Several Classes Of Non-smooth Numerical Methods For Image Denoising Problems

With the development of computer technology, image processing problems play an increasingly important role in daily life. However, in the course of the image formation, transmission and generation, some external factors led to de-crease in the image quality, so it is one of important research topics in image processing to effectively restore degraded image. Normally, the image restoration methods can be summarized into three categories:wavelet-based methods, based on probability and statistics methods, and methods based on partial differential equations (PDEs). Among these approaches, the image restoration models based on the PDEs have some advantages such as relatively strong self-adaptability, close to the image characteristics, etc., so that this class of approaches has been rapid development over the past ten years and has been extended to almost all fields in image processing.The image restoration problem is usually an ill-posed inverse problem so that a well-posed model is needed to build in the sense of the regularization. In order to build a model consistent with the characteristics of the image, the model is often required to satisfy certain mathematical properties so that the numerical difficulty is increased, therefore it is an important research topic in image processing to look for some rapid and effective numerical algorithms. Since the image denoising is an important part in the image restoration, in this dissertation, we focus on two basic and important denoising models—Rudin-Osher-Fatemi (ROF) model and Lysaker-Lundervold-Tai (LLT) model, and their rapid numerical methods. Our main work and the innovations are as follows:Since the optimality conditions for the dual problem of the ROF model or the LLT model contain a nonlinear complementarity problem(NCP), we can apply some mathematics qualities of the Fisher-Burmeister NCP function to transform this optimality condition into a system of semismooth equations. In order to deduce global convergence of the proposed algorithm, by introducing a merit func-tion, we propose to use the damped modified Gauss-Newton method to solve the system of semi-smooth equations. In addition, in the algorithm, we introduce a modified parameter to increase the search step length and use the preconditioned conjugate gradient method to improve the calculation speed. At the same time, we also give theoretical analyses about the global convergence and the Q-superlinear convergence rate for the proposed algorithm. Since the augmented Lagrangian approach combines the advantage of the Lagrangian method and penalty method, it is widely applied to solve the nons-mooth convex optimization problems. For the LLT model, we first convert it into a constrained problem and then attain the optimality conditions of the constrained problem based on the augmented Lagrangian approach. Moreover, we point out that the optimality conditions can be seen as a projected gradient method. So we propose to use the projected gradient method to solve the discretization LLT model and point out that the classic semi-implicit gradient descent method can be deduced from the projected gradient method. At the same time,, we extend the projected gradient method to solve a mixed model (mixing the ROF model and the LLT model) for texture extraction. In addition, we also apply the augmented Lagrange method to the image deblurring problem with nonnegative constraints and propose an active set strategy. Furthermore, we prove that the active set method can be fell into the framework of semismooth Newton methods.Recently, by using the ideas of the Bregman iterative algorithm, Goldstein and Osher proposed a split Bregman method to solve the image restoration problems. On this basis, we extend the split Bregman method to solve the anisotropic LLT model and the second step of the LOT model. Although the split Bregman method has certain advantages, it has to suffer from solving a PDE at each iterative so that the computation costs are increased. In order to overcome this drawback, based on the properties of the projection operator and the shrink operator, we propose a new rapid and efficient algorithm—the projection algorithm. In order to illustrate the effectiveness of the projection algorithm, we apply this algorithm to solve the anisotropic LLT model. Furthermore, the theoretical analysis about the convergence of this algorithm is also given. In particular, we point out that the projection algorithm based on the split Bregman method can be fell into the framework of the FBS algorithm.The dissertations is supported by the National Natural Science Foundation of China (No.60872129).