Iterative Methods For Image Deblurring With Sparsity Constraints

With the development of information technology in modern society, digital image has become an important source of human access to information. In practice, due to the presence of various factors in the process of forming and transmitting image, such as, the obtained image will be degraded by the blur and noise. In order to reduce this degradation effect and try to get the original image, image restoration technique emerges, which is a very important research area in real life and in possession of a wide range of applications.Among the image restoration field, we focus on the image deblurring problem which is the process of reconstructing an approximation of an image from blurred and noisy measurement. Due to the bluring matrix is severely ill-conditioned, image deblurring is a classic ill-posed problem, so regularization methods are required. The basic idea of regularization is to replace the original ill-posed problem with a nearby well-posed problem, whose solution approximates the true solution. One of the popular regularization techniques is the Tikhonov regularization, which can be simply computed, but the edges of restored image are usually over-smoothed. To overcome this unpleasant property, some researchers proposed iterative regularization method, like thresholding iterative method and Bregman iterative method.In this thesis, motivated by the fact that most real images usually have sparse approximations under some tight frame systems, we investigate the modified linearized Bregman algorithm(MLBA) for frame-based image deblurring problems, with a proper treatment of the boundary artifacts. We consider two standard approaches: the imposition of boundary conditions and the use of the rectangular blurring matrix.The convergence of the MLBA depends on a regularizing preconditioner which could be computationally expensive. Hence a block circulant circulant block(BCCB) matrix is usually chosen as a preconditioner since it can be diagonalized by discrete Fourier transform. We show that the standard approach based on the BCCB preconditioner may provide low quality restored images. Then we propose different preconditioning strategies in order to improving the quality of the restoration and saving some computational cost at the same time. Motivated by a recent nonstationary preconditioned iteration, we propose a new algorithm that combines such method with the MLBA. We prove that it is a regularizing and convergent method. A variant with a stationary preconditioner is also considered. Finally, a large number of numerical experiments show that our methods provide more accurate and faster restorations, when compared with the state of the art.