Research On Bound Constrained Image Restoration Problems And The Methods For Choosing The Regularization Parameter

Image restoration is an important branch of image processing, which is meaningful for us to research. Often the digital image we derived is blurry and noisy which is degenerate. It can be described by the model , where denote blurry operator, clean image, stochastic noise and degenerated image. Image restoration is to solve from , if unknown, it's a blind image restoration problem. In this paper K is known.Image restoration is a typical ill-posed problem, which is difficult to solve. But with the hard work of many researchers, the solved methods have been developed to several embranchments. In this paper we use the regularization methods, which have the following two classical regularization models. One is Tikhonov regularization model whose regularization term is quadratic. Its Euler equation is linear and easy to be solved. But it assumes that the restored image is smooth, which resulted to blurring the discontinuous region such as the edge or corner of the image. In order to improve it, Rudin et al proposed the total variation model(ROF). Its Euler equation is nonlinear, which can be solved by the popular primal-dual method. In this process we must choose an appropriate regularization parameter, which is difficult for us. Here we use an iterative method based on L-curve strategy to search it for the two models. At last we make program on MATLAB and compare the restored results, which show that the ROF model is prior than the traditional Tikhonov model at preserve edges.In order to improve quality of the restored images, we choose the regularization term and the regularization parameter adaptively according to the local information of image such as the gradient or variance. The general principle is that for the flat region of the image, noise is more sensitive than blur in vision, so we can choose an isotropic regularization term such as the Tikhonov term and let the parameter be a bit greater relatively. While for the edges or corner, deblurring is more important, we can choose the TV term and reduce the value of the parameter. For example, the variable exponent, linear growth function proposed by Levine, satisfy this rule. Actually, in order to get a restored image closer to the clean image, we should utilize the transcendent information such as nonnegative property sufficiently. In this paper, we add some bound constrains to the improved model, such as bound the norm of or limit the range of u . Then image restoration problem becomes a constrained minimization problem, which is complicated relatively. Here we use the primal-dual active-set method, essentially a semi-smooth Newton's method. At last program on MATLAB, the images restored by our methods are better than the ROF model, no matter the ISNR or in vision, which shows that our methods are feasible and valid.