Persistent Homology as Stopping-Criterion for Natural Neighbor Interpolation
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In this study the method of natural neighbours is used to interpolate data that has been drawn from a topological space with higher homology groups on its filtration. A particular difficulty with this algorithm are the boundary points. Its core is based on the Voronoi diagram, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fact and the persistent homology is therefore calculated after each iteration to capture the changing topology of the data. The Bottleneck and Wasserstein distance serve as a measure of quality between the original data and the interpolation. If the norm of two distances exceeds a heuristically determined threshold, the algorithm terminates. The theoretical basis for this procedure is given and the validity of this approach is justified with numerical experiments.
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