An Equivalence Relation between Morphological Dynamics and Persistent Homology in 1D

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

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

We state in this paper a strong relation existing between Mathematical Morphology and Discrete Morse Theory when we work with 1D Morse functions. Specifically, in Mathematical Morphology, a classic way to extract robust markers for segmentation purposes, is to use the dynamics. On the other hand, in Discrete Morse Theory, a well-known tool to simplify the Morse-Smale complexes representing the topo-logical information of a Morse function is the persistence. We show that pairing by persistence is equivalent to pairing by dynamics. Furthermore, self-duality and injectivity of these pairings are proved.

Domaines: Topologie algébrique [math.AT]

Nicolas Boutry, Thierry Géraud, Laurent Najman. An Equivalence Relation between Morphological Dynamics and Persistent Homology in 1D. 2019. ⟨hal-01978607⟩