The Study Of Optimization Methods For Inverse Problems And Its Applications In Image Restoration

Image restoration is a branch of image disposal, which belongs to inverse problems and can be solved by regularization and optimization methods. The combination of the two methods has wide applications in image restoration. In this thesis we propose a new model and a regularized active set method for image restoration problems. Numerical experiments on gray image restoration problems solved by the proposed method show that it is promising and reliable for image restoration.There are four chapters in this thesis.In Chapter 1, we introduce the definition of inverse problem, some knowledge of the first operator equation, as well as the concept of ill-posedness and its characteristics.In Chapter 2, we concentrate on the regularization methods and optimization methods for inverse problems. In Section 1, we first briefly review the choice method and the quasi-solution method, and then the Tikhonov regularization method. Moreover, the concepts of l_p, L_p and their relative knowledge are also presented in this section. In Section 2, we review some well-known optimization methods such as the gradient-type methods and Newton-type methods.Chapter 3 contains some basic knowledge of image restoration, and the main contribution of this thesis. We obtain a new model by improving the existing regularization model. Specifically, we add nonnegative constraints to represent that the pixels are nonnegative; we transform the general l_p—l_q problem into a constrained quadratic programming; and we use the active set method to solve the quadratic programming. To test the effectiveness of the new model and the proposed method, we use them to get restored images from the blurred images which are obtained by simulating the blurring and noise process. Numerical results demonstrate that our approach is effective, by noting that the computation time is largely shortened in comparison with the trust-truncate CG method.In Chapter 4, we give the conclusion of this thesis as well as some considerations for further research.