Computationally efficient method to compare the shape of planar Gaussian mixtures from point samples

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We discuss the problem of recognizing the shape of planar objects consisting of blobs that can be modeled as Gaussian mixture densities. We describe an empirical comparison method, assuming a large number of independent samples are given for each distribution. Instead of comparing the Gaussian mixtures directly, we compare the underlying distribution of distances of each mixture. Since distances are invariant under rotations and translations, this provides a workaround to the problem of aligning the objects before comparing them-thus speeding the comparison process. We prove that the distribution of distances is a lossless representation of the shape of generic Gaussian mixtures. Our numerical experiments indicate that, when all the components of the Gaussian mixtures are equally weighted and have the same standard deviation matrix, the proposed method is no less accurate than methods that compare the planar mixtures directly. The extension of our method to the problem of recognizing halftone patterns is briefly discussed.