Evaluation of linear models of surface spectral reflectance with small numbers of parameters

TLDR

Computational color‑vision models can achieve exact color constancy over limited lighting and surface ranges using linear models, but this depends on whether broad sets of surface spectral reflectances can be described by a linear model with only a few parameters. The paper investigates whether empirical surface spectral reflectances can be captured by linear models with few parameters, by testing three conjectures about parameter count, band‑limited functions, and the influence of human spectral‑sensitivity curves. To test these conjectures, the authors analyze two large empirical datasets of surface reflectances and theoretically derive the relationship between the number of photoreceptor classes and their filtering properties. The analysis shows that empirical reflectances fit linear models but require five to seven parameters, yet incorporating human visual sensitivity yields excellent fits with three to four parameters, indicating that both surface‑reflectance constraints and photoreceptor filtering contribute to color constancy and supporting a trichromatic, color‑constant visual system.

Abstract

Recent computational models of color vision demonstrate that it is possible to achieve exact color constancy over a limited range of lights and surfaces described by linear models. The success of these computational models hinges on whether any sizable range of surface spectral reflectances can be described by a linear model with about three parameters. In the first part of this paper, I analyze two large sets of empirical surface spectral reflectances and examine three conjectures concerning constraints on surface reflectance: (1) that empirical surface reflectances fall within a linear model with a small number of parameters, (2) that empirical surface reflectances fall within a linear model composed of band-limited functions with a small number of parameters, and (3) that the shape of the spectral-sensitivity curves of human vision enhance the fit between empirical surface reflectances and a linear model. I conclude that the first and second conjectures hold for the two sets of spectral reflectances analyzed but that the number of parameters required to model the spectral reflectances is five to seven, not three. A reanalysis of the empirical data that takes human visual sensitivity into account gives more promising results. The linear models derived provide excellent fits to the data with as few as three or four parameters, confirming the third conjecture. The results suggest that constraints on possible surface-reflectance functions and the "filtering" properties of the shapes of the spectral-sensitivity curves of photoreceptors can both contribute to color constancy. In the last part of the paper I derive the relation between the number of photoreceptor classes present in vision and the "filtering" properties of each class. The results of this analysis reverse a conclusion reached by Barlow: the "filtering" properties of human photoreceptors are consistent with a trichromatic visual system that is color constant.