Persistent homology transform for modeling shapes and surfaces
Persistent homology transform for modeling shapes and surfaces
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DOI：
10.1093/imaiai/iau011
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发表时间：
2014-12-01
影响因子：
1.6
通讯作者：
Boyer, Doug M.
中科院分区：
文献类型：
--
作者：
Turner, Katharine;Mukheriee, Sayan;Boyer, Doug M.
我们引入了一个统计，持久的同源变换（PHT），在R-3和R-2的形状模型表面。这个统计量是一个持久性拓扑摘要的集合，它是拓扑数据分析中广泛使用的多尺度拓扑摘要。我们使用PHT来表示形状和执行操作，如计算形状之间的距离或形状分类。我们提供了一个建设性的证明，从空间的单纯复形在R-3的空间到这个统计量所跨越的空间的映射是内射。这意味着我们可以用它来确定分段线性形状空间上的度量。稳定性的结果证明，我们可以近似这个度量使用许多持久性图。我们说明了这个统计模拟和真实的数据的效用。
We introduce a statistic, the persistent homology transform (PHT), to model surfaces in R-3 and shapes in R-2. This statistic is a collection of persistence diagrams-multiscale topological summaries used extensively in topological data analysis. We use the PHT to represent shapes and execute operations such as computing distances between shapes or classifying shapes. We provide a constructive proof that the map from the space of simplicial complexes in R-3 into the space spanned by this statistic is injective. This implies that we can use it to determine a metric on the space of piecewise linear shapes. Stability results justify that we can approximate this metric using finitely many persistence diagrams. We illustrate the utility of this statistic on simulated and real data.