Image Analysis Based on Differential Operators with Applications to Brain MRIs

Abstract

In differential geometry, computational diffeomorphism (smooth and invertible mapping) has become a fast-growing field in developing the theoretical frameworks and computational toolboxes for the tasks such as computer vision, movie production, gaming industry, medical imaging, etc. Mesh generation is one of components in computational diffeomorphism. In this dissertation, the deformation and variational methods (developed by Dr. Guojun Liao and his co-workers) for mesh generation are discussed, modified and generalized to 3D scenario. The former is based on the control of Jacobian determinant and the latter is based on the controls of both Jacobian determinant and curl vector of a diffeomorphism. In Brain Morphometry, image registration (identify a pixel-wise correspondent relationship of two images based on a dissimilarity measure) is a challenging problem, which demands a diffeomorphism to describe such pixel-wise correspondent relationship. The optimal control approach for image registration (developed by Dr. Guojun Liao and his co-workers) is revised and improved for cheaper computational costs and capability of 3D registration. A novel approach to averaging images is formulated based on averaging a given set of diffeomorphisms. This approach to averaging images is implemented by an algorithm which includes the variational method and the optimal control image registration.