Difference between revisions of "Publications/boutry.22.jmiv.2"
From LRDE
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| date = 2022-05-17 |
| date = 2022-05-17 |
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| authors = Nicolas Boutry, Laurent Najman, Thierry Géraud |
| authors = Nicolas Boutry, Laurent Najman, Thierry Géraud |
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− | | title = Some |
+ | | title = Some Equivalence Relation between Persistent Homology and Morphological Dynamics |
| journal = Journal of Mathematical Imaging and Vision |
| journal = Journal of Mathematical Imaging and Vision |
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+ | | volume = 64 |
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+ | | pages = 807 to 824 |
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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 related, paving the way for a more in-depth study of the relations between these two research fields. |
| 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. |
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| type = article |
| type = article |
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| id = boutry.22.jmiv.2 |
| id = boutry.22.jmiv.2 |
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+ | | identifier = doi:10.1007/s10851-022-01104-z |
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| bibtex = |
| bibtex = |
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@Article<nowiki>{</nowiki> boutry.22.jmiv.2, |
@Article<nowiki>{</nowiki> boutry.22.jmiv.2, |
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author = <nowiki>{</nowiki>Nicolas Boutry and Laurent Najman and Thierry G\'eraud<nowiki>}</nowiki>, |
author = <nowiki>{</nowiki>Nicolas Boutry and Laurent Najman and Thierry G\'eraud<nowiki>}</nowiki>, |
||
− | title = <nowiki>{</nowiki>Some |
+ | title = <nowiki>{</nowiki>Some Equivalence Relation between Persistent Homology and |
− | + | 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>, |
||
− | month = |
+ | month = sep, |
year = <nowiki>{</nowiki>2022<nowiki>}</nowiki>, |
year = <nowiki>{</nowiki>2022<nowiki>}</nowiki>, |
||
+ | volume = <nowiki>{</nowiki>64<nowiki>}</nowiki>, |
||
+ | pages = <nowiki>{</nowiki>807--824<nowiki>}</nowiki>, |
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abstract = <nowiki>{</nowiki>In Mathematical Morphology (MM), connected filters based |
abstract = <nowiki>{</nowiki>In Mathematical Morphology (MM), connected filters based |
||
on dynamics are used to filter the extrema of an image. |
on dynamics are used to filter the extrema of an image. |
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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>10.1007/s10851-022-01104-z<nowiki>}</nowiki> |
||
<nowiki>}</nowiki> |
<nowiki>}</nowiki> |
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Latest revision as of 14:54, 2 September 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 = sep, year = {2022}, volume = {64}, pages = {807--824}, 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.}, doi = {10.1007/s10851-022-01104-z} }