# An Equivalence Relation between Morphological Dynamics and Persistent Homology in n-D

### From LRDE

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## Abstract

In Mathematical Morphology (MM), dynamics are used to compute markers to proceed for example to watershed-based image decomposition. At the same time, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) and represents the stability of the extrema of a Morse function. Since these concepts are similar on Morse functions, we studied their relationship and we found, and proved, that they are equal on 1D Morse functions. Here, we propose to extend this proof to ${\displaystyle n}$-D, ${\displaystyle n\geq 2}$, showing that this equality can be applied to ${\displaystyle n}$-D images and not only to 1D functions. This is a step further to show how much MM and MT are related.

## Bibtex (lrde.bib)

```@InProceedings{	  boutry.21.dgmm.1,
author	= {Nicolas Boutry and Thierry G\'eraud and Laurent Najman},
title		= {An Equivalence Relation between Morphological Dynamics and
Persistent Homology in {\$n\$-D}},
booktitle	= {Proceedings of the IAPR International Conference on
Discrete Geometry and Mathematical Morphology (DGMM)},
year		= 2021,
month		= may,
series	= {Lecture Notes in Computer Science},
volume	= {12708},
publisher	= {Springer},
pages		= {525--537},
abstract	= {In Mathematical Morphology (MM), dynamics are used to
compute markers to proceed for example to watershed-based
image decomposition. At the same time, persistence is a
concept coming from Persistent Homology (PH) and Morse
Theory (MT) and represents the stability of the extrema of
a Morse function. Since these concepts are similar on Morse
functions, we studied their relationship and we found, and
proved, that they are equal on 1D Morse functions. Here, we
propose to extend this proof to \$n\$-D, \$n \geq 2\$, showing
that this equality can be applied to \$n\$-D images and not
only to 1D functions. This is a step further to show how
much MM and MT are related.},
doi		= {10.1007/978-3-030-76657-3_38}
}```