Illuminant-Dependence of Von Kries Type Quotients

A von Kries quotient is defined as the cone signal of a reflectance under some illuminant divided by the same cone signal of the illuminant. A von Kries type quotient is a similar ratio, the cone sensitivity being replaced with some linear combination of the CIE color matching functions P(λ). We study the illuminant-(in)dependent behavior of von Kries type quotients by means of an expansion consisting of one illuminant-independent term and a series of illuminant-dependent ones. It is proved that the series rapidly decreases and that the dominating first term is small if P(λ) is a narrow function of wavelength and the reflectance and spectral distribution functions are sufficiently broad-band, defined in the text. Von Kries type quotients have a favorable illuminant-independent behavior if and only if the reflectance and spectral distribution functions are smooth functions of wavelength with chromaticity coordinates in a restricted neighborhood of the achromatic point belonging to the equal-energy spectrum, dependent on the narrowness of P(λ), comprising the object color solid only if P(λ) were a delta-function