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Decomposition into low-rank plus additive matrices for background/foreground separation: a review for a comparative evaluation with a large-scale dataset. (English) Zbl 1398.68572

Summary: Background/foreground separation is the first step in video surveillance system to detect moving objects. Recent research on problem formulations based on decomposition into low-rank plus sparse matrices shows a suitable framework to separate moving objects from the background. The most representative problem formulation is the Robust Principal Component Analysis (RPCA) solved via Principal Component Pursuit (PCP) which decomposes a data matrix into a low-rank matrix and a sparse matrix. However, similar robust implicit or explicit decompositions can be made in the following problem formulations: Robust Non-negative Matrix Factorization (RNMF), Robust Matrix Completion (RMC), Robust Subspace Recovery (RSR), Robust Subspace Tracking (RST) and Robust Low-Rank Minimization (RLRM). The main goal of these similar problem formulations is to obtain explicitly or implicitly a decomposition into low-rank matrix plus additive matrices. These formulation problems differ from the implicit or explicit decomposition, the loss function, the optimization problem and the solvers. As the problem formulation can be NP-hard in its original formulation, and it can be convex or not following the constraints and the loss functions used, the key challenges concern the design of efficient relaxed models and solvers which have to be with iterations as few as possible, and as efficient as possible. In the application of background/foreground separation, constraints inherent to the specificities of the background and the foreground as the temporal and spatial properties need to be taken into account in the design of the problem formulation. Practically, the background sequence is then modeled by a low-rank subspace that can gradually change over time, while the moving foreground objects constitute the correlated sparse outliers. Although, many efforts have been made to develop methods for the decomposition into low-rank plus additive matrices that perform visually well in foreground detection with reducing their computational cost, no algorithm today seems to emerge and to be able to simultaneously address all the key challenges that accompany real-world videos. This is due, in part, to the absence of a rigorous quantitative evaluation with synthetic and realistic large-scale dataset with accurate ground truth providing a balanced coverage of the range of challenges present in the real world. In this context, this work aims to initiate a rigorous and comprehensive review of the similar problem formulations in robust subspace learning and tracking based on decomposition into low-rank plus additive matrices for testing and ranking existing algorithms for background/foreground separation. For this, we first provide a preliminary review of the recent developments in the different problem formulations which allows us to define a unified view that we called Decomposition into Low-rank plus Additive Matrices (DLAM). Then, we examine carefully each method in each robust subspace learning/tracking frameworks with their decomposition, their loss functions, their optimization problem and their solvers. Furthermore, we investigate if incremental algorithms and real-time implementations can be achieved for background/foreground separation. Finally, experimental results on a large-scale dataset called Background Models Challenge (BMC 2012) show the comparative performance of 32 different robust subspace learning/tracking methods.

MSC:

68T45 Machine vision and scene understanding
