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Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory

Authors Ulrich Bauer , Fabian Roll

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Theory of computation → Computational geometry
Keywords
  • persistent homology
  • discrete Morse theory
  • apparent pairs
  • Wrap complex
  • lexicographic optimal chains
  • shape reconstruction

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Abstract

We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we consider the construction of Wrap complexes, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the construction of lexicographic optimal homologous cycles, also considered by Cohen–Steiner, Lieutier, and Vuillamy in a similar setting. We show that for any cycle in a Delaunay complex for a given radius parameter, the lexicographically optimal homologous cycle is supported on the Wrap complex for the same parameter, thereby establishing a close connection between the two methods. We obtain this result by establishing a fundamental connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory.

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Ulrich Bauer and Fabian Roll. Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.15

Author Details

Ulrich Bauer
  • Department of Mathematics and Munich Data Science Institute, Technische Universität München, Germany
Fabian Roll
  • Department of Mathematics, Technische Universität München, Germany

Funding

This research has been supported by the Deutsche Forschungsgemeinschaft (DFG - German Research Foundation) - Project-ID 195170736 - TRR109 Discretization in Geometry and Dynamics.

Acknowledgements

We thank Herbert Edelsbrunner and Marian Mrozek for fruitful discussions about the connection between algebraic gradient flow theory and matrix reduction.

Supplementary Materials

References

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