Abstract
We state in this paper a strong relation existing between Mathematical Morphology and Morse Theory when we work with 1D \(\mathfrak {D}\)-Morse functions. Specifically, in Mathematical Morphology, a classic way to extract robust markers for segmentation purposes, is to use the dynamics. On the other hand, in Morse Theory, a well-known tool to simplify the Morse-Smale complexes representing the topological information of a \(\mathfrak {D}\)-Morse function is the persistence. We show that pairing by persistence is equivalent to pairing by dynamics.
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References
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Extending persistence using Poincaré and Lefschetz duality. Found. Comput. Math. 9(1), 79–103 (2009)
Čomić, L., De Floriani, L., Iuricich, F., Magillo, P.: Computing a discrete Morse gradient from a watershed decomposition. Comput. Graph. 58, 43–52 (2016)
Čomić, L., De Floriani, L., Magillo, P., Iuricich, F.: Morphological Modeling of Terrains and Volume Data. SCS. Springer, New York (2014). https://doi.org/10.1007/978-1-4939-2149-2
Cousty, J., Bertrand, G., Couprie, M., Najman, L.: Collapses and watersheds in pseudomanifolds of arbitrary dimension. J. Math. Imaging Vis. 50(3), 261–285 (2014)
Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: minimum spanning forests and the drop of water principle. IEEE Trans. Pattern Anal. Mach. Intell. 31(8), 1362–1374 (2009)
De Floriani, L., Iuricich, F., Magillo, P., Simari, P.: Discrete morse versus watershed decompositions of tessellated manifolds. In: Petrosino, A. (ed.) ICIAP 2013. LNCS, vol. 8157, pp. 339–348. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-41184-7_35
Edelsbrunner, H., Harer, J.: Persistent homology - a survey. Contemp. Math. 453, 257–282 (2008)
Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, New York (2010)
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Foundations of Computer Science, pp. 454–463. IEEE (2000)
Forman, R.: A user’s guide to Discrete Morse Theory. Séminaire Lotharingien de Combinatoire 48, 1–35 (2002)
Grimaud, M.: La géodésie numérique en Morphologie Mathématique. Application à la détection automatique des microcalcifications en mammographie numérique. Ph.D. thesis, École des Mines de Paris (1991)
Grimaud, M.: New measure of contrast: the dynamics. In: Image Algebra and Morphological Image Processing III, vol. 1769, pp. 292–306. International Society for Optics and Photonics (1992)
Milnor, J.W., Spivak, M., Wells, R.: Morse Theory. Princeton University Press, Princeton (1963)
Najman, L., Schmitt, M.: Watershed of a continuous function. Signal Process. 38(1), 99–112 (1994)
Najman, L., Schmitt, M.: Geodesic saliency of watershed contours and hierarchical segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 18(12), 1163–1173 (1996)
Najman, L., Talbot, H.: Mathematical Morphology: From Theory to Applications. Wiley, Hoboken (2013)
Serra, J.: Introduction to mathematical morphology. Comput. Vis. Graph. Image Process. 35(3), 283–305 (1986)
Serra, J., Soille, P.: Mathematical Morphology and its Applications to Image Processing, vol. 2. Springer Science & Business Media, Heidelberg (2012)
Vachier, C.: Extraction de caractéristiques, segmentation d’image et Morphologie Mathématique. Ph.D. thesis, École Nationale Supérieure des Mines de Paris (1995)
Vincent, L., Soille, P.: Watersheds in digital spaces: an efficient algorithm based on immersion simulations. IEEE Trans. Pattern Anal. Mach. Intell. 13(6), 583–598 (1991)
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Boutry, N., Géraud, T., Najman, L. (2019). An Equivalence Relation Between Morphological Dynamics and Persistent Homology in 1D. In: Burgeth, B., Kleefeld, A., Naegel, B., Passat, N., Perret, B. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2019. Lecture Notes in Computer Science(), vol 11564. Springer, Cham. https://doi.org/10.1007/978-3-030-20867-7_5
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DOI: https://doi.org/10.1007/978-3-030-20867-7_5
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