62H25 Factor analysis and principal components; correspondence analysis
65F30 Other matrix algorithms (MSC2010)
68-02 Research exposition (monographs, survey articles) pertaining to computer science
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Full Text: DOI arXiv

References:

[1] Bouwmans, T.; El Baf, F.; Vachon, B., Background modeling using mixture of Gaussians for foreground detection — a survey, RPCS, 1, 3, 219-237 (2008)
[2] Bouwmans, T., Subspace learning for background modeling: A survey, RPCS, 2, 3, 223-234 (2009)
[3] Bouwmans, T., Recent advanced statistical background modeling for foreground detection: A systematic survey, RPCS, 4, 3, 147-176 (2011)
[4] Bouwmans, T., Traditional and recent approaches in background modeling for foreground detection: An overview, Comput. Sci. Rev., 11, 31-66 (2014) · Zbl 1296.68170
[5] Bouwmans, T.; Porikli, F.; Hoferlin, B.; Vacavant, A., Handbook on Background Modeling and Foreground Detection for Video Surveillance (2015), Chapman and Hall/CRC · Zbl 1298.68023
[6] Shah, M.; Deng, J.; Woodford, B., Video background modeling: Recent approaches, issues and our solutions, Mach. Vis. Appl., 25, 5, 1105-1119 (2014)
[7] Sobral, A.; Bouwmans, T., BGS Library: a library framework for algorithms evaluation in foreground/background segmentation, (Handbook on Background Modeling and Foreground Detection for Video Surveillance: Traditional and Recent Approaches, Implementations, Benchmarking and Evaluation, 23 June (2014))
[10] De La Torre, F.; Black, M., A framework for robust subspace learning, Int. J. Comput. Vis., 117-142 (2003) · Zbl 1076.68058
[11] Candes, E.; Li, X.; Ma, Y.; Wright, J., Robust principal component analysis?, Int. J. ACM, 58, 3 (2011) · Zbl 1327.62369
[13] Chandrasekaran, V.; Sanghavi, S.; Parrilo, P.; Willsky, A., Ranksparsity incoherence for matrix decomposition, SIAM J. Optim., 21 (2011) · Zbl 1226.90067
[20] Cai, J.; Candes, E.; Shen, Z., A singular value thresholding algorithm for matrix completion, Int. J. ACM (2008)
[21] Yuan, X.; Yang, J., Sparse and low-rank matrix decomposition via alternating direction methods, (Optimization Online (2009))
[25] Liu, R.; Lin, Z.; Wei, S.; Su, Z., Solving principal component pursuit in linear time via \(l_1\) filtering, Int. J. Comput. Vis., 2011 (2011)
[28] Goldfarb, D.; Ma, S.; Scheinberg, K., Fast alternating linearization methods for minimizing the sum of two convex function, Math. Program. A (2010), Preprint
[45] Jiang, H.; Deng, W.; Shen, Z., Surveillance video processing using compressive sensing, Inverse Probl. Imaging, 6, 2, 201-214 (2012) · Zbl 1242.94004
[50] Li, S., Compressed Sensing in Resource-constrained Environments: From Sensing Mechanism Design to Recovery Algorithms (2015), University of Tennessee: University of Tennessee Knoxville, (thesis)
[51] Kang, B.; Zhu, W.; Yan, J., Object detection oriented video reconstruction using compressed sensing, EURASIP J. Adv. Signal Process. Sample (2015)
[52] Kang, B.; Zhu, W., Robust moving object detection using compressed sensing, IET Image Process. (2015)
[57] Hsu, D.; Kakade, S.; Zhang, T., Robust matrix decomposition with sparse corruptions, IEEE Trans. Inform. Theory, 57, 11, 7221-7234 (2011) · Zbl 1365.15018
[68] Han, G.; Wang, J.; Cai, X., Background subtraction based on modified online robust principal component analysis, Int. J. Mach. Learn. Cybernet., 1-14 (2016)
[70] Grosek, J., Robust Real-time Image Processing through Dynamic Mode Decomposition (2013), University of Washington: University of Washington USA, (Ph.D. thesis)
[73] Grosek, J.; Fu, X.; Brunton, S.; Kutz, J., Dynamic mode decomposition for robust PCA with applications to foreground/background subtraction in video streams, (Handbook on Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing (2016), CRC Press)
