Photoreceptor Nonlinearities Can Account for the MacAdam Ellipses

Colour perception is known to be nonlinear, as highlighted by D L MacAdam (1942 Journal of the Optical Society of America 32 247). He showed that the just noticeable difference (JND) between two colours is not constant and depends nonuniformly on the values of these two colours: under constant illumination, the constant-JND loci, drawn in the xy colour space around various centres, appear as ellipses of different orientations and eccentricities. Though this phenomenon encompasses the whole perception chain, from receptors to higher levels, we show that it can be accounted for mainly by the nonlinearity of receptor transduction. We start from the colorimetric space XYZ defined by the CIE (1931) and its linear transformation into the cone-excitation space LMS, using data from V C Smith and J Pokorny (1996 Color Research and Applications 21 5). Then we consider a perceptual colour space obtained by the nonlinear correspondence between the L, M, and S cone inputs and the l, m, and s cone outputs, through a compression law of Michaelis - Menten type, eg: l = L/(L + L0). In this space, the JNDs are circles, drawn on the surface corresponding to the nonlinear transformation of the constant-luminance plane in LMS, around the corresponding centres. By looking at the projections of those circles onto the constant-luminance plane in LMS, and their representations in the colorimetric space xy, we naturally obtain the ellipses of MacAdam. The formulas, fitted to the data, provide the parameters of the LMS-to-lms transformations.