Research For Optimization Model And Algorithm Of Matrix Completion Problem

In real life,the lack of image pixels will seriously affect the following image processing work,such as image segmentation,feature extraction and target detection.In order to meet the needs of higher-level image processing and improve the quality of image,image completion becomes a very important work in image preprocessing.The digital image can be represented by a matrix,so the image completion problem can be transformed into ma-trix completion problem.The continuous development of low-rank minimization theory has opened up a new research field for matrix completion.In this paper,we study two opti-mization models of matrix completion problem and their related algorithms,as well as their application in the field of image restoration.The work of the thesis is as follows:The first part of the thesis is introduction.It introduced the background and practi-cal significance of matrix completion problem,and the related researches on models and algorithms of existing matrix completion problems are introduced.In the second part of the thesis,in order to solve the imprecise rank problem of the nuclear norm in the general algorithm of matrix completion model,combining the advan-tages of capped l1 norm and nuclear norm,we proposed to use the capped nuclear norm to approximate rank,which improves the flexibility of nuclear norm,so as to make better use of the low rank property in matrix completion.Then,a matrix completion model based on rank minimization theory is constructed.In the model solution,a matrix completion algorithm based on DC algorithm framework is designed.Finally,a numerical examples are given to verify the performance of the algorithm in terms of the running time and relative error of the large-scale low rank matrix in synthetic data and real image restoration.The results show that the algorithm is superior to the traditional matrix completion algorithm.In the third part of the thesis,we study the matrix completion model with rank and linear constraints,and sequence convex approximation(SCA)algorithm is proposed to solve the model,by using the DC decomposition of nonconvex functions,the DC decomposition forms of three nonconvex approximations of rank functions are given:logarithm determinant function,capped nuclear norm and exponential penalty function,and we use SCA algorithm to solve the model.Finally,the SCA algorithm of three nonconvex approximation functions based on the above rank function is applied to image restoration.Through the numerical experiments on the completion problem of random generated matrix and real matrix,the performance of the three functions under the SCA algorithm is compared in terms of relative error and success rate.At the same time,it is compared with the existing ADMM algorithm based on truncated nuclear norm and the algorithm proposed in the second part of the paper The results show that the algorithm proposed in this part has higher completion accuracy and good performance of PSNR index,and has a certain application prospectFinally,we make a summary and put forward the further research direction of relevant issues.