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

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 article, the authors present computed geodesic curves and corresponding contours of the CIELAB (ΔEmath image ), the CIELUV (ΔEmath image ), the OSA-UCS (ΔEE) and an infinitesimal approximation of the CIEDE2000 (ΔE00) 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 (ΔE00) and the OSA-UCS (ΔEE) in the CIELAB color space are in general not better than the conventional CIELAB (ΔEmath image) and CIELUV (ΔEmath image) formulas.