Augmented Lagrangian Type Algorithms And Their Applications In Image Processing

Variational inequality (VI) finds important applications in diversified fields, such as economic equilibrium, transportation and engineering design, etc. A series of meth-ods are devised for handling this problem, e.g., alternating direction method (ADM), proximal point algorithm (PPA), projection contraction method, Newton method and interior point method. All of them are widely applicable in practical world.Recently, sparse optimization whose rationale is the optimal solution being sparse is hot-investigated in information science and statistics. It possesses some critical prop-erties:(1) large-scale, e.g., the problems in compressive sensing, image processing and machine learning.(2) ill-posed, e.g., the non-smoothness of the total variation norm, the ill-posed blurry matrix.(3) special structures, e.g., the separability in objective function or linear constraint.Based on the algorithms for solving VI, we develop some efficient algorithms for solving image processing problems, e.g., image restoration or reconstruction.In Chapter1, we summarize some basic and crucial inequalities for â…¥, such as the properties of projection, then present some definitions in image processing.In Chapter2, we consider the convex programming problems with separable struc-tures whose objective function is the sum of three functions without coupled variables. It's empirically effective to extend straightforwardly the ADM (EADM) for the prob-lem. But, the convergence of EADM is still ambiguous. We develop a variant splitting method with global convergence, which is empirically competitive to the EADM. We show the numerical efficiency of this variant to some problems in image processing and statistics. Chapter3generalizes the convex programming in Chapter2to the case that ob-jective function is sum of many individual functions without coupled variables. An algorithm is developed by splitting the augmented Lagrangian function in a parallel manner, thus it differs substantially from existing splitting methods in alternating style which require to solve the decomposed subproblems sequentially. We show wide ap-plicability and encouraging efficiency of the new algorithm in the area of image pro-cessing.Chapter4presents a customized application of the classical PPA to the convex programming with linear constraints. If the proximal parameter is chosen appropriate-ly, the application of PPA could be effective to exploit the simplicity of the objective function. The resulting subproblems could be easier than those of applying the aug-mented Lagrangian method, a benchmark method for convex programming under our consideration.In Chapter5, we develop a decomposition model to restore blurred image with missing pixel values. Our assumption is that the true image is the superposition of cartoon and texture. We use the total variation norm to regularize the cartoon and its dual norm to regularize the texture, respectively. The model not only gives the restored image, but also gives a decomposition of cartoon and texture. We apply the variable splitting method to solve the developed model and report some numerical results for image decomposition.