[76] Ding, X.; He, L.; Carin, L., Bayesian robust principal component analysis, IEEE Trans. Image Process. (2011) · Zbl 1381.62144
[77] Babacan, S.; Luessi, M.; Molina, R.; Katsaggelos, A., Sparse bayesian methods for low-rank matrix estimation, IEEE Trans. Signal Process., 60, 8, 3964-3977 (2012) · Zbl 1393.94670
[82] Nakajima, S.; Sugiyama, M.; Babacan, D., Variational bayesian sparse additive matrix factorization, Mach. Learn., 92, 319-347 (2013) · Zbl 1273.62136
[83] Chen, Z.; Babacan, S.; Molina, R.; Katsaggelos, A., Variational Bayesian methods for multimedia problems, IEEE Trans. Multimedia (2014)
[84] Guan, N.; Tao, D.; Luo, Z.; Shawe-Taylor, J., MahNMF: manhattan non-negative matrix factorization, J. Mach. Learn. Res. (2012)
[85] Kumar, A.; Sindhwani, V.; Kambadur, P., Fast conical hull algorithms for near-separable non-negative matrix factorization, J. Mach. Learn. Res., 28, 1, 231-239 (2013)
[95] Sobral, A.; Zahzah, E., Matrix and tensor completion algorithms for background model initialization: A comparative evaluation (Scene Background Modeling and Initialization), Pattern Recognit. Lett. (2016), (special issue)
[96] Wang, X.; Zhang, Z.; Ma, Y.; Bai, X.; Liu, W.; Tu, Z., Robust subspace discovery via relaxed rank minimization, Neural Comput. (2013)
[103] He, J.; Zhang, D.; Balzano, L.; Tao, T., Iterative grassmannian optimization for robust image alignment, Image Vis. Comput. (2013)
[105] Hage, C.; Kleinsteuber, M., Robust PCA and subspace tracking from incomplete observations using \(l_0\)-surrogates, Optim. Control (2012)
[106] Seidel, F.; Hage, C.; Kleinsteuber, M., pROST — a smoothed Lp-norm robust online subspace tracking method for realtime background subtraction in video (Background Modeling for Foreground Detection in Real-World Dynamic Scenes), Mach. Vis. Appl. (2013), (special issue)
[109] Zhou, X.; Yang, C.; Yu, W., Moving object detection by detecting contiguous outliers in the low-rank representation, IEEE Trans. Pattern Anal. Mach. Intell., 35, 597-610 (2013)
[115] Guyon, C.; Bouwmans, T.; Zahzah, E., Robust principal component analysis for background subtraction: Systematic evaluation and comparative analysis, (INTECH, Principal Component Analysis, Book 1, Chapter 12 (2012)), 223-238
[116] Deng, Y.; Dai, Q.; Liu, R.; Zhang, Z.; Hu, S., Low-rank structure learning via nonconvex heuristic recovery, IEEE Trans. Neural Netw. Learn. Syst., 24, 3 (2013)
[118] Shen, J.; Xu, H.; Li, P., Online optimization for max-norm regularization, Adv. Neural Inform. Process. Syst., 1718-1726 (2014)
[122] Huan, G.; Li, Y.; Song, Z., A novel robust principal component analysis method for image and video processing, Appl. Math., 197-214 (2016) · Zbl 1389.62096
[126] Javed, S.; Mahmood, A.; Bouwmans, T.; Jung, S., Spatiotemporal Low-rank Modeling for Complex Scene Background Initialization, IEEE Trans. Circuit. Syst. Video Technol. (2017)
[127] Li, P.; Bu, J.; Yu, J.; Chen, C., Towards robust subspace recovery via sparsity-constrained latent low-rank representation, J. Visual Commun. Image Rep. (2015)
[130] Chouvardas, S.; Kopsinis, Y.; Theodoridis, S., Robust subspace tracking with missing entries: a settheoretic approach, IEEE Trans. Signal Process. (2015) · Zbl 1394.94133
[134] Zhao, Q.; Meng, D.; Xu, Z.; Zuo, W.; Yan, Y., \(l_1\)-norm low-rank matrix factorization by variational Bayesian method, IEEE Trans. Neural Netw. Learn. Syst., 26, 4, 825-839 (2015)
[135] Kim, E.; Oh, S., Robust orthogonal matrix factorization for efficient subspace learning, Neurocomputing (2015)
[139] Dou, J.; Li, J.; Qin, Q.; Tu, Z., Moving object detection based on incremental learning low rank representation and spatial constraint, Neurocomputing (2015)
