Variational Image Segmentation and Restoration using Sobolev Gradients, Nonlocal and Iterative Regularization Methods

The variational partial differential equation (PDE) method in image processing has been studied extensively due to its strong mathematical theory and existence of state-of-the-art numerical PDE methods. In this dissertation, we present the analysis and computation of novel energy minimization models for image segmentation and object detection, as well as for image restoration. We first present joint image segmentation, deblurring and denoising models based on multilayer level set approach. For the computation of minimizers, we consider Sobolev H1and L2 gradient descents. Second, we propose a new degradation model with stochastic point spread function. We solve the image restoration problem in a variational framework, by formulating minimization problems based on the multiframe idea. Extensions of the degradation model to color image or real video restorations and to binary image segmentation are also illustrated. In the third place, we introduce a class of image restoration algorithms based on the Mumford-Shah model and nonlocal image information. Specifically, we propose new regularizers, non-local Mumford-Shah-like regularizers, and present several applications of the proposed regularizers to color image restoration problems such as deblurring in the presence of Gaussian or impulsive noise, inpainting with large missing regions, super-resolution of one still image, and color filter array demosaicing. Characterizations of minimizers are also given based on dual norm formulations. Finally, we present a generalized iterative regularization method to solve ill-posed inverse problems, with applications to image restoration problems such as deblurring in different types of noise models or deblurring via cartoon and texture decomposition. In addition to above, we present numerical experiments to show supporting evidence for all our work.