Abstract

The natural world is optically unconstrained. Surface properties may vary from one point to another, and reflected light may vary from one instant to the next. The aim of this work is to quantify some of the physical failures of color vision performance that result from uncertainty. In computational simulations with images of vegetated and nonvegetated outdoor scenes, it is shown that color provides an unreliable guide to surface identity. It is also shown that changes in illuminant may cause colors to no longer match and the relations between individual colors to vary. These failures are generally well described by a measure of the randomness of the colors in scenes, the Shannon entropy. Although uncertainty is intrinsic to the environment, its consequences for color vision can be predicted.

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References

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2017 (5)

R. J. Lee and H. E. Smithson, “Motion of glossy objects does not promote separation of lighting and surface colour,” R. Soc. Open Sci. 4, 171290 (2017).
[Crossref]

D. Weiss, C. Witzel, and K. Gegenfurtner, “Determinants of colour constancy and the blue bias,” I-Perception 8, 1–29 (2017).
[Crossref]

R. Lafer-Sousa and B. R. Conway, “#TheDress: categorical perception of an ambiguous color image,” J. Vis. 17(12), 25 (2017).
[Crossref]

A. Akbarinia and K. Gegenfurtner, “Metameric mismatching in natural and artificial reflectances,” J. Vis. 17(10), 390 (2017).
[Crossref]

P. Olsson and A. Kelber, “Relative colour cues improve colour constancy in birds,” J. Exp. Biol. 220, 1797–1802 (2017).
[Crossref]

2016 (2)

S. M. C. Nascimento, K. Amano, and D. H. Foster, “Spatial distributions of local illumination color in natural scenes,” Vision Res. 120, 39–44 (2016).
[Crossref]

D. H. Foster, K. Amano, and S. M. C. Nascimento, “Time-lapse ratios of cone excitations in natural scenes,” Vision Res. 120, 45–60 (2016).
[Crossref]

2014 (1)

A. Werner, “Spatial and temporal aspects of chromatic adaptation and their functional significance for colour constancy,” Vision Res. 104, 80–89 (2014).
[Crossref]

2013 (2)

I. Marín-Franch and D. H. Foster, “Estimating information from image colors: an application to digital cameras and natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 78–91 (2013).
[Crossref]

K. Chowdhary and P. Dupuis, “Distinguishing and integrating aleatoric and epistemic variation in uncertainty quantification,” ESAIM Math. Model. Numer. Anal. 47, 635–662 (2013).
[Crossref]

2012 (2)

2011 (1)

D. H. Foster, “Color constancy,” Vision Res. 51, 674–700 (2011).
[Crossref]

2010 (2)

J. Gert, “Color constancy, complexity, and counterfactual,” Noûs 44, 669–690 (2010).
[Crossref]

I. Marín-Franch and D. H. Foster, “Number of perceptually distinct surface colors in natural scenes,” J. Vis. 10(9), 9 (2010).
[Crossref]

2008 (3)

2007 (1)

2006 (5)

D. H. Foster, K. Amano, S. M. C. Nascimento, and M. J. Foster, “Frequency of metamerism in natural scenes,” J. Opt. Soc. Am. A 23, 2359–2372 (2006).
[Crossref]

D. H. Foster, K. Amano, and S. M. C. Nascimento, “Color constancy in natural scenes explained by global image statistics,” Vis. Neurosci. 23, 341–349 (2006).

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[Crossref]

M. Grendar, “Entropy and effective support size,” Entropy 8, 169–174 (2006).
[Crossref]

D. I. Warton, I. J. Wright, D. S. Falster, and M. Westoby, “Bivariate line-fitting methods for allometry,” Biol. Rev. 81, 259–291 (2006).
[Crossref]

2005 (4)

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on identification of natural surfaces by apparent colour,” Perception 34, 1003–1008 (2005).
[Crossref]

M. N. Goria, N. N. Leonenko, V. V. Mergel, and P. L. Novi Inverardi, “A new class of random vector entropy estimators and its applications in testing statistical hypotheses,” J. Nonparametr. Stat. 17, 277–297 (2005).
[Crossref]

K. Amano, D. H. Foster, and S. M. C. Nascimento, “Minimalist surface-colour matching,” Perception 34, 1009–1013 (2005).
[Crossref]

S. M. C. Nascimento, D. H. Foster, and K. Amano, “Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes,” J. Opt. Soc. Am. A 22, 1017–1022 (2005).
[Crossref]

2004 (2)

A. Kraskov, H. Stögbauer, and P. Grassberger, “Estimating mutual information,” Phys. Rev. E 69, 066138 (2004).
[Crossref]

T. Schürmann, “Bias analysis in entropy estimation,” J. Phys. A Math. Gen. 37, L295–L301 (2004).
[Crossref]

2003 (1)

M. R. Luo, C. J. Li, R. W. G. Hunt, B. Rigg, and K. J. Smith, “CMC 2002 colour inconstancy index: CMCCON02,” Color. Technol. 119, 280–285 (2003).
[Crossref]

2002 (1)

C. Li, M. R. Luo, B. Rigg, and R. W. G. Hunt, “CMC 2000 chromatic adaptation transform: CMCCAT2000,” Color Res. Appl. 27, 49–58 (2002).
[Crossref]

2001 (2)

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. 26, 340–350 (2001).
[Crossref]

