Abstract
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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Boutry, N., Géraud, T., Najman, L. (2019). An Equivalence Relation Between Morphological Dynamics and Persistent Homology in 1D. In: Burgeth, B., Kleefeld, A., Naegel, B., Passat, N., Perret, B. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2019. Lecture Notes in Computer Science(), vol 11564. Springer, Cham. https://doi.org/10.1007/978-3-030-20867-7_5
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DOI: https://doi.org/10.1007/978-3-030-20867-7_5
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