Abstract

The color rendering of superposed coloring components is often an issue either to predict or to simulate the appearance of colored surfaces. In graphical software, for example, transparence options are available to display different layouts on top of each other. With two colored layers, tuning the transparency of the top layer enables transitioning continuously from the color of this top layer to the color of the bottom layer. However, these options based very often only on additive color mixing offer limited transitions between two base colors. It would be advantageous to introduce more advanced options, providing, for example, realistic renderings of superposed paint layers. This is the aim of the present study, where simple models are proposed to create intermediate configurations between additive and subtractive color mixings. These models rely on the spectral power distribution of a finite set of primaries with given proportions. They may be extended to RGB color reflective or transmissive systems if the red, green, and blue wavebands do not overlap each other. An additional parameter is introduced to tune the proportions of additive and subtractive mixings, each type of mixing being based on its set of primaries. Various simulations of color mixings are presented, illustrating the possibilities offered by this model in addition to those permitted by the purely additive and subtractive mixings.

© 2013 Optical Society of America

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References

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  1. H. von Helmholtz, Handbuch der Physiologischen Optik (Voss, 1860), Band II, Sektion 20.
  2. R. G. Kuehni and A. Schwarz, Color Ordered: A Survey of Color Systems from Antiquity to the Present (Oxford University, 2008).
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  11. R. Levinson, P. Berdahl, and H. Akbari, “Solar spectral optical properties of pigments—Part II: survey of common colorants,” Sol. Energy Mater. Sol. Cells 89, 351–389 (2005).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  19. T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts, J. Johnson, ed. (Johnson, 1807).
  20. J. C. Maxwell, “Experiments on colour, as perceived by the eye, with remarks on colour-blindness,” Trans. R. Soc. Edinburgh 21, 275–297 (1855).
  21. J. C. Maxwell, “On the theory of compound colours, and the relations of the colours of the spectrum,” Philos. Trans. R. Soc. London 150, 57–84 (1860).
    [CrossRef]
  22. F. Metelli, “The perception of transparency,” Sci. Am. 230, 90–98 (1974).
    [CrossRef]
  23. M. D’Zmura, P. Colantoni, K. Knoblauch, and B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
    [CrossRef]

2009 (1)

2006 (1)

2005 (2)

R. Levinson, P. Berdahl, and H. Akbari, “Solar spectral optical properties of pigments—Part I: model for deriving scattering and absorption coefficients from transmittance and reflectance measurements,” Sol. Energy Mater. Sol. Cells 89, 319–349 (2005).

R. Levinson, P. Berdahl, and H. Akbari, “Solar spectral optical properties of pigments—Part II: survey of common colorants,” Sol. Energy Mater. Sol. Cells 89, 351–389 (2005).

1997 (1)

M. D’Zmura, P. Colantoni, K. Knoblauch, and B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
[CrossRef]

1985 (1)

J. A. S. Viggiano, “The color of halftone tints,” Proc. TAGA 37, 647–661 (1985).

1974 (1)

F. Metelli, “The perception of transparency,” Sci. Am. 230, 90–98 (1974).
[CrossRef]

1954 (1)

1951 (1)

J. Yule and W. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Proc. TAGA 3, 65–76 (1951).

1949 (1)

D. Duncan, “The colour of pigment mixtures,” Journal of Oil Colour Chemistry Association 32, 296–321 (1949).

1937 (1)

H. E. J. Neugebauer, “Die Theoretischen grundlagen Des Mehrfarbendruckes,” Zeitschrift für wissenschaftliche Photographie, Photophysik und Photochemie 36, 36–73 (1937).

1936 (1)

A. Murray, “Monochrome reproduction in photoengraving,” J. Franklin Inst. 221, 721–744 (1936).
[CrossRef]

1931 (1)

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeitschrift für technische Physik 12, 593–601 (1931).

1860 (1)

J. C. Maxwell, “On the theory of compound colours, and the relations of the colours of the spectrum,” Philos. Trans. R. Soc. London 150, 57–84 (1860).
[CrossRef]

1855 (1)

J. C. Maxwell, “Experiments on colour, as perceived by the eye, with remarks on colour-blindness,” Trans. R. Soc. Edinburgh 21, 275–297 (1855).