J. Hernández-Andrés, J. Romero, and J. L. Nieves, “Color and spectral analysis of daylight in southern Europe,” J. Opt. Soc. Am. A 18, 1325–1335 (2001).
[Crossref]

2000 (2)

1999 (1)

J. M. Kraft and D. H. Brainard, “Mechanisms of color constancy under nearly natural viewing,” Proc. Natl. Acad. Sci. USA 96, 307–312 (1999).
[Crossref]

1998 (1)

M. R. Pointer and G. G. Attridge, “The number of discernible colours,” Color Res. Appl. 23, 52–54 (1998).
[Crossref]

1997 (1)

S. M. C. Nascimento and D. H. Foster, “Detecting natural changes of cone-excitation ratios in simple and complex coloured images,” Proc. R. Soc. London Ser. B. Biol. Sci. 264, 1395–1402 (1997).
[Crossref]

1994 (1)

D. H. Foster and S. M. C. Nascimento, “Relational colour constancy from invariant cone-excitation ratios,” Proc. R. Soc. London Ser. B. Biol. Sci. 257, 115–121 (1994).
[Crossref]

1993 (1)

J. A. Endler, “The color of light in forests and its implications,” Ecol. Monogr. 63, 1–27 (1993).
[Crossref]

1992 (2)

B. J. Craven and D. H. Foster, “An operational approach to colour constancy,” Vision Res. 32, 1359–1366 (1992).
[Crossref]

D. H. Foster, B. J. Craven, and E. R. H. Sale, “Immediate colour constancy,” Ophthalmic Physiol. Opt. 12, 157–160 (1992).
[Crossref]

1987 (1)

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Probl. Inf. Transm. (Tr. Problemy Peredachi Informatsii) 23, 9–16, 95–101 (1987).

1986 (1)

1983 (1)

R. G. Kuehni, “Metamerism, exact and approximate,” Color Res. Appl. 8, 192 (1983).
[Crossref]

Akbarinia, A.

A. Akbarinia and K. Gegenfurtner, “Metameric mismatching in natural and artificial reflectances,” J. Vis. 17(10), 390 (2017).
[Crossref]

Amano, K.

S. M. C. Nascimento, K. Amano, and D. H. Foster, “Spatial distributions of local illumination color in natural scenes,” Vision Res. 120, 39–44 (2016).
[Crossref]

D. H. Foster, K. Amano, and S. M. C. Nascimento, “Time-lapse ratios of cone excitations in natural scenes,” Vision Res. 120, 45–60 (2016).
[Crossref]

A. J. Reeves, K. Amano, and D. H. Foster, “Color constancy: phenomenal or projective?” Percept. Psychophys. 70, 219–228 (2008).
[Crossref]

D. H. Foster, K. Amano, and S. M. C. Nascimento, “Color constancy in natural scenes explained by global image statistics,” Vis. Neurosci. 23, 341–349 (2006).

D. H. Foster, K. Amano, S. M. C. Nascimento, and M. J. Foster, “Frequency of metamerism in natural scenes,” J. Opt. Soc. Am. A 23, 2359–2372 (2006).
[Crossref]

S. M. C. Nascimento, D. H. Foster, and K. Amano, “Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes,” J. Opt. Soc. Am. A 22, 1017–1022 (2005).
[Crossref]

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on identification of natural surfaces by apparent colour,” Perception 34, 1003–1008 (2005).
[Crossref]

K. Amano, D. H. Foster, and S. M. C. Nascimento, “Minimalist surface-colour matching,” Perception 34, 1009–1013 (2005).
[Crossref]

Arend, L.

L. Arend and A. Reeves, “Simultaneous color constancy,” J. Opt. Soc. Am. A 3, 1743–1751 (1986).
[Crossref]

L. Arend, “Environmental challenges to color constancy,” in Human Vision and Electronic Imaging VI, B. E. Rogowitz and T. N. Pappas, eds. (SPIE, 2001), pp. 392–399.

Attridge, G. G.

M. R. Pointer and G. G. Attridge, “The number of discernible colours,” Color Res. Appl. 23, 52–54 (1998).
[Crossref]

Berns, R. S.

Brainard, D. H.

J. M. Kraft and D. H. Brainard, “Mechanisms of color constancy under nearly natural viewing,” Proc. Natl. Acad. Sci. USA 96, 307–312 (1999).
[Crossref]

Cheung, V.

S. Westland, C. Ripamonti, and V. Cheung, Computational Colour Science Using MATLAB, 2nd ed. (Wiley, 2012).

Chorro, E.

Chowdhary, K.

K. Chowdhary and P. Dupuis, “Distinguishing and integrating aleatoric and epistemic variation in uncertainty quantification,” ESAIM Math. Model. Numer. Anal. 47, 635–662 (2013).
[Crossref]

Conway, B. R.

R. Lafer-Sousa and B. R. Conway, “#TheDress: categorical perception of an ambiguous color image,” J. Vis. 17(12), 25 (2017).
[Crossref]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. (Wiley, 2006).

Cox, D. R.

D. R. Cox and E. J. Snell, The Analysis of Binary Data, 2nd ed. (Chapman and Hall/CRC, 1989).

Craven, B. J.

B. J. Craven and D. H. Foster, “An operational approach to colour constancy,” Vision Res. 32, 1359–1366 (1992).
[Crossref]

D. H. Foster, B. J. Craven, and E. R. H. Sale, “Immediate colour constancy,” Ophthalmic Physiol. Opt. 12, 157–160 (1992).
[Crossref]

Cui, G.

