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

For given tristimulus values X , Y , Z of the object with reflectance rho(lambda) viewed under an illuminant S (lambda) with tristimulus values X 0 , Y 0 , Z 0 , an earlier algorithm constructs the smoothest metameric estimate rho 0(lambda) under S (lambda) of rho(lambda), independent of the amplitude of S (lambda) . It satisfies a physical property of rho(lambda), i.e., 0 <= rho 0(lambda) <= 1, on the visual range. The second inequality secures the condition that for no lambda the corresponding patch returns more radiation from S (lambda) than is incident on it at lambda , i.e., rho 0(lambda) is a fundamental metameric estimate; rho 0(lambda) and rho(lambda) differ by an estimation error causing perceptual variables assigned to rho 0(lambda) and rho(lambda) under S (lambda) to differ under the universal reference illuminant E (lambda) = 1 for all lambda , 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 (lambda) 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 (lambda) unknown to the visual system as a function of lambda is replaced by its nonnegative smoothest metameric estimate S 0 (lambda) with tristimulus values made available in color rendering calculations, by specular reflection, or determined by any educated guess; rho(lambda) under S (lambda) is replaced by its corresponding color R 0 (lambda) under S 0 (lambda) like rho(lambda) independent of the amplitude of S 0 (lambda) . The visual system attributes to R 0 (lambda) E (lambda) one achromatic variable, in the CIE case defined by y (lambda)/ Y E , replaced by the narrowest middle wave function A 2 (lambda) normalized such that the integral of A 2 (lambda) E (lambda) over the visual range equals unity. It defines the achromatic variable xi 2 , A (lambda) , and xi 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. (c) 2024 Optica Publishing Group