Research On Nonsmooth Image Restoration Algorithm

Abstract Digital image restoration has a wide application in many fields including Navigation, Aerospace, Biomedicine and so on. In this technology, a mathematical model is firstly built by means of the degradation reasons of image such as boundary blur, and the original image is then restored along the inverse process of image degradation. However the problem often tends to be very ill-conditioned when the inverse process of blur system is only used to restore since the blur system (matrix) often has many singular values close to zero. Thus one of effective ways to solve these problems is to combine some priori information of the original image and define the regularization solution. In general, this regularization solution is just a minimum point of the energy function with regularization terms and these methods have better restoration performance. The regularization term often includes smooth and nonsmooth regularization. As a classical smooth regulariza-tion, the optimization problems including Tikhonov regularization have low computation-al cost, while it usually makes images over-smoothing in image restoration and could not protect boundaries very well. Even though many theoretical and numerical results show that the nonsmooth regularization could protect the image boundaries very effectively, they may cause several difficulties in numerical computation, and thus it is paid many attentions in recent years. Moreover, the nonsmooth image restoration problem could be solved by the subgradient method, augmented Lagrangian method, GNC method and so on, but it has very important influence on the image quality and algorithm efficiency to choose the compose of energy function such as data fitting term, potential function and so on. Therefore this thesis studies nonsmooth image restoration methods to improve the restoration performance. The main works can be stated as follows:1. For the images with neat boundaries and the additive noise satisfying the condition that its distribution is nonGaussian such as Uniform, Salt pepper, Laplace and so on, the energy function with l1-norm data fitting term and nonsmooth nonconvex potential func-tion has better restoration performance. However, the non-differentiability may cause some computational difficulties. Thus a GNC method is proposed to solve this problem by using the idea of fitting gradually and transforming variables. The proposed method makes an equivalent transformation of the original problem by introducing auxiliary vari-ables, and then optimizes alternatively for different objective variables.2. To overcome the drawbacks of the GNC method and augmented Lagrangian duali-ty method being used alone in nonsmooth image restoration, we propose a method on basis of the above two methods by transforming the original problem into equality constrained optimization problem, and strictly prove its dual convergence. The proposed method keeps the advantages of the GNC method and augmented Lagrangian duality method, which not only gets an effective initial value for original problem, but also ensures that the solutions of the nonsmooth optimization problem can approach to the dual solution when duality gap is zero. Moreover, an adaptive energy function is generated by the dual iterations. Experimental results show the better performance of image restoration for the proposed method.3. Since different objective variables are optimized alternatively, the search direc-tions generated by the alternating direction method are often not accurate enough in l1-TV and l2-TV image restoration problems. To overcome the drawback, a method is proposed to improve its optimization performance by using subspace optimization and its conver-gence is strictly proven. The proposed method corrects the current search direction by using the Taylor expansion of energy function and a linear combination of previous and current search directions effectively, and thus overcomes the inaccuracy of search direc-tions. Experimental results show that the proposed method could effectively enhance computing performance of the basic alternating direction method.4. By combining Tikhonov regularization with different order nonsmooth noncon-vex regularization, we represent a GNC image restoration method. The method could restore the smooth part and neat boundaries very well, and avoid to the over-smoothing and staircase artifacts. Some experimental results show the performance of the proposed method.5. We propose a method to construct approximate potential functions for GNC method, which produces more flexible similarity between approximate potential func-tions and original potential function. Experimental results show that new approximate potential functions could enhance the performance of GNC image restoration method.