[144] Bao, B.; Liu, G.; Xu, C.; Yan, S., Inductive robust principal component analysis, IEEE Trans. Image Process., 3794-3800 (2012) · Zbl 1381.62139
[146] Wang, J.; Wan, M.; Hu, X.; Yan, S., Image denoising with a unified schatten-\(p\) norm and \(l_q\) norm regularization, J. Optim. Theory Appl. (2014)
[147] Shao, W.; Ge, Q.; Gan, Z.; Deng, H.; Li, H., A generalized robust minimization framework for low-rank matrix recovery, Math. Prob. Eng. (2014) · Zbl 1407.90258
[150] Wen, J.; Xu, Y.; Tang, J.; Zhan, Y.; Lai, Z.; Guo, X., Joint video frame set division and low-rank decomposition for background subtraction, IEEE Trans. Circuit. Syst. Video Technol. (2014)
[151] Wang, S.; Feng, X., Optimization of the regularization in background and foreground modeling, J. Appl. Math. (2014) · Zbl 1474.68400
[152] He, R.; Tan, T.; Wang, L., Recovery of corrupted low-rank matrix by implicit regularizers, IEEE Trans. Pattern Anal. Mach. Intell. (2013)
[154] Sprechmann, P.; Bronstein, A.; Sapiro, G., Learning robust low-rank representations, Optim. Control (2012)
[156] Oreifej, O.; Li, X.; Shah, M., Simultaneous video stabilization and moving object detection in turbulence, IEEE Trans. Pattern Anal. Mach. Intell. (2012)
[158] Wang, S.; Feng, X.; Wang, W., Low-rank + dual model based dimensionality reduction, Neural Comput. (2015)
[162] Xin, B.; Kawahara, Y.; Wang, Y.; Hu, L.; Gao, W., Efficient generalized fused Lasso and its applications, ACM Trans. Intell. Syst. Technol., 7, 4 (2016)
[163] Han, L.; Bi, S.; Pan, S., Two-stage convex relaxation approach to least squares loss constrained low-rank plus sparsity optimization problems, Comput. Optim. Appl., 1-30 (2015)
[166] Hong, B.; We, L.; Hu, Y.; Cai, D.; He, X., Online robust principal component analysis via truncated nuclear norm regularization, Neurocomputing (2015)
[167] Zhang, Y.; Guo, J.; Zhao, J.; Wang, B., Robust principal component analysis via truncated nuclear norm minimization, J. Shanghai Jiaotong Univ., 21, 5, 576-583 (2016)
[168] Cao, F.; Chen, J.; Ye, H.; Zhao, J.; Zhou, Z., Recovering low-rank and sparse matrix based on the truncated nuclear norm, Neural Netw. (2016)
[169] Zhou, M.; Song, Z.; Han, N., Background subtraction based on low-rank approximation and structured sparsity, Signal Process.: Image Commun. (2016)
[171] Ye, X.; Yang, J.; Sun, X.; Li, K.; Hou, C.; Wang, Y., Foreground-background separation from video clips via motion-assisted matrix restoration, IEEE Trans. Circuit. Syst. Video Technol. (2015)
[172] Cao, X.; Yang, L.; Guo, X., Total variation regularized RPCA for irregularly moving object detection under dynamic background, IEEE Trans. Cybernet., 46, 4, 1014-1027 (2016)
[174] Ebadi, S. Erfanian; One, V. Guerra; Izquierdo, E., Approximated robust principal component analysis for improved general scene background subtraction, IEEE Trans. Image Process. (2015)
[178] Liu, X.; Zhao, G.; Yao, J.; Qi, C., Background subtraction based on low-rank model and structured sparse decomposition, IEEE Trans. Image Process. (2015) · Zbl 1408.94411
[180] Ebadi, S. Erfanian; Izquierdo, E., Foreground detection with dynamic tree-structured sparse RPCA, IEEE Trans. Pattern Anal. Mach. Intell. (2017)
[182] Brahma, P.; She, Y.; Li, S.; Wu, D., Reinforced robust principal component pursuit, IEEE Trans. Neural Netw. Learn. Syst. (2016)
[185] Chen, Z.; Molina, R.; Katsaggelos, A., A variational approach for sparse component estimation and low-rank matrix recovery, J. Commun., 8, 9 (2013)
[186] Chen, Z., Multidimensional Signal Processing for Sparse and Low-rank Problems (2014), Northwestern University: Northwestern University USA, (thesis)
[191] Yang, M.; An, Z., Video background modeling using low-rank matrix recovery, J. Nanjing Univ. Posts Telecommun. (2013)
