# Difference between revisions of "Publications/boutry.22.jmiv.2"

## 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 ${\displaystyle n}$-D Morse functions, ${\displaystyle n\geq 1}$. More exactlypairing a minimum with a ${\displaystyle 1}$-saddle by dynamics or pairing the same ${\displaystyle 1}$-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 relatedpaving the way for a more in-depth study of the relations between these two research fields.

## 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},
volume	= {},
number	= {},
pages		= {},
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