Study On The Improvement And Application Of Low Rank Matrix Approximation Model

In the era of data explosion,many problems in the fields of signal processing,image processing,and pattern recognition require processing high dimensional data sets.However,as the data dimension increases,the data processing difficulty grows exponentially.Therefore,minimizing the natural correlation in high-dimensional data while reducing the loss of data information has become a hot topic at home and abroad.Fortunately,high dimensional data matrices acquired in the field of image processing and the like tend to be low rank or approximately low rank.This paper mainly studies the application of low rank matrix approximation model in the fields of hyperspectral image denoising,background foreground segmentation,and color image reconstruction.The main work of the thesis is as follows:Firstly,the necessity of the application of low rank matrix approximation model and the research status of the model are introduced.Two algorithms for solving the new model in this paper are introduced:Accelerated proximal gradient method,Inexact augmented Lagrangian multipliers method.At the same time,the corresponding algorithm framework is given.Secondly,for the problem that the nuclear norm is a biased estimate and the amount of computation is large,a non-convex approximation model based on matrix decomposition is proposed.The low rank constraint is realized by matrix decomposition,which greatly reduces the computational cost.At the same time,using two non-convex functions instead of the nuclear norm improves the robustness and flexibility of the model,and uses the augmented Lagrangian multiplier method to solve the corresponding non-convex optimization model.Experiments on different real data sets demonstrate the effectiveness of the improved model in terms of image sharpness and computational efficiency.Thirdly,the non-convex approximation model based on total variation is proposed for the problem that the nuclear norm may cause the rank estimation is too large and the recovered image is too smooth.The gamma norm is used instead of the nuclear norm as the non-convex approximation of the matrix rank function.At the same time,the total variation regular term is introduced to preserve the true edge information and detail information of the image itself,which avoids the image from being excessively smooth.Compared to the nuclear norm,the gamma norm reduces the contribution of large singular values while making the contribution of smaller singular values close to zero.The augmented Lagrangian multiplier method is used to solve the new model,and the effectiveness of the algorithm is verified by experiments.Finally,the main contents of the article are summarized,and further research directions are proposed for the problems existing in the new model.