UCLA

This thesis is divided into two parts. In the first part, we will discuss the problem of restoring magnetic resonance (MR) images corrupted by blur and Rician noise. We discuss the formation of MR signals and how Rician noise is introduced into these images as a result of the MR acquisition process. Information about the Rician probability distribution and motivation for our proposed variational restoration model is then given. We show the existence of a minimizer and a comparison result. We also perform numerical experiments and comparisons using L^2 and H^1 gradient descent schemes to show the validity of our proposed model. This leads to a related second model that denoises High Angular Resolution Diffusion Imaging (HARDI) data, which is a modality of MR data that is used in reconstructing fiber pathways in the brain. HARDI data is vectorial data of dimension equal to the number of diffusion directions. This data can be used as input to calculate fractional anisotropy (FA) or orientation distribution functions (ODFs) which in turn are used to track fibers in the brain. Having denoised data may lead to more accurate fiber extractions. We test our proposed HARDI denoising model on various data sets, and various metrics are used to gauge improvements after denoising. In the second part of this thesis, we study the problem of restoring images distorted by atmospheric turbulence. Geometric distortions and blur are the two main components of degradations due to atmospheric turbulence, and prior work has been done to address these components separately. We propose a joint variational deblurring and geometric distortion correction model and give preliminary results.