Lossless Shape Representation using Invariant Statistics: the Case of Point-sets

Boutin and Kemper have shown that the set of unlabeled pairwise distances between the points of a generic point-set in Rⁿmiddot is a lossless representation of the shape of the point-set. In this paper, we extend this result to the case where each of the points observed is drawn from a similar spherical Gaussian distribution in R². More precisely, we consider the distribution of the (squared) distance between two points independently drawn from a mixture of spherical Gaussians, each Gaussian having the same variance sigma². We then show that two generic such mixtures of spherical Gaussians have the same shape (i.e., they are related by a rigid motion) if and only if their distribution of distances are the same.