Anatomical shape analysis problems are ubiquitous in medical imaging. In brain MRI imaging, the problem arises when comparing cortical features of functional importance, such as gray matter thickness, as well as when performing fine-grained analysis of sub-cortical structures. Two general types of approaches have been developed over the years to perform quantitative shape comparison of anatomy: the first, and more intuitive approach, attempts to bring surface models into dense point-to-point correspondence; the alternative approach avoids the need for dense registration by exploring intrinsic global properties of the shape. Each of these methods has their advantages and disadvantages, and I explore examples from each in applications to brain imaging. The general underlying theme of my work has been spherical parametric shape registration, and other descriptions based on spherical maps. Several times throughout this work I utilize the theory of spherical harmonics, both in their scalar and vector form as a means of parameterizing and describing brain anatomy. Using this theory, I develop several algorithms for rigid and non-rigid shape registration for a variety of brain shapes, which are described in Chapters 1-4.

In Chapter 5, I present an application of the methods developed in previous chapters to developing brain imaging biomarkers of Alzheimer's disease. I show that using the new surface-based features and statistical learning, it is possible to develop biomarkers which outperform all others to date in terms of sensitivity to disease-modifying effects and disease specificity.

A more recent field of application for shape-based analysis comes from human skull models generated from conical CT imaging. The general registration problems are similar in spirit to brain imaging. However, the much lower signal-to-noise ratio and greater topological variability of the shape models require a new set of tools to deal with these shapes in practice. In Chapter 6, I develop an approach to modify the topology of a skull surface while preserving useful features, and again return to the sphere for dense registration and the creation of shape atlases. Due to the noisy nature of the data, a fast non-local non-linear correspondence search is developed for pairs of skull shapes from different subjects. I show that this search is crucial for high-quality registration based on resulting population-based averages and individualized shape analyses.