A Numerical Framework for Sobolev Metrics on the Space of Curves
A Numerical Framework for Sobolev Metrics on the Space of Curves
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DOI：
10.1137/16m1066282
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发表时间：
2017-01-01
影响因子：
2.1
通讯作者：
Moller-Andersen, Jakob
中科院分区：
文献类型：
--
作者：
Bauer, Martin;Bruveris, Martins;Moller-Andersen, Jakob
统计形状分析可以在黎曼框架中通过赋予形状集合以黎曼度量来完成。参数化或非参数化曲线的形状空间上的二阶或更高阶的Sobolev度量具有低阶度量中不存在的几个理想性质，但它们的离散化仍然很大程度上缺失。在本文中，我们提出的算法来数值求解这些度量的测地线初值和边值问题。这些算法的组合，使一个计算Karcher手段在黎曼梯度为基础的优化方案，并执行主成分分析和聚类。我们的框架是足够普遍的，适用于广泛的一类指标。我们证明了我们的方法的有效性，通过分析代表HeLa细胞核的形状的集合。
Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is still largely missing. In this paper, we present algorithms to numerically solve the geodesic initial and boundary value problems for these metrics. The combination of these algorithms enables one to compute Karcher means in a Riemannian gradient-based optimization scheme and perform principal component analysis and clustering. Our framework is sufficiently general to be applicable to a wide class of metrics. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing HeLa cell nuclei.