Abstract

The volumetric theory of color constancy¹ provides illuminant-invariant estimation of reflectance spectra from reflected-light tristimulus values. The basis of the computation is the tristimulus volume ratio. Tristimulus volume ratios are illuminant-invariant when (a) the illuminant spectrum is a linear combination of N functions s_i(λ) free to vary only in their coefficients; and (b) each reflectance spectrum r(λ) is a linear combination of three basic functions r_k(λ) plus a residual term R(λ) which is orthogonal to the 3N products f_m(λ) of s_i and the tristimulus functions q_j(i.e., r must be orthogonal to a forbidden subspace spanned by f_m except for the projections of the functions r_k). The color-constant reflectance spectrum r′(λ) closest (in a least-squares sense) to r(λ) can be obtained by subtracting off the forbidden-subspace components of r(λ). The present paper displays for N = 3 the r′(λ) for several real reflectances, based on s_i and r_k drawn from principal-component analyses. We compute r′(λ) using the Gram-Schmidt orthogonalization on the forbidden-subspace functions. Each r′(λ) is compared with the function derived from the corresponding r(λ) by subtracting off the forbidden subspace of von Kries color constancy.² The comparison favors the volumetric theory.

© 1985 Optical Society of America