Spatial Coherence Enhancing Reconstructions for High Angular Resolution Diffusion MRI
von Christoph Rügge
Datum der mündl. Prüfung:2015-02-02
Betreuer:Prof. Dr. Thorsten Hohage
Gutachter:Prof. Dr. Thorsten Hohage
Gutachter:Prof. Dr. Jens Frahm
EnglischA main step in processing and analyzing data obtained from High Angular Resolution Diffusion MRI is the reconstruction of some measure of diffusion per spatial location and per direction in the volume under examination. This involves assumptions on the underlying physical diffusion process as well as on the mathematical structure of the solution. The latter is frequently expressed as a regularization penalty encoding an abstract form of smoothness. Incorparating suitable assumptions of this kind into the reconstruction can yield improved results for low SNR or coarsely sampled data sets. After outlining and analyzing some important physical modelling approaches, this thesis motivates and formally introduces a regularization penalty that enforces smoothness both inside a voxel and between neighbouring voxels, while taking into account the special structure of diffusion in the brain, which tends to be directed mainly along the nerve fibers, and can thus not be assumed to be smooth perpendicular to them. The method is formulated as a Tikhonov-type constrained quadratic optimization problem, and convergence for reconstructions from noisy, discrete data as the discretization becomes finer and the noise level goes to zero is proved. The result relies on a compact embedding property of a Sobolev-type space constructed from the regularization penalty. For the practical implementation of the method, both an exact and a faster approximate algorithm are described, and convergence of the approximate algorithm is shown. Both algorithms are tested on various artificial and in-vivo data sets.
Keywords: Inverse Problems; Magnetic Resonance Imaging; Diffusion MRI; Spatial Regularization