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

### From LRDE

Line 5: | Line 5: | ||

| title = Some equivalence relation between persistent homology and morphological dynamics |
| title = Some equivalence relation between persistent homology and morphological dynamics |
||

| journal = Journal of Mathematical Imaging and Vision |
| journal = Journal of Mathematical Imaging and Vision |
||

− | | volume = |
||

− | | number = |
||

− | | pages = |
||

| lrdeprojects = Olena |
| lrdeprojects = Olena |
||

− | | 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 <math>n</math>-D Morse functions, <math>n\geq 1</math>. More exactlypairing a minimum with a <math>1</math>-saddle by dynamics or pairing the same <math>1</math>-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 |
+ | | 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 <math>n</math>-D Morse functions, <math>n\geq 1</math>. More exactlypairing a minimum with a <math>1</math>-saddle by dynamics or pairing the same <math>1</math>-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. |

| lrdepaper = https://www.lrde.epita.fr/dload/papers/boutry.22.jmiv.2.pdf |
| lrdepaper = https://www.lrde.epita.fr/dload/papers/boutry.22.jmiv.2.pdf |
||

| lrdekeywords = Image |
| lrdekeywords = Image |
||

Line 15: | Line 12: | ||

| type = article |
| type = article |
||

| id = boutry.22.jmiv.2 |
| id = boutry.22.jmiv.2 |
||

− | | identifier = doi: |
||

| bibtex = |
| bibtex = |
||

@Article<nowiki>{</nowiki> boutry.22.jmiv.2, |
@Article<nowiki>{</nowiki> boutry.22.jmiv.2, |
||

Line 22: | Line 18: | ||

morphological dynamics<nowiki>}</nowiki>, |
morphological dynamics<nowiki>}</nowiki>, |
||

journal = <nowiki>{</nowiki>Journal of Mathematical Imaging and Vision<nowiki>}</nowiki>, |
journal = <nowiki>{</nowiki>Journal of Mathematical Imaging and Vision<nowiki>}</nowiki>, |
||

− | volume = <nowiki>{</nowiki><nowiki>}</nowiki>, |
||

− | number = <nowiki>{</nowiki><nowiki>}</nowiki>, |
||

− | pages = <nowiki>{</nowiki><nowiki>}</nowiki>, |
||

month = may, |
month = may, |
||

year = <nowiki>{</nowiki>2022<nowiki>}</nowiki>, |
year = <nowiki>{</nowiki>2022<nowiki>}</nowiki>, |
||

Line 42: | Line 35: | ||

analysis and mathematical morphology are related, paving |
analysis and mathematical morphology are related, paving |
||

the way for a more in-depth study of the relations between |
the way for a more in-depth study of the relations between |
||

− | these two research fields.<nowiki>}</nowiki> |
+ | these two research fields.<nowiki>}</nowiki> |

− | doi = <nowiki>{</nowiki><nowiki>}</nowiki> |
||

<nowiki>}</nowiki> |
<nowiki>}</nowiki> |
||

## Latest revision as of 16:37, 30 June 2022

- Authors
- Nicolas Boutry, Laurent Najman, Thierry Géraud
- Journal
- Journal of Mathematical Imaging and Vision
- Type
- article
- Projects
- Olena
- Keywords
- Image
- Date
- 2022-05-17

## 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.} }