An invariant metric on the manifold of second order moments

We introduce an invariant metric in the space of symmetric, positive definite matrices and illustrate the usage of this space together with this metric in color processing. For this metric closed-form expressions for the distances and the geodesics, (ie. the straight lines in this metric) are available and we show how to implement them in the case of matrices of size 2×2. In the first illustration we use the framework to investigate an interpolation problem related to the ellipses obtained in the measurements of just-noticeable-distances. For two such ellipses we use the metric to construct an interpolating sequence of ellipses between them. In the second application construct a texture descriptor for chromaticity distributions. We describe the probability distributions of chromaticity vectors by their matrices of second order moments. The distance between these matrices is independent under linear changes of the coordinate system in the chromaticity space and can therefore be used to define a distance between probability distributions that is independent of the coordinate system used. We illustrate this invariance, by way of an example, in the case of different white point corrections.