Abstract

This paper discusses a number of open problems in color constancy theory whose correct solution is a prerequisite for the theory of the phenomenon. Solutions employing suitable visually meaningful versus physically meaningful basis functions (principal components) are examined. In the former case the starting point is an estimate of the first derivative of the reflectance (illuminant), essential for defining color, instead of an estimate of the reflectance (illuminant), as in the latter. Conceptual consequences are discussed. Mathematical and physical constraints are identified. We compare the results of theories that do or do not ignore them. The following questions are considered. (1) Do unique solutions of the estimation problem exist everywhere in the object-color solid belonging to the illuminant? (2) Are they physically meaningful, i.e., at least nonnegative? (3) Are they representative for reflectance and spectral distribution functions? (4) What role plays metamerism?

© 2014 Optical Society of America

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References

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  1. C. E. Fröberg, Introduction to Numerical Analysis (Addison-Wesley, 1966), p. 297.
  2. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982).
  3. D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–35 (1990).
    [CrossRef]
  4. C. van Trigt, “Illuminant-dependence of von Kries type quotients,” Int. J. Comput. Vis. 61, 5–30 (2005).
    [CrossRef]
  5. C. van Trigt, “Von Kries versus color constancy,” Color Res. Appl.35, 164–183 (2010). A(λ)E(λ) and S(λ)/E(λ) in the present paper are denoted there by A(λ) and S(λ).
    [CrossRef]
  6. J. A. C. Yule, Principles of Color Reproduction (Wiley, 1967).
  7. H. E. Ives, “The relation between the color of the illuminant and the color of the illuminated object,” Trans. Illum. Eng. Soc. 7, 62–72 (1912).
  8. G. Polya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Band II (Springer-Verlag, 1964), aufgabe 63, p. 114.
  9. C. van Trigt, “Linear models in color constancy theory,” J. Opt. Soc. Am. A 24, 2684–2691 (2007).
    [CrossRef]
  10. D. Jameson and L. M. Hurvich, “Essay concerning colour constancy,” Ann. Rev. Psychol. 40, 1–22 (1989).
    [CrossRef]
  11. D. A. Foster, “Color constancy,” Vis. Res. 51, 674–700 (2011).
    [CrossRef]
  12. C. van Trigt, “Smoothest reflectance functions. I. Definition and main results,” J. Opt. Soc. Am. A 7, 1891–1904 (1990).
    [CrossRef]
  13. C. van Trigt, “Smoothest reflectance functions. II. Complete results,” J. Opt. Soc. Am. A 7, 2208–2222 (1990).
    [CrossRef]
  14. G. Szegö, Orthogonal Polynomials (American Mathematical Society, 1967).
  15. J. N. Lythgoe, The Ecology of Vision (Clarendon, 1997).
  16. CIE, Method of measuring and specifying colour rendering properties of light sources (Bureau Central de la CIE, 1974).
  17. J. M. Troost and C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Color Res. Appl. 17, 316–327 (1992).
    [CrossRef]
  18. G. Iverson and M. D’Zmura, “Criteria for color constancy in trichromatic bilinear models,” J. Opt. Soc. Am. A 11, 1970–1975 (1994).
    [CrossRef]
  19. J. I. Dannemiller, “Computational approaches to color constancy, adaptive and ontogenetic considerations,” Psychol. Rev. 96, 255–266 (1989).
    [CrossRef]
  20. J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
    [CrossRef]
  21. C. van Trigt, “Metameric blacks and estimating reflectance,” J. Opt. Soc. Am. A 11, 1003–1024 (1994).
    [CrossRef]
  22. D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives (Kluwer Academic, 1991). Wirtinger’s inequality occurs in two versions.
  23. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University, 1962).
  24. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions II (McGraw-Hill, 1953), p. 178.
  25. Reference [23], p. 161, example 1 on interval (−π, +π).
  26. Reference [24], p. 213, formula (7) with λ=1/2, take the imaginary part on both sides, apply p. 179, formula (3) and p. 78, formula (7).
  27. M. D’Zmura, “Color constancy: surface color from changing illumination,” J. Opt. Soc. Am. A 9, 490–493 (1992), Table 1.
    [CrossRef]
  28. J. P. S. Parkkinen, J. Hallikainen, and T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989), Fig. 1.
    [CrossRef]
  29. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982), p. 196.
  30. H. Lang, “Color vision theories in the nineteenth century Germany between idealism and empiricism,” Color Res. Appl. 12, 270–281 (1987).
    [CrossRef]
  31. L. M. Hurvich, Color Vision (Sinauer, 1981).
  32. E. Brunswick, “Zur Entwicklung der Albedowahrneming,” Z. Psychol. 64, 216–227 (1928).

