We propose an extension of the classical union-of-balls filtration of persistent homology: fixing a point q, we focus our attention to a ball centered at q whose radius is controlled by a second scale parameter. We discuss an absolute variant, where the union is just restricted to the q-ball, and a relative variant where the homology of the q-ball relative to its boundary is considered. Interestingly, these natural constructions lead to bifiltered simplicial complexes which are not k-critical for any finite k. Nevertheless, we demonstrate that these bifiltrations can be computed exactly and efficiently, and we provide a prototypical implementation using the CGAL library. We also argue that some of the recent algorithmic advances for 2-parameter persistence (which usually assume k-criticality for some finite k) carry over to the ∞-critical case.
@InProceedings{kerber_et_al:LIPIcs.SoCG.2023.45, author = {Kerber, Michael and S\"{o}ls, Matthias}, title = {{The Localized Union-Of-Balls Bifiltration}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {45:1--45:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.45}, URN = {urn:nbn:de:0030-drops-178953}, doi = {10.4230/LIPIcs.SoCG.2023.45}, annote = {Keywords: Topological Data Analysis, Multi-Parameter Persistence, Persistent Local Homology} }
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