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

### From LRDE

- Authors
- Nicolas Boutry, Thierry Géraud, Laurent Najman
- Where
- Mathematical Morphology and Its Application to Signal and Image Processing – Proceedings of the 14th International Symposium on Mathematical Morphology (ISMM)
- Place
- Saarbrücken, Germany
- Type
- inproceedings
- Publisher
- Springer
- Projects
- Olena
- Keywords
- Image
- Date
- 2019-03-13

## 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.

## Documents

## 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.} }