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[Crossref]

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. 26, 340–350 (2001).
[Crossref]

H. Wang, M. R. Luo, G. Cui, and H. Xu, “A comparison between perceptibility and acceptability methods,” in 11th Congress of the International Colour Association (AIC), D. Smith, P. Green-Armytage, M. A. Pope, and N. Harkness, eds. (AIC International Colour Association, 2009), pp. 1–7.

J. Liang, M. Georgoula, N. Zou, G. Cui, and M. R. Luo, “Colour difference evaluation using display colours,” Lighting Res. Technol., 1–13 (2017).
[Crossref]

de Fez, D.

Dupuis, P.

K. Chowdhary and P. Dupuis, “Distinguishing and integrating aleatoric and epistemic variation in uncertainty quantification,” ESAIM Math. Model. Numer. Anal. 47, 635–662 (2013).
[Crossref]

Ekroll, V.

F. Faul and V. Ekroll, “Transparent layer constancy,” J. Vis. 12(12), 7 (2012).
[Crossref]

Endler, J. A.

J. A. Endler, “The color of light in forests and its implications,” Ecol. Monogr. 63, 1–27 (1993).
[Crossref]

Falster, D. S.

D. I. Warton, I. J. Wright, D. S. Falster, and M. Westoby, “Bivariate line-fitting methods for allometry,” Biol. Rev. 81, 259–291 (2006).
[Crossref]

Faul, F.

F. Faul and V. Ekroll, “Transparent layer constancy,” J. Vis. 12(12), 7 (2012).
[Crossref]

Feng, G.

G. Feng and D. H. Foster, “Predicting frequency of metamerism in natural scenes by entropy of colors,” J. Opt. Soc. Am. A 29, A200–A208 (2012).
[Crossref]

D. H. Foster and G. Feng, “Visual and material identity in natural scenes: predicting how often indistinguishable surfaces become distinguishable,” in Predicting Perceptions: 3rd International Conference on Appearance (Lulu Press, 2012), pp. 79–81.

Foster, D. H.

S. M. C. Nascimento, K. Amano, and D. H. Foster, “Spatial distributions of local illumination color in natural scenes,” Vision Res. 120, 39–44 (2016).
[Crossref]

D. H. Foster, K. Amano, and S. M. C. Nascimento, “Time-lapse ratios of cone excitations in natural scenes,” Vision Res. 120, 45–60 (2016).
[Crossref]

I. Marín-Franch and D. H. Foster, “Estimating information from image colors: an application to digital cameras and natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 78–91 (2013).
[Crossref]

G. Feng and D. H. Foster, “Predicting frequency of metamerism in natural scenes by entropy of colors,” J. Opt. Soc. Am. A 29, A200–A208 (2012).
[Crossref]

D. H. Foster, “Color constancy,” Vision Res. 51, 674–700 (2011).
[Crossref]

I. Marín-Franch and D. H. Foster, “Number of perceptually distinct surface colors in natural scenes,” J. Vis. 10(9), 9 (2010).
[Crossref]

A. J. Reeves, K. Amano, and D. H. Foster, “Color constancy: phenomenal or projective?” Percept. Psychophys. 70, 219–228 (2008).
[Crossref]

D. H. Foster, K. Amano, and S. M. C. Nascimento, “Color constancy in natural scenes explained by global image statistics,” Vis. Neurosci. 23, 341–349 (2006).

D. H. Foster, K. Amano, S. M. C. Nascimento, and M. J. Foster, “Frequency of metamerism in natural scenes,” J. Opt. Soc. Am. A 23, 2359–2372 (2006).
[Crossref]

S. M. C. Nascimento, D. H. Foster, and K. Amano, “Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes,” J. Opt. Soc. Am. A 22, 1017–1022 (2005).
[Crossref]

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on identification of natural surfaces by apparent colour,” Perception 34, 1003–1008 (2005).
[Crossref]

K. Amano, D. H. Foster, and S. M. C. Nascimento, “Minimalist surface-colour matching,” Perception 34, 1009–1013 (2005).
[Crossref]

S. M. C. Nascimento and D. H. Foster, “Relational color constancy in achromatic and isoluminant images,” J. Opt. Soc. Am. A 17, 225–231 (2000).
[Crossref]

S. M. C. Nascimento and D. H. Foster, “Detecting natural changes of cone-excitation ratios in simple and complex coloured images,” Proc. R. Soc. London Ser. B. Biol. Sci. 264, 1395–1402 (1997).
[Crossref]

D. H. Foster and S. M. C. Nascimento, “Relational colour constancy from invariant cone-excitation ratios,” Proc. R. Soc. London Ser. B. Biol. Sci. 257, 115–121 (1994).
[Crossref]

B. J. Craven and D. H. Foster, “An operational approach to colour constancy,” Vision Res. 32, 1359–1366 (1992).
[Crossref]

D. H. Foster, B. J. Craven, and E. R. H. Sale, “Immediate colour constancy,” Ophthalmic Physiol. Opt. 12, 157–160 (1992).
[Crossref]

D. H. Foster and G. Feng, “Visual and material identity in natural scenes: predicting how often indistinguishable surfaces become distinguishable,” in Predicting Perceptions: 3rd International Conference on Appearance (Lulu Press, 2012), pp. 79–81.

