Describing Color Space as a Quotient Vector Space : Trichromatic System in Computer Graphics (1)

Abstract

This study was aimed at extending the trichromatic system (from geometric viewpoint) for use in computer graphics applications. First, color-matching experiment was mathematically described using the linear mapping Ψ^^- of n-vector representing a color stimulus onto a color space of additive color mixture system. The linear mapping is independent of bases used in both the color stimulus space and color space, while the color-matching matrix is dependent on both the former and the latter. The author proposes a linear operator named "sensor", which is independent of the basis of color stimulus space and dependent on that of the color space. It is shown that a metameric suite, (a suite of color stimuli that evoke an identical color sensation), mathematically organizes the equivalence class and becomes an element of quotient vector space Vn/Ker Ψ^^- where Ker Ψ^^- is the kernel of mapping Ψ^^- onto color space. Then, Grassmann's laws of additive color mixing can be demonstrated to reflect the properties of the mapping onto quotient vector space.