Persistent Homology as Stopping-Criterion for Voronoi Interpolation
This work addresses a stopping criterion issue in computational topology for researchers, but it is incremental as it applies existing methods to a specific interpolation task.
The paper tackles the problem of determining when to stop Voronoi interpolation by using persistent homology to monitor topological changes, terminating when the distance between original and interpolated sets exceeds a threshold, with results validated through numerical experiments.
In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration. The technique is based on Voronoi tessellation, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fact calculating the persistent homology on it after each iteration to capture the changing topology of the data. The boundary points are identified as critical. The Bottleneck and Wasserstein distance serve as a measure of quality between the original point set and the interpolation. If the norm of two distances exceeds a heuristically determined threshold, the algorithm terminates. We give the theoretical basis for this approach and justify its validity with numerical experiments.