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

From LRDE

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 topological 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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Bibtex (lrde.bib)

@InProceedings{	  boutry.19.ismm,
  author	= {Nicolas Boutry and Thierry G\'eraud and Laurent Najman},
  title		= {An Equivalence Relation between Morphological Dynamics and
		  Persistent Homology in {1D}},
  booktitle	= {Mathematical Morphology and Its Application to Signal and
		  Image Processing -- Proceedings of the 14th International
		  Symposium on Mathematical Morphology (ISMM)},
  year		= 2019,
  editor	= {},
  volume	= {},
  series	= {Lecture Notes in Computer Science Series},
  address	= {Saarbr\"ucken, Germany},
  publisher	= {Springer},
  pages		= {1--12},
  month		= {July},
  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
		  topological 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.}
}