Document Open Access Logo

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

Authors Ulrich Bauer , Fabian Roll

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2024.15.pdf
  • Filesize: 2.57 MB
  • 16 pages

Document Identifiers

Related Versions

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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.

Cite As Get BibTex

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

  1. N. Amenta and M. Bern. Surface reconstruction by Voronoi filtering. Discrete Comput. Geom., 22(4):481-504, 1999. 14th Annual ACM Symposium on Computational Geometry (Minneapolis, MN, 1998). URL: https://doi.org/10.1007/PL00009475.
  2. Nina Amenta, Sunghee Choi, Tamal K. Dey, and Naveen Leekha. A simple algorithm for homeomorphic surface reconstruction. Internat. J. Comput. Geom. Appl., 12(1-2):125-141, 2002. 16th Annual Symposium on Computational Geometry (Kowloon, 2000). URL: https://doi.org/10.1142/S0218195902000773.
  3. Dominique Attali, Hana Dal Poz Kouřimská, Christopher Fillmore, Ishika Ghosh, André Lieutier, Elizabeth Stephenson, and Mathijs Wintraecken. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds. In 40th International Symposium on Computational Geometry (SoCG 2024), volume 224 of Leibniz International Proceedings in Informatics (LIPIcs), 2024. https://arxiv.org/abs/2206.10485, URL: https://doi.org/10.4230/LIPIcs.SoCG.2024.12.
  4. Dominique Attali and André Lieutier. Delaunay-like triangulation of smooth orientable submanifolds by 𝓁₁-norm minimization. In 38th International Symposium on Computational Geometry, volume 224 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 8, 16. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2022. URL: https://doi.org/10.4230/lipics.socg.2022.8.
  5. Ulrich Bauer. Ripser: efficient computation of Vietoris-Rips persistence barcodes. J. Appl. Comput. Topol., 5(3):391-423, 2021. URL: https://doi.org/10.1007/s41468-021-00071-5.
  6. Ulrich Bauer and Herbert Edelsbrunner. The Morse theory of Čech and Delaunay complexes. Trans. Amer. Math. Soc., 369(5):3741-3762, 2017. URL: https://doi.org/10.1090/tran/6991.
  7. Ulrich Bauer, Michael Kerber, Fabian Roll, and Alexander Rolle. A unified view on the functorial nerve theorem and its variations. Expo. Math., 41(4):125503, 2023. URL: https://doi.org/10.1016/j.exmath.2023.04.005.
  8. Ulrich Bauer, Carsten Lange, and Max Wardetzky. Optimal topological simplification of discrete functions on surfaces. Discrete Comput. Geom., 47(2):347-377, 2012. URL: https://doi.org/10.1007/s00454-011-9350-z.
  9. Ulrich Bauer and Fabian Roll. Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations. In 38th International Symposium on Computational Geometry (SoCG 2022), volume 224 of Leibniz International Proceedings in Informatics (LIPIcs), pages 15:1-15:15, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.15.
  10. Ulrich Bauer and Fabian Roll. Wrapping cycles in Delaunay complexes: Bridging persistent homology and discrete Morse theory. Preprint, 2023. URL: https://arxiv.org/abs/2212.02345.
  11. Ulrich Bauer and Fabian Roll. WrappingCycles, 2023. Executable directly by running docker build -o output github.com/fabian-roll/wrappingcycles on any machine with Docker installed and configured with sufficient memory (16GB recommended). URL: https://github.com/fabian-roll/wrappingcycles.
  12. Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec. Geometric and topological inference. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2018. URL: https://doi.org/10.1017/9781108297806.
  13. Jean-Daniel Boissonnat and Arijit Ghosh. Manifold reconstruction using tangential Delaunay complexes. Discrete Comput. Geom., 51(1):221-267, 2014. URL: https://doi.org/10.1007/s00454-013-9557-2.
  14. Siu-Wing Cheng, Tamal K. Dey, Herbert Edelsbrunner, Michael A. Facello, and Shang-Hua Teng. Sliver exudation. J. ACM, 47(5):883-904, 2000. URL: https://doi.org/10.1145/355483.355487.
  15. Siu-Wing Cheng, Tamal K. Dey, and Edgar A. Ramos. Manifold reconstruction from point samples. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1018-1027. ACM, New York, 2005. URL: https://doi.org/10.5555/1070432.1070579.
  16. David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete Comput. Geom., 37(1), 2007. URL: https://doi.org/10.1007/s00454-006-1276-5.
  17. David Cohen-Steiner, Herbert Edelsbrunner, and Dmitriy Morozov. Vines and vineyards by updating persistence in linear time. In Computational geometry (SCG'06), pages 119-126. ACM, New York, 2006. URL: https://doi.org/10.1145/1137856.1137877.
  18. David Cohen-Steiner, André Lieutier, and Julien Vuillamy. Lexicographic optimal chains and manifold triangulations. Research report, INRIA Sophia-Antipolis Méditerranée, December 2019. URL: https://hal.science/hal-02391190.
