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Persistent Homology Computation Using Combinatorial Map Simplification

  • Guillaume Damiand
  • Rocio Gonzalez-Diaz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11382)

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 \(\delta \).

Keywords

Persistent homology computation 2D combinatorial map Mesh simplification 

Notes

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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Univ. Lyon, CNRS, LIRIS, UMR5205Villeurbanne CedexFrance
  2. 2.Dpto. de Matemática Aplicada IUniversidad de SevillaSevilleSpain

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