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

From LRDE

 
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| 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 relatedpaving 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.
 
| 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
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| 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,
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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>,
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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><nowiki>}</nowiki>
 
 
<nowiki>}</nowiki>
 
<nowiki>}</nowiki>
   

Latest revision as of 16:37, 30 June 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 -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.}
}