Faithful Shape Representation for 2D Gaussian Mixtures

It has been recently discovered that a faithful representation for the shape of some simple distributions can be constructed using invariant statistics [1,2]. In this paper, we consider the more general case of a Gaussian mixture model. We show that the shape of generic Gaussian mixtures can be represented without any loss by the distribution of the distance between two points independently drawn from this mixture. In other words, we show that if their respective distributions of distances are the same, then there exists a rigid transformation mapping one Gaussian mixture onto the other. Our main motivation is the problem of recognizing the shape of an object represented by points given noisy measurements of these points which can be modeled as a Gaussian mixture.