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Riemannian metric tensors of color difference formulas are derived from the line elements in a color space. The shortest curve between two points in a color space can be calculated from the metric tensors. This shortest curve is called a geodesic. In this paper, the authors present computed geodesic curves and corresponding contours of the CIELAB (∆E * ab ), the CIELUV (∆E * uv ), the OSA-UCS (∆EE) and an infinitesimal approximation of the CIEDE2000 (∆E00) color difference metrics in the CIELAB color space. At a fixed value of lightness L * , geodesic curves originating from the achromatic point and their corresponding contours of the above four formulas in the CIELAB color space can be described as hue geodesics and chroma contours. The Munsell chromas and hue circles at the Munsell values 3, 5 and 7 are compared with computed hue geodesics and chroma contours of these formulas at three different fixed lightness values. It is found that the Munsell chromas and hue circles do not the match the computed hue geodesics and chroma contours of above mentioned formulas at different Munsell values. The results also show that the distribution of color stimuli predicted by the infinitesimal approximation of CIEDE2000 (∆E00) and the OSA-UCS (∆EE) in the CIELAB color space are in general not better than the conventional CIELAB (∆E * ab ) and CIELUV (∆E * uv ) formulas.

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 .

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