Practical non-parametric density estimation on a transformation group for vision

It is now common practice in machine vision to define the variability in an object's appearance in a factored manner, as a combination of shape and texture transformations. In this context, we present a simple and practical method for estimating non-parametric probability densities over a group of linear shape deformations. Samples drawn from such a distribution do not lie in a Euclidean space, and standard kernel density estimates may perform poorly. While variable kernel estimators may mitigate this problem to some extent, the geometry of the underlying configuration space ultimately demands a kernel, which accommodates its group structure. In this perspective, we propose a suitable invariant estimator on the linear group of non-singular matrices with positive determinant. We illustrate this approach by modeling image transformations in digit recognition problems, and present results showing the superiority of our estimator to comparable Euclidean estimators in this domain.