Abstract

Color modeling of translucent and opaque media commonly uses two-constant Kubelka–Munk (KM) turbid media theory. KM theory is designed for isotropic color systems that rely on absorption and scatter to produce an overall reflected color. KM theory has previously been considered inadequate to use with interference pigments (IPs) due to their specular reflected, angle-dependent color and anisotropic behavior. If, however, an IP’s reflected color is considered to contribute to the background reflectance and not as a colorant in a mixture with a conventional colorant, KM theory can be used. KM theory was successfully implemented to predict the goniospectrophotometric, normalized spectral reflectance of conventional colorants and IP mixtures.

© 2013 Optical Society of America

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References

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  1. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef]
  2. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part II: nonhomogeneous layers,” J. Opt. Soc. Am. 44, 330–335 (1954).
    [CrossRef]
  3. D. R. Duncan, “The colour of pigment mixtures,” Proc. Phys. Soc. London 52, 390–401 (1940).
    [CrossRef]
  4. J. W. Ryde, “The scattering of light by turbid media: part I,” Proc. R. Soc. A 131, 451–464 (1931).
    [CrossRef]
  5. J. L. Saunderson, “Calculation of the color of pigmented plastics,” J. Opt. Soc. Am. 32, 727–735 (1942).
    [CrossRef]
  6. G. A. Klein, Industrial Color Physics (Springer, 2010), pp. 337–340.
  7. A. Rodrigues, “Color technology and paint,” in Association Internationale de la Couleur (AIC) Color and Paints, Interim Meeting of the International Color Association Proceedings, 2004, pp. 103–108.
  8. J. D. T. Kruschwitz and R. J. Berns, “Color modeling of paint mixtures containing conventional and interference pigments,” Proceedings from the OSA Optical Interference Coating Topical Meeting,  2013, paper MB8.
  9. R. Berns, “A generic approach to color modeling,” Color Res. Appl. 22, 318–325 (1997).
    [CrossRef]
  10. Y. Okumura, “Developing a spectral and colorimetric database of artist paint materials,” M.S. Thesis (Rochester Institute of Technology, College of Science, Center for Imaging Science, September 2005).

1997 (1)

R. Berns, “A generic approach to color modeling,” Color Res. Appl. 22, 318–325 (1997).
[CrossRef]

1954 (1)

1948 (1)

1942 (1)

1940 (1)

D. R. Duncan, “The colour of pigment mixtures,” Proc. Phys. Soc. London 52, 390–401 (1940).
[CrossRef]

1931 (1)

J. W. Ryde, “The scattering of light by turbid media: part I,” Proc. R. Soc. A 131, 451–464 (1931).
[CrossRef]

Berns, R.

R. Berns, “A generic approach to color modeling,” Color Res. Appl. 22, 318–325 (1997).
[CrossRef]

Berns, R. J.

J. D. T. Kruschwitz and R. J. Berns, “Color modeling of paint mixtures containing conventional and interference pigments,” Proceedings from the OSA Optical Interference Coating Topical Meeting,  2013, paper MB8.

Duncan, D. R.

D. R. Duncan, “The colour of pigment mixtures,” Proc. Phys. Soc. London 52, 390–401 (1940).
[CrossRef]

Klein, G. A.

G. A. Klein, Industrial Color Physics (Springer, 2010), pp. 337–340.

Kruschwitz, J. D. T.

J. D. T. Kruschwitz and R. J. Berns, “Color modeling of paint mixtures containing conventional and interference pigments,” Proceedings from the OSA Optical Interference Coating Topical Meeting,  2013, paper MB8.

Kubelka, P.

Okumura, Y.

Y. Okumura, “Developing a spectral and colorimetric database of artist paint materials,” M.S. Thesis (Rochester Institute of Technology, College of Science, Center for Imaging Science, September 2005).

Rodrigues, A.

A. Rodrigues, “Color technology and paint,” in Association Internationale de la Couleur (AIC) Color and Paints, Interim Meeting of the International Color Association Proceedings, 2004, pp. 103–108.

Ryde, J. W.

J. W. Ryde, “The scattering of light by turbid media: part I,” Proc. R. Soc. A 131, 451–464 (1931).
[CrossRef]

Saunderson, J. L.

Color Res. Appl. (1)

R. Berns, “A generic approach to color modeling,” Color Res. Appl. 22, 318–325 (1997).
[CrossRef]

J. Opt. Soc. Am. (3)

Proc. Phys. Soc. London (1)

D. R. Duncan, “The colour of pigment mixtures,” Proc. Phys. Soc. London 52, 390–401 (1940).
[CrossRef]

Proc. R. Soc. A (1)

J. W. Ryde, “The scattering of light by turbid media: part I,” Proc. R. Soc. A 131, 451–464 (1931).
[CrossRef]

Other (4)

G. A. Klein, Industrial Color Physics (Springer, 2010), pp. 337–340.

A. Rodrigues, “Color technology and paint,” in Association Internationale de la Couleur (AIC) Color and Paints, Interim Meeting of the International Color Association Proceedings, 2004, pp. 103–108.

