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Enhanced tensor RPCA and its Application

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2021年第6期43卷 2133-2140页

作者： Gao, Quanxue Zhang, Pu Xia, Wei Xie, Deyan Gao, Xinbo Tao, Dacheng Xidian Univ Integrated Serv Networks Xian 710071 Peoples R China Northwestern Polytech Univ Unmanned Syst Res Inst Xian 710069 Peoples R China Xidian Univ Sch Elect Engn Xian 710071 Peoples R China Chongqing Univ Posts & Telecommun Chongqing Key Lab Image Cognit Chongqing 400065 Peoples R China Univ Sydney Fac Engn & Informat Technol UBTECH Sydney Artificial Intelligence Ctr Darlington NSW 2008 Australia Univ Sydney Fac Engn & Informat Technol Sch Informat Technol Darlington NSW 2008 Australia

Despite the promising results, tensor robust principal component analysis (TRPCA), which aims to recover underlying low-rank structure of clean tensor data corrupted with noise/outliers by shrinking all singular values equally, cannot well preserve the salient content of image. The major reason is that, in real applications, there is a salient difference information between all singular values of a tensor image, and the larger singular values are generally associated with some salient parts in the image. Thus, the singular values should be treated differently. Inspired by this observation, we investigate whether there is a better alternative solution when using tensor rank minimization. In this paper, we develop an enhanced TRPCA (ETRPCA) which explicitly considers the salient difference information between singular values of tensor data by the weighted tensor Schatten p-norm minimization, and then propose an efficient algorithm, which has a good convergence, to solve ETRPCA. Extensive experimental results reveal that the proposed method ETRPCA is superior to several state-of-the-art variant RPCA methods in terms of performance.

关键词： tensor singular value decomposition robust principal component analysis multidimensional image denoising

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tensor discriminant analysis on grassmann manifold with application to video based human action recognition

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INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS 2024年第8期15卷 3353-3365页

作者： Ozdemir, Cagri Hoover, Randy C. Caudle, Kyle Braman, Karen South Dakota Mines Dept Comp Sci & Engn 501 E St Joseph St Rapid City SD 57701 USA South Dakota Mines Dept Math 501 E St Joseph St Rapid City SD 57701 USA

Representing videos as linear subspaces on Grassmann manifolds has made great strides in action recognition problems. Recent studies have explored the convenience of discriminant analysis by making use of Grassmann kernels. However, traditional methods rely on the matrix representation of videos based on the temporal dimension and suffer from not considering the two spatial dimensions. To overcome this problem, we keep the natural form of videos by representing video inputs as multidimensional arrays known as tensors and propose a tensor discriminant analysis approach on Grassmannian manifolds. Because matrix algebra does not handle tensor data, we introduce a new Grassmann projection kernel based on the tensor-tensor decomposition and product. Experiments with human action databases show that the proposed method performs well compared with the state-of-the-art algorithms.

关键词： Grassmann Discriminant Analysis Action recognition tensor singular value decomposition tensor eigendecomposition

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Generalized Visual Information Analysis Via tensorial Algebra

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JOURNAL OF MATHEMATICAL IMAGING AND VISION 2020年第4期62卷 560-584页

作者： Liao, Liang Maybank, Stephen John Zhongyuan Univ Technol Sch Elect & Informat Zhengzhou 450000 Peoples R China Univ London Birkbeck Coll London WC1E 7HX England

High-order data are modeled using matrices whose entries are numerical arrays of a fixed size. These arrays, called t-scalars, form a commutative ring under the convolution product. Matrices with elements in the ring of t-scalars are referred to as t-matrices. The t-matrices can be scaled, added and multiplied in the usual way. There are t-matrix generalizations of positive matrices, orthogonal matrices and Hermitian symmetric matrices. With the t-matrix model, it is possible to generalize many well-known matrix algorithms. In particular, the t-matrices are used to generalize the singular value decomposition (SVD), high-order SVD (HOSVD), principal component analysis (PCA), two-dimensional PCA (2DPCA) and Grassmannian component analysis (GCA). The generalized t-matrix algorithms, namely TSVD, THOSVD, TPCA, T2DPCA and TGCA, are applied to low-rank approximation, reconstruction and supervised classification of images. Experiments show that the t-matrix algorithms compare favorably with standard matrix algorithms.

关键词： Commutative ring Grassmannian manifold Image classification tensor singular value decomposition tensors

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A Fast tensor Completion Method Based on tensor QR decomposition and tensor Nuclear Norm Minimization

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IEEE TRANSACTIONS ON COMPUTATIONAL IMAGING 2021年 7卷 1267-1277页

作者： Wu, Fengsheng Li, Yaotang Li, Chaoqian Wu, Ying Yunnan Univ Sch Math & Stat Kunming 650091 Yunnan Peoples R China

Currently, the tensor completion problem has been paid high attention in the machine learning, especially in the field of computer vision and image processing. The low-rank tensor completion methods based on the tensor singular value decomposition and the tensor nuclear norm minimization has been proposed. However, they have limitations in computing speed, since they are SVD-based methods and need high computational cost for high dimensional tensor. In this paper, based on the tensor QR decomposition and the tensor nuclear norm, a fast low-rank tensor completion method is proposed. By reducing the dimensions of the tensor in the nuclear norm regularization term, the performance of the completion is substantially improved. Numerical experiments for color images, MRI and videos demonstrate that the effectiveness of the proposed method.

