Some equivalence relation between persistent homology and morphological dynamics

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Abstract

In Mathematical Morphology (MM), connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) that represents the stability of the extrema of a Morse function. Since these two concepts seem to be closely related, in this paper we examine their relationship, and we prove that they are equal on -D Morse functions, . More exactlypairing a minimum with a -saddle by dynamics or pairing the same -saddle with a minimum by persistence leads exactly to the same pairing, assuming that the critical values of the studied Morse function are unique. This result is a step further to show how much topological data analysis and mathematical morphology are related, paving the way for a more in-depth study of the relations between these two research fields.

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Bibtex (lrde.bib)

@Article{	  boutry.22.jmiv.2,
  author	= {Nicolas Boutry and Laurent Najman and Thierry G\'eraud},
  title		= {Some equivalence relation between persistent homology and
		  morphological dynamics},
  journal	= {Journal of Mathematical Imaging and Vision},
  month		= may,
  year		= {2022},
  abstract	= {In Mathematical Morphology (MM), connected filters based
		  on dynamics are used to filter the extrema of an image.
		  Similarly, persistence is a concept coming from Persistent
		  Homology (PH) and Morse Theory (MT) that represents the
		  stability of the extrema of a Morse function. Since these
		  two concepts seem to be closely related, in this paper we
		  examine their relationship, and we prove that they are
		  equal on $n$-D Morse functions, $n\geq 1$. More exactly,
		  pairing a minimum with a $1$-saddle by dynamics or pairing
		  the same $1$-saddle with a minimum by persistence leads
		  exactly to the same pairing, assuming that the critical
		  values of the studied Morse function are unique. This
		  result is a step further to show how much topological data
		  analysis and mathematical morphology are related, paving
		  the way for a more in-depth study of the relations between
		  these two research fields.}
}