Diffeomorphic surface matching via currents and tangent space representations for statistics on diffeomorphisms

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.