For a finite set of balls of radius r, the k-fold cover is the space covered by at least k balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the k-fold filtration of the centers. For k = 1, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger k, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the k-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case k = 1, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points.
@InProceedings{buchet_et_al:LIPIcs.SoCG.2023.20, author = {Buchet, Micka\"{e}l and B. Dornelas, Bianca and Kerber, Michael}, title = {{Sparse Higher Order \v{C}ech Filtrations}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {20:1--20:17}, 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.2023.20}, URN = {urn:nbn:de:0030-drops-178709}, doi = {10.4230/LIPIcs.SoCG.2023.20}, annote = {Keywords: Sparsification, k-fold cover, Higher order \v{C}ech complexes} }
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