Identification of an algebraic domain in two dimensions from a finite number of its generalized polarization tensors

TL;DR

This paper demonstrates how a finite set of generalized polarization tensors can uniquely identify the shape of a planar algebraic domain by deriving its minimal polynomial, leading to a new shape recognition algorithm.

Contribution

It introduces a method to determine the boundary of algebraic domains from polarization tensors and proposes an efficient shape classification algorithm.

Findings

The minimal polynomial of the boundary can be identified from polarization tensors.

The size of the relevant matrix depends polynomially on the boundary degree.

A new shape recognition algorithm is proposed with promising efficiency.

Abstract

This paper aims at studying how finitely many generalized polarization tensors of an algebraic domain can be used to determine its shape. Precisely, given a planar set with real algebraic boundary, it is shown that the minimal polynomial with real coefficients vanishing on the boundary can be identified as the generator of a one dimensional kernel of a matrix whose entries are obtained from a finite number of generalized polarization tensors. The size of the matrix depends polynomially on the degree of the boundary of the algebraic domain. The density with respect to Hausdorff distance of algebraic domains among all bounded domains invites to extend via approximation our reconstruction procedure beyond its natural context. Based on this, a new algorithm for shape recognition/classification is proposed with some strong hints about its efficiency.