D. H. Foster, “Estimating limits on colour vision performance in natural scenes,” in AIC Colour 2013, 12th Congress of the International Colour Association, L. MacDonald, S. Westland, and S. Wuerger, eds. (AIC International Colour Association, 2013), pp. 633–636.

Foster, M. J.

Gegenfurtner, K.

A. Akbarinia and K. Gegenfurtner, “Metameric mismatching in natural and artificial reflectances,” J. Vis. 17(10), 390 (2017).
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D. Weiss, C. Witzel, and K. Gegenfurtner, “Determinants of colour constancy and the blue bias,” I-Perception 8, 1–29 (2017).
[Crossref]

Georgoula, M.

J. Liang, M. Georgoula, N. Zou, G. Cui, and M. R. Luo, “Colour difference evaluation using display colours,” Lighting Res. Technol., 1–13 (2017).
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Gert, J.

J. Gert, “Color constancy, complexity, and counterfactual,” Noûs 44, 669–690 (2010).
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Gilabert, E.

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (Institute for Basic Standards, National Bureau of Standards, 1997).

Goria, M. N.

M. N. Goria, N. N. Leonenko, V. V. Mergel, and P. L. Novi Inverardi, “A new class of random vector entropy estimators and its applications in testing statistical hypotheses,” J. Nonparametr. Stat. 17, 277–297 (2005).
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Grassberger, P.

A. Kraskov, H. Stögbauer, and P. Grassberger, “Estimating mutual information,” Phys. Rev. E 69, 066138 (2004).
[Crossref]

Grendar, M.

M. Grendar, “Entropy and effective support size,” Entropy 8, 169–174 (2006).
[Crossref]

Hernández-Andrés, J.

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (Institute for Basic Standards, National Bureau of Standards, 1997).

Huertas, R.

Hunt, R. W. G.

M. R. Luo, C. J. Li, R. W. G. Hunt, B. Rigg, and K. J. Smith, “CMC 2002 colour inconstancy index: CMCCON02,” Color. Technol. 119, 280–285 (2003).
[Crossref]

C. Li, M. R. Luo, B. Rigg, and R. W. G. Hunt, “CMC 2000 chromatic adaptation transform: CMCCAT2000,” Color Res. Appl. 27, 49–58 (2002).
[Crossref]

Kelber, A.

P. Olsson and A. Kelber, “Relative colour cues improve colour constancy in birds,” J. Exp. Biol. 220, 1797–1802 (2017).
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Kontoyiannis, I.

I. Kontoyiannis and M. Madiman, “Sumset inequalities for differential entropy and mutual information,” in IEEE International Symposium on Information Theory (ISIT) (IEEE, 2012).

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L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Probl. Inf. Transm. (Tr. Problemy Peredachi Informatsii) 23, 9–16, 95–101 (1987).

Kraft, J. M.

J. M. Kraft and D. H. Brainard, “Mechanisms of color constancy under nearly natural viewing,” Proc. Natl. Acad. Sci. USA 96, 307–312 (1999).
[Crossref]

Kraskov, A.

A. Kraskov, H. Stögbauer, and P. Grassberger, “Estimating mutual information,” Phys. Rev. E 69, 066138 (2004).
[Crossref]

Kuehni, R. G.

R. G. Kuehni, “Metamerism, exact and approximate,” Color Res. Appl. 8, 192 (1983).
[Crossref]

Lafer-Sousa, R.

R. Lafer-Sousa and B. R. Conway, “#TheDress: categorical perception of an ambiguous color image,” J. Vis. 17(12), 25 (2017).
[Crossref]

Lee, R. J.

R. J. Lee and H. E. Smithson, “Motion of glossy objects does not promote separation of lighting and surface colour,” R. Soc. Open Sci. 4, 171290 (2017).
[Crossref]

Leonenko, N. N.

M. N. Goria, N. N. Leonenko, V. V. Mergel, and P. L. Novi Inverardi, “A new class of random vector entropy estimators and its applications in testing statistical hypotheses,” J. Nonparametr. Stat. 17, 277–297 (2005).
[Crossref]

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Probl. Inf. Transm. (Tr. Problemy Peredachi Informatsii) 23, 9–16, 95–101 (1987).

Li, C.

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[Crossref]

C. Li, M. R. Luo, B. Rigg, and R. W. G. Hunt, “CMC 2000 chromatic adaptation transform: CMCCAT2000,” Color Res. Appl. 27, 49–58 (2002).
[Crossref]

Li, C. J.

M. R. Luo, C. J. Li, R. W. G. Hunt, B. Rigg, and K. J. Smith, “CMC 2002 colour inconstancy index: CMCCON02,” Color. Technol. 119, 280–285 (2003).
[Crossref]

Liang, J.

J. Liang, M. Georgoula, N. Zou, G. Cui, and M. R. Luo, “Colour difference evaluation using display colours,” Lighting Res. Technol., 1–13 (2017).
[Crossref]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (Institute for Basic Standards, National Bureau of Standards, 1997).

Linhares, J. M. M.

Luo, M. R.

