Color Similarity Represented as a Metric of Color Space
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The traditional vector space representation of colors is based on the empirical conditions of Grassmann’s (1853) laws: If the data of color matching exhibit a Grassmann structure, the corresponding representation theorem is interpretable in the color laboratory. The theory encompasses the phenomena of metameric stimuli, color matching, and visually invariant change of primary colors. It does not apply to the phenomena of color adaptation and of visual similarity of pairs of colors. The present study develops a representation for visual similarity of colors by specializing the Grassmann structure. If, in such a structure, (i) pairs of stimuli form an additive conjoint structure with respect to detectability, if (ii) the structure contains a smallest element, if (iii) (temporal) adaptation to color forms a group with respect to composition, and if (iv) the spectral locus is an invariant with respect to color adaptation, then there exists a mapping from pairs of colors into the real numbers which has the properties of a metric and is invariant with respect to projective hyperbolic transformations. The map is interpreted as dissimilarity of colors. The proof of the map’s existence leads to analytic expressions for color matching functions. These are used to identify the analytic expression for the surface of the color solid. This color cone, in turn, is employed as a geometrical invariant with respect to the automorphisms of color space. They are interpreted as the effects of color adaptation. The major result states that color space possesses a projective hyperbolic metric. Since all assumptions are formulated in empirical terms, they can be tested experimentally. Available data do not contradict these assertions.
KeywordsColor Space Color Vision Visual Similarity Cross Ratio Color Match
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