Spherical Model of Color and Brightness Discrimination

The most important problem confronting color science is the construction of a uniform color space, i.e., a geometrical model of color discrimination in which Euclidean distances between the points representing colors are proportional to perceived color differences. The traditional approach to the construction of a metric color space is based on the integration of just-noticeable color differences (Wyszecki & Stiles, 1982). Experimental data show, however, that the integral of justnoticeable differences between colors does not coincide with direct estimations of the subjective differences between the colors (Judd, 1967; Izmailov, 1980). We suggest another way to construct a uniform color space, namely, to analyze large color differences by multidimensional scaling. This paper reports three groups of experimental data of the measurement of large color differences. Based on these data, we suggest a new color space model taking into account nontraditional relations between threshold and suprathreshold differences. The first group of data includes the results of research on color discrimination for a set of equibright monochromatic lights. The second group includes data on the discrimination of achromatic light stimuli resulting from different relations between test and background luminances. The third group consists of results of color-naming classification of lights varying in chromaticity and brightness. Chromatic differences between spectral stimuli of equal brightness, varying in hue and saturation, and differences in brightness between achromatic lights varying in luminance were analyzed separately. The results are compared with a general color space of colors of different hue, saturation, and brightness. The color spaces were constructed by the same multidimensional scaling technique. An important advantage of multidimensional scaling is that it offers the possibility of finding the dimensionality of a color space directly from experimental data, as we demonstrate for the analysis of color discrimination data for equibright stimuli. SPHERICAL MODEL FOR CHROMATIC STIMULI The traditional concept of dimensionality of a chromatic subspace of a color space is based on two sensory characteristics of light: either color hue and saturation, or red-green and blueyellow color-opponent systems. A typical model describing color discrimination in these terms is the Euclidean plane, where color hue and saturation are two polar coordinates of a point (horizontal angle and radius), and the opponent systems are two Cartesian coordinates of the same point. Discrimination data, however, prove the color space to be significantly non-Euclidean. First, local color discrimination data obtained by Mac Adam, and Brown and Mac Adam (see Wyszecki & Stiles, 1982) show that differential sensitivity areas in an equibright color space cannot be represented as a Euclidean quadratic form, but only as a surface having a nonzero Gaussian curvature. Second, research on the relation between barely just noticeable and suprathreshold differences (Judd, 1967; Izmailov, 1980) show the interrelation to be nonlinear due to nonadditivity of color discrimination. Third, data on large color differences (Shepard and Carroll, 1966; Izmailov, 1980) indicate that global linearity of color discrimination space with respect to perceived differences between colors increases the dimensionality of the Euclidean color space. Substantial proof of this hypothesis can be found in Shepard and Carroll (1966). These authors consider the problem of finding the dimensionality of the subjective color discrimination space for equiluminance colors. Their theoretical analysis is based on data reported by Boynton and Gordon (1965). Boynton and Gordon (1965) studied the dependence of color discrimination on stimulus luminance with three normal subjects by a color-naming technique. They presented 23 monochromatic stimuli with wavelength ranging from 440 to 660 nm (at a step of 10 nm) to subjects who were to divide the stimuli into four color classes bearing the names blue, green, yellow, and red. If a stimulus was considered intermediate between two classes, it was given a double name, such as blue-green, with the name of the color perceived as more manifest coming first. Weight was assigned to each class by the following rules: If a stimulus belonged to one class only, the class received the weight 3; if a stimulus had a double name, as in our example with blue-green, then the first class (blue) was given the weight 2 and the second class (green) the weight 1. Stimuli were presented 25 times each. The weighted frequency of the attribution of a stimulus to each class was assumed as a measure of the stimulus subjective estimate. Marking stimulus wavelength on the abscissa axis and the subjective estimate on the ordinate axis, Boynton and Gordon obtained color-naming curves. By changing stimulus luminance at the levels of 100 and 1,000 trolands, Boynton and Gordon measured the dependence of color discrimination on brightness. In order to make the data suitable for multidimensional sealAddress correspondence to Dr. E.N. Sokolov, Marx Avenue, 18/5, Moscow, USSR 103009. VOL. 2, NO. 4, JULY 1991 Copyright © 1991 American Psychological Society 249 PSYCHOLOGICAL SCIENCE Spherical Discrimination Model ing, Shepard and Carroll treated the responses to each stimulus as a naming vector. The vector's components were weighted frequencies of the assigned names. The number of components thus depends on the number of classes, and consequently, all the vectors were four-component. The distance measure between vectors was taken to be the city-block measure in one case, and the Euclidean measure in the other. Distances between all pairs of vectors formed an n(n l)/2 matrix of distances averaged for three subjects on luminance level 100 trolands (n is the number of stimuli). The matrix was processed by several multidimensional scaling techniques. The spatial model was shown to be independent of the chosen distance measure between reaction vectors. The solution was identical when the differences were interpreted in Euclidean metrics and cityblock metrics. Moreover, the spatial model did not depend on the multidimensional scaling algorithm. The results led the authors to a conclusion about the rigidity of the color discrimination space structure. It was found that the dimensionality of the obtained space changed according to the choice of the approximation criterion of interpoint distances to initial distance measures. When global linearity was postulated for initial distance measures, the minimal dimensionality of the Euclidean space was three. In a three-dimensional space the 23 points representing monochromatic olors were located so that the line connecting points that represent stimuli from the first to the 23rd was a one-dimensional curve with bends in blue, green, and yellow areas. When the relation between initial data and interpoint distances was required to be globally monotone instead of linear, Shepard and Carroll obtained a two-dimensional space for the same data. The points were located on a curve in this case also, with the same bends in blue, green, and yellow areas. When the condition was changed from a globally monotone relation to a locally monotone one, the points became located on a straight line. The single dimension could be interpreted as a substantial change of wavelength. Analyzing the data, Shepard and Carroll noticed a reciprocal relation between the simplicity of the spatial presentation of the initial data, and the simplicity of the presentation's connection with the data. The more complex three-dimensional solution has the advantage of a linear relation with initial data, while the simplest solution compensates for its efficiency by a nonlinear relation with data. There is another substantive criterion for determining the true dimensionality of the subjective space that must be taken into consideration, namely, the neurophysiological interpretation of obtained data. The existence of a single sensory mechanism with the complicated principles suggested by the onedimensional solution does not seem probable. Numerous data on the performance of the color analyzer point to the existence of several simple similarly performing mechanisms conforming to the first solution. The three-dimensional solution meets, however, with serious difficulties in giving a traditional interpretation to subjective characteristics of aperture colors. Agreement among the various data becomes possible in a spherical color discrimination space obtained by Sokolov, Izmailov, and Schonebeck (1982), Izmailov et al. (1989), Izmailov (1982), and Sokolov and Izmailov (1983). We now present two experimental runs in above-threshold discrimination and their analysis by multidimensional scaling.