The Algorithm Research Of Low-Rank Matrix Reconstruction For Image Restoration

With the development of science and technology, multimedia, computer networks and communication technology have close relationship with people's daily work and life.People often need to store, analyze and deal with larger scaled data, which has higher dimensional structure. Thus, how to restore the original image or data from them with noise or missing elements has become a hotspot in the fields of image processing,pattern recognition, computer vision, machine learning and data mining.In recent years, low-rank matrix reconstruction, as a traditional sparse representation in the case of low-rank matrix, has become a new tool of high-dimensional data analysis. It has been wildly concerned in mang field by researchers. The typic reconsturction models can be divied into two types. One is the low-rank sparse decomposition,which decomposes the observed data matrix as the sum of an approximately low-rank structure and the sparse error matrix when the observed data matrix itself has a low-rank structure or an approximate low-rank structure. The other is matrix completion, which achieves the missing element complement through its inherent low-rank structure when the observed data matrix contains missing elements.However, in practical applications,the above two types of models are still shortcomings, and should be further improved. Firstly, the traditional low-rank matrix reconstruction model is utilized only for matrix with lower rank. It is difficult to achieve the desired results when matrix with higher rank or more complex structure. Secondly,the traditional low-rank matrix reconstruction model is mainly used for single matrix.To solve practical problems, a large matrix often be constructed by pulling a lot of small matrices into the column vectors. And then it can be dealed by traditional low-rank matrix reconstruction model. However, it will result in high algorithm complexity for larger matrix. In addition, it is easy to destroy the original two-dimensional structure of the data. Finnaly, the current mainstream reconstruction algorithms adopt the iterative method, and each iteration process often contains the singular value decomposition of matrix. When matrix is large, the complexity of the algorithm is very high. To solve the above problems, we focus on low-rank matrix reconstruction for image restoration, such as model improvement, algorithm design and so on. The main work is summarized as follows:1. We proposed two low-rank and sparse matrix decomposition models and their algorithms for image restoration(1) Combining the TV norm to enhance the smoothness of the image or the local area, a reconstruction model of the structure smoothness and reweighted low-rank matrix decomposition is proposed. The model incorporates the reweighted low-rank matrix recovery model and the TV norm model. The former is used to capture column correlation,that is,low-rank structure. The later is used to enhance the local smoothness structure of the matrix. Like the traditional low-rank matrix restoration model, the l1 norm is used to measure the concentration of large sparse noise.Furthermore, in all iterations, each pixel value of the restored image must be constrainted by the interval of [0, 255]. However, in existing methods, this constraint is only applied to the final output of low-rank matrix analysis. Finally, in order to solve the optimization problem, an iterative algorithm based on the inexact Lagrangian multiplier method is proposed. Finally, the experimental results show that the proposed algorithm not only has a small reconstruction error, but also has good robustness to large sparse and stripe noise.(2) Inspired by the low-rank matrix completion algorithm based on smooth rank function, a new low-rank and sparse matrix decomposition method based on smooth and differentiable rank function and l0 norm function of matrices. Since the objective function is one smooth function, the classical gradient descent method can be applied to the optimization problem of the proposed model, and the convergence analysis of the algorithm is given theoretically. The experimental results show that the proposed low-rank and sparse decomposition algorithm based on smooth function has good performance. Especially for Gaussian noise, as well as Gaussian and salt and pepper mixed noise has better robustness.2. We proposed the robust generalized low-rank decomposition of multimatrices and its algorithm for images restorationThe generalized low-rank matrix approximation method focuses on the minimization of reconstruction errors, regardless of the influence of the rank of the approximate matrices. However, the rank of the approximate matrix has closely relationship with the details of the output image. In general, the higher the matrix with rank, the more detail it retains. Large sparse noise often corresponds to the detail feature of the image, to achieve the purpose of removing large sparse noise, the rank minimization of the approximate matrix should be considered in the optimization model of the generalized low-rank matrix approximation. Therefore, a new multi-matrices robust generalized low-rank sparse decomposition model is proposed. The model incorporates the matrix rank into the original generalized low-rank matrix model and extends the inexact Lagrangian multiplier to the solution of the problem. The experimental results show that the model has good robustness to large sparse noise.3. We proposed two matrix completion models and their algorithms for image restoration(1) A fast matrix completion algorithm based on smooth rank function is proposed to solve the problem that the existing low-rank matrix completion algorithms have long running time. Firstly, the three-factor approximation replaces the original matrix.Thus, matrix singular value decomposition can be done in the subspace of the projection, and the running time of the proposed algorithm is greatly reduced.Secondly, a rank adaptive fast algorithm for matrix completion is proposed by proper rank estimation. The experimental results show that compared with the traditional matrix completion algorithms,the proposed algorithm has a faster speed,and it has better filling precision in most cases.(2) In order to solve the problem that the rank should be given for matrix completion method based on atomic decomposition, we propose a rank adaptive atomic decomposition matrix completion algorithm. The method can effectively solve the problem by the double threshold. In addition, in order to improve the speed and precision of the algorithm, first, the algorithm adopts the lager step, then adopts smaller step iteration by iteration, until the iteration is terminated. The experimental results show that the proposed matrix completion algorithm can well estimate the rank of the matrix and has better reconstruction accuracy.