the construction of canonical variables, decreases. Therefore, applying the proposed weighting matrix to the original dataset before implementation of standard CCR provides the capability to construct a suitable dynamic variable which varies from one sample to another. Eab
3.
Experimental Procedure
We downloaded a dataset consisting of 1269 reflectance spectra of the matt Munsell color chips.23) The spectra set was measured with a Perkin Elmer Lambda 18 spectrophotometer and the wavelength range was from 380 to 800 nm with 1 nm interval. In the current research, the reflectance data were fixed between 400 to 700 nm at 10 nm intervals. The reflectance values of 237 samples of Colorchecker DC from GretagMacbeth were also measured personally. The Colorchecker samples were only considered for the reconstruction sequence. The color measurements for later samples were carried out using a GretagMacbeth Color-Eye 7000A spectrophotometer with d/8 geometry. The reflectance values were measured at 10 nm intervals from 400 to 700 nm with a specular component excluded. The XYZ tristimulus values of all samples were calculated under illuminant D65 and CIE1964 standard observer. In order to provide balanced dimensions between the spectral and colorimetric data and increase the number of input sets in the CCR and wCCR methods, it was necessary to extend the colorimetric data by adding different cross product terms of tristimulus values. In fact, twenty-eight cross products of colorimetric data were used besides the original XYZ tristimulus values as an input variable. This operation, i.e., increasing the number of canonical variables, decreases reconstruction error; so, the dimension of matrix X
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was increased to n 31. The expanded version of matrix X was called COL and is shown in eq. (14): COL ¼ ½X; Y; Z; XY; XZ; YZ; X 2 ; Y 2 ; Z 2 ; X 3 ; Y 3 ; Z 3 ; XYZ; X 2 Y; X 2 Z; Y 2 Z; XY 2 ; XZ 2 ; Z 2 Y; XYZ 2 ; XZY 2 ; YZX 2 ; X 3 Y; X 3 Z; Y 3 X; Y 3 Z; Z 3 Y; Z 3 X; X 4 ; Y 4 ; Z 4 :
ð14Þ
The collected data were used in different manners. In the case of classical canonical correlation regression and classical PCA, the Munsell dataset was used for determining the canonical variables and the most important eigenvectors. The extracted functions were simply used for the reconstruction of reflectance spectra of both Munsell and Colorchecker sets. For the CCR method, eq. (8) was used for the reconstruction of spectral data and the spectral reflectance was estimated from the expanded input variables expressed in eq. (15): RE^ F ¼ COL ½A cc Bþ ;
ð15Þ
where A and B are two coefficient matrices, cc is a matrix of canonical correlations between two sets of canonical variables, COL is an expanded colorimetric matrix shown in ^ F is reproduced reflectance values. eq. (14) and RE In the case of weighted versions of CCR and PCA, two different sets of unknowns were considered. Firstly, one of the Munsell specimens was considered a sample at whose reflection reconstruction was aimed. The spectral and colorimetric specifications of this sample were removed from the dataset and then the excluded spectral and colorimetric data were weighted. In fact, the color difference values between this sample and the other specimens of Munsell dataset were calculated and the weighting diagonal matrix Wðn1Þðn1Þ was created as shown in eq. (13). Then, the reflectance data and enhanced XYZ tristimulus values (the cross products) of the remaining Munsell dataset were multiplied by the calculated weighting matrix as shown in eqs. (11) and (12). By extracting of the weighted canonical variables, i.e. Xw and Yw and weighted eigenvectors the reflectance spectra were simply reconstructed similarly to classical CCR or PCA routines. This procedure was repeated for all Munsell samples one by one. The Colorchecker DC was considered as a second set of unknowns and the wCCR and wPCA methods were used to examine on this dataset.
Hence, matrix Wnn was formed according to the color difference values of each Colorchecker sample and all Munsell chips and used for the next modification of the dataset. Then, the reconstructed reflectance values of Colorchecker samples were calculated following the previous procedure. Matlab, the mathematical software package from Mathwork, was employed for all computational calculations.24) 4.
