| Image information is an important source for human to know our world. However, due to the bad conditions in the process of imaging or transferring, the quality of image could be degraded, thus deteriorating its use and post-processing. How to restore a clear, content-rich image from degraded one is a problem of focus. This is exactly what image restoration does.Image restoration is one of the most important research in the field of image processing. Image recovery problem is usually an ill-posed inverse problem,priors can regularize these problems to yield high-quality results. Meanwhile, natural image statistics indicate that image marginal distributions is neither exactly Gaussian distribution nor exactly Laplacian distribution, is nearly analogous to a hyper-Laplacian, i.e. priors are nonconvex. This thesis is just to build some nonconvex and nonsmooth optimization regularization models with the help of various nonconvex potential functions which are Lipschitz or non-Lipschitz. These models can be solved by alternating minimization methods, and their convergence analysis is given. Our main work and innovations are as follows:Aiming at Lipschitz and nonconvex potential functions and additive noise, we build the novel nonconvex energy model composed of L2 data fitting and Lipschitz regularization functions. About computational methods, we firstly introduce graduated nonconvexity(GNC) approach to dispose nonconvex object energy function,generate some corresponding graduated nonconvexity energy functions which is closing to object energy function as variable coefficient is enlarging. When the coefficient is fixed, the surrogate function is separately solved by four types of alternating minimization methods, which can be summed up in one kind of method.So one method is selected for discussing convergence analysis, simultaneously, in order to obtain a positive definite matrix, we only consider the definite part of the Hessian matrix of energy functions. At last, combined with Kurdyka- Lojasiewicz inequality, we successfully analyze the convergence of algorithms, especially when the surrogate function closing to the nonconvex object energy function along with the coefficient increasing.Aiming at non-Lipschitz quasinorm ?p(0 < p < 1) and multiplicative noise,we build the novel nonconvex energy model composed of L1 data fitting and nonLipschitz regularization functions, which is disposed by variable separation and alternating minimization methods. For a subproblem including T Vpterm(i.e. denoising model), we firstly deal with quasinorm ?pusing Huber function, again obtain dual vectors by solving its corresponding euler equation with the help of the primal-dual Newton method, and then gain a generalized positive definite Hessian matrix, next find the optimal solution of the subproblem using the trust-region method, and the global as well as local superlinear convergence toward its stationary point is acquired. Finally, we achieve the convergence analysis of the whole algorithm, using the theory of Kurdyka- Lojasiewicz inequality.On the other hand, we give three nonconvex models applied in image processing or signal processing in recent years, such as nonconvex minimization model with box-constraints, p-shrinkage(0 < p < 1) and the modified nonconvex minimization,which are solved by their corresponding alternating minimization methods according to different kinds of noises, and obtain their dependent convergence.In the experiments, we verify the effectiveness and efficiency of the various algorithms. Specifically, we analyze the interval of variable coefficients in Chapter3, and the maximum of PSNR is obtained at p =12when the regularization function?pis selected to deal with the multiplicative noise in Chapter 4, as well as the special effects of others nonconvex models in Chapter 5, such as, the nonconvex minimization model with box-constraints have better restoration effect as image regions have a lot of extreme pixels, and the modified nonconvex minimization have faster convergence rate.
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