Some equivalence relation between persistent homology and morphological dynamics

Nicolas Boutry (1) , Laurent Najman (2) , Thierry Géraud (1)

1 LRDE - Laboratoire de Recherche et de Développement de l'EPITA
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

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 ≥ 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.

Mots clés: Mathematical Morphology; Morse Theory; Computational Topology; Persistent Homology; Dynamics; Persistence

Domaines: Géométrie différentielle [math.DG]; Vision par ordinateur et reconnaissance de formes [cs.CV]; Topologie algébrique [math.AT]; Traitement du signal et de l'image [eess.SP]; Traitement des images [eess.IV]

Nicolas Boutry, Laurent Najman, Thierry Géraud. Some equivalence relation between persistent homology and morphological dynamics. Journal of Mathematical Imaging and Vision, 2022, 64, pp.807--824. ⟨10.1007/s10851-022-01104-z⟩. ⟨hal-03676854⟩