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Persistent Homology Computation Using Combinatorial Map Simplification

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Computational Topology in Image Context (CTIC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11382))

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Abstract

We propose an algorithm for persistence homology computation of orientable 2-dimensional (2D) manifolds with or without boundary (meshes) represented by 2D combinatorial maps. Having as an input a real function h on the vertices of the mesh, we first compute persistent homology of filtrations obtained by adding cells incident to each vertex of the mesh, The cells to add are controlled by both the function h and a parameter \(\delta \). The parameter \(\delta \) is used to control the number of cells added to each level of the filtration. Bigger \(\delta \) produces less levels in the filtration and consequently more cells in each level. We then simplify each level (cluster) by merging faces of the same cluster. Our experiments demonstrate that our method allows fast computation of persistent homology of big meshes and it is persistent-homology aware in the sense that persistent homology does not change in the simplification process when fixing \(\delta \).

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Notes

  1. 1.

    The \(L_{\infty }\)-distance between points \(u = (u_1, u_2)\) and \(v = (v_1, v_2)\) in the extended plane is \( \max \{|u_1 - v_1|, |u_2 - v_2|\}\).

  2. 2.

    https://cran.r-project.org/web/packages/TDA/vignettes/article.pdf.

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Acknowledgments

This research has been partially supported by MINECO, FEDER/UE under grant MTM2015-67072-P. We thank the anonymous reviewers for their valuable comments.

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Authors and Affiliations

  1. Univ. Lyon, CNRS, LIRIS, UMR5205, 69622, Villeurbanne Cedex, France

    Guillaume Damiand

  2. Dpto. de Matemática Aplicada I, Universidad de Sevilla, 41012, Seville, Spain

    Rocio Gonzalez-Diaz

Authors
  1. Guillaume Damiand
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  2. Rocio Gonzalez-Diaz
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Corresponding author

Correspondence to Guillaume Damiand .

Editor information

Editors and Affiliations

  1. Universidad de Málaga, Málaga, Spain

    Rebeca Marfil

  2. Universidad de Málaga, Málaga, Spain

    Mariletty Calderón

  3. University of Seville, Sevilla, Spain

    Fernando Díaz del Río

  4. University of Seville, Sevilla, Spain

    Pedro Real

  5. Universidad de Málaga, Málaga, Spain

    Antonio Bandera

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Damiand, G., Gonzalez-Diaz, R. (2019). Persistent Homology Computation Using Combinatorial Map Simplification. In: Marfil, R., Calderón, M., Díaz del Río, F., Real, P., Bandera, A. (eds) Computational Topology in Image Context. CTIC 2019. Lecture Notes in Computer Science(), vol 11382. Springer, Cham. https://doi.org/10.1007/978-3-030-10828-1_3

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  • DOI: https://doi.org/10.1007/978-3-030-10828-1_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10827-4

  • Online ISBN: 978-3-030-10828-1

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