  19. David Cohen-Steiner, André Lieutier, and Julien Vuillamy. Lexicographic optimal homologous chains and applications to point cloud triangulations. Discrete Comput. Geom., 68(4):1155-1174, 2022. Special Issue: 36th Annual Symposium on Computational Geometry (SoCG 2020). URL: https://doi.org/10.1007/s00454-022-00432-6.
  20. David Cohen-Steiner, André Lieutier, and Julien Vuillamy. Delaunay and regular triangulations as lexicographic optimal chains. Discrete Comput. Geom., 70(1):1-50, 2023. URL: https://doi.org/10.1007/s00454-023-00485-1.
  21. Vin de Silva, Dmitriy Morozov, and Mikael Vejdemo-Johansson. Dualities in persistent (co)homology. Inverse Problems, 27(12):124003, 17, 2011. URL: https://doi.org/10.1088/0266-5611/27/12/124003.
  22. Tamal K. Dey. Curve and surface reconstruction: algorithms with mathematical analysis, volume 23 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2007. URL: https://doi.org/10.1017/CBO9780511546860.
  23. Tamal K. Dey, Joachim Giesen, Edgar A. Ramos, and Bardia Sadri. Critical points of the distance to an epsilon-sampling of a surface and flow-complex-based surface reconstruction. In Computational geometry (SCG'05), pages 218-227. ACM, New York, 2005. URL: https://doi.org/10.1145/1064092.1064126.
  24. Herbert Edelsbrunner. The union of balls and its dual shape. Discrete Comput. Geom., 13(3-4):415-440, 1995. URL: https://doi.org/10.1007/BF02574053.
  25. Herbert Edelsbrunner. Surface reconstruction by wrapping finite sets in space. In Discrete and computational geometry, volume 25 of Algorithms Combin., pages 379-404. Springer, Berlin, 2003. URL: https://doi.org/10.1007/978-3-642-55566-4_17.
  26. Herbert Edelsbrunner and John L. Harer. Computational topology. American Mathematical Society, Providence, RI, 2010. An introduction. URL: https://doi.org/10.1090/mbk/069.
  27. Herbert Edelsbrunner and Katharina Ölsböck. Tri-partitions and bases of an ordered complex. Discrete Comput. Geom., 64(3):759-775, 2020. URL: https://doi.org/10.1007/s00454-020-00188-x.
  28. Herbert Edelsbrunner and Afra Zomorodian. Computing linking numbers of a filtration. Homology Homotopy Appl., 5(2):19-37, 2003. Algebraic topological methods in computer science (Stanford, CA, 2001). URL: https://doi.org/10.4310/HHA.2003.v5.n2.a2.
  29. Robin Forman. Morse theory for cell complexes. Adv. Math., 134(1):90-145, 1998. URL: https://doi.org/10.1006/aima.1997.1650.
  30. Ragnar Freij. Equivariant discrete Morse theory. Discrete Math., 309(12):3821-3829, 2009. URL: https://doi.org/10.1016/j.disc.2008.10.029.
  31. Michael Jöllenbeck and Volkmar Welker. Minimal resolutions via algebraic discrete Morse theory. Mem. Amer. Math. Soc., 197(923):vi+74, 2009. URL: https://doi.org/10.1090/memo/0923.
  32. Dmitry Kozlov. Combinatorial algebraic topology, volume 21 of Algorithms and Computation in Mathematics. Springer, Berlin, 2008. URL: https://doi.org/10.1007/978-3-540-71962-5.
  33. Stanford Computer Graphics Laboratory. The Stanford 3D Scanning Repository. URL: http://graphics.stanford.edu/data/3Dscanrep/.
  34. Leon Lampret. Chain complex reduction via fast digraph traversal. Preprint, 2020. URL: https://arxiv.org/abs/1903.00783.
  35. S. Lefschetz. Algebraic Topology. AMS books online. American Mathematical Society, 1942. Google Scholar
  36. Konstantin Mischaikow and Vidit Nanda. Morse theory for filtrations and efficient computation of persistent homology. Discrete Comput. Geom., 50(2):330-353, 2013. URL: https://doi.org/10.1007/s00454-013-9529-6.
  37. James R. Munkres. Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park, CA, 1984. Google Scholar
  38. Partha Niyogi, Stephen Smale, and Shmuel Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom., 39(1-3):419-441, 2008. URL: https://doi.org/10.1007/s00454-008-9053-2.
  39. Cédric Portaneri, Mael Rouxel-Labbé, Michael Hemmer, David Cohen-Steiner, and Pierre Alliez. Alpha wrapping with an offset. ACM Trans. Graph., 41(4):127:1-127:22, 2022. URL: https://doi.org/10.1145/3528223.3530152.
  40. Edgar A. Ramos and Bardia Sadri. Geometric and topological guarantees for the WRAP reconstruction algorithm. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1086-1095. ACM, New York, 2007. Google Scholar
  41. Bardia Sadri. Manifold homotopy via the flow complex. Comput. Graph. Forum, 28(5):1361-1370, 2009. URL: https://doi.org/10.1111/J.1467-8659.2009.01513.X.
  42. Emil Sköldberg. Morse theory from an algebraic viewpoint. Trans. Amer. Math. Soc., 358(1):115-129, 2006. URL: https://doi.org/10.1090/S0002-9947-05-04079-1.
  43. Julien Vuillamy. Planimetric simplification and lexicographic optimal chains for 3D urban scene reconstruction. Theses, Université Côte d'Azur, September 2021. URL: https://hal.science/tel-03339931.
  44. Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994. URL: https://doi.org/10.1017/CBO9781139644136.
  45. Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete Comput. Geom., 33(2):249-274, 2005. URL: https://doi.org/10.1007/s00454-004-1146-y.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact