Research On Low-rank Matrix Recovery Methods And Its Application

With the increasing popularity of the internet,rapid development of mobile communication,and emergence of various multimedia services,high-dimensional data with more complex structures are becoming very ubiquitous across many areas of science and engineering,including pattern recognition,machine learning,data mining and computer vision.Besides,these data often suffer from the problem of deficiency,loss,or corrupted with noise or outliers.How to make use of the sparsity and redundancy of high-dimensional data and to recover the original data correctly and efficiently has been the focus of intensive research of signal and image processing fields,and is also the main research content of this thesis.Compressed sensing based on convex optimization and related matrix rank minimization and low-rank matrix recovery are important tools of high-dimensional data analysis.This thesis reviews the fundamental theories about compressed sensing,matrix rank minimization,low-rank matrix recovery,and structure sparse matrix convex optimization algorithms,which are applicable to large-scale problems.The existing matrix recovery algorithms use a l1-norm to constrain the sparse matrix.The l1-norm assumes that each pixel is independently corrupted,but the distributions of sparse outliers in real scenes are often not only pixel-wised sparsity but also structured sparsity.Besides,in many applications,the rank function can not be approximated accurately.To relieve this issue,this thesis focuses on matrix decomposition based on low-rank approximation and structured sparse.Specially,based on the background video prior,this thesis presents an algorithm to approximate the rank.Afterwards,with the better target rank,this thesis further proposes an efficient iterative minimization method to solve the convex optimization problem with structured sparsity-inducing norm(l1,2-norm).This method could directly substrate background and foreground from the surveillance video sequence,overcoming the disadvantages of traditional background modeling.Finally,this thesis studys a novel inexact augmented Lagrange multiplier(IALM)algorithm,with faster convergence in the framework of augmented Lagrange multiplier method.The original IALM suffers from high computation cost of multiple singular value decompositions(SVDs).This thesis uses block Lanczos and warm starttechnique to compute the partial SVDs.Experiments on real video surveillance show that our approach is capable of achieving dramatically more accurate background recovery than other algorithms.These confirm that the new algorithm can solve practical background modeling problems more concisely and efficiently.