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[Crossref]

M. R. Luo, C. J. Li, R. W. G. Hunt, B. Rigg, and K. J. Smith, “CMC 2002 colour inconstancy index: CMCCON02,” Color. Technol. 119, 280–285 (2003).
[Crossref]

C. Li, M. R. Luo, B. Rigg, and R. W. G. Hunt, “CMC 2000 chromatic adaptation transform: CMCCAT2000,” Color Res. Appl. 27, 49–58 (2002).
[Crossref]

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. 26, 340–350 (2001).
[Crossref]

H. Wang, M. R. Luo, G. Cui, and H. Xu, “A comparison between perceptibility and acceptability methods,” in 11th Congress of the International Colour Association (AIC), D. Smith, P. Green-Armytage, M. A. Pope, and N. Harkness, eds. (AIC International Colour Association, 2009), pp. 1–7.

J. Liang, M. Georgoula, N. Zou, G. Cui, and M. R. Luo, “Colour difference evaluation using display colours,” Lighting Res. Technol., 1–13 (2017).
[Crossref]

Madiman, M.

I. Kontoyiannis and M. Madiman, “Sumset inequalities for differential entropy and mutual information,” in IEEE International Symposium on Information Theory (ISIT) (IEEE, 2012).

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I. Marín-Franch and D. H. Foster, “Estimating information from image colors: an application to digital cameras and natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 78–91 (2013).
[Crossref]

I. Marín-Franch and D. H. Foster, “Number of perceptually distinct surface colors in natural scenes,” J. Vis. 10(9), 9 (2010).
[Crossref]

Martínez-Verdú, F.

Melgosa, M.

Mergel, V. V.

M. N. Goria, N. N. Leonenko, V. V. Mergel, and P. L. Novi Inverardi, “A new class of random vector entropy estimators and its applications in testing statistical hypotheses,” J. Nonparametr. Stat. 17, 277–297 (2005).
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Morovic, J.

J. Morovič, Color Gamut Mapping (Wiley, 2008).

P.-L. Sun and J. Morovic, “Inter-relating colour difference metrics,” in Tenth Color Imaging Conference: Color Science and Engineering Systems, Technologies, Applications (Society for Imaging Science and Technology, 2002), pp. 55–60.

Nascimento, S. M. C.

S. M. C. Nascimento, K. Amano, and D. H. Foster, “Spatial distributions of local illumination color in natural scenes,” Vision Res. 120, 39–44 (2016).
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D. H. Foster, K. Amano, and S. M. C. Nascimento, “Time-lapse ratios of cone excitations in natural scenes,” Vision Res. 120, 45–60 (2016).
[Crossref]

J. M. M. Linhares, P. D. Pinto, and S. M. C. Nascimento, “The number of discernible colors in natural scenes,” J. Opt. Soc. Am. A 25, 2918–2924 (2008).
[Crossref]

D. H. Foster, K. Amano, and S. M. C. Nascimento, “Color constancy in natural scenes explained by global image statistics,” Vis. Neurosci. 23, 341–349 (2006).

D. H. Foster, K. Amano, S. M. C. Nascimento, and M. J. Foster, “Frequency of metamerism in natural scenes,” J. Opt. Soc. Am. A 23, 2359–2372 (2006).
[Crossref]

S. M. C. Nascimento, D. H. Foster, and K. Amano, “Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes,” J. Opt. Soc. Am. A 22, 1017–1022 (2005).
[Crossref]

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on identification of natural surfaces by apparent colour,” Perception 34, 1003–1008 (2005).
[Crossref]

K. Amano, D. H. Foster, and S. M. C. Nascimento, “Minimalist surface-colour matching,” Perception 34, 1009–1013 (2005).
[Crossref]

S. M. C. Nascimento and D. H. Foster, “Relational color constancy in achromatic and isoluminant images,” J. Opt. Soc. Am. A 17, 225–231 (2000).
[Crossref]

S. M. C. Nascimento and D. H. Foster, “Detecting natural changes of cone-excitation ratios in simple and complex coloured images,” Proc. R. Soc. London Ser. B. Biol. Sci. 264, 1395–1402 (1997).
[Crossref]

D. H. Foster and S. M. C. Nascimento, “Relational colour constancy from invariant cone-excitation ratios,” Proc. R. Soc. London Ser. B. Biol. Sci. 257, 115–121 (1994).
[Crossref]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (Institute for Basic Standards, National Bureau of Standards, 1997).

Nieves, J. L.

Novi Inverardi, P. L.

M. N. Goria, N. N. Leonenko, V. V. Mergel, and P. L. Novi Inverardi, “A new class of random vector entropy estimators and its applications in testing statistical hypotheses,” J. Nonparametr. Stat. 17, 277–297 (2005).
[Crossref]

Olsson, P.

P. Olsson and A. Kelber, “Relative colour cues improve colour constancy in birds,” J. Exp. Biol. 220, 1797–1802 (2017).
[Crossref]

Perales, E.

Pinto, P. D.

Pointer, M. R.

M. R. Pointer and G. G. Attridge, “The number of discernible colours,” Color Res. Appl. 23, 52–54 (1998).
[Crossref]

Reeves, A.

Reeves, A. J.

A. J. Reeves, K. Amano, and D. H. Foster, “Color constancy: phenomenal or projective?” Percept. Psychophys. 70, 219–228 (2008).
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Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (Institute for Basic Standards, National Bureau of Standards, 1997).

Rigg, B.

