Year: 2015
Type: preprint
Abstract: A key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an ℝ-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an ℝ²-valued function, also satisfying an analogous weak equivalence. For a point cloud X ⊂ ℝ^d, our trifiltration has size O(|X|^{⌈(d+1)/2⌉+1}). We present an algorithm that computes this trifiltration in time O(|X|^{⌈d/2⌉+2}), together with an implementation. Our experiments demonstrate that the implementation can handle thousands of points in ℝ³, with memory growth that is nearly linear.
Source: arXiv (Cornell University)
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: Austrian Science Fund
Award: G8590477180
OpenAlex ID: W2963574704