Results and Discussion
The results of spectral reflectance reproduction of Munsell and Colorchecker datasets from their XYZ tristimulus values using CCR, PCA, wCCR, and wPCA techniques are summarized in Tables 1 and 2. The spectral and colorimetric accuracies of methods were quantified by the calculation of root mean square (RMS) error between the actual and estimated spectra of samples as well as mean, maximum and color difference values under the standard deviation of Eab illuminants A and TL84 for CIE1964 standard observer. Since illuminant D65 was used for calculation of the estimation matrix, the color difference values of all employed methods under this illuminant were zero. Therefore, the colorimetric performances of the applied methods for D65 illuminant are not reported in the tables. Moreover, in order to demonstrate the performance of each method, the recovered reflectance data of eight randomly selected samples of Munsell chips and Colorchecker DC obtained
Table 1. The spectral accuracies of reflectance estimation shown by RMS error using wCCR, CCR, wPCA, and PCA methods. Dataset
RMS Error
Method Mean
Max
SD
Munsell
wCCR CCR wPCA PCA
0.0083 0.0222 0.0125 0.0243
0.1151 0.1271 0.1253 0.1599
0.0094 0.0114 0.0121 0.0186
Colorchecker
wCCR CCR wPCA PCA
0.0150 0.0283 0.0167 0.0289
0.0896 0.1269 0.1152 0.2044
0.0138 0.0193 0.0179 0.0296
SD: standard deviation
Table 2. The colorimetric accuracy of spectral estimation by wCCR, CCR, PCA, and wPCA. Sample
ETL84
EA
Method
Mean
Max
SD
0.55 1.05 1.05 1.61
0.69 1.16 1.16 2.18
8.20 7.90 10.67 12.74
0.93 1.05 1.44 2.10
0.60 1.04 1.13 2.17
0.89 1.32 1.23 2.32
5.31 9.36 10.57 13.56
0.90 1.31 1.66 2.43
Mean
Max
SD
Munsell
wCCR CCR wPCA PCA
0.44 1.18 0.82 1.68
4.27 8.16 7.51 11.17
Colorchecker
wCCR CCR wPCA PCA
0.67 1.47 0.97 2.02
3.68 5.86 6.67 10.81
SD: standard deviation
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(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 1. (Color online) Results of spectral recovery of eight randomly selected samples of Munsell chips from their XYZ tristimulus values using wCCR, CCR, wPCA, and PCA methods. STD represents target data.
from the methods along with the original ones are illustrated in Figs. 1 and 2. As Table 1 shows, implementation of the weighted version of CCR and PCA leads to a considerable decrease
in RMS error in comparison to the standard techniques. The explanation is that the weighted methods weigh the samples of the original dataset according to their distance from the given target and modify datasets so that the influence of
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(b)
(c)
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(e)
(f)
(g)
(h)
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Fig. 2. (Color online) Results of spectral recovery of eight randomly selected samples of Colorchecker DC from their XYZ tristimulus values using wCCR, CCR, wPCA, and PCA methods. STD represents target data.
samples which are similar or closer to the proposed sample become more significant. As Table 1 shows, the reconstruction results by classical CCR technique are obviously better than by the PCA method. It should be emphasized that the
CCR method incorporates thirty-one color specification terms (including XYZ tristimulus values and their cross products) in the simulation process while the classical PCA routine deals with three basic colorimetric coordinates. The
where Ract ðj Þ is the measured original spectral data at the wavelength j and Rest ðj Þ is the estimated spectral data at the wavelength j .25) The mean, maximum and the standard deviation of GFC values obtained from different techniques are shown in Table 3. GFC values were suggested as metrics to evaluate reproduced results. Figures 3 and 4 show the frequencies of poor (GFC < 0:995), good (GFC 0:995), very good (GFC 0:999) and excellent (GFC 0:9999) reconstruction results of samples of Munsell chips and Colorchecker DC which were achieved by different techniques. These figures reconfirm the previous metrics and show a type of priority for the wCCR method in comparison with other recovery techniques.
