Image Rsetoration Based On ADMM Algorithm

In our daily life,images are affected by various factors,such as distortion,blurring,noise,etc,and consequently,the significant characteristics of the images are lost and reduce the image quality.As an important problem in image processing,image restoration is an ill-posed problem.The general method solving this problem is to introduce the relevant regularization items and to establish a regularization model based on the a-priori information about the original image.In this paper,we address the image restoration in terms of an optimization problem with l₁ regularized term instead of l₂ penalty term.For the restoration of noisy images,the former has better denoising effect as compared with the l₂ regularization model.There already haven been several iterative algorithms for the optimization problems,such as dual ascent method,conjugate gradient method,augmented Lagrange method,etc.We apply the alternating direction multiplier method(ADMM)to solve this opti-mization problem.This method combines the decomposability of the dual method with the better convergence of the augmented Lagrangian method.ADMM is an effective method for solving large-scale convex optimization problems with separable structure.Taking advantage of the separability of the objective functional,the original problem is decomposed into several sub-problems to for efficiently solving.Nowadays,ADMM has been successfully applied to image deblurring,which can achieve linear convergence for the fixed penalty parameter ? with a convergence rate of o(1/n).In order to improve the convergence speed of the iteration algorithm,we propose an adaptive penalty step?k based on the variational form,and a new iteration scheme is given with rigorous con-vergence analysis.On the other hand,image deblurring is essentially a de-convolution process.In order to reduce the computation time and the storage in the deconvolution process,we consider an image model with cyclic boundary conditions and use the discrete Fourier transform(fft)to solve the first sub-problem.The diagonalization of the coeffi-cient matrix greatly reduces the cost of the calculation and then avoid the trouble caused by the large size of the fuzzy matrix.