[195] Chartrand, R., Non convex splitting for regularized low-rank and sparse decomposition, IEEE Trans. Signal Process. (2012)
[196] Zhang, H.; Liu, L., Recovering low-rank and sparse components of matrices for object detection, Electron. Lett., 49, 2 (2013)
[197] Zhu, W.; Shu, S.; Cheng, L., Proximity point algorithm for low-rank matrix recovery from sparse noise corrupted data, Appl. Math. Mech., 35, 2, 259-268 (2014) · Zbl 1283.49025
[198] Kim, E.; Lee, M.; Choi, C.; Kwak, N.; Oh, S., Efficient \(l_1\)-norm-based low-rank matrix approximations for large-scale problems using alternating rectified gradient method, IEEE Trans. Neural Netw. Learn. Syst. (2014)
[201] Liu, R.; Lin, Z.; Su, Z.; Gao, J., Linear time principal component pursuit and its extensions using \(l_1\) filtering, Neurocomputing (2014)
[203] Liu, Y.; Jiao, L.; Shang, F., An efficient matrix factorization based low-rank representation for subspace clustering, Pattern Recognit., 46, 284-292 (2013) · Zbl 1248.68411
[204] Liu, Y.; Jiao, L.; Shang, F., A fast tri-factorization method for low-rank matrix recovery and completion, Pattern Recognit., 46, 163-173 (2012) · Zbl 1248.68437
[205] Orabona, F.; Argyriou, A.; Srebro, N., PRISMA: PRoximal Iterative Smoothing Algorithm, Optim. Control (2012)
[207] Yang, M.; Wang, Y., Fast alternating direction method of multipliers for robust PCA, J. Nanjing Univ., 34, 2, 83-88 (2014)
[212] Tao, M.; Yuan, X., Recovering low-rank and sparse components of matrices from incomplete and noisy observations, SIAM J. Optim., 21, 1, 57-81 (2011) · Zbl 1218.90115
[214] Aybat, N.; Iyengar, G., An augmented Lagrangian method for conic convex programming, Math. Program. J. A (2012) · Zbl 1251.90303
[217] Aybat, N.; Iyengar, G., An alternating direction method with increasing penalty for stable principal component pursuit, Comput. Optim. Appl. (2014) · Zbl 1326.90061
[218] Hintermuller, M.; Wu, T., Robust principal component pursuit via inexact alternating minimization on matrix manifolds, J. Math. Imaging Vis. (2014)
[219] Hou, L.; He, H.; Yang, J., A partially parallel splitting method for multiple-block separable convex programming with applications to robust PCA, Comput. Optim. Appl. (2015)
[221] Goldstein, T.; Osher, S., The split Bregman method for \(l_1\)-regularized problems, SIAM J. Image Sci., 2, 2, 323-343 (2009) · Zbl 1177.65088
[222] Zou, H.; Hastie, T.; Tibshirani, T., Sparse principal component analysis, J. Comput. Graph. Statist., 15, 2, 265-286 (2006)
[223] Fazel, M., Matrix Rank Minimization with Applications (2002), Stanford University, (Ph.D. thesis)
[224] Lange, K.; Hunter, D.; Yang, I., Optimization transfer using surrogate objective functions, J. Comput. Graph. Statist., 9, 1-59 (2000)
[225] Robert, C.; Casella, G., (Monte Carlo Statistical Methods (2004), Springer: Springer New York) · Zbl 1096.62003
[226] Beal, M., Variational Algorithms for Approximate Bayesian Inference (2003), University of London, (Ph.D. thesis)
[227] Nesterov, Y., Smooth minimization of non-smooth functions, Math. Program., 103, 1, 127-152 (2004) · Zbl 1079.90102
[228] Boyd, S.; Parikh, N.; Chu, E.; Peleato, B.; Eckstein, J., Distribute optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3, 1, 1-123 (2011)
[229] Berg, E.; Friedlander, M., Probing the Pareto frontier for basis pursuit solutions, SIAM J. Sci. Comput., 31, 2, 890-912 (2008) · Zbl 1193.49033
[230] She, Y.; Owen, A., Outlier detection using nonconvex penalized regression, J. Amer. Stat. Assoc., 106, 494, 626-639 (2011) · Zbl 1232.62068
[231] Edelman, A.; Arias, T.; Smith, S., The geometry of algorithms with orthogonality constraints, IAM J. Matrix Anal. Appl., 20, 2, 303-353 (1998) · Zbl 0928.65050
