亏格为零曲面弹性形状分析的SRNF映射数值反演
Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces
摘要：
弹性形状分析（ESA）的最新发展的动机是，它提供了全面的框架，同时注册，变形和比较的形状。这些方法使用将不变弹性度量转换为欧几里得度量的某些平方根表示来实现计算效率，从而允许标准算法和统计工具的应用。为了分析$\mathbb{S}^2$在$\mathbb{R}^3$中嵌入的形状，Jermyn等人引入了平方根正规场（SRNF），其将具有期望不变性质的弹性度量转换为$\mathbb{L}^2$度量。这些SRNF本质上是由无穷小面积元素的平方根缩放的表面法线。形状分析中的一个关键需求是将SRNF空间中计算的解决方案（变形，平均值，变化模式等）反转回原始表面空间以进行可视化和推断。由于缺乏理解SRNF映射及其逆的理论，我们采取数值方法，并推导出一个有效的多分辨率算法，基于解决表面空间中的优化问题，估计对应于给定SRNF的表面。该解决方案被发现是有效的，即使对于复杂的形状，例如人体和动物，经历显著的变形，包括弯曲和拉伸。具体来说，我们使用这种反演计算弹性形状变形，转移变形，总结形状，并在给定的集合中找到变化模式，同时注册的表面。我们证明了所提出的算法，使用统计分析的人体形状，分类的通用表面和大脑结构的分析。
英文摘要：
Recent developments in elastic shape analysis (ESA) are motivated by the fact that it provides comprehensive frameworks for simultaneous registration, deformation, and comparison of shapes. These methods achieve computational efficiency using certain square-root representations that transform invariant elastic metrics into Euclidean metrics, allowing for applications of standard algorithms and statistical tools. For analyzing shapes of embeddings of $\mathbb{S}^2$ in $\mathbb{R}^3$, Jermyn et al. introduced square-root normal fields (SRNFs) that transformed an elastic metric, with desirable invariant properties, into the $\mathbb{L}^2$ metric. These SRNFs are essentially surface normals scaled by square-roots of infinitesimal area elements. A critical need in shape analysis is to invert solutions (deformations, averages, modes of variations, etc) computed in the SRNF space, back to the original surface space for visualizations and inferences. Due to the lack of theory for understanding SRNFs maps and their inverses, we take a numerical approach and derive an efficient multiresolution algorithm, based on solving an optimization problem in the surface space, that estimates surfaces corresponding to given SRNFs. This solution is found effective, even for complex shapes, e.g. human bodies and animals, that undergo significant deformations including bending and stretching. Specifically, we use this inversion for computing elastic shape deformations, transferring deformations, summarizing shapes, and for finding modes of variability in a given collection, while simultaneously registering the surfaces. We demonstrate the proposed algorithms using a statistical analysis of human body shapes, classification of generic surfaces and analysis of brain structures.
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