License:
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.58
URN: urn:nbn:de:0030-drops-138579
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13857/
Sheehy, Donald R.
A Sparse Delaunay Filtration
Abstract
We show how a filtration of Delaunay complexes can be used to approximate the persistence diagram of the distance to a point set in ℝ^d. Whereas the full Delaunay complex can be used to compute this persistence diagram exactly, it may have size O(n^⌈d/2⌉). In contrast, our construction uses only O(n) simplices. The central idea is to connect Delaunay complexes on progressively denser subsamples by considering the flips in an incremental construction as simplices in d+1 dimensions. This approach leads to a very simple and straightforward proof of correctness in geometric terms, because the final filtration is dual to a (d+1)-dimensional Voronoi construction similar to the standard Delaunay filtration. We also, show how this complex can be efficiently constructed.
BibTeX - Entry
@InProceedings{sheehy:LIPIcs.SoCG.2021.58,
author = {Sheehy, Donald R.},
title = {{A Sparse Delaunay Filtration}},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
pages = {58:1--58:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-184-9},
ISSN = {1868-8969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13857},
URN = {urn:nbn:de:0030-drops-138579},
doi = {10.4230/LIPIcs.SoCG.2021.58},
annote = {Keywords: Delaunay Triangulation, Persistent Homology, Sparse Filtrations}
}
Keywords: |
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Delaunay Triangulation, Persistent Homology, Sparse Filtrations |
Collection: |
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37th International Symposium on Computational Geometry (SoCG 2021) |
Issue Date: |
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2021 |
Date of publication: |
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02.06.2021 |