Some equivalence relation between persistent homology and morphological dynamics

From LRDE

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.

Documents

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.}
}