Principal Component Pursuit with Weighted Nuclear Norm

2014 ◽  
Vol 513-517 ◽  
pp. 1722-1726
Author(s):  
Qing Shan You ◽  
Qun Wan
Keyword(s):  
Sparse Matrix ◽  
Principal Component ◽  
Low Rank ◽  
Linear Measurements ◽  
Alternating Direction ◽  
Rank Matrix ◽  
Small Set ◽  
Principal Component Pursuit ◽  
Low Dimensional ◽  
Low Rank Matrix

Principal Component Pursuit (PCP) recovers low-dimensional structures from a small set of linear measurements, such as low rank matrix and sparse matrix. Pervious works mainly focus on exact recovery without additional noise. However, in many applications the observed measurements are corrupted by an additional white Gaussian noise (AWGN). In this paper, we model the recovered matrix the sum a low-rank matrix, a sparse matrix and an AWGN. We propose a weighted PCP for the recovery matrix, which is solved by alternating direction method. Numerical results show that the reconstructions performance of weighted PCP outperforms the classical PCP in term of accuracy.

scholarly journals Restricted Isometry Property of Principal Component Pursuit with Reduced Linear Measurements

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Qingshan You ◽  
Qun Wan ◽  
Haiwen Xu
Keyword(s):  
Sparse Matrix ◽  
Principal Component ◽  
Restricted Isometry Property ◽  
Low Rank ◽  
Linear Measurements ◽  
Matrix Recovery ◽  
Strongly Convex ◽  
Rank Matrix ◽  
Principal Component Pursuit ◽  
Low Rank Matrix Recovery

The principal component prsuit with reduced linear measurements (PCP_RLM) has gained great attention in applications, such as machine learning, video, and aligning multiple images. The recent research shows that strongly convex optimization for compressive principal component pursuit can guarantee the exact low-rank matrix recovery and sparse matrix recovery as well. In this paper, we prove that the operator of PCP_RLM satisfies restricted isometry property (RIP) with high probability. In addition, we derive the bound of parameters depending only on observed quantities based on RIP property, which will guide us how to choose suitable parameters in strongly convex programming.

scholarly journals Robust Principal Component Thermography for Defect Detection in Composites

Sensors ◽  
2021 ◽  
Vol 21 (8) ◽  
pp. 2682
Author(s):  
Samira Ebrahimi ◽  
Julien Fleuret ◽  
Matthieu Klein ◽  
Louis-Daniel Théroux ◽  
Marc Georges ◽  
...  
Keyword(s):  
Defect Detection ◽  
Sparse Matrix ◽  
Research Effort ◽  
Principal Component ◽  
Low Rank ◽  
Computational Time ◽  
Rank Matrix ◽  
Detection Capabilities ◽  
Signal Processing Methods ◽  
Low Rank Matrix

Pulsed Thermography (PT) data are usually affected by noise and as such most of the research effort in the last few years has been directed towards the development of advanced signal processing methods to improve defect detection. Among the numerous techniques that have been proposed, principal component thermography (PCT)—based on principal component analysis (PCA)—is one of the most effective in terms of defect contrast enhancement and data compression. However, it is well-known that PCA can be significantly affected in the presence of corrupted data (e.g., noise and outliers). Robust PCA (RPCA) has been recently proposed as an alternative statistical method that handles noisy data more properly by decomposing the input data into a low-rank matrix and a sparse matrix. We propose to process PT data by RPCA instead of PCA in order to improve defect detectability. The performance of the resulting approach, Robust Principal Component Thermography (RPCT)—based on RPCA, was evaluated with respect to PCT—based on PCA, using a CFRP sample containing artificially produced defects. We compared results quantitatively based on two metrics, Contrast-to-Noise Ratio (CNR), for defect detection capabilities, and the Jaccard similarity coefficient, for defect segmentation potential. CNR results were on average 40% higher for RPCT than for PCT, and the Jaccard index was slightly higher for RPCT (0.7395) than for PCT (0.7010). In terms of computational time, however, PCT was 11.5 times faster than RPCT. Further investigations are needed to assess RPCT performance on a wider range of materials and to optimize computational time.

