An analytical model of the influence of cone sensitivity and numerosity on the Rayleigh match

The Rayleigh match is defined by the range of mixtures of red and green lights that appear the same as an intensity-adjustable monochromatic yellow light. The perceptual match indicates that the red-green mixture and the yellow light have evoked the same respective cone absorptions in the L- and M-cone pathways. Going beyond the existing models, the Poisson noise in cone absorptions is proposed to make the matching proportion of red-green mixtures span a finite range because any mixture in that range evokes cone absorptions that do not differ from those by a yellow light by more than the variations in the absorption noise. We derive a mathematical formula linking the match midpoint or match range with the sensitivities and numerosities of the two cones. The noise-free, exact, matching point, close to the midpoint of the matching range, depends only on the L- and M-cone sensitivities to each of the red, green, and yellow lights [these sensitivities, in turn, depend on the preferred wavelengths (λmax) and optical densities of the cone pigments and the properties of prereceptoral light filtering]. Meanwhile, the matching range depends on both these cone sensitivities and the relative numerosity of the L and M cones. The model predicts that, in normal trichromats, all other things being equal, the match range is smallest when the ratio r between L and M cone densities is r=R(-1/2) with R as the ratio between the sensitivities of the L and M cones to the yellow light, i.e., when L and M cones are similarly abundant in typical cases, and, as r departs from R(-1/2), the match range increases. For example, when one cone type is 10 times more numerous, the match range increases two- to threefold, depending on the sensitivities of the cones. Testing these model predictions requires either a large data set to identify the effect of one factor (e.g., cone numerosity) while averaging out the effects of the other factors (e.g., cone sensitivities) or for all factors to be known. A corollary of this prediction is that, because they are more likely than usual to have L:M cone ratios skewed, the matching ranges of normal female trichromats who are carriers of dichromacy (but not anomalous trichromacy) are likely to have a larger matching range than usual, particularly for the deutan carriers. In addition, the model predicts that, in strong tetrachromats (whose four dimensions of color are preserved post-receptorally), either the Rayleigh matching is impossible or the matching range is typically smaller than usual.