Persistent Homology as Stopping-Criterion for Voronoi Interpolation
Persistent Homology as Stopping-Criterion for Voronoi Interpolation
复制标题
持久同源作为 Voronoi 插值的停止标准
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
R. Lenz
中科院分区：
文献类型：
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作者：
Luciano Melodia;R. Lenz
本文利用Voronoi插值法对滤子上具有较高同调群的拓扑空间中的一组点进行插补。该技术是基于Voronoi细分的，它引入了到Delaunay三角剖分的自然对偶映射。利用这一事实，在每次迭代之后计算其上的持久同调，以捕获数据不断变化的拓扑。边界点被识别为关键。瓶颈和沃瑟斯坦距离作为原始点集和插补之间的质量度量。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，并用数值实验证明了它的有效性。
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.