Abstract

Estimating the frequency of metameric surfaces in natural scenes usually requires many comparisons of surface colors to determine which are visually indistinguishable under one light but distinguishable—by a certain criterion degree—under another. The aim here was to test the predictive power of a simpler approach to estimation based on the entropy of colors. In simulations with 50 hyperspectral images of natural scenes, the logarithm of the observed relative frequency of metamerism in each scene under two successive daylights was regressed on combinations of the estimated Shannon differential entropies of the colors of the scene under the same two daylights. The regression was strong, and it remained so when restricted to the estimated differential entropy under just the first daylight, providing that the criterion degree of metamerism was limited. When the criterion degree was made more extreme, however, the restricted regression failed. A possible explanation of the predictive power of differential entropy is briefly considered.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2010 (2)

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

A. M. Bakke, I. Farup, and J. Y. Hardeberg, “Evaluation of algorithms for the determination of color gamut boundaries,” J. Imaging Sci. Technol. 54, 050502 (2010).
[CrossRef]

2009 (3)

2008 (2)

2006 (2)

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]

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]

2005 (2)

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]

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. Nonparametric Stat. 17, 277–297 (2005).
[CrossRef]

2004 (1)

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

2002 (3)

C.-J. 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]

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (2002).
[CrossRef]

S. M. C. Nascimento, F. P. Ferreira, and D. H. Foster, “Statistics of spatial cone-excitation ratios in natural scenes,” J. Opt. Soc. Am. A 19, 1484–1490 (2002).
[CrossRef]

2001 (3)

T. Wachtler, T.-W. Lee, and T. J. Sejnowski, “Chromatic structure of natural scenes,” J. Opt. Soc. Am. A 18, 65–77 (2001).
[CrossRef]

J. Morovic and M. R. Luo, “The fundamentals of gamut mapping: a survey,” J. Imaging Sci. Technol. 45, 283–290 (2001).

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]

1987 (1)

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Prob. Peredachi Inf. 23, 9–16 (1987).

1983 (1)

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

1973 (1)

D. B. Montgomery and D. G. Morrison, “A note on adjusting R2,” J. Finance 28, 1009–1013 (1973).
[CrossRef]

Amano, K.

Bakke, A. M.

A. M. Bakke, I. Farup, and J. Y. Hardeberg, “Evaluation of algorithms for the determination of color gamut boundaries,” J. Imaging Sci. Technol. 54, 050502 (2010).
[CrossRef]

Benton, C. P.

T. Troscianko, C. P. Benton, P. G. Lovell, D. J. Tolhurst, and Z. Pizlo, “Camouflage and visual perception,” Phil. Trans. R. Soc. B 364, 449–461 (2009).
[CrossRef]

Berns, R. S.

Clark, J. J.

Cover, T. M.

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

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]

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (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]

Efron, B.

B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap (Chapman & Hall, 1993).

Farup, I.

A. M. Bakke, I. Farup, and J. Y. Hardeberg, “Evaluation of algorithms for the determination of color gamut boundaries,” J. Imaging Sci. Technol. 54, 050502 (2010).
[CrossRef]

Ferreira, F. P.

Foster, D. H.

Foster, M. J.

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. Nonparametric Stat. 17, 277–297 (2005).
[CrossRef]

Grassberger, P.

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

Hardeberg, J. Y.

A. M. Bakke, I. Farup, and J. Y. Hardeberg, “Evaluation of algorithms for the determination of color gamut boundaries,” J. Imaging Sci. Technol. 54, 050502 (2010).
[CrossRef]

Huertas, R.

Hunt, R. W. G.

C.-J. 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]

R. W. G. Hunt, Measuring Colour, 3rd ed. (Fountain Press, 1998).

Kozachenko, L. F.

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Prob. Peredachi Inf. 23, 9–16 (1987).

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]

Lee, T.-W.

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. Nonparametric Stat. 17, 277–297 (2005).
[CrossRef]

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Prob. Peredachi Inf. 23, 9–16 (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]

Li, C.-J.

C.-J. 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]

Linhares, J. M. M.

