Weak feature size and persistent homology: computing homology of solids in Rn from noisy data samples
Weak feature size and persistent homology: computing homology of solids in Rn from noisy data samples
复制标题
弱特征大小和持久同源性：从噪声数据样本中计算 Rn 中固体的同源性
DOI：
10.1145/1064092.1064132
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
A. Lieutier
中科院分区：
文献类型：
--
作者：
F. Chazal;A. Lieutier
在这项工作中，一个证明，在相当一般的假设下，可以推导出一个有界的开集在Rn的拓扑结构从它的Hausdorff距离近似。为此，一个介绍了弱特征大小（wfs），推广的概念，局部特征大小。我们的结果适用于具有正wfs的开集，其中包括许多边界不光滑甚至无处光滑的集。这类也包括分段解析开集，它涵盖了实际应用中遇到的许多情况。证明基于对闭集的距离函数及其临界点的研究。作为一个应用，一个算法的方法，由于持久的同源技术，计算的同源群的开集从噪声样本的点在其边界上。
In this work, one proves that under quite general assumptions one can deduce the topology of a bounded open set in Rn from a Hausdorff distance approximation of it. For this, one introduces the weak feature size (wfs) that generalizes the notion of local feature size. Our results apply to open sets with positive wfs, which include many sets whose boundaries are not smooth and even nowhere smooth. This class includes also the piecewise analytic open sets which cover many cases encountered in practical applications. The proofs are based on the study of distance functions to closed sets and their critical points. As an application, one gives an algorithmic way, thanks to persistent homology techniques, to compute the homology groups of open sets from noisy samples of points on their boundary.