The vector-space representation opens the doors for utilizing the well-developed mathematical fields of linear algebra and matrix theory. Cohen, Horn, and Trussell, among others, have elegantly put forth the fundamental properties of the vector-space color representation. They showed the existence of the color match, the color additivity, the identity, the transformation of primaries, the equivalent primaries, the metameric relationship, the effect of illumination, and many imaging constraints. The abilities of representing spectral sensitivity of human vision and related visual phenomena in matrix-vector form provide the foundation for a major branch of computational color technology, namely, the studies of the phenomenon of surface reflection. This approach deals with signals reflected from an object surface under a given illumination into a human visual pathway; it has no interest in the physical and chemical interactions within the object and substrate. On the other hand, the physical interactions of the light with objects and substrates form the basis for another major branch of the computational color technology, namely, the physical color-mixing models that are used primarily in the printing industry.
With the vector-space representation and matrix theory, in this chapter, we lay the groundwork for these computational approaches by revisiting the Grassmann's law of color mixing and reexamining color matching as well as other properties.
2.1 Visual Sensitivity and Color-Matching Functions
Visual spectral sensitivity of the eye measured as the spectral absorption characteristics of human cones is given in Fig. 2.1. The sampled visual spectral sensitivity is represented by a matrix of three vectors V = [V1, V2, V3], one for each cone type, where Vi is a vector of n elements. Compared to the CMF curves of Fig. 1.2, they differ in several ways: first, the visual spectral sensitivity has no negative elements; second, the overlapping of green (middle-wavelength) and red (long-wavelength) components is much stronger.
The sensor responses to the object spectrum Î·(Î») can be represented by Î³=V T Î⋅.
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