Research On Modelling And Sparse Optimization Algorithms For Image Restoration

Digital image processing has a wide range of applications in many fields of science and engineering. Image restoration is one of the essential topics in image processing, and the related problems are ill-posed inverse problems. Due to the ill-posedness, an effective strategy to obtain a reliable solution is to use some regularization techniques. According to the problem type, one constructs a regularization model incorporating some prior knowledge of the original image.There are two parts:in the first part, novel regularization models for some image restoration problems are proposed and efficient algorithms are designed to solve the pro-posed models based on sparse optimization; in the second part, a class of regularization models with overlapping group sparsity (OGS) are studied, and then a new direct method is proposed, called inexact explicit shrinkage formulas, which can improve the computa-tional efficiency for OGS problems.One popular regularization method for image restoration is the total variation (TV) regularization method, because it preserves sharp edges (image regions where pixel val-ues change greatly) well. However, it processes smooth regions into piecewise constants, i.e., the staircase effects. In order to alleviate the staircase effects, a hybrid model combin-ing the TV regularizer and the high-order TV regularizer with an L1-norm fidelity term is proposed for blurred and impulse noise corrupted image restoration. To efficiently solve the proposed model, we develop an algorithm under the framework of alternating direction method of multipliers (ADMM). In addition, a spatially adapted regularization parameter selection scheme is also employed. Numerical results show that the quality of restored images by the proposed methods is competitive with other existing TV methods with overcoming the staircase effects.In order to overcome the staircase effects of TV, another new model is proposed for restoring blurred images under impulse noise. The model consists of an L1-norm fidelity term and a TV with OGS (OGS-TV) regularization term. Moreover, a box con-straint is imposed to the proposed model for improving the solution accuracy. Based the framework of ADMM and the majorization minimization (MM) method, an effective al-gorithm is designed. Extensive numerical results illustrate that the established model can significantly enhance the restoration quality on PSNR values and the relative error with edge-preserving property and less artifacts compared with other TV-based methods, such as TV, HOTV, TGV.Incorporating the advantages of the OGS regularization methods, an MM algorithm for weighted OGS problems is proposed. Then, a weighted OGS-TV regularization term is given. We propose a uniform framework under OGS-TV for image restoration and consider applications of various image processing problems, e.g., image inpainting, im-age zooming, mixed noise removal. Extensive experiments illustrate that the proposed method is competitive with some existing methods on PSNR and SSIM.Because of the difficulty on "overlapping" in OGS functionals, iterative methods, for instance, the MM method, are popular methods for solving OGS problems. How-ever, the computational speed of those methods is slow. A new direct method for one class of OGS problems is designed, called inexact explicit shrinkage formulas, which can improve the computational efficiency. Numerical results show that the proposed method can obtain very similar results as the MM method with saving much computational time. The proposed method can be effectively applied to solve the OGS subproblems in some applications, for instance, OGS-TV.