A New Analysis of the Iterative Threshold Algorithm for RPCA by Primal-Dual Method

2014 ◽  
Vol 989-994 ◽  
pp. 2462-2466 ◽  
Author(s):  
Ru Ya Fan ◽  
Hong Xia Wang ◽  
Hui Zhang
Keyword(s):  
Gradient Method ◽  
Principal Component ◽  
Low Rank ◽  
Dual Method ◽  
Theoretical Derivation ◽  
Threshold Algorithm ◽  
Rank Matrix ◽  
Primal Dual ◽  
Low Rank Matrix ◽  
Iterative Threshold

This paper studies the iterative threshold algorithm (ITA) for solving the Robust Principal Component Analysis (RPCA) problems, which is to recover a low-rank matrix with a fraction of its entries being arbitrarily corrupted. By utilizing the primal-dual method, we analyze the ITA in a new way and prove that the ITA is essentially equivalent to gradient method applying to a dual problem. In the original ITA, it is hard to choose the parameters and hence it converges very slowly. Now, based on the new insight, existing techniques of the gradient method can be used to accelerate the ITA. We combine the theoretical derivation with the numerical simulation experiments to give an empirical guidance to set the parameters. As illustration, background modeling problem is solved by the ITA with optimal parameters.

Small Infrared Target Detection Based on Low-Rank and Sparse Matrix Decomposition

2012 ◽  
Vol 239-240 ◽  
pp. 214-218 ◽  
Author(s):  
Cheng Yong Zheng ◽  
Hong Li
Keyword(s):  
Target Detection ◽  
Sparse Matrix ◽  
Infrared Image ◽  
Matrix Decomposition ◽  
Low Rank ◽  
Target Component ◽  
Lagrange Method ◽  
Rank Matrix ◽  
Augmented Lagrange Method ◽  
Low Rank Matrix

Sparse and low-rank matrix decomposition (SLMD) tries to decompose a matrix into a low-rank matrix and a sparse matrix, it has recently attached much research interest and has good applications in many fields. An infrared image with small target usually has slowly transitional background, it can be seen as the sum of low-rank background component and sparse target component. So by SLMD, the sparse target component can be separated from the infrared image and then be used for small infrared target detection (SITD). The augmented Lagrange method, which is currently the most efficient algorithm used for solving SLMD, was applied in this paper for SITD, some parameters were analyzed and adjusted for SITD. Experimental results show our algorithm is fast and reliable.

scholarly journals Background modeling from video sequences via online motion-aware RPCA

2021 ◽  
pp. 29-29
Author(s):  
Xu Weiyao ◽  
Xia Ting ◽  
Jing Changqiang
Keyword(s):  
Principal Component ◽  
Background Modeling ◽  
Low Rank ◽  
Video Frame ◽  
Robust Principal Component Analysis ◽  
Online Application ◽  
Rank Matrix ◽  
Computer Vision Applications ◽  
Low Rank Matrix ◽  
Nuclear Norm Regularization

Background modeling of video frame sequences is a prerequisite for computer vision applications. Robust principal component analysis(RPCA), which aims to recover low rank matrix in applications of data mining and machine learning, has shown improved background modeling performance. Unfortunately, The traditional RPCA method considers the batch recovery of low rank matrix of all samples, which leads to higher storage cost. This paper proposes a novel online motion-aware RPCA algorithm, named OM-RPCAT, which adopt truncated nuclear norm regularization as an approximation method for of low rank constraint. And then, Two methods are employed to obtain the motion estimation matrix, the optical flow and the frame selection, which are merged into the data items to separate the foreground and background. Finally, an efficient alternating optimization algorithm is designed in an online manner. Experimental evaluations of challenging sequences demonstrate promising results over state-of-the-art methods in online application.

scholarly journals ROUTE: Robust Outlier Estimation for Low Rank Matrix Recovery

2017 ◽  
Author(s):  
Xiaojie Guo ◽  
Zhouchen Lin
Keyword(s):  
Real Data ◽  
Low Rank ◽  
Underlying Structure ◽  
The Arts ◽  
Rank Matrix ◽  
Low Dimensional ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix ◽  
Convergence And Optimality ◽  
Very High

