UC San Diego

Image restoration is an inverse problem where the goal is to recover an image from a blurry and noisy observation. An image restoration problem can be formulated as a total variation regularized least-squares minimization where the objective function is the l2-norm squares of the residue between the observation and the prediction. Since the total variation norm is not differentiable, existing methods are inefficient. In this dissertation, a fast numerical optimization method is proposed to solve total variation image restoration problems. The method transforms the original unconstrained problem to an equivalent constrained problem and uses an augmented Lagrangian method to handle the constraints. The transformation allows the differentiable and non- differentiable parts of the objective function to be separated into different subproblems where each subproblem may be solved efficiently. An alternating strategy is then used to combine the subproblem solutions. The image restoration method is extended to handle video restoration problems. The proposed method considers a video as a space -time volume, and introduces a three-dimensional total variation regularization function to enhance the spatial and temporal consistency. The new video restoration framework opens a wide range of applications, including video deblurring and denoising, disparity map refinement, and hot-air turbulence removal. Practical image and video restoration methods need to take into account spatially variant blur and blind deconvolution issues. Therefore, spectral properties of the spatially variant convolution matrices are studied. A fast and robust blind deconvolution method for single image spatially variant out-of-focus blur removal is proposed