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

For given tristimulus values X, Y, Z of the object with reflectance ρ(λ) viewed under an illuminant S(λ) with tristimulus values X ₀, Y ₀, Z ₀, an earlier algorithm constructs the smoothest metameric estimate ρ ₀(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≤ρ ₀(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., ρ ₀(λ) is a fundamental metameric estimate; ρ ₀(λ) and ρ(λ) differ by an estimation error causing perceptual variables assigned to ρ ₀(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 S ₀(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 R ₀(λ) under S ₀(λ) like ρ(λ) independent of the amplitude of S ₀(λ). The visual system attributes to R ₀(λ)E(λ) one achromatic variable, in the CIE case defined by y(λ)/Y _E, replaced by the narrowest middle wave function A ₂(λ) normalized such that the integral of A ₂(λ)E(λ) over the visual range equals unity. It defines the achromatic variable ξ ₂, 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.