Lovell, P. G.

T. Troscianko, C. P. Benton, P. G. Lovell, D. J. Tolhurst, and Z. Pizlo, “Camouflage and visual perception,” Phil. Trans. R. Soc. B 364, 449–461 (2009).
[CrossRef]

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]

C.-J. 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]

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (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]

J. Morovic and M. R. Luo, “The fundamentals of gamut mapping: a survey,” J. Imaging Sci. Technol. 45, 283–290 (2001).

Marín-Franch, I.

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

D. H. Foster, I. Marín-Franch, K. Amano, and S. M. C. Nascimento, “Approaching ideal observer efficiency in using color to retrieve information from natural scenes,” J. Opt. Soc. Am. A 26, B14–B24 (2009).
[CrossRef]

I. Marín-Franch and D. H. Foster, “Estimating information from color signals: an application to digital cameras and natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. (submitted).

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. Nonparametric Stat. 17, 277–297 (2005).
[CrossRef]

Montgomery, D. B.

D. B. Montgomery and D. G. Morrison, “A note on adjusting R2,” J. Finance 28, 1009–1013 (1973).
[CrossRef]

Morovic, J.

J. Morovic and M. R. Luo, “The fundamentals of gamut mapping: a survey,” J. Imaging Sci. Technol. 45, 283–290 (2001).

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

Morrison, D. G.

D. B. Montgomery and D. G. Morrison, “A note on adjusting R2,” J. Finance 28, 1009–1013 (1973).
[CrossRef]

Nascimento, S. M. C.

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. Nonparametric Stat. 17, 277–297 (2005).
[CrossRef]

Pinto, P. D.

Pizlo, Z.

T. Troscianko, C. P. Benton, P. G. Lovell, D. J. Tolhurst, and Z. Pizlo, “Camouflage and visual perception,” Phil. Trans. R. Soc. B 364, 449–461 (2009).
[CrossRef]

Rigg, B.

C.-J. 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]

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (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]

Roesler, G.

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (2002).
[CrossRef]

Sejnowski, T. J.

Skaff, S.

Stiles, W. S.

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

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas (Wiley, 1967).

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 Proceedings of Tenth Color Imaging Conference: Color Science and Engineering Systems, Technologies, Applications (Society for Imaging Science and Technology, 2002) pp. 55–60.

Thomas, J. A.

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

Tibshirani, R. J.

B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap (Chapman & Hall, 1993).

Tolhurst, D. J.

T. Troscianko, C. P. Benton, P. G. Lovell, D. J. Tolhurst, and Z. Pizlo, “Camouflage and visual perception,” Phil. Trans. R. Soc. B 364, 449–461 (2009).
[CrossRef]

Troscianko, T.

T. Troscianko, C. P. Benton, P. G. Lovell, D. J. Tolhurst, and Z. Pizlo, “Camouflage and visual perception,” Phil. Trans. R. Soc. B 364, 449–461 (2009).
[CrossRef]

Wachtler, T.

Witt, K.

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (2002).
[CrossRef]

Wyszecki, G.

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

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas (Wiley, 1967).

Color Res. Appl. (5)

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

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, “Uniform colour spaces based on the DIN99 colour-difference formula,” Color Res. Appl. 27, 282–290 (2002).
[CrossRef]

C.-J. 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 C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[CrossRef]

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

I. Marín-Franch and D. H. Foster, “Estimating information from color signals: an application to digital cameras and natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. (submitted).

J. Finance (1)

D. B. Montgomery and D. G. Morrison, “A note on adjusting R2,” J. Finance 28, 1009–1013 (1973).
[CrossRef]

J. Imaging Sci. Technol. (2)

J. Morovic and M. R. Luo, “The fundamentals of gamut mapping: a survey,” J. Imaging Sci. Technol. 45, 283–290 (2001).

A. M. Bakke, I. Farup, and J. Y. Hardeberg, “Evaluation of algorithms for the determination of color gamut boundaries,” J. Imaging Sci. Technol. 54, 050502 (2010).
[CrossRef]

J. Nonparametric 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. Nonparametric Stat. 17, 277–297 (2005).
[CrossRef]

J. Opt. Soc. Am. A (8)

J. Vision (1)

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

Phil. Trans. R. Soc. B (1)

T. Troscianko, C. P. Benton, P. G. Lovell, D. J. Tolhurst, and Z. Pizlo, “Camouflage and visual perception,” Phil. Trans. R. Soc. B 364, 449–461 (2009).
[CrossRef]

Phys. Rev. E (1)

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

Prob. Peredachi Inf. (1)

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Prob. Peredachi Inf. 23, 9–16 (1987).

