Hartman Emmanuel, Sukurdeep Yashil, Klassen Eric, Charon Nicolas, Bauer Martin
Department of Mathematics, Florida State University, Tallahassee, USA.
Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, USA.
Int J Comput Vis. 2023;131(5):1183-1209. doi: 10.1007/s11263-022-01743-0. Epub 2023 Jan 21.
This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real.
The online version contains supplementary material available at 10.1007/s11263-022-01743-0.
本文介绍了一组用于在不变（弹性）二阶索伯列夫度量设置下对三维表面进行黎曼形状分析的数值方法。更具体地说，我们处理表示为三维网格的参数化或非参数化浸入曲面之间测地线和测地距离的计算。在此基础上，我们开发了用于曲面集统计形状分析的工具，包括估计卡尔彻均值和对形状总体进行切空间主成分分析的方法，以及用于沿曲面路径计算平行移动的方法。我们提出的方法从根本上依赖于通过使用变分曲面保真项对测地线匹配问题进行松弛变分公式化，这使我们在计算非参数化曲面之间的测地线时能够强制实现重新参数化独立性，同时还产生了通用算法，使我们能够比较具有不同采样或网格结构的曲面。重要的是，我们展示了如何扩展我们的松弛变分框架来处理部分观测数据。我们的数值流程的不同优势在各种合成和真实示例中得到了说明。
在线版本包含可在10.1007/s11263-022-01743-0获取的补充材料。
