In this thesis, we develop a metric on manifolds---specializing to hypersurfaces of R3 ---that is motivated by the need for an appropriate matching criterion for finding optimal transformations between manifolds. Our main contribution is a novel metric via representations of manifolds by objects called currents from geometric measure theory [1]. Surfaces are important in modeling and studying the shape of anatomical structures, which motivates the need to develop techniques for comparison and statistical analysis. This endeavor falls under the discipline called Computational Anatomy, whose overriding goal seeks to find relationships between anatomical shape changes and disease states. The underlying paradigm---recognized as early as 1917 by D'Arcy Thompson and given mathematical rigor by Ulf Grenander---is to study shape variation through transformations. The metric we design can be used as a matching criterion in many different approaches for finding transformations between surfaces. In this thesis, we study the matching problem under the diffeomorphic matching framework of Miller/Trouve/Younes [2].;The development of the metric parallels the distribution theoretic approach of Glaunes et al. in which point sets are represented by a sum of dirac measures. Currents are another appropriate representation because they encode local geometry via the Gauss Map (normal field) on the surface, and inherit natural geometric transformation properties from differential forms. They have vector space structure, and we impose a Hilbert space structure---dual to a reproducing kernel Hilbert space of differential forms---from which the metric is induced.;The matching problem---designed to find a diffeomorphism &phis; from a source surface S to a target surface T---is presented as a variational optimization problem in the general framework of Miller/Trouve/Younes diffeomorphic matching. We follow the usual program of deriving the Frechet derivative of the matching energy which provides a gradient for carrying out a gradient descent algorithmic solution. We implement once such gradient descent algorithm in C++, detailing the implementation challenges, and we provide results and validation on facial surfaces, as well as surfaces important in computational anatomy such as the hippocampus and planum temporale of the human brain.;A second focus of this thesis is toward advancing statistical methods for shape comparisons under the large deformation diffeomorphism framework. This track is based on a recent discovery [3] of a fundamental property of diffeomorphic flow that enables a concise linear representation of diffeomorphic transformations. The space of representations becomes a natural space in which to focus statistical modeling. We have been motivated to pursue this statistical setting because of the dimensionality reduction afforded by representing entire diffeomorphic flows at a single instant in time, and because of the powerful capability these representations enable in providing a simple linear statistical setting for a highly non-linear shape space. We specialize to the landmark matching setting and derive a new variational problem---parameterized by the initial momentum---and implement a numerical gradient algorithm. Finally, we detail the implementation of principal component analysis (PCA) in this setting. Results of the optimization algorithm and a PCA analysis of 3D face and hippocampus surfaces are presented in the final section.
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机译：在本文中，我们开发了一个关于流形的度量-专门针对R3的超曲面-的动机是，需要一个合适的匹配标准来寻找流形之间的最佳转换。我们的主要贡献是一种新颖的度量标准，它通过用几何度量理论[1]来表示被称为电流的对象来表示歧管。表面对于建模和研究解剖结构的形状很重要，这激发了开发用于比较和统计分析的技术的需求。这项工作属于计算解剖学学科，该学科的首要目标是寻找解剖学形状变化与疾病状态之间的关系。潜在的范式-通过变换研究形状变化，最早在1917年被达西·汤普森（D'Arcy Thompson）认可，而数学严格性则由乌尔夫·格林纳德（Ulf Grenander）赋予。我们设计的度量可以在许多不同的方法中用作匹配标准，以查找曲面之间的变换。在本文中，我们研究了在Miller / Trouve / Younes的微分匹配框架下的匹配问题[2]。度量的发展与Glaunes等人的分布理论方法相似。其中点集由狄拉克测度的总和表示。电流是另一种合适的表示形式，因为它们通过表面上的高斯图（法线场）对局部几何进行编码，并从微分形式继承自然的几何变换特性。它们具有向量空间结构，我们强加了一个希尔伯特空间结构，它是微分形式的再生希尔伯特空间的对偶，由此引入了度量。从源表面S到目标表面T ---在Miller / Trouve / Younes微分匹配的一般框架中作为变分优化问题提出。我们遵循导出匹配能量的Frechet导数的常规程序，该程序为执行梯度下降算法解决方案提供了梯度。我们曾经在C ++中实现过这种梯度下降算法，详细说明了实现方面的挑战，并提供了面部表面以及在计算解剖学中重要的表面（例如人脑的海马和颞上皮）上的结果和验证。本文致力于在大变形亚同构框架下发展用于形状比较的统计方法。该轨迹基于最近发现的微晶流基本属性[3]，该基本属性使得能够精确地线性表示微晶变换。表示的空间成为关注统计建模的自然空间。我们之所以追求这种统计设置，是因为通过在单个时刻表示整个微分形流可以降低维数，并且由于这些表示的强大功能，它们可以为高度非线性的形状提供简单的线性统计设置空间。我们专注于地标匹配设置，并得出了一个新的变分问题-由初始动量进行了参数化-并实现了数值梯度算法。最后，我们详细介绍了在这种情况下主成分分析（PCA）的实现。最后一部分介绍了优化算法的结果以及3D面部和海马表面的PCA分析。
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