M. R. Luo, C. J. Li, R. W. G. Hunt, B. Rigg, and K. J. Smith, “CMC 2002 colour inconstancy index: CMCCON02,” Color. Technol. 119, 280–285 (2003).
[Crossref]

C. Li, M. R. Luo, B. Rigg, and R. W. G. Hunt, “CMC 2000 chromatic adaptation transform: CMCCAT2000,” Color Res. Appl. 27, 49–58 (2002).
[Crossref]

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. 26, 340–350 (2001).
[Crossref]

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S. Westland and C. Ripamonti, “Invariant cone-excitation ratios may predict transparency,” J. Opt. Soc. Am. A 17, 255–264 (2000).
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S. Westland, C. Ripamonti, and V. Cheung, Computational Colour Science Using MATLAB, 2nd ed. (Wiley, 2012).

Romero, J.

Sale, E. R. H.

D. H. Foster, B. J. Craven, and E. R. H. Sale, “Immediate colour constancy,” Ophthalmic Physiol. Opt. 12, 157–160 (1992).
[Crossref]

Schürmann, T.

T. Schürmann, “Bias analysis in entropy estimation,” J. Phys. A Math. Gen. 37, L295–L301 (2004).
[Crossref]

Smith, K. J.

M. R. Luo, C. J. Li, R. W. G. Hunt, B. Rigg, and K. J. Smith, “CMC 2002 colour inconstancy index: CMCCON02,” Color. Technol. 119, 280–285 (2003).
[Crossref]

Smithson, H. E.

R. J. Lee and H. E. Smithson, “Motion of glossy objects does not promote separation of lighting and surface colour,” R. Soc. Open Sci. 4, 171290 (2017).
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D. R. Cox and E. J. Snell, The Analysis of Binary Data, 2nd ed. (Chapman and Hall/CRC, 1989).

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C. Soize, Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering, Vol. 47 of Interdisciplinary Applied Mathematics (Springer Nature, 2017).

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

Stögbauer, H.

A. Kraskov, H. Stögbauer, and P. Grassberger, “Estimating mutual information,” Phys. Rev. E 69, 066138 (2004).
[Crossref]

Sun, P.-L.

P.-L. Sun and J. Morovic, “Inter-relating colour difference metrics,” in Tenth Color Imaging Conference: Color Science and Engineering Systems, Technologies, Applications (Society for Imaging Science and Technology, 2002), pp. 55–60.

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T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. (Wiley, 2006).

Viqueira, V.

Wang, H.

H. Wang, M. R. Luo, G. Cui, and H. Xu, “A comparison between perceptibility and acceptability methods,” in 11th Congress of the International Colour Association (AIC), D. Smith, P. Green-Armytage, M. A. Pope, and N. Harkness, eds. (AIC International Colour Association, 2009), pp. 1–7.

Warton, D. I.

D. I. Warton, I. J. Wright, D. S. Falster, and M. Westoby, “Bivariate line-fitting methods for allometry,” Biol. Rev. 81, 259–291 (2006).
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Weiss, D.

D. Weiss, C. Witzel, and K. Gegenfurtner, “Determinants of colour constancy and the blue bias,” I-Perception 8, 1–29 (2017).
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A. Werner, “Spatial and temporal aspects of chromatic adaptation and their functional significance for colour constancy,” Vision Res. 104, 80–89 (2014).
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S. Westland and C. Ripamonti, “Invariant cone-excitation ratios may predict transparency,” J. Opt. Soc. Am. A 17, 255–264 (2000).
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S. Westland, C. Ripamonti, and V. Cheung, Computational Colour Science Using MATLAB, 2nd ed. (Wiley, 2012).

Westoby, M.

D. I. Warton, I. J. Wright, D. S. Falster, and M. Westoby, “Bivariate line-fitting methods for allometry,” Biol. Rev. 81, 259–291 (2006).
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Witzel, C.

D. Weiss, C. Witzel, and K. Gegenfurtner, “Determinants of colour constancy and the blue bias,” I-Perception 8, 1–29 (2017).
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Wright, I. J.

D. I. Warton, I. J. Wright, D. S. Falster, and M. Westoby, “Bivariate line-fitting methods for allometry,” Biol. Rev. 81, 259–291 (2006).
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Wyszecki, G.

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

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H. Wang, M. R. Luo, G. Cui, and H. Xu, “A comparison between perceptibility and acceptability methods,” in 11th Congress of the International Colour Association (AIC), D. Smith, P. Green-Armytage, M. A. Pope, and N. Harkness, eds. (AIC International Colour Association, 2009), pp. 1–7.

Zou, N.

J. Liang, M. Georgoula, N. Zou, G. Cui, and M. R. Luo, “Colour difference evaluation using display colours,” Lighting Res. Technol., 1–13 (2017).
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Biol. Rev. (1)

D. I. Warton, I. J. Wright, D. S. Falster, and M. Westoby, “Bivariate line-fitting methods for allometry,” Biol. Rev. 81, 259–291 (2006).
[Crossref]

Color Res. Appl. (5)

R. G. Kuehni, “Metamerism, exact and approximate,” Color Res. Appl. 8, 192 (1983).
[Crossref]

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[Crossref]

C. Li, M. R. Luo, B. Rigg, and R. W. G. Hunt, “CMC 2000 chromatic adaptation transform: CMCCAT2000,” Color Res. Appl. 27, 49–58 (2002).
[Crossref]

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. 26, 340–350 (2001).
[Crossref]

M. R. Pointer and G. G. Attridge, “The number of discernible colours,” Color Res. Appl. 23, 52–54 (1998).
[Crossref]

Color. Technol. (1)

M. R. Luo, C. J. Li, R. W. G. Hunt, B. Rigg, and K. J. Smith, “CMC 2002 colour inconstancy index: CMCCON02,” Color. Technol. 119, 280–285 (2003).
[Crossref]

Ecol. Monogr. (1)

J. A. Endler, “The color of light in forests and its implications,” Ecol. Monogr. 63, 1–27 (1993).
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Entropy (1)

M. Grendar, “Entropy and effective support size,” Entropy 8, 169–174 (2006).
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ESAIM Math. Model. Numer. Anal. (1)

K. Chowdhary and P. Dupuis, “Distinguishing and integrating aleatoric and epistemic variation in uncertainty quantification,” ESAIM Math. Model. Numer. Anal. 47, 635–662 (2013).
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I-Perception (1)

D. Weiss, C. Witzel, and K. Gegenfurtner, “Determinants of colour constancy and the blue bias,” I-Perception 8, 1–29 (2017).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

I. Marín-Franch and D. H. Foster, “Estimating information from image colors: an application to digital cameras and natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 78–91 (2013).
[Crossref]

J. Exp. Biol. (1)

P. Olsson and A. Kelber, “Relative colour cues improve colour constancy in birds,” J. Exp. Biol. 220, 1797–1802 (2017).
[Crossref]

J. Nonparametr. Stat. (1)

M. N. Goria, N. N. Leonenko, V. V. Mergel, and P. L. Novi Inverardi, “A new class of random vector entropy estimators and its applications in testing statistical hypotheses,” J. Nonparametr. Stat. 17, 277–297 (2005).
[Crossref]

J. Opt. Soc. Am. A (10)

D. H. Foster, K. Amano, S. M. C. Nascimento, and M. J. Foster, “Frequency of metamerism in natural scenes,” J. Opt. Soc. Am. A 23, 2359–2372 (2006).
[Crossref]

S. M. C. Nascimento, D. H. Foster, and K. Amano, “Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes,” J. Opt. Soc. Am. A 22, 1017–1022 (2005).
[Crossref]

G. Feng and D. H. Foster, “Predicting frequency of metamerism in natural scenes by entropy of colors,” J. Opt. Soc. Am. A 29, A200–A208 (2012).
[Crossref]

S. Westland and C. Ripamonti, “Invariant cone-excitation ratios may predict transparency,” J. Opt. Soc. Am. A 17, 255–264 (2000).
[Crossref]

S. M. C. Nascimento and D. H. Foster, “Relational color constancy in achromatic and isoluminant images,” J. Opt. Soc. Am. A 17, 225–231 (2000).
[Crossref]

L. Arend and A. Reeves, “Simultaneous color constancy,” J. Opt. Soc. Am. A 3, 1743–1751 (1986).
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J. Hernández-Andrés, J. Romero, and J. L. Nieves, “Color and spectral analysis of daylight in southern Europe,” J. Opt. Soc. Am. A 18, 1325–1335 (2001).
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M. Melgosa, R. Huertas, and R. S. Berns, “Performance of recent advanced color-difference formulas using the standardized residual sum of squares index,” J. Opt. Soc. Am. A 25, 1828–1834 (2008).
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F. Martínez-Verdú, E. Perales, E. Chorro, D. de Fez, V. Viqueira, and E. Gilabert, “Computation and visualization of the MacAdam limits for any lightness, hue angle, and light source,” J. Opt. Soc. Am. A 24, 1501–1515 (2007).
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J. M. M. Linhares, P. D. Pinto, and S. M. C. Nascimento, “The number of discernible colors in natural scenes,” J. Opt. Soc. Am. A 25, 2918–2924 (2008).
[Crossref]

J. Phys. A Math. Gen. (1)

T. Schürmann, “Bias analysis in entropy estimation,” J. Phys. A Math. Gen. 37, L295–L301 (2004).
[Crossref]

J. Vis. (4)

R. Lafer-Sousa and B. R. Conway, “#TheDress: categorical perception of an ambiguous color image,” J. Vis. 17(12), 25 (2017).
[Crossref]

A. Akbarinia and K. Gegenfurtner, “Metameric mismatching in natural and artificial reflectances,” J. Vis. 17(10), 390 (2017).
[Crossref]

F. Faul and V. Ekroll, “Transparent layer constancy,” J. Vis. 12(12), 7 (2012).
[Crossref]

I. Marín-Franch and D. H. Foster, “Number of perceptually distinct surface colors in natural scenes,” J. Vis. 10(9), 9 (2010).
[Crossref]

Noûs (1)

J. Gert, “Color constancy, complexity, and counterfactual,” Noûs 44, 669–690 (2010).
[Crossref]

Ophthalmic Physiol. Opt. (1)

D. H. Foster, B. J. Craven, and E. R. H. Sale, “Immediate colour constancy,” Ophthalmic Physiol. Opt. 12, 157–160 (1992).
[Crossref]

Percept. Psychophys. (1)

A. J. Reeves, K. Amano, and D. H. Foster, “Color constancy: phenomenal or projective?” Percept. Psychophys. 70, 219–228 (2008).
[Crossref]

Perception (2)

K. Amano, D. H. Foster, and S. M. C. Nascimento, “Minimalist surface-colour matching,” Perception 34, 1009–1013 (2005).
[Crossref]

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on identification of natural surfaces by apparent colour,” Perception 34, 1003–1008 (2005).
[Crossref]

Phys. Rev. E (1)

A. Kraskov, H. Stögbauer, and P. Grassberger, “Estimating mutual information,” Phys. Rev. E 69, 066138 (2004).
[Crossref]

Probl. Inf. Transm. (Tr. Problemy Peredachi Informatsii) (1)

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Probl. Inf. Transm. (Tr. Problemy Peredachi Informatsii) 23, 9–16, 95–101 (1987).