[232] Mazumder, R.; Hastie, T.; Tibshirani, R., Spectral regularization algorithms for learning large incomplete matrices, J. Mach. Learn., 11, 2287-2322 (2010) · Zbl 1242.68237
[235] Huang, X.; Huang, P.; Cao, Y.; Yan, H., A block-sparse RPCA algorithm for moving object detection based on PCP, J. East China, Jiaotong Univ., 5, 30-36 (2013)
[237] Chen, C.; Li, S.; Qin, H.; Hao, A., Robust salient motion detection in non-stationary videos via novel integrated strategies of spatio-temporal coherency clues and low-rank analysis, Pattern Recognit. (2015)
[242] Peng, Y.; Ganesh, A.; Wright, J.; Xu, W.; Ma, Y., RASL: Robust Alignment by Sparse and Low-rank decomposition for linearly correlated images, IEEE Trans. Pattern Anal. Mach. Intell., 34, 11, 2233-2246 (2012)
[247] Xu, X., Online Robust Principal Component Analysis for Background Subtraction: A System Evaluation on Toyota Car Data (2014), University of Illinois: University of Illinois Urbana-Champaign, USA, (Master thesis)
[248] Chen, C.; Cai, J.; Lin, W.; Shi, G., Incremental low-rank and sparse decomposition for compressing videos captured by fixed cameras, J. Vis. Commun. Image Represent. (2014)
[253] Song, W.; Zhu, J.; Li, Y.; Chen, C., Image alignment by online robust PCA via stochastic gradient descent, IEEE Trans. Circuits Syst. Video Technol. (2015)
[256] Javed, S.; Oh, S.; Bouwmans, T.; Jung, S., Robust background subtraction to global illumination changes via multiple features based OR-PCA with MRF, J. Electron. Imaging (2015)
[258] Lee, H.; Lee, J., Online update techniques for projection based robust principal component analysis, ICT Exp. (2015)
[261] Stagliano, A.; Noceti, N.; Verri, A.; Odone, F., Online space-variant background modeling with sparse coding, IEEE Trans. Image Process. (2015) · Zbl 1408.94611
[263] Aharon, M.; Elad, M.; Bruckstein, A., The K-SVD: An algorithm for designing of overcomplete dictionaries for sparse representation, IEEE Trans. Signal Process., 24, 11, 4311-4322 (2006) · Zbl 1375.94040
[269] Sang, N.; Zhang, T.; Li, B.; Wu, X., Dictionary-based background subtraction, J. Huazhong Univ. Sci. Technol., 41, 9, 28-31 (2013)
[270] Zhao, C.; Wang, X.; Cham, W., Background subtraction via robust dictionary learning, EURASIP J. Image Video Process. (2011)
[272] Lu, C.; Shi, J.; Jia, J., Online robust dictionary learning, EURASIP J. Image Video Process. (2011)
[276] .Tropp, J.; Gilbert, A., Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inform. Theory, 53, 4655-4666 (2007) · Zbl 1288.94022
[277] Tibshirani, R., Regression shrinkage and selection via the Lasso, J. Roy. Statist. Soc., 58, 267-288 (1996) · Zbl 0850.62538
[278] Efron, B.; Hastie, T.; Johnstone, I.; Tibshirani, R., Least angle regression, Ann. Statist., 32, 2, 407-499 (2004) · Zbl 1091.62054
[279] Xiao, H.; Liu, Y.; Tan, S.; Duan, J.; Zhang, M., A noisy videos background subtraction algorithm based on dictionary learning, KSII Trans. Internet Inform. Syst., 1946-1963 (2014)
[283] Dikmen, M., A Foreground Detection System for Automatic Surveillanc (2010), University of Illinois: University of Illinois Urbana-Champaign, (Ph.D. thesis)
[285] Xue, G.; Song, L.; Sun, J., Foreground estimation based on linear regression model with fused sparsity on outliers, IEEE Trans. Circuits Syst. Video Technol. (2013)
[289] He, J.; Gao, M.; Zhang, L.; Wu, H., Sparse signal recovery from fixed low-rank subspace via compressive measurement, Algorithms 2013, 6, 4, 871-882 (2008) · Zbl 07042192
[290] Li, J.; Wang, J.; Shen, W., Moving object detection in framework of compressive sampling, J. Syst. Eng. Electron., 5, 740-745 (2010)
[291] Wang, X.; Liu, F.; Ye, Z., Background modeling in compressed sensing scheme, ESEP 2011, 13, 4776-4783 (2011)
