Difference between revisions of "Publications/boutry.22.jmiv.2"
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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 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 relatedpaving the way for a more in-depth study of the relations between these two research fields. |
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+ | | 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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| lrdenewsdate = 2022-05-17 |
| lrdenewsdate = 2022-05-17 |
Revision as of 04:58, 19 May 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 relatedpaving 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}, volume = {}, number = {}, pages = {}, 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.}, doi = {} }