Maximum possible contrast level for silent substitution: a theoretical model applied to melanopsin stimulation

For any given set of light sources stimulating the photoreceptors of the retina, the theoretical levels of illumination producing the smallest and the largest expression of one photoreceptor with fixed stimulation for the others are analytically computed. The cases of four, five, and more light sources are studied. We show that, for contrast optimization, only as many light sources as photoreceptors do matter and that, in the case of four light sources, the maximum contrast achievable for melanopsin lies at the intersection of the lines joining the sources in the CIE xy chromaticity diagram. This result is used to obtain the optimal position of four Gaussian primaries of equal bandwidth. In addition, we derive a procedure to construct level maps for melanopsin contrast overlying the diagram. In the second part of the paper, the interpersonal variability of the perceived stimulation is shown to be globally reduced if the bandwidth of the light sources is increased and, under some assumptions, if a light source is added.