Akbari, H.

R. Levinson, P. Berdahl, and H. Akbari, “Solar spectral optical properties of pigments—Part I: model for deriving scattering and absorption coefficients from transmittance and reflectance measurements,” Sol. Energy Mater. Sol. Cells 89, 319–349 (2005).

R. Levinson, P. Berdahl, and H. Akbari, “Solar spectral optical properties of pigments—Part II: survey of common colorants,” Sol. Energy Mater. Sol. Cells 89, 351–389 (2005).

Bartleson, C. J.

F. Grum and C. J. Bartleson, Optical Radiation Measurement (Academic, 1980), Vol. 2.

Berdahl, P.

R. Levinson, P. Berdahl, and H. Akbari, “Solar spectral optical properties of pigments—Part I: model for deriving scattering and absorption coefficients from transmittance and reflectance measurements,” Sol. Energy Mater. Sol. Cells 89, 319–349 (2005).

R. Levinson, P. Berdahl, and H. Akbari, “Solar spectral optical properties of pigments—Part II: survey of common colorants,” Sol. Energy Mater. Sol. Cells 89, 351–389 (2005).

Byrne, G.

Colantoni, P.

M. D’Zmura, P. Colantoni, K. Knoblauch, and B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
[CrossRef]

D’Zmura, M.

M. D’Zmura, P. Colantoni, K. Knoblauch, and B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
[CrossRef]

Dorffner, G.

Duncan, D.

D. Duncan, “The colour of pigment mixtures,” Journal of Oil Colour Chemistry Association 32, 296–321 (1949).

Elias, M.

Frigerio, J. M.

Grum, F.

F. Grum and C. J. Bartleson, Optical Radiation Measurement (Academic, 1980), Vol. 2.

Knoblauch, K.

M. D’Zmura, P. Colantoni, K. Knoblauch, and B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
[CrossRef]

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensively light scattering material. Part II: non-homogeneous layers,” J. Opt. Soc. Am. 44, 330–335 (1954).
[CrossRef]

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeitschrift für technische Physik 12, 593–601 (1931).

Kuehni, R. G.

R. G. Kuehni and A. Schwarz, Color Ordered: A Survey of Color Systems from Antiquity to the Present (Oxford University, 2008).

Laget, B.

M. D’Zmura, P. Colantoni, K. Knoblauch, and B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
[CrossRef]

Lambert, J. H.

J. H. Lambert, Beschreibung einer mit dem Calaunischen Wachse ausgemalten Farbenpyramide, Haude and Spener, eds. (Haude and Spener, 1772).

Latour, G.

Levinson, R.

R. Levinson, P. Berdahl, and H. Akbari, “Solar spectral optical properties of pigments—Part I: model for deriving scattering and absorption coefficients from transmittance and reflectance measurements,” Sol. Energy Mater. Sol. Cells 89, 319–349 (2005).

R. Levinson, P. Berdahl, and H. Akbari, “Solar spectral optical properties of pigments—Part II: survey of common colorants,” Sol. Energy Mater. Sol. Cells 89, 351–389 (2005).

Lewandowski, A.

Ludl, M.

Maxwell, J. C.

J. C. Maxwell, “On the theory of compound colours, and the relations of the colours of the spectrum,” Philos. Trans. R. Soc. London 150, 57–84 (1860).
[CrossRef]

J. C. Maxwell, “Experiments on colour, as perceived by the eye, with remarks on colour-blindness,” Trans. R. Soc. Edinburgh 21, 275–297 (1855).

Mayer, T.

T. Mayer, De affinitate colorum commentatio, in Opera inedita Tobiae Mayeri, G. C. Lichtenberg, ed. (Göttingen, 1775).

Metelli, F.

F. Metelli, “The perception of transparency,” Sci. Am. 230, 90–98 (1974).
[CrossRef]

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeitschrift für technische Physik 12, 593–601 (1931).

Murray, A.

A. Murray, “Monochrome reproduction in photoengraving,” J. Franklin Inst. 221, 721–744 (1936).
[CrossRef]

Neugebauer, H. E. J.

H. E. J. Neugebauer, “Die Theoretischen grundlagen Des Mehrfarbendruckes,” Zeitschrift für wissenschaftliche Photographie, Photophysik und Photochemie 36, 36–73 (1937).

Nielsen, W.

J. Yule and W. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Proc. TAGA 3, 65–76 (1951).

Schwarz, A.

R. G. Kuehni and A. Schwarz, Color Ordered: A Survey of Color Systems from Antiquity to the Present (Oxford University, 2008).

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

Viggiano, J. A. S.

J. A. S. Viggiano, “The color of halftone tints,” Proc. TAGA 37, 647–661 (1985).

von Helmholtz, H.

H. von Helmholtz, Handbuch der Physiologischen Optik (Voss, 1860), Band II, Sektion 20.

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

Young, T.

T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts, J. Johnson, ed. (Johnson, 1807).

Yule, J.

J. Yule and W. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Proc. TAGA 3, 65–76 (1951).

Appl. Spectrosc. (1)

J. Franklin Inst. (1)

A. Murray, “Monochrome reproduction in photoengraving,” J. Franklin Inst. 221, 721–744 (1936).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Journal of Oil Colour Chemistry Association (1)

D. Duncan, “The colour of pigment mixtures,” Journal of Oil Colour Chemistry Association 32, 296–321 (1949).

Perception (1)

M. D’Zmura, P. Colantoni, K. Knoblauch, and B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
[CrossRef]

Philos. Trans. R. Soc. London (1)

J. C. Maxwell, “On the theory of compound colours, and the relations of the colours of the spectrum,” Philos. Trans. R. Soc. London 150, 57–84 (1860).
[CrossRef]

Proc. TAGA (2)

J. A. S. Viggiano, “The color of halftone tints,” Proc. TAGA 37, 647–661 (1985).

J. Yule and W. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Proc. TAGA 3, 65–76 (1951).

Sci. Am. (1)

F. Metelli, “The perception of transparency,” Sci. Am. 230, 90–98 (1974).
[CrossRef]

Sol. Energy Mater. Sol. Cells (2)

R. Levinson, P. Berdahl, and H. Akbari, “Solar spectral optical properties of pigments—Part I: model for deriving scattering and absorption coefficients from transmittance and reflectance measurements,” Sol. Energy Mater. Sol. Cells 89, 319–349 (2005).

R. Levinson, P. Berdahl, and H. Akbari, “Solar spectral optical properties of pigments—Part II: survey of common colorants,” Sol. Energy Mater. Sol. Cells 89, 351–389 (2005).

Trans. R. Soc. Edinburgh (1)

J. C. Maxwell, “Experiments on colour, as perceived by the eye, with remarks on colour-blindness,” Trans. R. Soc. Edinburgh 21, 275–297 (1855).

Zeitschrift für technische Physik (1)

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeitschrift für technische Physik 12, 593–601 (1931).

Zeitschrift für wissenschaftliche Photographie, Photophysik und Photochemie (1)

H. E. J. Neugebauer, “Die Theoretischen grundlagen Des Mehrfarbendruckes,” Zeitschrift für wissenschaftliche Photographie, Photophysik und Photochemie 36, 36–73 (1937).

Other (8)

T. Mayer, De affinitate colorum commentatio, in Opera inedita Tobiae Mayeri, G. C. Lichtenberg, ed. (Göttingen, 1775).

J. H. Lambert, Beschreibung einer mit dem Calaunischen Wachse ausgemalten Farbenpyramide, Haude and Spener, eds. (Haude and Spener, 1772).

T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts, J. Johnson, ed. (Johnson, 1807).

F. Grum and C. J. Bartleson, Optical Radiation Measurement (Academic, 1980), Vol. 2.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

CIE, “Colorimetry,” 3rd ed., (CIE, 2004).

