Achromatic solutions of the color constancy problem: the Helmholtz–Kohlrausch effect explained

Abstract

For given tristimulus values X, Y, Z of the object with reflectance ρ(λ) viewed under an illuminant S(λ) with tristimulus values X0, Y0, Z0, an earlier algorithm constructs the smoothest metameric estimate ρ0(λ) under S(λ) of ρ(λ), independent of the amplitude of S(λ). It satisfies a physical property of ρ(λ), i.e., 0≤ρ0(λ)≤1, on the visual range. The second inequality secures the condition that for no λ the corresponding patch returns more radiation from S(λ) than is incident on it at λ, i.e., ρ0(λ) is a fundamental metameric estimate; ρ0(λ) and ρ(λ) differ by an estimation error causing perceptual variables assigned to ρ0(λ) and ρ(λ) under S(λ) to differ under the universal reference illuminant E(λ)=1 for all λ, tristimulus values X E , Y E , Z E . This color constancy error is suppressed but not nullified by three narrowest nonnegative achromatic response functions A i (λ) defined in this paper, replacing the cone sensitivities and invariant under any nonsingular transformation T of the color matching functions, a demand from theoretical physics. They coincide with three functions numerically constructed by Yule apart from an error corrected here. S(λ) unknown to the visual system as a function of λ is replaced by its nonnegative smoothest metameric estimate S0(λ) with tristimulus values made available in color rendering calculations, by specular reflection, or determined by any educated guess; ρ(λ) under S(λ) is replaced by its corresponding color R0(λ) under S0(λ) like ρ(λ) independent of the amplitude of S0(λ). The visual system attributes to R0(λ)E(λ) one achromatic variable, in the CIE case defined by y(λ)/Y E , replaced by the narrowest middle wave function A2(λ) normalized such that the integral of A2(λ)E(λ) over the visual range equals unity. It defines the achromatic variable ξ2, A(λ), and ξ as described in the paper. The associated definition of present luminance explains the Helmholtz–Kohlrausch effect in the last figure of the paper and rejects CIE 1924 luminance that fails to do so. It can be understood without the mathematical details.