Mathematical limitations when choosing psychophysical methods: geometric versus linear grey scales

Abstract

The grey scale method is commonly used for investigating differences in material appearance. Specifically, for testing color difference equations, perceived color differences between sample pairs are obtained by visually comparing to differences in a series of achromatic sample pairs. Two types of grey scales are known: linear and geometric. Their instrumental color differences vary linearly or geometrically (i.e., exponentially), respectively. Geometric grey scales are used in ISO standards and standard procedures of the textile industries. We compared both types of grey scale in a psychophysical study. Color patches were shown on a color-calibrated display. Ten observers assessed color differences in sample pairs, with color differences between ΔEab = 0.13 and 2.50. Assessments were scored by comparison to either a linear or a geometric grey scale, both consisting of six achromatic pairs. For the linear scale we used color differences ΔEab = 0.0, 0.6, 1.2,..., 3.0. For the geometric scale this was ΔEab=0.0, 0.4, 0.8, 1.6, 3.2, 6.4. Our results show that for the geometric scale, data from visual scores clutter at the low end of the scale and do not match the ΔEab range of the grey scale pairs. We explain why this happens, and why this is mathematically inevitable when studying small color differences with geometric grey scales. Our analysis explains why previous studies showed larger observer variability for geometric than for linear scales.