Variational methods: Theory and its applications to image deblurring and denoising problems

Image deblurring and denoising problems are well known inverse problems. In this dissertation, we develop new variational methods for image restoration. In the first part, we notice that the homogeneous Sobolev spaces W˙alpha,p are good candidates in modeling oscillations in images as was observed in the work of [42] when the authors investigated their image decomposition method. We recover a clean image from its noisy blurred version by searching the cartoon part (piece-wise smooth component) and the texture part (oscillatory component) separately in two different spaces. As was already done in [76], the cartoon part is modeled in the space of functions of bounded variation. The texture part is modeled in the homogeneous Sobolev spaces. Using these function spaces, we formulate a minimization problem whose minimizer is considered to be a recovered clean image. We present both theoretical and computational results such as the existence of a minimizer, characterization of minimizers, and the numerical scheme we have used to compute a minimizer. Several experimental results on blurry images and on noisy-blurry images are shown. Since we have a family of homogeneous Sobolev spaces with parameters alpha and p, we analyze the role of these parameters in recovering oscillations.;In the second part of the dissertation, we propose two image denoising methods which can be applied to noisy HARDI (High Angular Resolution Diffusion Imaging) data arising in medical imaging of the brain. This data set has the structure R3xS2 xR , thus at every position in R3 and at every direction on S2, an intensity value of the signal is observed. This represents the local diffusivity of water molecules in the tissue. The observed data is usually degraded by random noise following a Rician distribution. There are also constraints that arise from the data acquisition model, which led us to use the logarithmic barrier method. Denoising process involves smoothing process which led us to use vectorial total variation regularization. We again investigate both theoretical and computational questions regarding the two minimization problems we propose. We compare our results with those provided in McGraw et al. [64] and notice that our methods outperform those methods presented in [64] visually and quantitatively.