Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory

TL;DR

This paper establishes a theoretical connection between discrete Morse theory and persistent homology by showing that lexicographically optimal homologous cycles are supported on Wrap complexes within Delaunay complexes, linking shape reconstruction methods.

Contribution

It demonstrates that for any cycle in a Delaunay complex, the optimal homologous cycle resides within the Wrap complex, bridging two key shape analysis techniques.

Findings

Lexicographically optimal cycles are supported on Wrap complexes.

A fundamental link between persistent homology reduction and discrete Morse gradient flows.

Supports shape reconstruction by connecting two topological methods.

Abstract

We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we consider the construction of Wrap complexes, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the construction of lexicographic optimal homologous cycles, also considered by Cohen-Steiner, Lieutier, and Vuillamy in a similar setting. We show that for any cycle in a Delaunay complex for a given radius parameter, the lexicographically optimal homologous cycle is supported on the Wrap complex for the same parameter, thereby establishing a close connection between the two methods. We obtain this result by establishing a fundamental connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory.