Riemannian Shape Analysis of Curves and Surfaces

Shape analysis of curves and surfaces is a very important tool in many applications ranging from computer vision to bioinformatics and medical imaging. There are many difficulties when analyzing shapes of parameterized curves and surfaces. Firstly, it is important to develop representations and metrics such that the analysis is invariant to parameterization in addition to the standard transformations (rigid motion and scaling). Furthermore, under the chosen representations and metrics, the analysis must be performed on infinite-dimensional and sometimes non-linear spaces, which poses an additional difficulty. In this work, we develop and apply methods which address these issues. We begin by defining a framework for shape analysis of parameterized open curves and extend these ideas to shape analysis of surfaces. We utilize the presented frameworks in various classification experiments spanning multiple application areas. In the case of curves, we consider the problem of clustering DT-MRI brain fibers, classification of protein backbones, modeling and segmentation of signatures and statistical analysis of biosignals. In the case of surfaces, we perform disease classification using 3D anatomical structures in the brain, classification of handwritten digits by viewing images as quadrilateral surfaces, and finally classification of cropped facial surfaces. We provide two additional extensions of the general shape analysis frameworks that are the focus of this dissertation. The first one considers shape analysis of marked spherical surfaces where in addition to the surface information we are given a set of manually or automatically generated landmarks. This requires additional constraints on the definition of the re-parameterization group and is applicable in many domains, especially medical imaging and graphics. Second, we consider reflection symmetry analysis of planar closed curves and spherical surfaces. Here, we also provide an example of disease detection based on brain asymmetry measures. We close with a brief summary and a discussion of open problems, which we plan on exploring in the future.