Research On Optimization Model And Algorithm For Image Restoration And Low-rank Tensor Completion

In recent decades,digital image processing has been widely used in medical,communication and other fields with the rapid development of computer technology.Therefore,it is of great practical significance to research this kind of problem.Image restoration is the most basic problem in digital image processing,the purpose is to restore the real image from the observed one,which is the cornerstone of other advanced image processing problems.However,the emergence of highdimensional data such as videos and hyperspectral images makes tensor research extremely important.Low-rank tensor completion is an important field of tensor research,the main task is to restore the lost tensor information based on some existing observation data,and it has been applied in the fields of artificial intelligence and machine learning.This paper mainly focuses on image restoration and low-rank tensor completion.The main work is as follows:Firstly,the image deblurring with mixed Gaussian impulse noise problem is studied.We mainly consider how to choose the spatially adapted parameters in the image model,compared with the scalar parameters,the spatially adapted parameters can better control the smoothness required by the image at different scales.Chapter 3 proposes a joint optimization model,which can simultaneously estimate the spatially adapted parameters and the restored image.To overcome the staircase effect brought by total variation regularization,hybrid total variation is introduced to obtain a better recovery effect.For this model,a block coordinate descent algorithm is proposed and the alternating direction method of multiplier is used to solve subproblems related to the image.In addition,the existence of the solution of the proposed model and the convergence of the algorithm are theoretically proved.The relevant numerical results show that the proposed method is superior to some existing algorithms.Secondly,according to the statistical characteristics of the mixed Gaussian impulse noise,Chapter 4 proposes a multi-scale l1-l2-TV model.By utilizing this model and related local constraint problem,an update algorithm for the spatially adapted parameters is given and solve the discrete multi-scale l1-l2-TV optimization problem by the inexact alternating direction method.The existence and uniqueness of the solution of local constraint problem and the convergence of algorithms are theoretically investigated.The related numerical experiments verify the effectiveness of the proposed algorithm,and when the impulse noise level is high,this method can better restore images than the joint optimization algorithm.Secondly,the impulse noise image deblurring problem is studied.To obtain a better regularization effect,we construct a l2,p(0<p<1)regularized total variation with overlapping group sparsity prior,which can not only overcome the staircase effect brought by the total variation but also promote the sparsity at the group level.To solve the constructed model,a proximal alternate minimization algorithm is proposed,which can transform the original complex optimization model into several subproblems that are relatively easy to solve,and for the constituted l2-l2,p subproblem in the algorithm framework,the half-quadratic technique and the alternating direction method are adopted.In addition,the convergence guarantee of the algorithm is theoretically given.The relevant numerical experiments verify the feasibility and effectiveness of the method.Finally,the low-rank tensor completion problem is studied.To better describe the low-rank structure of tensors,we construct a partial sum of multi-directions tensor nuclear norm,which is a combination of the weighted sum of the tensor nuclear norm and the partial sum of the tensor nuclear norm,not only can handle the correlation between different models of high-order tensors but also can better approximate the rank of tensor.At the same time,to enhance the piecewise smoothness of high-dimensional images in the spatial dimension,total variation regularization is introduced while considering the low-rank structure of the tensor.The alternating direction method of multiplier is used to solve the model and the subproblems obtained by this method have closed-form solutions.The relevant numerical experiments show that the proposed method has a good filling effect.