Some Equivalence Relation between Persistent Homology and Morphological Dynamics

Abstract

In mathematical morphology, connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from persistent homology and Morse theory 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.