H. von Helmholtz, Handbuch der Physiologischen Optik (Voss, 1860), Band II, Sektion 20.

R. G. Kuehni and A. Schwarz, Color Ordered: A Survey of Color Systems from Antiquity to the Present (Oxford University, 2008).

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Figures (9)

Fig. 1.
Fig. 1.

Color cells describing the different mixing laws.

Fig. 2.
Fig. 2.

Spectra resulting in binary color mixing (Nc=2) between blue and yellow. (a) Additive mixing, (b) subtractive mixing, (c) additive–subtractive mixing (with τ=0.5), and (d) subtractive–additive mixing (with τ=0.5). The proportion varies from 0 to 1 by steps of 0.1.

Fig. 3.
Fig. 3.

CIELab coordinates resulting from different binary color mixings between yellow, blue, and red. (a) Variation of the lightness L* in terms of the proportion and (b) variation in the (a*, b*) plane. The proportion varies from 0 to 1 by steps of 0.1.

Fig. 4.
Fig. 4.

Variation as a function of the mixing parameter τ of the color difference ΔE between the Yule–Nielsen modified spectral Neugebauer mixing laws and the additive–subtractive (gray line) or subtractive–additive (black line) mixing law, for a binary color mixing with c=0.5 between yellow and blue primaries.

Fig. 5.
Fig. 5.

Flowchart for the color mixing calculation from RGB values with three primaries.

Fig. 6.
Fig. 6.

Binary color mixing in RGB between yellow, blue, and red by using the additive–subtractive mixing law [Eq. (21) with Nc=2]. The proportion c varies from 0 to 1 by steps of 0.1, and the mixing parameter τ varies from 0 to 1 by steps of 0.25. The RGB values are written in each box.

Fig. 7.
Fig. 7.

Color mixing triangle. (a) Representation of Mayer triangle by Lichtenberg from [16]. (b)–(d) Ternary color mixing in RGB between yellow, blue, and red by using the additive–subtractive mixing law [Eq. (21) with Nc=3] for τ=0, 0.5, and 1, respectively. The RGB values are written in each box.

Fig. 8.
Fig. 8.

Binary color mixing computed from the RGB values of blue, purple, and white by using the additive–subtractive mixing law [Eq. (21) with Nc=2]. The proportion c varies from 0 to 1 by steps of 0.1, and the mixing parameter τ varies from 0 to 1 by steps of 0.25. The RGB values are indicated in each color patch.

Fig. 9.
Fig. 9.

Color transparency in RGB color system. Foreground color: yellow (R=240, G=200, B=20) with a transparency rate c=0.5. Additive–subtractive mixing [Eq. (26)] for different values of τ. Background colors: black, blue, and white.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

0Pλ1λ.
{X=kλPλSλx¯λY=kλPλSλy¯λZ=kλPλSλz¯λ,
0ci1
i=1Nci=1.
i=1Ncai=1,
Pλ=i=1NcciPi,λ
i=1Ncti=1.
Pλ=i=1NcPi,λci
0<Pλ1λ.
KS=(1R)22R.
KS=iNcciKiiNcciSi,
Rλ=R1,λ+T1,λ2R2,λ1R1,λR2,λ,
Tλ=T1,λT2,λ1R1,λR2,λ.
Pλ=(i=1NcciPi,λ1/n)n,
Pλ=(i=1NcciPi,λτ)1/τ.
τ=i=1Ncai,
i=1Ncti=1.
ci=ai+(1τ)ti.
{ai=τciti=ci.
Pλ=i=1NcaiPi,λ+(1i=1Ncai)i=1NcPi,λti,
Pλ=τi=1NcciPi,λ+(1τ)i=1NcPi,λci.
τ=1i=1Ncti,
i=1Ncai=1.
ci=ti+τai.
{ai=citi=(1τ)ci.
Pλ=(i=1NcaiPi,λ(1i=1Ncti)).i=1NcPi,λti,
Pλ=(i=1NcciPi,λτ).i=1NcPi,λci(1τ).
X=i=1NcciXiwithX=R,GorB.
x=X+1256,

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