An Equivalence Relation between Morphological Dynamics and Persistent Homology in n-D

From LRDE

Abstract

In Mathematical Morphology (MM), dynamics are used to compute markers to proceed for example to watershed-based image decomposition. At the same time, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) and represents the stability of the extrema of a Morse function. Since these concepts are similar on Morse functions, we studied their relationship and we found, and proved, that they are equal on 1D Morse functions. Here, we propose to extend this proof to -D, showing that this equality can be applied to -D images and not only to 1D functions. This is a step further to show how much MM and MT are related.


Bibtex (lrde.bib)

@InProceedings{	  boutry.21.dgmm.1,
  author	= {Nicolas Boutry and Thierry G\'eraud and Laurent Najman},
  title		= {An Equivalence Relation between Morphological Dynamics and
		  Persistent Homology in {$n$-D}},
  booktitle	= {Proceedings of the IAPR International Conference on
		  Discrete Geometry and Mathematical Morphology (DGMM)},
  year		= 2021,
  month		= {May},
  address	= {Uppsala, Sweden},
  abstract	= {In Mathematical Morphology (MM), dynamics are used to
		  compute markers to proceed for example to watershed-based
		  image decomposition. At the same time, persistence is a
		  concept coming from Persistent Homology (PH) and Morse
		  Theory (MT) and represents the stability of the extrema of
		  a Morse function. Since these concepts are similar on Morse
		  functions, we studied their relationship and we found, and
		  proved, that they are equal on 1D Morse functions. Here, we
		  propose to extend this proof to $n$-D, $n \geq 2$, showing
		  that this equality can be applied to $n$-D images and not
		  only to 1D functions. This is a step further to show how
		  much MM and MT are related.},
  note		= {To appear}
}