Research On Regularization Techniques Applied To Image Restoration

This thesis will research into image reconstruction problems, which belong to 2-D deconvolution problems and are special cases of inverse problems. Inverse problems commonly are ill-posed, in order to obtain stable approximate solutions, some regularization techniques are required.The main purpose of this paper is to construct computational methods for treating 2-D linear deconvolution problems. Discretization of these problems leads to structured matrix problems with a block Toeplitz coefficient matrix, and the dimensions of the matrix can be extremely large. The main focus is on numerical deconvolution algorithms that exploit the Toeplitz structure of the matrix problems, and introduce the Kronecker product and show how it is used in the discretization and solution of 2-D deconvolution problems whose variables separate. Since traditional regularization methods only can deal with the small scale problems. The difficulty and main primary content in the thesis is how to construct regularization methods with Kronecker products to solve large scale problems.Fist, the math model and methodologies for image reconstruction are introduced. The difficulties of deblurring image are discussed. Then, the discretization of the continuous problem and how to blur image are described in detail. The special structure of the coefficient matrix and three basic blurring are set out. Finally, the analysis of the difficult to solve the 2-D linear deconvolution problems using 1-D regularization methods which can't deal with the puzzle about time and storage is given, then based on the characteristic of the coefficient matrix, a new technique- regularization methods with Kronecker products is proposed, such as 2-D TSVD, 2-D Tikhonov, PP-TSVD and so on.. All of algorithms do not only apply to image reconstruction problem, but all other two-dimensional ill-posed problems as well.We achieve all the numerical experiments on personal computer. Numerical experiments and the analysis of the results shown that, the 2-D regularization techniques adopted in thispaper are effective and feasible in reconstructing the degenerate image. It makes solving large scale problems become true.