[292] Wang, Y.; Lu, Q.; Wang, D.; Liu, W., Compressive background modeling for foreground extraction, J. Electric. Comput. Eng. (2015)
[295] Chen, S.; Donoho, D.; Saunders, M., Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20, 33 (1998)
[297] Candes, E., Compressive sampling, Internat. Congress Math. (2006) · Zbl 1130.94013
[299] Xu, M.; Lu, J., K-cluster-valued compressive sensing for imaging, EURASIP J. Adv. Signal Process. (2011)
[301] Needell, D.; Tropp, J., CoSaMP: iterative signal recovery from incomplete and inaccurate samples, Appl. Comput. Harmon. Anal. (2008) · Zbl 1163.94003
[303] Huang, J.; Zhang, T.; Metaxas, D., Learning with structured sparsity, J. Mach. Learn., 12, 3371-3412 (2011) · Zbl 1280.68169
[304] Huang, J., Structured Sparsity: Theorems, Algorithms and Applications (2011), University of New Jersey, (Ph.D. thesis)
[308] Liu, X.; Yao, J.; Hong, X.; Huang, X.; Zhou, Z.; Qi, C.; Zhao, G., Background subtraction using spatio-temporal group sparsity recovery, IEEE Trans. Circuits Syst. Video Technol. (2017)
[310] Lee, K.; Bresler, Y., ADMIRA: atomic decomposition for minimum rank approximation, IEEE Trans. Inform. Theory, 55, 9, 4402-4416 (2010) · Zbl 1366.94112
[312] Bouwmans, T.; Aybat, N.; Zahzah, E., Handbook on Robust Low-rank and Sparse Matrix Decomposition: Applications in Image and Video Processing (2016), CRC Press, Taylor and Francis Group · Zbl 1339.68002
[313] Candes, E.; Soltanolkotabi, M., Discussion of latent variable graphical model selection via convex optimization, Ann. Statist., 40, 4 (2012) · Zbl 1288.62084
[315] Chandrasekaran, V.; Parillo, P.; Willsky, A., Latent variable graphical model selection via convex optimization, Ann. Statist., 40, 4, 1935-1967 (2012) · Zbl 1257.62061
[316] Ma, S.; Xue, L.; Zou, H., Alternating direction methods for latent variable Gaussian graphical model selection, Neural Comput., 25, 2172-2198 (2013) · Zbl 1418.62234
[319] Bhardwaj, A.; Raman, S., Robust PCA-based solution to image composition using augmented lagrange multiplier (alm), Visual Comput. (2015)
[323] Wan, T.; Zhu, C.; Qin, Z., Multifocus image fusion based on robust principal component analysis, Pattern Recognit. Lett., 34, 9, 1001-1008 (2013)
[324] Wright, J.; Yang, A.; Ganesh, A.; Sastry, S.; Ma, Y., Robust face recognition via sparse representation, IEEE Trans. Pattern Anal. Mach. Intell. (2009)
[325] Huang, S.; Ye, J.; Wang, T.; Jiang, L.; Wu, X.; Li, Y., Extracting refined low-rank features of robust PCA for human action recognition, Arab. J. Sci. Eng., 40, 2, 1427-1441 (2015)
[328] Zhao, M.; Jiao, L.; Ma, W.; Liu, H.; Yang, S., Classification and saliency detection by semi-supervised low-rank representation, Pattern Recognit. (2015)
[329] Chen, C.; Li, S.; Qin, H.; Hao, A., Robust salient motion detection in non-stationary videos via novel integrated strategies of spatio-temporal coherency clues and low-rank analysis, Pattern Recognit. (2015)
[335] Ji, H.; Huang, S.; Shen, Z.; Xu, Y., Robust video restoration by joint sparse and low rank matrix approximation, SIAM J. Imaging Sci., 4, 4, 1122-1142 (2011) · Zbl 1234.68451
[343] Wu, L.; Wang, Y.; Liu, Y.; Wang, Y., Robust structure from motion with affine camera via low-rank matrix recovery, China Inform. Sci., 56, 11, 1-10 (2015)
[345] Bouwmans, T.; Porikli, F.; Horferlin, B.; Vacavant, A., Handbook on Background Modeling and Foreground Detection for Video Surveillance (2014), CRC Press, Taylor and Francis Group
[350] Li, L.; Huang, W.; Gu, I.; Tian, Q., Statistical modeling of complex backgrounds for foreground object detection, IEEE Trans. Image Process., 1459-1472 (2004)
[351] Sheikh, Y.; Shah, M., Bayesian modeling of dynamic scenes for object detection, IEEE Trans. Pattern Anal. Mach. Intell., 27, 1778-1792 (2005)