J. D. T. Kruschwitz and R. J. Berns, “Color modeling of paint mixtures containing conventional and interference pigments,” Proceedings from the OSA Optical Interference Coating Topical Meeting,  2013, paper MB8.

Y. Okumura, “Developing a spectral and colorimetric database of artist paint materials,” M.S. Thesis (Rochester Institute of Technology, College of Science, Center for Imaging Science, September 2005).

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Figures (10)

Fig. 1.
Fig. 1.

Aspecular angles related to the specular reflected beam (top) and the correlated refracted angles exiting the medium compared to the surface normal (bottom). The sun symbol indicates the incident light, and each numbered beam represents a detector in a gonio device.

Fig. 2.
Fig. 2.

Illustration of how the IP layer appears in reality (upper left) and how it can be represented (upper right) to use in the KM model. Also included is the IP/colorant layer and how it is represented in reality (lower left) and how it can be represented (lower right) to fit into the KM model in Eq. (4).

Fig. 3.
Fig. 3.

Normalized reflectance factor of 0%–20% chromium oxide green colorant with blue IP paint mixtures measured at the 45°/15° collection angle.

Fig. 4.
Fig. 4.

Actual concentrations versus predicted concentrations of the four conventional colorants used in the optimization process to calculate k and s. Also included is the linear 11 relationship between actual and predicted to compare the linearity of the data.

Fig. 5.
Fig. 5.

Scaled absorption coefficients for the four conventional colorants calculated from traditional tint ladders using titanium white paint.

Fig. 6.
Fig. 6.

Unit k and unit s for four colorants. Each k was calculated over the wavelength region of 400–700 nm and over six detection angles from the MA98. Each s was calculated for the same wavelength range but for only two groups of three detection angles (low or high angles). Colorants are (a) anthraquinone blue, (b) chromium oxide green, (c) Hansa yellow opaque, and (d) quinacridone magenta.

Fig. 7.
Fig. 7.

Color difference (ΔE00) calculations for measured versus predicted spectral reflectance of single colorant/IP mixtures for four concentrations over six aspecular angles.

Fig. 8.
Fig. 8.

Measured and predicted normalized reflectance for anthraquinone blue and chromium oxide green colorants mixed with the blue IP. Each plot represents four tints (1%, 5%, 10%, and 20%) for each of six detection angle for the MA98.

Fig. 9.
Fig. 9.

Measured and predicted normalized reflectance for Hansa yellow opaque and quinacridone magenta colorants mixed with the blue IP. Each plot represents four tints (1%, 5%, 10%, and 20%) for each of six detection angle for the MA98.

Fig. 10.
Fig. 10.

(Top row) Predicted versus measured normalized reflectance for a 10% tint of Hansa yellow opaque and chromium oxide green (60%–40%) with blue IP and a 12% tint of anthraquinone blue and chromium oxide green (50%–50%) with blue IP for six aspecular angles. (Bottom row) CIELAB color coordinates calculated under D65, 2° observer, for predicted and measured normalized reflectance for both colorant/IP mixtures over six aspecular angles.

Tables (2)

Tables Icon

Table 1. Angle Designations and Their Appropriate Saunderson Correction Constants

Tables Icon

Table 2. ΔE00 and RMS for Two-Colorant/IP Mixtures under D65, 2° Observer, Predicted Versus Measured Data for Normalized Reflectance

Equations (24)

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R=1Rg(abcoth(bsX))aRg+bcoth(bsX),
a=(s+k)s,
b=a21,
R=Rm(1K1)(1K2)+K2Rm.
R=1Rg,(abcoth(bctints))aRg,+bcoth(bctints).
K=i=1Nciki,
S=i=1Ncisi,
cmix=i=1Nci=1,
ctint+cIP Medium=1.
R=1Rg,(abcoth(bSctint))aRg,+bcoth(bSctint),
a=(S+K)S,
b=a21.
Z=1N1λj=1Nλ(R^,λ,i,jR,λ,i,j)2,
0ci0.25,
s0.03.
K1=((non1)(no+n1))2,
ηp,0=n0/cosθ0,
ηp,1=n1/cosθ1,
ηs,0=n0cosθ0,
ηs,1=n1cosθ1,
n0sinθ0=n1sinθ1,
Rp=((ηp,0ηp,1)(ηp,0+ηp,1))2,
Rs=((ηs,oηs,1)(ηs,o+ηs,1))2,
K1=K2=Rave=(Rp+Rs)/2.

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