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.
@InProceedings{bauer_et_al:LIPIcs.SoCG.2024.15, author = {Bauer, Ulrich and Roll, Fabian}, title = {{Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {15:1--15:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.15}, URN = {urn:nbn:de:0030-drops-199600}, doi = {10.4230/LIPIcs.SoCG.2024.15}, annote = {Keywords: persistent homology, discrete Morse theory, apparent pairs, Wrap complex, lexicographic optimal chains, shape reconstruction} }
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