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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-01978607
Contributor : Laurent Najman <>
Submitted on : Friday, January 11, 2019 - 4:05:11 PM
Last modification on : Tuesday, March 19, 2019 - 11:43:25 PM

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

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