Research Of Image Restoration Algorithm Based On Variational Method

It is acknowledged that about 75%of the external information received by the human perception system comes from vision.Nowadays,the study on the vision has been attracting more and more attention in the research areas,and important applications have been developed in a variety of practical scenes.One of the key issues in the study or application of vision is the information degradation of the images,which might be caused by several reasons such as production,transmission and storage.Degradation would negatively affect the study on images,such as edge detection,feature extraction and so on.Therefore,a quality recovery of the images is crucial for the further study and applications.In this thesis,towards the quality image recovery,we focus on the mathematical theories,regularization,denoising and deblurring in image restoration.The innovations and contributions of our work can be concluded as follows.Firstly,the mathematical foundations of image processing are discussed.The mathematical theories of operators and functional,variational and convex optimization used in image processing are introduced systematically.The relations among functional variation,subdifferential and Gateaux derivative are pointed out,which will help to find out the Euler-Lagrange equation of the optimization problem accurately.In order to simplify the description of geometric properties of subdifferential for a convex function,three concepts of left tangent line,right tangent line and subtangent line are given.Secondly,the regularization problems of image processing is studied.The application of three regularization terms in image processing are summarized,from the point of view of mathematics.Firstly,the derivation process,advantages and disadvantages of the early restoration methods such as inverse filtering and Wiener filtering are described.Then,the regularized Tikhonov method,the regularized total variational denoising theory and the regularized gradient L0 norm smoothing are discussed.It is found that the Tikhonov method can remove the additive white Gaussian noise,but it also can bring ambiguity.The total variational method can not only effectively remove the additive Gaussian white noise on the image,but also preserve the edges and detail information of image effectively.At the same time,it also brings a certain degree of blurring of the edges and detail information.The L0 norm method of image gradient can not only smooth out the details of the image effectively,but also preserve the edge information of the image without blurring the edges,which is helpful to extract the edges of the image.Numerical experiments show that,different regularization terms can produce different processing effects.Therefore,it is very important for image processing to select the appropriate regularization terms according to the requirements.Thirdly,the algorithms of removing additive Gaussian white noise are analyzed.Two useful image denoising algorithms,Chambolle A's dual denoising algorithm and Osher's Bregman iterative denoising algorithm are discussed.Based on these two kinds of denoising methods,a weighted denoising algorithm is proposed for additive white Gaussian noise.The weighted denoising algorithm is based on the Bregman iteration method,and each iteration is solved by the dual method.Unlike the Bregman iterative denoising algorithm,the weighted denoising algorithm assigns a weight coefficient to the "taking-back-noise" term,and the size of weight coefficient is adjusted according to the noise level.The greater the noise is,the smaller the weight coefficient is.The denoising experiments of one-dimensional signal images,two-dimensional gray images and three-dimensional color images with different noise levels demonstrate the effectiveness of the weighted denoising algorithm.Fourthly,non-blind deblurring algorithms are studied.An alternating direction iterative image restoration algorithm based on the total variation and a split Bregman deblurring model is presented.For degraded images with given blur kernel and noise information,an alternating direction iterative restoration algorithm based on the total bounded variation is proposed.Each iteration of the restoration algorithm only needs to solve two smaller sub-problems.Under the assumption of periodic boundary,the sub-problems can be solved quickly by the soft-shrinkage algorithm and the fast Fourier transforms method,respectively.The new method can recover blurred images(motion blur or Gaussian blur)effectively,and can also restore the blurred images with additive white Gaussian noise effectively.Our algorithm is not only suitable for the the restoration of natural images,but also for the restoration of text images such as license plates and characters images.A large number of numerical experiments show the effectiveness of the proposed method.