Analytical Approximations for Color Metric Coefficients*

Abstract

On the assumption of a color space of constant negative curvature, von Schelling has given simple formulas for the color metric coefficients. An attempt has been made to fit such formulas to newly available color-discrimination data for 12 normal observers. A least-squares analysis indicates that Y should be replaced with the quantity, Y′=Y−0.085X+0.011Z, in the special role assigned to luminance by von Schelling. Corresponding to this change, coordinates x′=X/(X+Y′+Z), y′=Y′/(X+Y′+Z) yield the simplest formulas for the coefficients of the expression for color difference: ΔS=(g11Δx′2+2g12Δx′Δy′+g22Δy′2+g33ΔY2/Y2)12.Least-squares fitting of the experimental data to von Schelling’s formulas yielded: g11=g33/(10x′2-6.886x′+2.083+2.5y′-16.66x′y′+19.6y′2).g12=g11(0.3443-x′)/y′.g22=g11[0.09+(0.3443-x′)2]/y′2.These formulas are the best approximations obtainable on the assumption of a color space of constant negative or zero curvature. However, they give such poor approximations to the experimental data that it is concluded that the assumption of a color space inherently more complicated than a space of constant curvature would be required to fit the data to a closeness commensurate with the experimental reproducibility.