Comparison of Persistent Homologies for Vector Functions: from continuous to discrete and back

TL;DR

This paper develops a stability-preserving method to compare rank invariants of vector functions, bridging the gap between continuous and discrete multidimensional persistent homology, and confirming its effectiveness in shape comparison tasks.

Contribution

A novel method that maintains stability when comparing discrete and continuous data in multidimensional persistent homology, enhancing shape analysis applications.

Findings

The method effectively compares rank invariants across data types.

Numerical tests validate the approach's stability and applicability.

Multidimensional persistent homology is confirmed as suitable for shape comparison.

Abstract

The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of multidimensional persistence have been proved to hold when topological spaces are filtered by continuous functions, i.e. for continuous data. This paper aims to provide a bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out. More precisely, a stability preserving method is developed to compare rank invariants of vector functions obtained from discrete data. These advances confirm that multidimensional persistent homology is an appropriate tool for shape comparison in computer vision and computer graphics applications. The results are supported by numerical tests.