Color formation as a Hilbert space operator: metamer mismatching and the limits of chromatic adaptation.

Abstract

Building on the Hilbert space operator framework of color formation developed in Morovic and Morovic [J. Opt. Soc. Am. A43, 1614 (2026)JOAOD60740-323210.1364/JOSAA.606555], we extend the single-operator algebra to two color-formation operators W₁ and W₂ simultaneously, one for each of two viewing conditions-a condition being a choice of illuminant and observer-capturing metamer mismatching and chromatic adaptation. We define the mismatch operatorR₂₁=W₂Π_ker⁡(W1), the composition of projection onto the metameric-black subspace of the first condition with color formation under the second. Its operator norm is the worst-case mismatch magnitude; its singular value decomposition identifies the spectral directions of greatest mismatch susceptibility; and the image under W₂ of the physically realizable metamers of a color under W₁ is a compact convex polytope-the metamer mismatch volume of Morovic ["Metamer sets," Ph.D. thesis, 2002], recovered intrinsically from the columns of the two operators. The central result is that for any linear chromatic adaptation transform (CAT) M∈R^C×C, where C is the number of sensor classes of the observer (C=3 for a trichromat), the Frobenius-norm residual ‖W₂-MW₁‖_F is bounded below by ‖R₂₁‖_F, and that this limit is achieved uniquely by the optimal CATM_opt=W₂W1+. The bound holds in any sensor basis, but its numerical value does not: an invertible sensor transform T carries R₂₁ to TR₂₁, so it is the vanishing of the mismatch operator, and not its magnitude, that is basis-independent. Magnitudes below are quoted in the CIE 1931 sensor basis. Von Kries scaling is recovered as the special case in which W₁ and W₂ share a right singular subspace and differ on it by a factor diagonal in the sensor basis. A worked example with the CIE 1931 observer under D65 and illuminant A demonstrates the theory: ‖R₂₁‖_F=5.225 in that basis exactly bounds CAT residuals (von Kries, Bradford, CMCCAT2000, and sharp all exceed this limit by factors of 1.4-2.2). Fundamental limits are then tabulated for all ordered pairs drawn from a panel of nine illuminants (daylights, incandescent, fluorescent, and CIE LED sources), showing that the limit is strongly asymmetric-predicting a narrowband source from a smooth one is much harder than the reverse-and that it predicts in advance whether the choice of CAT matters at all. A multi-condition extension shows how the universal metameric-black subspace shrinks as illuminants are added and quantifies what each added condition contributes: the CIE daylight locus is shown to saturate at three conditions, and the 12 CIE fluorescents to drive the universal kernel to zero while leaving directions that are colorimetrically negligible. The construction connects to the spectral sharpening problem of Finlayson et al. [J. Opt. Soc. Am. A11, 1553 (1994)JOAOD60740-323210.1364/JOSAA.11.001553].