Some equivalence relation between persistent homology and morphological dynamics

TL;DR

This paper establishes a precise equivalence between persistent homology and morphological dynamics in Morse functions, revealing a fundamental connection between topological data analysis and mathematical morphology.

Contribution

It proves that persistence and dynamics-based pairings are identical for n-D Morse functions with unique critical values, bridging two research fields.

Findings

Persistence and dynamics pairings coincide for Morse functions.

The result applies to n-D Morse functions with unique critical values.

This work advances understanding of the relationship between TDA and MM.

Abstract

In Mathematical Morphology (MM), connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) that represents the stability of the extrema of a Morse function. Since these two concepts seem to be closely related, in this paper we examine their relationship, and we prove that they are equal on n-D Morse functions, n $\ge$ 1. More exactly, pairing a minimum with a 1-saddle by dynamics or pairing the same 1-saddle with a minimum by persistence leads exactly to the same pairing, assuming that the critical values of the studied Morse function are unique. This result is a step further to show how much topological data analysis and mathematical morphology are related, paving the way for a more in-depth study of the relations between these two research fields.