A Sparse Delaunay Filtration
share
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 complex. We also, show how this complex can be efficiently constructed.
READ FULL TEXT