关键词： tensor completion tensor singular value decomposition tensor Qatar Riyal decomposition tensor nuclear norm

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tensor robust principal component analysis with total generalized variation for high-dimensional data recovery

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APPLIED MATHEMATICS AND COMPUTATION 2024年 483卷

作者： Xu, Zhi Yang, Jing-Hua Wang, Chuan-long Wang, Fusheng Yan, Xi-hong Taiyuan Normal Univ Sch Math & Stat Jinzhong 030619 Shanxi Peoples R China Taiyuan Normal Univ Shanxi Key Lab Intelligent Optimizat Comp & Blockc Jinzhong 030619 Shanxi Peoples R China Southwest Jiaotong Univ Sch Informat Sci & Technol Chengdu 611756 Sichuan Peoples R China

In the past few years, tensor robust principal component analysis (TRPCA) which is based on tensor singular value decomposition (t-SVD) has got a lot of attention in recovering low-rank tensor corrupted by sparse noise. However, most TRPCA methods only consider the global structure of the image, ignoring the local details and sharp edge information of the image, resulting in the unsatisfactory restoration results. In this paper, to fully preserve the local details and edge information of the image, we propose a new TRPCA method by introducing a total generalized variation (TGV) regularization. The proposed method can simultaneously explore the global and local prior information of high-dimensional data. Specifically, the tensor nuclear norm (TNN) is employed to develop the global structure feature. Moreover, we introduce the TGV, a higher-order generalization of total variation (TV), to preserve the local details and edges of the underlying image. Subsequently, the alternating direction method of multiplier (ADMM) algorithm is introduced to solve the proposed model. Sufficient experiments on color images and videos have demonstrated that our method is superior to other comparison methods.

关键词： tensor robust principal component analysis Total generalized variation tensor singular value decomposition tensor nuclear norm The alternating direction method of multiplier algorithm

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Low-Rank tensor Completion for Image and Video Recovery via Capped Nuclear Norm

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IEEE ACCESS 2019年 7卷 112142-112153页

作者： Chen, Xi Li, Jie Song, Yun Li, Feng Chen, Jianjun Yang, Kun Changsha Univ Sci & Technol Hunan Prov Key Lab Intelligent Proc Big Data Tran Changsha 410114 Hunan Peoples R China Changsha Univ Sci & Technol Sch Comp & Commun Engn Changsha 410114 Hunan Peoples R China Chinese Acad Sci Inst Informat Engn Natl Engn Lab Informat Secur Technol Beijing Peoples R China Univ Chinese Acad Sci Sch Cyber Secur Beijing 100093 Peoples R China Univ Essex Sch Comp Sci & Elect Engn Colchester CO4 3SQ Essex England

Inspired by the robustness and efficiency of the capped nuclear norm, in this paper, we apply it to 3D tensor applications and propose a novel low-rank tensor completion method via tensor singular value decomposition (t-SVD) for image and video recovery. The proposed tensor capped nuclear norm model (TCNN) handles corrupted low-rank tensors by sparsity enhancement via truncating its partial singular values dynamically. We also develop a simple and efficient algorithm to solve the proposed nonconvex and nonsmooth optimization problem using the Majorization-Minimization (MM) framework. Since the proposed algorithm admits a closed-form solution by optimizing a well-selected approximate version of the original objective function at each iteration, it is very efficient. Experimental results on both synthetic and real-world datasets, clearly demonstrate the superior performance of the proposed method.

关键词： Low-rank tensor completion tensor singular value decomposition capped nuclear norm visual data completion

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A fast correction approach to tensor robust principal component analysis

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APPLIED MATHEMATICAL MODELLING 2024年 128卷 195-219页

作者： Zhang, Zhechen Liu, Sanyang Lin, Zhiping Xue, Jize Liu, Lixia Xidian Univ Sch Math & Stat Xian 710126 Peoples R China Nanyang Technol Univ Elect & Elect Engn Singapore 639798 Singapore Xian Univ Posts & Telecommun Sch Commun & Informat Engn Xian 710121 Shaanxi Peoples R China

tensor robust principal component analysis (TRPCA) is a useful approach for obtaining low-rank data corrupted by noise or outliers. However, existing TRPCA methods face certain challenges when it comes to estimating the tensor rank and the sparsity accurately. The commonly used tensor nuclear norm (TNN) may lead to sub-optimal solutions due to the gap between TNN and the tensor rank. Additionally, the C1-norm is not an ideal estimation of the C0-norm, and solving TNN minimization can be computationally intensive because of the tensor singular value thresholding (t-SVT) scheme. To address these issues, a method called fast correction TNN (FCTNN) is proposed for TRPCA. In contrast to existing methods, FC-TNN introduces a correction term to bridge the gap between TNN and the tensor rank. Furthermore, a new correction term is employed for the C1-norm to achieve the desired solution. To improve computational efficiency, the Chebyshev polynomial approximation (CPA) technique is presented for computing t-SVT without requiring tensor singular value decomposition (t-SVD). The CPA technique is incorporated into the alternating direction method of multipliers (ADMM) algorithm to solve the proposed model effectively. Theoretical analysis demonstrates that FC-TNN offers a lower error bound compared to TNN under certain conditions. Extensive experiments conducted on various tensor-based datasets illustrate that the proposed method outperforms several state-ofthe-art methods.

