Research On Sparse Optimization Models And Algorithms For Inverse Problems In Image Processing

Many practical image processing tasks are inverse problems that estimate the true data from the observed ones,such as image restoration?including denoising,deblurring,inpainting,etc.?,image decomposition,image reconstruction,etc.Most of them are illposed,due to the inversion of real physical processes,and stable and effective solutions usually rely on prior knowledge of the true image.In recent years,sparse optimization methods,based on the sparsity prior of the true image,have achieved great success in image processing.These methods assume that the true image has nearly sparse expressions under suitable bases,and then estimate the true image by imposing its sparse representations.This dissertation focuses on two central issues in the solution process:the construction of sparse optimization models and the design of solving algorithms.The main contents and contributions are listed below.Firstly,we propose two sparse optimization models for image denoising.The proposed models use spatially dependent regularization parameters to control the local degree of image filtering,so that the representation coefficients approach the empirical distribution,and the sparsity of image representation gets enhanced.We use the alternating minimization algorithm to decompose the difficult original problems into two simple subproblems,and establish the convergence of the proposed numerical scheme.Secondly,we propose a sparse optimization method based on nonlocal image representation for cartoon-texture image decomposition.We stack similar image patches into 3-D groups,and propose a group-based decomposition framework.The proposed framework achieves an enhanced sparse image representation by exploiting the nonlocal self-similarity image prior.We choose suitable representation bases for the cartoon and texture components,and construct a sparse optimization model for the decomposition of each group.The proposed model characterizes both the local sparsity and the nonlocal self-similarity of cartoon and texture.We use the alternating direction method of multipliers to solve the proposed model efficiently,and design an adaptive scheme for the selection of the regularization parameter.Thirdly,based on the second part,we apply nonlocal image representation to address image deblurring with an inaccurate blur kernel.We propose an enhanced regularized structured total least squares model to estimate the image and the blur kernel simultaneously.The two variables are characterized by a group-based low-rank prior,which combines the priors of local sparsity and nonlocal self-similarity.We use the alternating minimization algorithm to decompose the original nonconvex problem into two convex subproblems,and establish the convergence of the proposed numerical scheme.Fourthly,we propose a sparsity metric based on the truncated-?_1-2 quasi-norm,with applications to image reconstruction.By discarding large magnitude entries/singular values in penalization,the proposed metric achieves a nearly unbiased approximation of the vector sparsity/matrix rank.Theoretically,we establish reconstruction conditions of truncated-?_1-2 minimization and properties of its minimizers.Computationally,we prove that the proposed model can be expressed as a difference of two convex functions,and apply the difference of convex functions algorithm to decompose the original nonconvex problem into a series of convex subproblems.We also develop an adaptive scheme for selecting the number of truncated entries/singular values.Fifthly,based on the second,third,and fourth parts,we combine nonlocal image representation and the truncated-?_1-2 quasi-norm to address image inpainting.We propose a group-based truncated-?_1-2 minimization model,incorporating the advantages of nonlocal image representation in image characterization and those of the truncated-?_1-2quasi-norm in sparsity promotion.We decompose the proposed model into a difference of two convex functions,and design a numerical scheme based on the difference of convex functions algorithm.We also develop an adaptive scheme for selecting the number of truncated singular values of each group.