Approximating Loops in a Shortest Homology Basis from Point Data

Abstract

Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of loops from a point data that presumably sample a smooth manifold $M\subset {\mathbb{R}}^d$. These loops approximate a {\em shortest} basis of the one dimensional homology group $H_1(M)$ over coefficients in finite field ${\mathbb{Z}}_2$. Previous results addressed the issue of computing the rank of the homology groups from point data, but there is no result on approximating the shortest basis of a manifold from its point sample. In arriving our result, we also present a polynomial time algorithm for computing a shortest basis of $H_1(K)$ for any finite {\em simplicial complex} $K$ whose edges have non-negative weights.