Evaluating Color Difference Formulae by Riemannian Metric

For precision color matching, visual sensitivity to small color difference is an essential factor. Small color differences can be measured by the just noticeable difference (JND) ellipses. The points on the ellipse represent colours that are just noticably different from the colour of the centre point. Mathematically, such an ellipse can be described by a positive definite quadratic differential form, which is also known as the Riemannian metric. In this paper, we propose a method which makes use of the Riemannian metric and Jacobean transformations to transform JND ellipses between different colour spaces. As an example, we compute the JND ellipses of the CIELAB and CIELUV color difference formulae in the xy chromaticity diagram. We also propose a measure for comparing the similarity of a pair of ellipses and use that measure to compare the CIELAB and CIELUV ellipses to two previously established experimental sets of ellipses. The proposed measure takes into account the size, shape and orientation. The technique works by calculating the ratio of the area of the intersection and the area of the union of a pair of ellipses. The method developed can in principle be applied for comparing the performance of any color difference formula and experimentally obtained sets of colour discrimination ellipses.