Polynomial Eulerian shape distributions

TL;DR

This paper introduces a new shape distribution based on Euler hypergeometric functions, connecting shape invariants with canonical correlations, and provides an analytical computational method for exact inference in shape analysis.

Contribution

It develops a novel shape distribution involving Euler hypergeometric functions and offers an analytical approach for exact inference using polynomial distributions.

Findings

Derived a new shape distribution using Euler hypergeometric functions

Established a connection with canonical correlations for shape analysis

Proposed a method for exact inference and applied it to handwritten differentiation

Abstract

In this paper a new approach is derived in the context of shape theory. The implemented methodology is motivated in an open problem proposed in \citet{GM93} about the construction of certain shape density involving Euler hypergeometric functions of matrix arguments. The associated distribution is obtained by establishing a connection between the required shape invariants and a known result on canonical correlations available since 1963; as usual in statistical shape theory and the addressed result, the densities are expressed in terms of infinite series of zonal polynomials which involves considerable difficulties in inference. Then the work proceeds to solve analytically the problem of computation by using the Eulerian matrix relation of two matrix argument for deriving the corresponding polynomial distribution in certain parametric space which allows to perform exact inference based on exact distributions characterized for polynomials of very low degree. A methodology for comparing correlation shape structure is proposed and applied in handwritten differentiation.