Shape theory via polar decomposition

TL;DR

This paper introduces a novel statistical shape model using polar decomposition, enabling easier computation of shape distributions and improved inference, demonstrated through biological data analysis with model selection and shape comparison.

Contribution

It develops new polar shape distributions avoiding complex invariant polynomials, facilitating exact inference in statistical shape analysis.

Findings

Polar shape distributions are computationally efficient.

The method successfully distinguishes between different shape models.

Application to biological data demonstrates practical utility.

Abstract

This work proposes a new model in the context of statistical theory of shape, based on the polar decomposition. The non isotropic noncentral elliptical shape distributions via polar decomposition is derived in the context of zonal polynomials, avoiding the invariant polynomials and the open problems for their computation. The new polar shape distributions are easily computable and then the inference procedure can be studied under exact densities. As an example of the technique, a classical application in Biology is studied under three models, the usual Gaussian and two non normal Kotz models; the best model is selected by a modified BIC criterion, then a test for equality in polar shapes is performed.