Persistent Homology as Stopping-Criterion for Voronoi Interpolation

TL;DR

This paper introduces a novel stopping criterion for Voronoi interpolation based on persistent homology, capturing topological changes during the process to improve interpolation quality.

Contribution

It presents a new method that uses persistent homology to determine when to stop Voronoi interpolation, supported by theoretical analysis and numerical experiments.

Findings

Effective topological stopping criterion established

Persistent homology captures critical boundary points

Numerical experiments validate the approach

Abstract

In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration. The technique is based on Voronoi tessellation, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fact calculating the persistent homology on it after each iteration to capture the changing topology of the data. The boundary points are identified as critical. The Bottleneck and Wasserstein distance serve as a measure of quality between the original point set and the interpolation. If the norm of two distances exceeds a heuristically determined threshold, the algorithm terminates. We give the theoretical basis for this approach and justify its validity with numerical experiments.