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.

David H. Foster, Iván Marín-Franch, Kinjiro Amano, and Sérgio M. C. Nascimento J. Opt. Soc. Am. A 26(11) B14-B24 (2009)

References

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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

Nominal threshold for color differences under first daylight, giving ${N}_{0}$ pairs out of $N\approx 1.25\times {10}^{9}$ pairs chosen randomly from each of the 50 scenes
Multiple of $\mathrm{\Delta}{E}^{\text{thr}}$ for color differences under second daylight, giving ${N}_{1}$ pairs out of ${N}_{0}$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{0}/N$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{1}/{N}_{0}$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{1}/N$.
Proportion of variance accounted for by restricted linear model (7), adjusted for degrees of freedom.
Proportion of variance accounted for by restricted linear model (8), adjusted for degrees of freedom.
Proportion of variance accounted for by full linear model (6), adjusted for degrees of freedom.
Adjustment for degrees of freedom omitted, as invalid.
Results for other daylight pair or pairs omitted as ${N}_{1}=0$ for one or more scenes.

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

Nominal threshold for color differences under first daylight, giving ${N}_{0}$ pairs out of $N\approx 1.25\times {10}^{9}$ pairs chosen randomly from each of the 50 scenes.
Multiple of $\mathrm{\Delta}{E}^{\text{thr}}$ for color differences under second daylight, giving ${N}_{1}$ pairs out of ${N}_{0}$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{0}/N$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{1}/{N}_{0}$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{1}/N$.
Proportion of variance accounted for by restricted linear model (7), adjusted for degrees of freedom.
Proportion of variance accounted for by restricted linear model (8), adjusted for degrees of freedom.
Proportion of variance accounted for by full linear model (6), adjusted for degrees of freedom.
Results for other daylight pair or pairs omitted as ${N}_{1}=0$ for one or more scenes.
Adjustment for degrees of freedom omitted, as invalid.

Tables (2)

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

Nominal threshold for color differences under first daylight, giving ${N}_{0}$ pairs out of $N\approx 1.25\times {10}^{9}$ pairs chosen randomly from each of the 50 scenes
Multiple of $\mathrm{\Delta}{E}^{\text{thr}}$ for color differences under second daylight, giving ${N}_{1}$ pairs out of ${N}_{0}$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{0}/N$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{1}/{N}_{0}$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{1}/N$.
Proportion of variance accounted for by restricted linear model (7), adjusted for degrees of freedom.
Proportion of variance accounted for by restricted linear model (8), adjusted for degrees of freedom.
Proportion of variance accounted for by full linear model (6), adjusted for degrees of freedom.
Adjustment for degrees of freedom omitted, as invalid.
Results for other daylight pair or pairs omitted as ${N}_{1}=0$ for one or more scenes.

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

Nominal threshold for color differences under first daylight, giving ${N}_{0}$ pairs out of $N\approx 1.25\times {10}^{9}$ pairs chosen randomly from each of the 50 scenes.
Multiple of $\mathrm{\Delta}{E}^{\text{thr}}$ for color differences under second daylight, giving ${N}_{1}$ pairs out of ${N}_{0}$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{0}/N$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{1}/{N}_{0}$.
Difference between maximum and minimum of $\mathrm{log}\text{\hspace{0.17em}}{N}_{1}/N$.
Proportion of variance accounted for by restricted linear model (7), adjusted for degrees of freedom.
Proportion of variance accounted for by restricted linear model (8), adjusted for degrees of freedom.
Proportion of variance accounted for by full linear model (6), adjusted for degrees of freedom.
Results for other daylight pair or pairs omitted as ${N}_{1}=0$ for one or more scenes.
Adjustment for degrees of freedom omitted, as invalid.