In brain imaging study, 3D surface-based algorithms may provide more advantages over volume-based methods, due to their sub-voxel accuracy to represent subtle subregional changes and solid mathematical foundations on which global shape analyses can be achieved on complicated topological structures, such as the convoluted cortical surfaces. On the other hand, given the enormous amount of data being generated daily, it is still challenging to develop effective and efficient surface-based methods to analyze brain shape morphometry. There are two major problems in surface-based shape analysis research: correspondence and similarity. This dissertation covers both topics by proposing novel surface registration and indexing algorithms based on conformal geometry for brain morphometry analysis.

First, I propose a surface fluid registration system, which extends the traditional image fluid registration to surfaces. With surface conformal parameterization, the complexity of the proposed registration formula has been greatly reduced, compared to prior methods. Inverse consistency is also incorporated to drive a symmetric correspondence between surfaces. After registration, the multivariate tensor-based morphometry (mTBM) is computed to measure local shape deformations. The algorithm was applied to study hippocampal atrophy associated with Alzheimer's disease (AD).

Next, I propose a ventricular surface registration algorithm based on hyperbolic Ricci flow, which computes a global conformal parameterization for each ventricular surface without introducing any singularity. Furthermore, in the parameter space, unique hyperbolic geodesic curves are introduced to guide consistent correspondences across subjects, a technique called geodesic curve lifting. Tensor-based morphometry (TBM) statistic is computed from the registration to measure shape changes. This algorithm was applied to study ventricular enlargement in mild cognitive impatient (MCI) converters.

Finally, a new shape index, the hyperbolic Wasserstein distance, is introduced. This algorithm computes the Wasserstein distance between general topological surfaces as a shape similarity measure of different surfaces. It is based on hyperbolic Ricci flow, hyperbolic harmonic map, and optimal mass transportation map, which is extended to hyperbolic space. This method fills a gap in the Wasserstein distance study, where prior work only dealt with images or genus-0 closed surfaces. The algorithm was applied in an AD vs. control cortical shape classification study and achieved promising accuracy rate.