Papers › Delaunay Bifiltrations of Functions on Point Clouds

Delaunay Bifiltrations of Functions on Point Clouds

24 Oct 2023arXiv:2310.15902links table onlyarchive 2025-07-28

Ángel Javier Alonso, Michael Kerber, Tung Lam, Michael Lesnick

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The Delaunay filtration 𝒟_∙(X) of a point cloud X⊂ℝᵈ is a central tool of computational topology. Its use is justified by the topological equivalence of 𝒟_∙(X) and the offset (i.e., union-of-balls) filtration of X. Given a function γ: X →ℝ, we introduce a Delaunay bifiltration 𝒟𝒞_∙(γ) that satisfies an analogous topological equivalence, ensuring that 𝒟𝒞_∙(γ) topologically encodes the offset filtrations of all sublevel sets of γ, as well as the topological relations between them. 𝒟𝒞_∙(γ) is of size O(|X|^(⌈(d+1)/2⌉)), which for d odd matches the worst-case size of 𝒟_∙(X). Adapting the Bowyer-Watson algorithm for computing Delaunay triangulations, we give a simple, practical algorithm to compute 𝒟𝒞_∙(γ) in time O(|X|^(⌈d/2⌉+1)). Our implementation, based on CGAL, computes 𝒟𝒞_∙(γ) with modest overhead compared to computing 𝒟_∙(X), and handles tens of thousands of points in ℝ³ within seconds.

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