2011

D. A. Foster, “Color constancy,” Vis. Res. 51, 674–700 (2011).
[CrossRef]

2007

2005

C. van Trigt, “Illuminant-dependence of von Kries type quotients,” Int. J. Comput. Vis. 61, 5–30 (2005).
[CrossRef]

1994

1992

M. D’Zmura, “Color constancy: surface color from changing illumination,” J. Opt. Soc. Am. A 9, 490–493 (1992), Table 1.
[CrossRef]

J. M. Troost and C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Color Res. Appl. 17, 316–327 (1992).
[CrossRef]

1990

1989

D. Jameson and L. M. Hurvich, “Essay concerning colour constancy,” Ann. Rev. Psychol. 40, 1–22 (1989).
[CrossRef]

J. P. S. Parkkinen, J. Hallikainen, and T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989), Fig. 1.
[CrossRef]

J. I. Dannemiller, “Computational approaches to color constancy, adaptive and ontogenetic considerations,” Psychol. Rev. 96, 255–266 (1989).
[CrossRef]

1987

H. Lang, “Color vision theories in the nineteenth century Germany between idealism and empiricism,” Color Res. Appl. 12, 270–281 (1987).
[CrossRef]

1982

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef]

1928

E. Brunswick, “Zur Entwicklung der Albedowahrneming,” Z. Psychol. 64, 216–227 (1928).

1912

H. E. Ives, “The relation between the color of the illuminant and the color of the illuminated object,” Trans. Illum. Eng. Soc. 7, 62–72 (1912).

Brunswick, E.

E. Brunswick, “Zur Entwicklung der Albedowahrneming,” Z. Psychol. 64, 216–227 (1928).

Cohen, J. B.

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef]

D’Zmura, M.

Dannemiller, J. I.

J. I. Dannemiller, “Computational approaches to color constancy, adaptive and ontogenetic considerations,” Psychol. Rev. 96, 255–266 (1989).
[CrossRef]

de Weert, C. M. M.

J. M. Troost and C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Color Res. Appl. 17, 316–327 (1992).
[CrossRef]

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions II (McGraw-Hill, 1953), p. 178.

Fink, A. M.

D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives (Kluwer Academic, 1991). Wirtinger’s inequality occurs in two versions.

Forsyth, D. A.

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–35 (1990).
[CrossRef]

Foster, D. A.

D. A. Foster, “Color constancy,” Vis. Res. 51, 674–700 (2011).
[CrossRef]

Fröberg, C. E.

C. E. Fröberg, Introduction to Numerical Analysis (Addison-Wesley, 1966), p. 297.

Hallikainen, J.

Hurvich, L. M.

D. Jameson and L. M. Hurvich, “Essay concerning colour constancy,” Ann. Rev. Psychol. 40, 1–22 (1989).
[CrossRef]

L. M. Hurvich, Color Vision (Sinauer, 1981).

Iverson, G.

Ives, H. E.

H. E. Ives, “The relation between the color of the illuminant and the color of the illuminated object,” Trans. Illum. Eng. Soc. 7, 62–72 (1912).

Jaaskelainen, T.

Jameson, D.

D. Jameson and L. M. Hurvich, “Essay concerning colour constancy,” Ann. Rev. Psychol. 40, 1–22 (1989).
[CrossRef]

Kappauf, W. E.

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef]

Lang, H.

