Riemannian formulation and comparison of color difference formulas

Abstract

Study of various color difference formulas by the Riemannian approach is useful. By this approach, it is possible to evaluate the performance of various color difference formulas having different color spaces for measuring visual color difference. In this article, the authors present mathematical formulations of CIELAB (ΔEmath image), CIELUV (ΔEmath image), OSA-UCS (ΔEE) and infinitesimal approximation of CIEDE2000 (ΔE00) as Riemannian metric tensors in a color space. It is shown how such metrics are transformed in other color spaces by means of Jacobian matrices. The coefficients of such metrics give equi-distance ellipsoids in three dimensions and ellipses in two dimensions. A method is also proposed for comparing the similarity between a pair of ellipses. The technique works by calculating the ratio of the area of intersection and the area of union of a pair of ellipses. The performance of these four color difference formulas is evaluated by comparing computed ellipses with experimentally observed ellipses in the xy chromaticity diagram. The result shows that there is no significant difference between the Riemannized ΔE00 and the ΔEE at small color difference, but they are both notably better than ΔEmath image and ΔEmath image.