Proc. Natl. Acad. Sci. USA (1)

J. M. Kraft and D. H. Brainard, “Mechanisms of color constancy under nearly natural viewing,” Proc. Natl. Acad. Sci. USA 96, 307–312 (1999).
[Crossref]

Proc. R. Soc. London Ser. B. Biol. Sci. (2)

S. M. C. Nascimento and D. H. Foster, “Detecting natural changes of cone-excitation ratios in simple and complex coloured images,” Proc. R. Soc. London Ser. B. Biol. Sci. 264, 1395–1402 (1997).
[Crossref]

D. H. Foster and S. M. C. Nascimento, “Relational colour constancy from invariant cone-excitation ratios,” Proc. R. Soc. London Ser. B. Biol. Sci. 257, 115–121 (1994).
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Figures (8)

Fig. 1.
Fig. 1. Translating a color-constancy discrimination task to a natural scene. The images were rendered from a hyperspectral reflectance image of a woodland scene under an illuminant representing on the left, direct sunlight with correlated color temperature 4000 K, and on the right, a mixture of sunlight and skylight with correlated color temperature 6700 K. The test object is a small sphere attached to the trunk of a tree, center right [10]. The scene is from Sameiro, Braga, Portugal [11].
Fig. 2.
Fig. 2. Example scene to illustrate counting colors and surfaces. The color image on the left was rendered from a hyperspectral radiance image of a woodland scene. The image on the right was obtained by histogram equalization of the gray-scale rendering of the hyperspectral image. The scene is from Ruivães, Vieira do Minho, Portugal [11].
Fig. 3.
Fig. 3. Histograms of relative frequencies of lightness values J (top left) and red-green and yellow-blue chroma component values a C (bottom left) and b C (bottom right) of the CIECAM02 representation of the scene in Fig. 2, left. The histogram of relative frequencies of adjusted lightness values J (top right) is from the histogram-equalized gray-scale rendering of the scene in Fig. 2, right.
Fig. 4.
Fig. 4. Number of discriminable surfaces and colors. The logarithm of the number of surfaces discriminable by their color is plotted against the logarithm of the number of discriminable colors. Each point represents data from one of 50 outdoor scenes, two of which are pictured [33,41]. The dashed line shows a linear regression, with R 2 = 73 % . The oblique solid line represents the upper limit on the number of discriminable surfaces for each scene. Unpublished data from [43].
Fig. 5.
Fig. 5. Relative frequency of metamerism and entropy of colors. The logarithm of the relative frequency N 1 / N under 4000 K and 25,000 K daylight illuminants is plotted against the entropy of colors h ( U ) under the 4000 K illuminant. Each point represents data from one of 50 outdoor scenes [33,41]. The dashed line shows a linear regression, with R 2 = 90 % . Earlier versions of this plot appear in [50,51].
Fig. 6.
Fig. 6. Conditional relative frequency of metamerism and conditional entropy of vector color differences. The logit conditional relative frequency N 1 / N 0 is plotted against the conditional entropy h ( Δ V | Δ U ) of the vector color difference Δ V under a 25,000 K daylight illuminant given the subthreshold vector color difference Δ U under a 4000 K daylight illuminant. Each point represents data from one of 50 outdoor scenes [33,41]. The dashed line shows a linear regression, with R 2 = 90 % .
Fig. 7.
Fig. 7. Magnitude of metamerism and conditional entropy of vector color differences. The logarithm of the median total color difference Δ E t under a 25,000 K daylight illuminant is plotted against the conditional entropy of the vector color difference Δ V under a 25,000 K daylight illuminant given the subthreshold vector color difference Δ U under a 4000 K daylight illuminant. Each point represents data from one of 50 outdoor scenes [33,41]. The dashed line shows a linear regression, with R 2 = 88 % .
Fig. 8.
Fig. 8. Magnitude of generalized metamerism and conditional entropy of vector color differences. The logarithm of the median total color difference Δ E r , t under a 25,000 K daylight illuminant is plotted against the conditional entropy h ( Δ V | Δ U ) of the vector color difference Δ V under a 25,000 K daylight illuminant given the vector color difference Δ U under a 4000 K daylight illuminant. Each point represents data from one of 50 outdoor scenes [33,41]. The dashed line shows a linear regression, with R 2 = 75 % . Notice the difference between the vertical scale and that in Fig. 7.

Equations (5)

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h ( U ) = f ( u ) log f ( u ) d u ,
N 1 N = N 1 N 0 · N 0 N .
h ( V | U ) = f ( v , u ) log f ( v | u ) d v d u ,
h ( V | U ) = h ( V , U ) h ( U ) ,
Δ E r , t = [ ( Δ J t Δ J r ) 2 + ( Δ a C , t Δ a C , r ) 2 + ( Δ b C , t Δ b C , r ) 2 ] 1 / 2 .

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