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Some Equivalence Relation between Persistent Homology and Morphological Dynamics

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Abstract

In mathematical morphology, connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from persistent homology and Morse theory 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, \(n\ge 1\). 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.

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Acknowledgements

The authors would like to thank Julien Tierny for many interesting discussions and for providing us Fig. 17.

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Authors and Affiliations

  1. EPITA Research and Development Laboratory (LRDE), 14-16 rue Voltaire, FR-94276, Le Kremlin-Bicêtre, France

    Nicolas Boutry & Thierry Géraud

  2. Université Gustave Eiffel, LIGM, Équipe A3SI, ESIEE, Champs-sur-Marne, France

    Laurent Najman

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  1. Nicolas Boutry
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  3. Thierry Géraud
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Correspondence to Nicolas Boutry.

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A Ambiguities Occurring When Values are not Unique

A Ambiguities Occurring When Values are not Unique

Fig. 19
figure 19

Ambiguities can occur when critical values are not unique for pairing by dynamics and for pairing by persistence

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As depicted in Fig. 19, the abscissa of the blue point can be paired by persistence to the abscissas of the orange and/or the red points. The same thing appears when we want to pair the abscissa of the pink point to the abscissas of the green and/or blue points. This shows how much it is important to have unique critical values on Morse functions. This point is discussed in detail in [3], where it is shown that a strict total order relation on the set of minima allows for a good definition of the dynamics.

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Boutry, N., Najman, L. & Géraud, T. Some Equivalence Relation between Persistent Homology and Morphological Dynamics. J Math Imaging Vis 64, 807–824 (2022). https://doi.org/10.1007/s10851-022-01104-z

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