Other (8)

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

B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap (Chapman & Hall, 1993).

CIE, “A colour appearance model for colour management systems: CIECAM02,” CIE Publication 159:2004 (CIE Central Bureau, 2004).

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

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas (Wiley, 1967).

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

R. W. G. Hunt, Measuring Colour, 3rd ed. (Fountain Press, 1998).

CIE, “Colorimetry,” 3rd ed., CIE Publication 15:2004 (CIE Central Bureau, 2004).

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Figures (3)

Fig. 1.
Fig. 1.

Prediction of relative frequency of metamerism by combinations of differential entropy. In each panel, the logarithm of the observed relative frequency of metamerism for daylights with CCTs of 4000 K and 25000 K and nominal color-difference threshold ΔEthr of 0.5 is plotted against the combination of estimated entropies defined by Eq. (6′). The dotted lines represent perfect fits. Each panel is based on data for 50 natural scenes with a different criterion degree of metamerism n of 1, 2, 3, and 4, as indicated. Notice the different abscissa and ordinate ranges. The images above each panel show the scenes with the lowest and highest estimated relative frequencies of metamerism.

Fig. 2.
Fig. 2.

Dependence of goodness of fit of the full entropy model on the difference in relative frequency ranges. Proportion R2 of variance accounted for by Eq. (6) is plotted against the difference in ranges of logN0/N and logN1/N0, values taken from Table 1. Each point represents data from 50 scenes under different pairs of daylights, with different nominal color-difference thresholds, and different criterion degrees of metamerism n. The dotted line is a linear regression.

Fig. 3.
Fig. 3.

Dependence of goodness of fit of the restricted entropy model on the difference in relative frequency ranges. Proportion R2 of variance accounted for by Eq. (7) is plotted against the difference in ranges of logN0/N and logN1/N0, values taken from Table 1. Each point represents data from 50 scenes under different pairs of daylights, with different nominal color-difference thresholds and different criterion degrees of metamerism n. The dotted line is a piecewise linear regression.

Tables (2)

Tables Icon

Table 1. Regression over 50 Scenes of the Logarithm of Observed Relative Frequency of Metamerism on Estimated Differential Entropy Calculated within the Color Space CIECAM02 [12] for Various Nominal Color-Difference Thresholds, Criterion Degrees of Metamerism, and Daylights with Different Correlated Color Temperatures

Tables Icon

Table 2. Regression over 50 Scenes of the Logarithm of Observed Relative Frequency of Metamerism on Estimated Differential Entropy Calculated within the Color Space CIELAB with Color-Difference Formula CIEDE2000 [4] for Various Nominal Color-Difference Thresholds, Criterion Degrees of Metamerism, and Daylights with Different Correlated Color Temperatures

Equations (9)

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

X=x¯(λ)e(λ)r(λ;x,y)dλ,Y=y¯(λ)e(λ)r(λ;x,y)dλ,Z=z¯(λ)e(λ)r(λ;x,y)dλ,
h(U)=f(u)logf(u)du,
h(U2|U1)=h(U2,U1)h(U1).
E[log(N0/N)]=β1h(U1)+α1,
E[log(N1/N0)]=β2h(U2|U1)+α2,
E[log(N1/N)]=β1h(U1)+β2h(U2|U1)+α0,
E[log(N1/N)]=β0[β1h(U1)β2h(U2|U1)]+α0,
E[log(N1/N)]=β1h(U1)+α1,
E[log(N1/N)]=β2h(U2|U1)+α2.

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