A Generalized Regularization Method For Image Restoration

In particular, the images we observed, are the convolution of blurring operator and real images, with corrupted by random noise. One of the keys about effectiveness of the image restoration depends on the precision of the image degradation model. Because of the blurring matrix is ill-posed, substantially, we solve the model of image restoration for such a scale matrix as computing an approximate solution of linear ill-posed inverse problem. If we want to compute the linear inverse problems directly, there may be a lot of disadvantages, and the solutions of this method are nonsense commonly. A kind of good methods for the inverse problem is Tikhonov regularization which is one of the most popular methods. This article introduces a Tikhonov regularization method with a general linear regularization operator for large-scale ill-posed problems, and the method of regularization parameter selection is studied.For general large scale ill-conditioned problems, we introduce projective methods which are iterative bidiagonalization process and QR decomposition. When dealing with discrete ill-posed problems, the large-scale regularization problem is decomposed into low rank matrix regularization model. It is only took a few steps of Lanczos bidiagonalization process, we can obtain a well low rank approximation for the original problem. In the theory the Lanczos decomposition is a kind of very popular and generalized solution method, and not limited to special structured matrices which is the most important reason. The time complexity of matrix-vector multiplication can be carried out in O(n) operations. Then we chose QR decomposition method based on the subspace from the Lanczos projection process. The approximate solutions of original problem are projected onto the Krylov subspace, and the new algorithm reduces the calculation cost of time.The choice of a suitable value of μ is an essential part of Tikhonov regularization. The stand or fall of parameters determine that how sensitive the solution is to the error e and how close the real solution. The method of parameter selection is studied in this paper. According to the probability distribution, the augmented Tikhonov functional and the value function are introduced in the case of unknown noise variance. Through the convexity and differentiability of the value function and balance principle, we constructed the new rule. By minimizing the new rule and balancing principle, the general expression of solving regular parameters is deduced.We give an alternating iterative algorithm can get effective implementation to the new method of parameter selection and the total model. By the way of the curves and the graphs illustrate the convergence of the experimental solutions clearly and high efficiency of the selected method is presented in this paper. We compared the proposed method with other parameter selection methods, such as quasi-optimally criterion, L curve method and the optimal solution. The numerical experiment results indicate the reliability and stability of the method. By comparing with other methods indicate that the parameter selection method presented in this article is significant and more effective than the other methods.