Since brain is one of the most important organ of human being, it is hot topic of medical science. With the rapid development of electrophysiology and neuroimaging techniques, people could obtain many kinds of data recorded from the brain. To deal with these data effectively will help us to understand the brain. At the same time, with the improvement of geometry theory, especially the research of digital geometry technique, it has great application foreground of introducing digital geometry to brain structure shape analysis. The main works and contributions of this dissertation are as follows: (1) We introduced a spherical surface parameterization method based on discrete Euclidean Ricci flow. By improving the calculation of Gaussian curvature, we use Euclidean geometry to approximate the spherical case, which avoid the convex problem of spherical Ricci flow. The experimental results show that our method can achieve the Ricci flow spherical surface parameterization efficiently, while keeping the properties of Ricci flow like conformal, intrinsic and robustness. Compared with traditional Ricci flow, our method improved the convergence speed while keeping the accuracy of the results, which provide the reliability for later analysis. (2) We proposed a multi-scale feature extraction and registration method based on Ricci energy. We first construct a scale space of the surface based on Ricci energy. Then, the multi-scale features are extracted from the shape space and used in the surface registration. The surface registration combined local multi-scale feature information and global Ricci energy together. The experimental results show that the multi-scale space features can successfully represent the shape geometry information with properties of intrinsic, robustness, etc. Besides, Ricci energy not only keep the Gaussian curvature information but also provide the conformal factor at the same time, which can represent more information of the shape. Thus, combining local feature and global Ricci energy together can make the registration improving the local accuracy around feature points while keeping the global registration result. Compared with traditional registration method with curvature and sulci, our method can improve the registration obviously. (3) We use spherical harmonic coordinate to construct shape manifold, and introduce kernel regression to the shape manifold. The regression method is applied to study the hippocampus changes ...
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