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 fourmlas having different color spaces for measuring visual color difference. In this paper, the authors present mathematical formulations of CIELAB (∆E * ab ), CIELUV (∆E * uv ), OSA-UCS (∆EE) and infinitesimal approximation of CIEDE2000 (∆E00) 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 ∆E00 and the ∆EE at small colour difference, but they are both notably better than ∆E * ab and ∆E * uv .

Geodesic calculation of color difference formulas and comparison with the munsell color order system

“…The detailed discussion about difficulty for getting Riemannian metric of the CIEDE2000 can be found in the article of Pant and Farup. 29 Geodesic grids of the Riemannized DE 00 and their comparison with the Munsell hue and chromas at the different Munsell values are shown in Figs. 2(c), 3(c), and 4(c).…”

“…The detailed explanation about this as well as the derivation of Riemannian metrics of above color difference formulas can be found in the authors' previous article. 29…”

“…For a detailed derivation of the Jacobians involved, it is referred to the authors' previous paper. 29 The Geodesic Equation…”

A multilevel investigation supported by multivariate analysis for tomato product formulation

“…The a*/b* ratio was computed to describe redness index [24,25]. Differences of color between control (TDC with no ingredients) and samples with added ingredients were evaluated using the ΔE value [26].…”

Multi-level contrast filtering in image difference metrics

“…The chroma log compression is required in the passage from large to small color differences. Because the image quality is evaluated in the visual situation of the small-medium color differences, we think this formula is a good candidate for the color image metrics [22,23].…”