Color formation as a Hilbert space operator: quotient structures and limits of metamerism.

Abstract

Color formation, the mapping from spectral reflectances through an illuminant and observer to color responses, is a bounded linear operator from a Hilbert space of spectral functions to the space R^C of color responses, where C is the number of sensor classes of the observer (C=3 for a trichromat). This paper develops the algebraic ecosystem of this framework and uses it to establish two results on the limits of metamerism. Metameric equivalence classes are cosets of the kernel, making color space the quotient L²/ker(W); the pseudoinverse is the canonical section selecting minimum-norm representatives; physical constraints cut each coset to a compact convex body whose image is the object-color gamut (a zonotope); and two complementary types of domain restriction, linear-model bases and convex-combination "naturalness" domains, transfer the algebraic structure to a finite-dimensional effective operator. Within this framework we prove: (i) non-trivial absolute metamers, reflectance pairs that match under every possible illuminant, cannot exist as long as every wavelength excites at least one photoreceptor class; (ii) a two-sided dimensional threshold for metamers stable across a restricted family of illuminants: such metamers exist whenever the illuminant family spans fewer than ⌈S/C⌉ dimensions, where S is the number of sampled wavelengths, and cannot exist once the stacked operator attains full rank-e.g., for a trichromatic observer (C=3) and the CIE daylight family, whose span has dimension d=2 or 3, in an S=31-dimensional spectral representation this threshold is far from being reached. The existence half of (ii) was given in discrete form by Burns, Cohen, and Kuznetsov and in dual form by Brainard, Wandell, and Cowan, both in 1989. What is added here is the impossibility direction, the two-sided threshold, and its transport to a restricted domain. A worked example with the CIE 1931 observer under D65, the CIE daylight basis, and a 10-dimensional PCA basis of the SOCS dataset illustrates the theory and constructs daylight-absolute metamer pairs by linear programming. A companion paper [J. Opt. Soc. Am. A (to be published)] extends the framework to two operators, treating metamer mismatching and chromatic adaptation.