Table 3. Mean, maximum and the standard deviation of GFC values of different methods. Sample
GFC Value
Method Mean
Max
SD
Munsell
wCCR CCR wPCA PCA
0.9988 0.9901 0.9973 0.9935
1.0000 1.0000 1.0000 0.9999
0.0051 0.0196 0.0081 0.0121
Colorchecker
wCCR CCR wPCA PCA
0.9967 0.9880 0.9964 0.9889
0.9999 0.9999 0.9999 0.9998
0.0060 0.0179 0.0083 0.0166
SD: standard deviation
700 600
Frequency
comparison between PCA and CCA shows that CCA can achieve two goals simultaneously. Firstly, it can give the maximum correlation between two datasets, i.e. colorimetric and spectral variables. Secondly, like PCA, it can give the optimal explanation of variability within the subgroup of variables.16) In fact in CCA, the between-set covariance matrices are normalized with respect to the within-set covariances in both the X and the Y spaces. On the other hand, PCA only concerns one set of variables and reduces the dimensionality of this dataset.18) Table 2 reconfirms these finding from the colorimetric point of view. Clearly, the color difference values under different illuminants are smaller for the weighted versions of CCR and PCA methods in comparison with classical ones. Again, the CCR methods lead to a higher degree of precision in comparison with the PCA technique. As Table 2 shows, the color difference values for the Munsell chips are smaller than Colorchecker samples since the same group of samples was employed to obtain fundamental bases. Figures 1 and 2 show the recovery results for eight randomly selected samples of Munsell chips and Colorchecker DC, respectively. As these figures show, the wCCR method almost duplicates the actual reflectance curves and benefits from the highest accuracy for both datasets. Method performance was also examined by calculation of the goodness-of-fit coefficient (GFC) of the recovered reflectance spectrum, using eq. (16): X Ract ðj ÞRest ðj Þ j ð16Þ GFC ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u uX uX t ½Ract ðj Þ2 t ½Rest ðj Þ2 j j
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500
Excellent
400
Very Good Good
300
Poor 200 100 0
CCR
wCCR
PCA
wPCA
Fig. 3. Frequencies of poor, good, very good and excellent reconstruction results of Munsell chips by different methods.
140 120
Frequency
302
100
Excellent Very Good
80
Good
60
Poor 40 20 0
CCR
wCCR
PCA
wPCA
Fig. 4. Frequencies of poor, good, very good and excellent reconstruction results of Colorchecker DC by different methods.
Conclusions
A method based on implementation of weighted canonical correlation regression was presented for recovery of spectral data from the corresponding tristimulus values. Instead of using a fixed set of input variables in the CCR technique, a series of dynamic variables which vary with the XYZ tristimulus values of a desired sample were used for each target sample. The color differences between a target and samples of an original dataset under illuminant D65 and 1964 standard observer, i.e. Euclidean distance in CIELAB color space, were considered a criterion for determination of
weighting factors. The desired weights for samples in a dataset were applied on the reflectance and colorimetric data of a Munsell dataset prior to determination of canonical coefficients and correlations. Therefore, different canonical coefficients and correlations were extracted depending on the color specifications of a target. The performance of the suggested method was evaluated for reproduction of the reflectance data of 1269 Matt Munsell color chips as well as 237 colored samples of GretagMacbeth Colorchecker DC. To estimate the reconstruction accuracy of color spectra, the
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color difference values under illuminants A and TL84 and the RMS errors between the targets and the estimated spectra were calculated; the goodness-of-fit coefficients were also determined. The obtained results from the suggested modification showed a significant improvement from a colorimetric and spectrophotometric point of view, in comparison to the standard CCR, PCA, and wPCA approaches. References 1) D. Dupont: Color Res. Appl. 27 (2002) 88. 2) G. Sharma and S. Wang: Proc. SPIE 4663 (2002) 8. 3) D. H. Marimont and B. A. Wandell: J. Opt. Soc. Am. A 9 (1992) 1905. 4) H. S. Fairman and M. H. Brill: Color Res. Appl. 29 (2004) 104. 5) D. Connah, S. Westland, and M. G. A. Thomson: Color Technol. 117 (2001) 309. 6) L. T. Maloney: J. Opt. Soc. Am. A 3 (1986) 29. 7) T. Jaaskelainen, J. Parkkinen, and S. Toyooka: J. Opt. Soc. Am. A 7 (1990) 725. 8) M. J. Vrhel, R. Gershon, and L. S. Iwan: Color Res. Appl. 19 (1994) 4. 9) J. Romero, A. Garcia-Beltran, and J. Hernandez-Andres: J. Opt. Soc. Am. A 14 (1997) 1007. 10) A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andres, and J. Romero: Color Res. Appl. 23 (1998) 39.
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