Measurement of small color differences.

standpoint is assumed. . . . Many blind alleys can be avoided if strings of equations are replaced by simple abstract arguments. Thirdly, one obtains a new perspective on the interrelation between color measurement and color theories. This makes it much easier to survey the plethora of starting points for various extant color systems or theories, and to formulate more clearly some of the main unresolved empirical problems in color perception, (p. 283) Just as the exposition provided by Krantz has already proved useful in understanding color matching, I believe that the new exposition for color-difference measurement 288 BRIAN A. WANDELL has already had certain benefits. The algebraic description emphasizes the crucial assumptions required to test whether an important class of metrics may be applied to discrimination. I develop the formal notation with the following argument in mind: The color-matching theorem includes two kinds of results. First, we may map lights to the points of a vector space in a way that captures both the psychological relationship of metamerism and the additivity properties called Grassman's laws. This result is called the representation result. Second, all linear transformations of the vector space are equivalent in the sense that any linear transformation of the space will preserve both the metamerism relationship and the additivity relationship. This result is called the uniqueness result. Together these results prompt the following question: Is there a distance formula for color discriminability with the property that small color discriminability may be predicted by the vector difference between two lights? This question includes as a special case the examples listed above where the distance formula is Euclidean. Before considering the general question, I develop the theory for this important, special case. Invariances Across Linear Transformations Suppose that when the observer is adapted to a light, a, there exists a set of basis coordinates in which small color differences may be represented as Euclidean vector distances. Let the space satisfying this property be denoted as Va where the subscript denotes the point in space from which the discriminability of nearby points may be measured by Euclidean distance. In this definition the color matching coordinates, which we refer to as C, are a linear transformation of the space Va, but the metric properties of discriminability only hold for a small region of the space, near the adapting point, a. We denote the linear transformation that maps the coordinate system Va into the coordinates of C as and we denote the action of A on a vector, V(a), as V(a)A. We may treat A as a three by three, nonsingular (i.e., possessing inverse) matrix. As a computational example, in the space Va we measure the Euclidean distance between two not very different row vectors, a and a '+' 8, by first computing the vector difference, V(5), and then taking the inner product of the row vector V(5) and its transpose (column vector) ¥(6)': ds = [V(« '+' 5) V(fl)] •