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A Sparse Delaunay Filtration

Author Donald R. Sheehy

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Donald R. Sheehy
  • North Carolina State University, Raleigh, NC, USA

Funding

This research was supported by the NSF under grant CCF-2017980.

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Donald R. Sheehy. A Sparse Delaunay Filtration. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.58

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Delaunay Triangulation
  • Persistent Homology
  • Sparse Filtrations

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