Theoretical Research Of Robust Principal Component Analysis Based On Rank Estimate And Its Applications

The rapid development of signal acquisition technology prompts the quantity and dimension of the sampled data.Robust Principal Component Analysis（RPCA）,due to its excellent performance for the low-rank structure of data,has been widely used in the field of signal processing to analyze the low-rank structure information of high-dimensional signals.It is noted that the existing RPCA techniques require a clear low-rank boundary as a prior.Nevertheless,the low-rank boundary in a specific scenario is unknown,which limits RPCA applications and results in an inaccurate low-rank matrix.To this end,a low-rank estimation methodology is proposed in this dissertation through the in-depth study of the matrix eigenvalue positioning set theory.Specifically,a novel RPCA model is proposed according to the proposed rank estimation algorithm.By applying the new RPCA model to the fields of one-dimensional signal processing and two-dimensional image processing,the universality of the proposed model is effectively verified.The main contributions of this dissertation are summarized as follows:1)Motivated by the problem that the low-rank is unknown due to the unclear positioning of the low-rank boundary in the current RPCA methods,a novel methodology is presented in this dissertation to determine the boundary between the low-rank eigenvalue positioning set and the sparse eigenvalue positioning set by studying the matrix eigenvalue positioning set theory.The decision rule of the boundary is established,and the accurate positioning of the low-rank boundary is found.In this dissertation,a pioneering algorithm for estimating the rank of a low-rank matrix is proposed,which achieves 95%accuracy with a corruption rate of 0.1.The rank estimation algorithm is further improved by introducing the unequal shrinkage transformation theory,and the accuracy of the rank estimation is 100%when the corruption rate is 0.1.2)To address the problem of information loss in solving the nuclear norm minimization andl₁norm minimization subproblems by the equivalent shrinkage in the current RPCA methods,this dissertation presents to re-weights the rank andl₁norm of the estimated low-rank matrix,on which an adaptive double-weighted RPCA model based on improved rank estimation.Through rigorous derivation,a closed-form solution is developed in the proposed optimization,which greatly simplifies the solving process.The proposed method can adaptively decompose the required low-rank matrix in unknown scenarios.The experiments on background modeling demonstrate that the proposed algorithm achieves the best performance assessed not only in visual effects,but also in objective metrics in terms of six evaluation indicators.Then,the proposed methods are verified by one-dimensional sound direction of arrival（DOA）estimation and two-dimensional image signal denoising.To deal with the mixed interference of Gaussian noise and reverberation,which yields the problem of low accuracy and low robustness in DOA estimation,a novel rank-estimation-based adaptive double-weighted RPCA is presented in this dissertation by introducing the ideas of conjugate reconstruction and random sample consistency.Experiments on the data with a signal-to-noise ratio of-15d B show that the DOA error by the proposed method is greatly reduced by 71%.Regarding image denoising,the traditional RPCA method may remove noise as well as the content structural information due to the lack of intensity constraint.In this dissertation,the rank-estimation-based adaptive double-weighted RPCA is employed by introducing the total variation regular term and the value range constraint.The proposed method not only successfully preserves the structural information in the original image but also effectively reduces the loss of image information.Experiments show that even if 40%of the original image is corrupted,the proposed method can recover the corrupted data,and achieves good visual effects and the best objective evaluation in comparison with the state-of-the-art methods.This dissertation offers a theoretical methodology as well as a powerful tool for applications.