In practice, even very high-dimensional data are typically sampled from low-dimensional subspaces but with intrusion of outliers and/or noises. Recovering the underlying structure and the pollution from the observations is key to understanding and processing such data. Besides properly modeling the low-rank structure of subspace, how to handle the pollution, is core regarding the performance of recovery. Often, the observed data is posed as a superimposition of the clean data and residual, while the residual can be roughly divided into two groups, including small dense noises and gross sparse outliers. Compared with small noises, outliers more likely ruin the recovery, as they can be arbitrarily large. By considering the above, this paper designs a method for recovering the low rank matrix with robust outlier estimation, termed as ROUTE, in a unified manner. Theoretical analysis on convergence and optimality, and experimental results on both synthetic and real data are provided to demonstrate the efficacy of our proposed method and show its superiority over other state-of-the-arts.

scholarly journals Guarantees of Fast Band Restricted Thresholding Algorithm for Low-Rank Matrix Recovery Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Fujun Zhao ◽  
Jigen Peng ◽  
Kai Sun ◽  
Angang Cui
Keyword(s):  
Combinatorial Problem ◽  
Low Rank ◽  
Linear Measurements ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
Restricted Isometry Constant ◽  
Wide Range ◽  
Thresholding Algorithm ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

Affine matrix rank minimization problem is a famous problem with a wide range of application backgrounds. This problem is a combinatorial problem and deemed to be NP-hard. In this paper, we propose a family of fast band restricted thresholding (FBRT) algorithms for low rank matrix recovery from a small number of linear measurements. Characterized via restricted isometry constant, we elaborate the theoretical guarantees in both noise-free and noisy cases. Two thresholding operators are discussed and numerical demonstrations show that FBRT algorithms have better performances than some state-of-the-art methods. Particularly, the running time of FBRT algorithms is much faster than the commonly singular value thresholding algorithms.

Research on Background Modeling Based on Low-Rank Matrix Recovery

2014 ◽  
Vol 635-637 ◽  
pp. 1056-1059 ◽  
Author(s):  
Bao Yan Wang ◽  
Xin Gang Wang
Keyword(s):  
Sparse Matrix ◽  
Background Modeling ◽  
Low Rank ◽  
Complex Scene ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
Training Samples ◽  
The Individual ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

Key and difficult points of background subtraction method lie in looking for an ideal background modeling under complex scene. Stacking the individual frames as columns of a big matrix, background parts can be viewed as a low-rank background matrix because of large similarity among individual frames, yet foreground parts can be viewed as a sparse matrix as foreground parts play a small role in individual frames. Thus the process of video background modeling is in fact a process of low-rank matrix recovery. Background modeling based on low-rank matrix recovery can separate foreground images from background at the same time without pre-training samples, besides, the approach is robust to illumination changes. However, there exist some shortcomings in background modeling based on low-rank matrix recovery by analyzing numerical experiments, which is developed from three aspects.

scholarly journals Multichannel Color Image Denoising via Weighted Schatten p-norm Minimization

2020 ◽  
Author(s):  
Xinjian Huang ◽  
Bo Du ◽  
Weiwei Liu
Keyword(s):  
Image Denoising ◽  
Color Image ◽  
Low Rank ◽  
Norm Minimization ◽  
Alternating Direction ◽  
Rank Matrix ◽  
Color Image Denoising ◽  
Non Local ◽  
Comparison Algorithms ◽  
Low Rank Matrix

The R, G and B channels of a color image generally have different noise statistical properties or noise strengths. It is thus problematic to apply grayscale image denoising algorithms to color image denoising. In this paper, based on the non-local self-similarity of an image and the different noise strength across each channel, we propose a MultiChannel Weighted Schatten p-Norm Minimization (MCWSNM) model for RGB color image denoising. More specifically, considering a small local RGB patch in a noisy image, we first find its nonlocal similar cubic patches in a search window with an appropriate size. These similar cubic patches are then vectorized and grouped to construct a noisy low-rank matrix, which can be recovered using the Schatten p-norm minimization framework. Moreover, a weight matrix is introduced to balance each channel’s contribution to the final denoising results. The proposed MCWSNM can be solved via the alternating direction method of multipliers. Convergence property of the proposed method are also theoretically analyzed . Experiments conducted on both synthetic and real noisy color image datasets demonstrate highly competitive denoising performance, outperforming comparison algorithms, including several methods based on neural networks.

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