H. Lang, “Color vision theories in the nineteenth century Germany between idealism and empiricism,” Color Res. Appl. 12, 270–281 (1987).
[CrossRef]

Lythgoe, J. N.

J. N. Lythgoe, The Ecology of Vision (Clarendon, 1997).

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions II (McGraw-Hill, 1953), p. 178.

Mitrinovic, D. S.

D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives (Kluwer Academic, 1991). Wirtinger’s inequality occurs in two versions.

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions II (McGraw-Hill, 1953), p. 178.

Parkkinen, J. P. S.

Pecaric, J. E.

D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives (Kluwer Academic, 1991). Wirtinger’s inequality occurs in two versions.

Polya, G.

G. Polya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Band II (Springer-Verlag, 1964), aufgabe 63, p. 114.

Stiles, W. S.

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

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982), p. 196.

Szegö, G.

G. Szegö, Orthogonal Polynomials (American Mathematical Society, 1967).

G. Polya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Band II (Springer-Verlag, 1964), aufgabe 63, p. 114.

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions II (McGraw-Hill, 1953), p. 178.

Troost, J. M.

J. M. Troost and C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Color Res. Appl. 17, 316–327 (1992).
[CrossRef]

van Trigt, C.

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University, 1962).

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University, 1962).

Wyszecki, G.

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

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982), p. 196.

Yule, J. A. C.

J. A. C. Yule, Principles of Color Reproduction (Wiley, 1967).

Am. J. Psychol.

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef]

Ann. Rev. Psychol.

D. Jameson and L. M. Hurvich, “Essay concerning colour constancy,” Ann. Rev. Psychol. 40, 1–22 (1989).
[CrossRef]

Color Res. Appl.

J. M. Troost and C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Color Res. Appl. 17, 316–327 (1992).
[CrossRef]

H. Lang, “Color vision theories in the nineteenth century Germany between idealism and empiricism,” Color Res. Appl. 12, 270–281 (1987).
[CrossRef]

Int. J. Comput. Vis.

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–35 (1990).
[CrossRef]

C. van Trigt, “Illuminant-dependence of von Kries type quotients,” Int. J. Comput. Vis. 61, 5–30 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Psychol. Rev.

J. I. Dannemiller, “Computational approaches to color constancy, adaptive and ontogenetic considerations,” Psychol. Rev. 96, 255–266 (1989).
[CrossRef]

Trans. Illum. Eng. Soc.

H. E. Ives, “The relation between the color of the illuminant and the color of the illuminated object,” Trans. Illum. Eng. Soc. 7, 62–72 (1912).

Vis. Res.

D. A. Foster, “Color constancy,” Vis. Res. 51, 674–700 (2011).
[CrossRef]

Z. Psychol.

E. Brunswick, “Zur Entwicklung der Albedowahrneming,” Z. Psychol. 64, 216–227 (1928).

Other

D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives (Kluwer Academic, 1991). Wirtinger’s inequality occurs in two versions.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University, 1962).

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions II (McGraw-Hill, 1953), p. 178.

Reference [23], p. 161, example 1 on interval (−π, +π).

Reference [24], p. 213, formula (7) with λ=1/2, take the imaginary part on both sides, apply p. 179, formula (3) and p. 78, formula (7).

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982), p. 196.

L. M. Hurvich, Color Vision (Sinauer, 1981).

G. Szegö, Orthogonal Polynomials (American Mathematical Society, 1967).

J. N. Lythgoe, The Ecology of Vision (Clarendon, 1997).

CIE, Method of measuring and specifying colour rendering properties of light sources (Bureau Central de la CIE, 1974).

