Symmetric polynomials for 2D shape representation

Abstract

We discuss the usage of symmetric polynomials for representing 2D shapes in their most general form, i.e., arbitrary sets of unlabeled points in the plane. Although particular families of these polynomials have been used in the past, we present general results that pave the way for the development of new representations that exhibit key properties for shape recognition. We show that all monomial symmetric polynomials satisfy homogeneity, which enables leveraging on previous work on spectral invariants to obtain invariance/completeness with respect to shape orientation. Representations based on symmetric polynomials are invariant to shape point relabeling. We single out elementary symmetric polynomials and power sums as particular families of polynomials that further enable obtaining completeness with respect to point labeling. We discuss the efficient computation of these polynomials and study how perturbations in the shape point coordinates affect their values.