Delaunay Bifiltrations of Functions on Point Clouds

The Delaunay filtration D•(X) of a point cloud X ⊂ R^d 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 (equation presented), 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 (equation presented). 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.