Research On Fast Algorithms For Solving The Weight Total Variation Image Denoising Models

Image denoising is one of the most basic and the most important research topics in the field of digital image processing, so that the research on it has very important theoretical and practical significance. The total variation model (ROF model) proposed by Rudin and Osher can keep the image edges and details while removing noise and is a classic denoising model in the field of image restoration, but ROF model would produce the "staircase" effect in a flat area. In order to overcome this drawback and improve the denoising effect of this model, Lixia Chen et al. improved ROF model, proposed a nonlinear weighted variational model and a linear weighted variational model, and gave a numerical algorithm for solving this model, i.e., the gradient descent algorithm.In order to establish fast and effective algorithms for image denoising, in this dissertation, we consider to apply the multigrid algorithm to solve num-erically the weighted variational models. The structure of this dissertation is a rranged as follows:In the first chapter, we first introduce the development history, application fields and three levels of the image processing technology. Then, we introduce the basic knowledges and application advantages on image restoration technology and variational PDEs-based image restoration technology.Finally, the research contents and the arrangement of chapters in this dissertation are presented.In the second chapter, we review the basic knowledge on the functional analysis and the optimization theory related to this dissertation, the BV space closely related to the total variation model, as well as the discrete schemes of the operators which will be used in the latter chapters.In the Chapter3, we first introduce the framework of image denoising models based on PDEs, and the ROF model. Then, we introduce the Euler-Lagrange equation of the ROF model and the corresponding explicit time marching method, and the Chambolle’s dual algorithm based on ROF model. Finally, we introduce the nonlinear weighted total variation model and its gradient descent method, and the linear weighted total variation model and its Chambolle’s dual algorithm.In the Chapter4, we first review the multigrid method, we then suggest to apply the multigrid method to solve the nonlinear weighted model and the linear weighted model, respectively. According to the differences between the nonlinear weighted model and the linear weighted model, we deal with them separately:The linear weighted model, starting from the dual equation, is solved by using the multigrid method, while the nonlinear weighted model, starting from the Euler-Lagrange equation, is solved by combining the fixed point method with the multigrid method.We implement the numerical experiments with the algorithms mentioned in Chapters3and4and give the analysis of the experimental results in the Chapter5. Finally, we summarize the whole dissertation, and analyze the feasibility and effectiveness of the models studied and algorithms proposed in this dissertation.