Delaunay Bifiltrations of Functions on Point Clouds

The Delaunay filtration Dbullet(X) of a point cloud Xsubset ℝd is a central tool of computational topology. Its use is justified by the topological equivalence of Dbullet(X) and the offset (i.e., union-of-balls) filtration of X. Given a function γ: X → ℝ, we introduce a Delaunay bifiltration DCbullet(γ) that satisfies an analogous topological equivalence, ensuring that DCbullet(γ) topologically encodes the offset filtrations of all sublevel sets of γ, as well as the topological relations between them. DCbullet(γ) is of size O(|X|^lceilfracd+12rceil), which for d odd matches the worst-case size of Dbullet(X). Adapting the Bowyer-Watson algorithm for computing Delaunay triangulations, we give a simple, practical algorithm to compute DCbullet(γ) in time O(|X|^lceil fracd2rceil +1). Our implementation, based on CGAL, computes DCbullet(γ) with modest overhead compared to computing Dbullet(X), and handles tens of thousands of points in ℝ3 within seconds.