Research On Nonconvex Rank Approximations Based Low-rank Matrix Recovery Models And Algorithms For Image Processing

In recent years,the theories and applications of low-rank matrix recovery have attracted many researchers,and have obtained the outstanding achievements in data analysis,image processing and so on.How to exactly recover the low-rank structures contained in the high-dimensional data is a key problem in applying low-rank matrix recovery theories into the various fields.To recover low-rank matrix mainly depends on the low-rank constraint,and minimizing the rank function is a NP-hard problem owing to the fact that the rank function is discrete,although the traditional models adopt the nuclear norm as rank approximation,which is the minimum convex envelope of rank function,and have achieved success.However,in many applications,it is shown that the convex rank approximation based on the nuclear norm has obvious drawbacks in terms of approximating the true rank of the low-rank matrix.This thesis studies more accurate nonconvex rank approximations for the low-rank matirx,proposes according nonconvex optimization models for several image processing problems in different fields,then solves the associated models by appropriate algorithms,the main contributions are outlined as follows:1.The development of the theories of low-rank matrix recovery are overviewed.The drawbacks of using the nuclear norm to approximate the rank of low-rank matrix in the current low-rank matrix recovery models are concluded,for the problem that the nuclear norm is not an accurate rank approximation,how to construct the nonconvex rank function is demonstrated,and based on which,the modified low-rank matrix recovery models are proposed.2.For the reconstruction of dynamic magnetic resonance imaging(MRI)in the medical image processing field,a nonconvex rank approximation based optimization model for the MRI reconstruction is constructed,where two nonconvex rank approximations,Gamma norm and Laplace norm are used to surrogate the rank function of the low-rank matrix,instead of the nuclear norm in the current low-rank matrix recovery models.These two rank approximations are proved to have some satisfying properties and more accurately approximate the true rank.By relaxing the original model into the unconstrained problem,the alternating direction method(ADM)is utilized to solve all the subproblems.The convergence analysis guarantees the algorithm can obtain the local minimizer of the original problem.The experimental results on two MR images:cardiac perfusion and cardiac cine,show that the low-rank component can be recovered more quilkly and accurately owing to the more accurate rank approximations,at the same time,the obtained sparse component is more complete.Because the sparse matrix is related to the nidus of disease,by our proposed method in this chapter,the nidus of disease in the images can be seen more quickly and clearly.3.For the hyperspectral image denoising problem,a log-determinant function is used as the nonconvex rank approximation of the low-rank matrix in the low-rank matrix recovery model,at the same time,a low-rank matrix factorization strategy is exploited in this model,and then the associated HSI denoising model is formulated based on the nonconvex rank approximation and matrix factorization.More accurate nonconvex rank approximation improves the accuracy of the algorithm to obtain better low-rank matrix recovery,and the matrix factorization simplifies the computation for the singular value decomposition(SVD)of the low-rank matrix and the computational complexity is reduced greatly to improve the efficiency of the algorithm.The method of augmented Lagrangian multipliers(ALM)is utilized and all the variants are updated alternately,finally the optimal solutions for all the subproblems are obtained.Both the simulated and real data experiments show that the proposed algorithm has better denoising performance when compared with several state-of-the-art algorithms.The improvement is due to more accurate rank approximation used in our denoising model,which makes the recovered low-rank matrix from the original data more accurate,meaning that the noises in data can be removed more effectively for the reason that the low-rank matrix is related to clean data.An outstanding result is that our proposed algorithm can remove the dead lines and stripes excellently,covering the shortage of the other state-of-the-art methods.4.For the following image recognition problems,face recognition and motion segmentation,the subspace clustering method is adopted.In the general low-rank representation based subspace clustering model,two novel nonconvex rank approximations are proposed and used,based on exponential function and Gaussian distribution function,hence,a nonconvex rank approximation based subspace clustering model is constructed.These two rank approximation functions are the better rank approximations than the nuclear norm,and the advantages are more significantly especially when there exist large singular values.By integrating the proposed rank approximations with low-rank based models,the low-rank representation of the data is found to show the latent structures of the data more clearly.The converted model is solved by the method of ALM,and an affinity matrix is constructed by the solved coefficient matrix,which is finally used in the subspace clustering.This coefficient matrix is the forementioned low-rank reprensentation matrix,and to find which is to recover the low-rank matrix.The more accurate rank approximations make the recovered low-rank matrix better to help the following clustering.The experimental results show the effectiveness of the proposed algorithm,moreover,it has advantages over the state-of-the-art algorithms.Especially,the proposed algorithm is more applicable to the clustering problem with large number of objects in terms of face clustering problem,showing the robustness of the algorithm.