A method for recognizing the shape of a Gaussian mixture from a sparse sample set

The motivating application for this research is the problem of recognizing a planar object consisting of points from a noisy observation of that object. Given is a planar Gaussian mixture model ρT (x) representing an object along with a noise model for the observation process (the template). Also given are points representing the observation of the object (the query). We propose a method to determine if these points were drawn from a Gaussian mixture ρQ(x) with the same shape as the template. The method consists in comparing samples from the distribution of distances of ρT (x) and ρQ(x), respectively. The distribution of distances is a faithful representation of the shape of generic Gaussian mixtures. Since it is invariant under rotations and translations of the Gaussian mixture, it provides a workaround to the problem of aligning objects before recognizing their shape without sacrificing accuracy. Experiments using synthetic data show a robust performance against type I errors, and few type II errors when the given template Gaussian mixtures are well distinguished.