G. Polya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Band II (Springer-Verlag, 1964), aufgabe 63, p. 114.

C. E. Fröberg, Introduction to Numerical Analysis (Addison-Wesley, 1966), p. 297.

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

C. van Trigt, “Von Kries versus color constancy,” Color Res. Appl.35, 164–183 (2010). A(λ)E(λ) and S(λ)/E(λ) in the present paper are denoted there by A(λ) and S(λ).
[CrossRef]

J. A. C. Yule, Principles of Color Reproduction (Wiley, 1967).

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Equations (39)

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dS0/dλ=s1[ϕ1(λ)ϕ3(λ)]+s2[ϕ2(λ)ϕ3(λ)].
[ϕ1(λ),ϕ2(λ),ϕ3(λ)]=λbλ[x(λ)/XE,y(λ)/YE,z(λ)/ZE]E(λ)dλ
(dS/dλdS0/dλ)2dλ,
[X0/XEZ0/ZE,Y0/YEZ0/ZE]*=C[s1,s2]*,
ci,j=[ϕi(λ)ϕ3(λ)][ϕj(λ)ϕ3(λ)]dλ.
ρ(λ)=ρ0(λ)+r(λ),
ρ0(λ)E(λ)A(λ)dλ=ρ(λ)E(λ)A(λ)dλr(λ)E(λ)A(λ)dλ.
Q(ρ,S)=ρ(λ)S(λ)A(λ)dλρ(λ)E(λ)A(λ)dλS(λ)A(λ)dλ.
01+x=1+2n(1)n+1sin(nπx)/(πn).
01+sin(πx)=1+2n(1)n(2n+3/2)J2n+3/2(π)P2n+1(x),
1+x=1+2sin(πx)/π+residual Fourier error
1+sin(πx)=1+3x/π+residual polynomial error
R(λ,λ)=mρm(λ)ρm(λ)/M
ρ(λ)S(λ)A(λ)dλ=ρ0(ξ;λ)S0(ξ;λ)A(λ)dλ.
ρ0(ξ;λ)S0(ξ;λ)A(λ)dλ=Yλ1λ2S0(ξ;λ)A(λ)dλ/Ys.
TS[X/X0,Y/Y0,Z/Z0]*=[X/XE,Y/YE,Z/ZE]*,
ρ(λ)S(λ)Ai(λ)dλ/S(λ)Ai(λ)dλ=ρ(λ)E(λ)Ai(λ)dλ.
ρ0(λ)E(λ)Ai(λ)dλ=ρ(λ)E(λ)Ai(λ)dλ,
ρe(λ)=ρ0(λ)+AR1(λ)+BR2(λ)
0ρ(λ)1,
ρ(λ)S(λ)[x(λ),y(λ),z(λ)]dλ=[X,Y,Z],
(dρ/dλ)2w(λ)dλ=minimal.
D[w(λ)Dρ]+[μ1x(λ)/X0+μ2y(λ)/Y0+μ3z(λ)/Z0]S(λ)=0,
w(λ)dρ/dλ=0atλbandλe.
[f1(λ),f2(λ),f3(λ)]=λbλ[x(λ)/X0,y(λ)/Y0,z(λ)/Z0]S(λ)dλ.
w(λ)dρ/dλ=jμjfj(λ),
ρ(λ)=ρ(λe)+jμjλλefj(λ)dλ/w(λ).
μ3=(μ1+μ2),
w(λ)dρ0/dλ=μ1[f1(λ)f3(λ)]+μ2[f2(λ)f3(λ)].
x(λ)+y(λ)+z(λ)A(λλb)α
f1(λ)(λλb)x(λ)S(λ)/[(α+1)X0].
w(λ)=E(λ)D(λ)/D>0;λλbandλe.
M2(λ)=0.5088x(λ)/XE+1.4088y(λ)/YE+0.1000z(λ)/ZE.
[X/X0Z/Z0,Y/Y0Z/Z0]*=C[μ1,μ2]*,
ci,j=[fi(λ)f3(λ)][fj(λ)f3(λ)]dλ/w(λ).
ρ(λe)=ν1X/X0+ν2Y/Y0+ν3Z/Z0;jνj=1.
[ν1,ν2]*=C1[f1,f2]*;fi=[fi(λ)f3(λ)]f3(λ)dλ/w(λ).
ρ(λb)=ν1X/X0+ν2Y/Y0+ν3Z/Z0;jνj=1,
[ν1,ν2]*=C1[g1,g2]*;gi=[gi(λ)g3(λ)]g3(λ)dλ/w(λ).

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