From tristimulus color space to cone space

A mathematically simple transformation from the Judd (1951) modified CIE space into the space spanned by the cones is derived. The transformation is based on geometrical intuition (Pitt 1945). Light fluxes that initiate vision are transmitted into electrical signals by the photoreceptors. In day light these are the retinal cones. Most humans have three types of cones with spectral sensitivities in the short (S), middle (M) and long (L) part of the visible spectrum, and hence are called trichromats. Absorption of a photon leads to a structural change of photo pigment, which through an enzymatic cascade generates the electrical cone signal. In this process information about the wavelength of the photon is lost. Thus the cone signals processed further in the retina just depend on the numbers of photons that are absorbed in the respective type. Therefore colors can be described in a threedimensional space formed by the excitations of the three cone types (MacLeod & Boynton, 1973). When color metrics was developed, absorption spectra of cones were not known yet. Color perception had to be quantified using other means. Three primary colors (primaries) were defined, each chosen from a different range of colors. Human observers can match any color by varying the intensities of the primaries and hence define this color within the chosen metric (Judd & Wysecki 1975). In some cases one of the primaries has to be added to the color being analyzed to make it appear less saturated. This procedure is the reason for the appearance of negative color mixture values. Colors perceived due to occurrence of energy in just one line of the spectrum are called spectral colors. The intensities of the three primaries needed to match spectral colors plotted over the wavelength are called color matching functions (CMF). CMFs can be transformed from one set of primaries to another using linear algebra. Obviously, each primary has a defined overlap with the absorption spectra of the cones. Thus changing the intensities of the primaries leads to proportional changes in the excitations of the cones. This means that we can find a linear transformation between the CMF and the spectral sensitivities of the cones. One way to derive these is based on the specific deficits of dichromats in discriminating between certain colors. Dichromats are naturally occurring mutants in (mostly male) humans, each of whom has only two of the three cone pigments found in trichromats. Though frequently used in color vision, transformations based on these findings still lack a simple interpretation. The aim of this paper is to give a simple geometric interpretation of this crucial step in color metrics from conventional color space (CIE) into cone space. We elaborate an earlier proposal (Pitt, 1945). We avoid technical details and emphasize the rather simple steps of the transformation. Elementary mathematical tools required are provided in the appendix. Brief describtion of CIE space Since the starting point of our transformation is the modified version of CIE-xy space (Judd, 1951), denoted x’y’-space, we will mention a few features of it. In 1931 the CIE (Commission Internationale de l‘Eclairage) recommended a special set of CMFs x(λ), y(λ), z(λ) as the standard colorimetric system. These functions were transformations of the results of color mixture experiments into a space of primaries which does not have a physical realisation. The reason for this choice was that all these CMFs have positive values and that the y(λ) function is proportional to Vλ, the photopic luminous-efficiency function (of the 1924 standard human observer). Vλ is the spectral distribution connected to the perception of brightness. The three numbers that characterize a color in CIE space are called the tristimulus values X, Y and Z, where Y corresponds to the luminance of the color. X, Y and Z are calculated by integrating the spectral radiance over the CMF. The CIE chromaticity diagram (CIE-xy space) is a normalized representation of the colors perceived by the standard observer formed by the tristimulus values X, Y and Z. It is calculated by the fractions of X and Y in terms of X+Y+Z. The xand z-values are called chromaticity coordinates. The 1924 Vλ is known to be in error in the short wavelength region of the spectrum. Judd (1951) proposed a modified version of the luminous-efficiency function, which also implies the use of a modified set of CMF, denoted x’(λ), y’(λ) and z’(λ). Color mixture data show a large variability between different observers. The question which set of CMF is to be preferred over the others is discussed in detail by Stockman et al. (1993ab). In psychophysical experiments (see next section) these authors measured the spectral sensitivities of the S-, Mand L-cones in human observers using a transient chromatic adaptation method.