Difference between revisions of "Publications/boutry.22.jmiv.2"
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| title = Some equivalence relation between persistent homology and morphological dynamics |
| title = Some equivalence relation between persistent homology and morphological dynamics |
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| journal = Journal of Mathematical Imaging and Vision |
| journal = Journal of Mathematical Imaging and Vision |
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| lrdeprojects = Olena |
| lrdeprojects = Olena |
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| − | | 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 |
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| lrdekeywords = Image |
| lrdekeywords = Image |
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| type = article |
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| id = boutry.22.jmiv.2 |
| id = boutry.22.jmiv.2 |
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| bibtex = |
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@Article<nowiki>{</nowiki> boutry.22.jmiv.2, |
@Article<nowiki>{</nowiki> boutry.22.jmiv.2, |
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morphological dynamics<nowiki>}</nowiki>, |
morphological dynamics<nowiki>}</nowiki>, |
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journal = <nowiki>{</nowiki>Journal of Mathematical Imaging and Vision<nowiki>}</nowiki>, |
journal = <nowiki>{</nowiki>Journal of Mathematical Imaging and Vision<nowiki>}</nowiki>, |
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| − | number = <nowiki>{</nowiki><nowiki>}</nowiki>, |
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| − | pages = <nowiki>{</nowiki><nowiki>}</nowiki>, |
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month = may, |
month = may, |
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year = <nowiki>{</nowiki>2022<nowiki>}</nowiki>, |
year = <nowiki>{</nowiki>2022<nowiki>}</nowiki>, |
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analysis and mathematical morphology are related, paving |
analysis and mathematical morphology are related, paving |
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the way for a more in-depth study of the relations between |
the way for a more in-depth study of the relations between |
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| − | these two research fields.<nowiki>}</nowiki> |
+ | these two research fields.<nowiki>}</nowiki> |
| − | doi = <nowiki>{</nowiki><nowiki>}</nowiki> |
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<nowiki>}</nowiki> |
<nowiki>}</nowiki> |
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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.}
}