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

We state in this paper a strong relation existing between Mathematical Morphology and Morse Theory when we work with 1D \(\mathfrak {D}\)-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 Morse Theory, a well-known tool to simplify the Morse-Smale complexes representing the topological information of a \(\mathfrak {D}\)-Morse function is the persistence. We show that pairing by persistence is equivalent to pairing by dynamics.

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