Jun 29, 2004 ... Kyungpook National University. Billmeyer and Saltzman's. Principles of Color
Technology. Wiley-Interscience Publication. Roy S. Berns ...
Billmeyer and Saltzman’s Principles of Color Technology Wiley-Interscience Publication Roy S. Berns
School of Electrical Engineering and Computer Science Kyungpook National University
Simple-Subtractive Mixing Subtractive
mixing
– Referring to the removal by an object of part of the light coming from a source Simple-subtractive
mixing
– Only absorption without scattering – Colored filters (glass, gelatin, and plastic) and colored liquid Complex-subtractive
mixing
– Scattering as well as absorption 2 / 26
Appropriate
color-mixing law
– Linear system • The spectral properties of each colorant are scalable • Mixtures are an additive combination of individual components
The
series of speciments
– Evaluating mixing law – Varying the amount of a single colorant – Tint ladder, concentration, thickness series, or color ramp – Two methods
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A single colorant is used to produce a color ramp either by varying concentration at a fixed thickness or by varying thickness at a fixed concentration. 4 / 26
Mixtures
– Can be made by gluing different colored glasses into “sandwiches”
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Tλ ,i = tλ
b
Tλ ,i is the internal spectral transmittance
tλ
is Tλ ,i at unit thickness (i.e., 1cm)
b
is thickness
∝
absorption
Bouguer, and later Lambert, discovered that there was an exponential relationship between thickness and spectral transmittance.
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Tλ ,i = tλ
c
c is concentration
Beer found that changes in concentration also had an exponential effect on transmittance.
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Bouguer-Beer
law or Lambert-Beer law
– Linear mixing law for colored materials that do not scatter light 1 = Aλ = aλ bc − log10 (Tλ ,i ) = log10 Tλ ,i Aλ is absorbance aλ is the colorant’s absorptivity
b is the thickness of the sample
c is the concentration of the colorant
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– For color mixtures Aλ ,mix = Aλ ,1 + Aλ , 2 + Aλ ,3 + ... = aλ ,1b1c1 + aλ , 2b2 c2 + aλ ,3b3c3 + ...
– Internal transmittance Tλ ,i ,mix =
1 antilog10 ( Aλ ,mix )
= 10
− Aλ ,mix
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The
relationship between measured transmittance and internal transmittance 1(1 − K1 ) 2 [(1 − K1 ) 4 + 4 K1 Tλ ,m ]1/ 2 2
Tλ ,i =
2
2
2 K1 Tλ ,m
Tλ ,m is measured transmittance K1 is the Fresnel external first surface reflectance
(n − 1) 2 K1 = (n + 1) 2 n is the material’s refractive index
Tλ ,m =
(1 − K1 ) 2 Tλ ,i 1 − K1 Tλ ,i 2
2
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The scalability requirement is met when the normalized absorbance curves (Aλ/A at λ maximum) are nearly identical or when the shapes of the log-absorbance curves are independent of thickness or concentration.
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The additivity requirement is met when the spectral absorbance of a mixture (solid line) can be reproduced by the combination (dotted line) of each colorant’s unit absorptivity.
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The unit absorptivity and absorbance curves from the color ramp are used to calculate scalars, thicknesses, or effective concentrations. beffective=(A/a)at λ maximum beffective=ΣλAλΔ λ /ΣλaλΔ λ Least squares can be used where the optimal scalar is one in which the sum-ofsquares spectral differences between Aλ and aλ are minimized. When the scalability requirement is very well met, all three methods are equivalent. 13 / 26
In this example, there is a linear relationship between theoretical and effective thickness. Least squares is used to fit the line. It is rare not to find a one-to-one relationship between theoretical and effective thickness for materials appropriately modeled using Bougure’s law.
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The steps required in order to convert from a filter’s thickness to its color, defined via CIELAB.
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Numerical
Example
– Finding Dye Concentrations in a Transparent Sample by the Use of Bouguer-Beer’s Law – Refer to p.p. 184
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Complex-Subtractive Mixing Scattering
as well as absorbing light
– The most common type of color mixing we experience – More complex laws than those of simple subtractive mixing – Simplifying assumption
Kubelka-Munk theory assumes that the light flux within a translucent absorbing and scattering layer travels either “up” or “down”.
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Kubelka-Munk
theory
– Considering a translucent colorant layer on top of an opaque background – The key assumption that the light within the colorant layer is completely diffuse – Two-flux theory – Solving a pair of differential equations – Predicting internal reflectance from knowledge of the background – Predicting Absorption and scattering properties of the colorant layer – Predicting thickness of the colorant layer 18 / 26
Translucent plastics, printing inks with appreciable scattering, and paint films not at complete hiding
Photo graphic paper and continuous-tone prints using thermal-transfer technologies
Textiles, paint films and plastics at complete hiding, and dyed paper
Kubelka-Munk theory is most often applied to translucent materials, transparent colored layers on a opaque, scattering support, or opaque materials. 19 / 26
Reflectance
for opaque systems
– The ratio of absorption, K, to scattering, S, (K/S)λ – “two-constant Kubelka-Munk theory” • The scalability and additivity requirements apply to the individual absorption and scattering properties for each colorant • Least-squares techniques are used most often, in which the two coefficients are estimated simultaneously
– “single-constant Kubelka-Munk theory” • For materials where the colorants have negligible scattering properties • Only the K/S ratio is used to characterize a colorant
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– For opaque materials,
Rλ ,i
2 K K K = 1 + − + 2 S λ S λ S λ
1/ 2
(1 − Rλ ,i ) 2 K = 2 Rλ ,i S λ
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– The (K/S)λ ratio of a mixture K K = λ ,mix S λ ,mix S λ ,mix k + c k + c k + c k + ... = λ ,t 1 λ ,1 2 λ , 2 3 λ ,3 sλ ,t + c1sλ ,1 + c2 sλ , 2 + c3 sλ ,3 + ... kλ is the colorant’s unit absorptivity c is the effective concentration Subscript t means the absorption and scattering of the substrate
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– For materials where the colorants do not scatter (e.g. textiles) K K K K K = + c1 + c2 + c3 + ... S λ ,3 S λ ,1 S λ ,2 S λ ,mix S λ ,t
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– The relationship between internal reflectance and measured reflectance using the Saunderson correction Rλ ,i =
Rλ ,m − K1 1 − K1 − K 2 + K 2 Rλ ,m
Rλ ,m = K1 +
(1 − K1 )(1 − K 2 ) Rλ ,i 1 − K 2 Rλ ,i
K1 is the Fresnel reflection coefficient for collimated light K2 is the Fresnel reflection coefficient for diffuse light striking the surface from the inside
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Theoretical
concentration
– Concentration measured by a user such as the concentration of a dye in a dye-bath – This is equivalent to the “user controls” of a generic color model Effective
concentration
– Concentration determined from colorant measurements of the colored material – This is equivalent to the scalars of a generic color model
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Numerical
Example
– Finding Pigment Concentrations in an Opaque Sample by the Use of the Kubelka-Munk Law – Refer to p.p.188
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