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.

中文翻译：

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持久同源性作为 Voronoi 插值的停止准则

在这项研究中，Voronoi 插值用于对从拓扑空间中抽取的一组点进行插值，该拓扑空间的过滤具有更高的同调群。该技术基于 Voronoi 曲面细分，可将自然对偶映射引入 Delaunay 三角剖分。优点是在每次迭代后计算其上的持久同源性以捕获数据的变化拓扑。边界点被确定为关键点。瓶颈和 Wasserstein 距离用作原始点集和插值之间的质量度量。如果两个距离的范数超过启发式确定的阈值，则算法终止。我们给出了这种方法的理论基础，并通过数值实验证明了其有效性。