关键词： TRPCA Low-rank tensor nuclear norm Correction tensor singular value decomposition Chebyshev polynomial approximation

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tensor total variation regularised low-rank approximation framework for video deraining

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IET IMAGE PROCESSING 2020年第14期14卷 3602-3612页

作者： Baiju, P. S. Jayan, P. Deepak George, Sudhish N. Natl Inst Technol Calicut Dept Elect & Commun Engn Calicut Kerala India

Outdoor monitoring systems are known to exhibit better performance under normal weather conditions, while it lacks effectiveness under inclement conditions. Often video footage captured by the camera under rainy conditions comprises several visual distortions. It eventually leads to flaws when handled with succeeding computer vision algorithms, namely the object identification and tracking. Additionally, eliminating such unpleasant rainy effects is essential prior to the processing of video footage by suitable algorithms. The present work attempts to formulate a new low-rank tensor recovery based deraining algorithm that enables to remove the rain streaks from video footage. The proposed method detects the rain streaks by adopting optical flow estimation along with the brightness features inherent with the rain streaks. A unified framework comprised of tensor singular value decomposition (t-SVD) based weighted nuclear norm minimisation and tensor total variation (TTV) regularisation effectively removes rain streaks and recovers the original rain-free data from the available rainy data. The use of t-SVD enforces the concept of low rankness and also exploits the temporal redundancy among the video frames. Furthermore, TTV regularisation facilitates to promote the temporal continuity for discriminating most of the natural image contents from sparse rain streaks by preserving piece-wise smoothness of video frames. Comprehensive experimental findings based on real and synthetic data with dynamic background show that the rain streaks are more efficaciously eliminated by adopting the proposed method without much loss in the information.

关键词： image motion analysis computer vision minimisation image segmentation tensors rain singular value decomposition image representation video signal processing image sequences tensor total variation low-rank approximation framework video deraining outdoor monitoring systems normal weather conditions inclement conditions video footage rainy conditions computer vision algorithms unpleasant rainy effects suitable algorithms low-rank tensor recovery tensor singular value decomposition original rain-free data low rankness video frames TTV regularisation facilitates sparse rain streaks

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CONTINUITY, DIFFERENTIABILITY AND SEMISMOOTHNESS OF GENERALIZED tensor FUNCTIONS

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JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 2021年第6期17卷 3525-3550页

作者： Li, Xia Wang, Yong Huang, Zheng-Hai Tianjin Univ Sch Math Tianjin 300350 Peoples R China

A large number of real-world problems can be transformed into mathematical problems by means of third-order real tensors. Recently, as an extension of the generalized matrix function, the generalized tensor function over the third-order real tensor space was introduced with the aid of a scalar function based on the T-pro duct for third-order tensors and the tensor singular value decomposition;and some useful algebraic properties of the function were investigated. In this paper, we show that the generalized tensor function can inherit a lot of good properties from the associated scalar function, including continuity, directional differentiability, Fre ' chet differentiability, Lipschitz continuity and semismoothness. These properties provide an important theoretical basis for the studies of various mathematical problems with generalized tensor functions, and particularly, for the studies of tensor optimization problems with generalized tensor functions.

关键词： T-pro duct tensor singular value decomposition generalized tensor function semismoothness differentiability

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Improved Robust tensor Principal Component Analysis via Low-Rank Core Matrix

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IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING 2018年第6期12卷 1378-1389页

作者： Liu, Yipeng Chen, Longxi Zhu, Ce Univ Elect Sci & Technol China Sch Informat & Commun Engn Chengdu 611731 Sichuan Peoples R China

Robust principal component analysis (RPCA) has been widely used for many data analysis problems in matrix data. Robust tensor principal component analysis (RTPCA) aims to extract the low rank and sparse components of multidimensional data, which is a generation of RPCA. The current RTPCAmethods are directly based on tensor singular value decomposition (t-SVD), which is a new tensor decomposition method similar to singular value decomposition (SVD) in matrices. These methods focus on utilizing different sparse constraints for real applications andmake less analysis for tensor nuclear norm (TNN) defined in t-SVD. However, we find low-rank structure still exists in the core tensor and existing methods can not fully extract the low-rank structure of tensor data. To further exploit the low-rank structures in multiway data, we extract low-rank component for the core matrix whose entries are from the diagonal elements of the core tensor. Based on this idea, we have defined a new TNN that extends TNN with core matrix and propose a creative algorithm to deal with RTPCA problems. The results of numerical experiments show that the proposed method outperforms state-of-the-art methods in terms of both accuracy and computational complexity.

关键词： Robust tensor principal component analysis tensor singular value decomposition low rank tensor approximation image denoising background modeling

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