[356] Yang, X.; Gao, X.; Tao, D.; Li, X.; Han, B.; Li, J., Shape-constrained sparse and low-rank decomposition for auroral substorm detection, IEEE Trans. Neural Netw. Learn. Syst. (2015)
[357] Zhang, F.; Yang, J.; Tai, Y.; Tang, J., Double nuclear norm-based matrix decomposition for occluded image recovery and background modeling, IEEE Trans. Image Process., 24, 6, 1956-1966 (2015) · Zbl 1408.94794
[358] Zhou, Z.; Jin, Z., Robust principal component analysis for image disocclusion and object detection, Neurocomputing (2016)
[361] Han, L.; Zhang, Q., Multi-stage convex relaxation method for low-rank and sparse matrix separation problem, Appl. Math. Comput., 284, 175-184 (2016) · Zbl 1410.65133
[362] Li, C.; Wang, X.; Zhang, L.; Tang, J.; Wu, H.; Lin, L., WELD: weighted low-rank decomposition for robust grayscale thermal foreground detection, IEEE Trans. Circuits Syst. Video Technol. (2016)
[363] Shi, J.; Zheng, X.; Yang, W., Regularized approach for incomplete robust component analysis and its application to background modeling, J. Computer Appl. (2016)
[368] Chan, T.; Yang, Y., Polar n-complex and n-bicomplex singular value decomposition and principal component pursuit, IEEE Trans. Signal Process. (2016) · Zbl 1414.94108
[370] Gandy, S.; Yamada, I., Convex optimization techniques for the efficient recovery of a sparsely corrupted low-rank matrix, J. Math-for-Ind., 2, 147-156 (2010) · Zbl 1208.90211
[372] Fan, R.; Wang, H.; Zhang, H., A new analysis of the iterative threshold algorithm for RPCA by primal-dual method, Adv. Mater. Res., 989-994 (2014)
[376] Gu, G.; He, B.; Yang, J., Inexact alternating direction based contraction methods for separable linearly constrained convex programming, J. Optim. Theory Appl. (2013)
[382] Gan, C.; Wang, Y.; Wang, X., Multi-feature robust principal component analysis for video moving object segmentation, J. Image Graph., 18, 9 (2013)
[383] Wang, X.; Wan, W., Motion segmentation via multi-task robust principal component analysis, J. Appl. Sci., Electron. Inform. Eng., 32, 5, 473-480 (2014)
[393] Lin, T.; Ma, S.; Zhang, S., On the sublinear convergence rate of multi-block ADMM, J. Oper. Res. Soc. China (2015) · Zbl 1323.90052
[394] Lin, T.; Ma, S.; Zhang, S., On the global linear convergence of the ADMM with multi-block variables, SIAM J. Optim. (2015)
[396] Sun, Y.; Tao, X.; Li, Y.; Lu, J., Robust two-dimensional principal component analysis: A structured sparsity regularized approach, IEEE Trans. Image Process. (2015)
[398] Qiu, C.; Vaswani, N.; Lois, B.; Hogben, L., Recursive robust pca or recursive sparse recovery in large but structured noise, IEEE Trans. Inform. Theory (2014) · Zbl 1360.94093
[406] Guo, H.; Qiu, C.; Vaswani, N., An online algorithm for separating sparse and low-dimensional signal sequences from their sum, IEEE Trans. Signal Process. (2014) · Zbl 1394.94216
[407] Rodriguez, P.; Wohlberg, B., Incremental principal component pursuit for video background modeling, Springer J. Math. Imaging Vis. (2015) · Zbl 1334.68284
[411] Hastie, T.; Tibshirani, R.; Friedman, J., (The Elements of Statistical Learning: Data Mining, Inference and Prediction (2009), Springer) · Zbl 1273.62005
[412] Biao, Y.; Lin, Z., Robust foreground detection using block based RPCA, Opt. - Int. J. Light Electron. Opt. (2015)
[416] Li, H.; Zhang, Y.; Wang, J.; Xu, Y.; Li, Y.; Pan, Z., Inequality-constrained RPCA for shadow removal and foreground detection, IEICE Trans. Inf. Syst., 98, 6, 1256-1259 (2015)
[417] Cheng, D.; Yang, J.; Wang, J.; D.; Liu, X., Double-noise-dual-problem approach to the augmented lagrange multiplier method for robust principal component analysis, Soft Comput. (2015)
[418] Huai, K.; Ni, M.; Ma, F.; Yu, Z., A customized proximal point algorithm for stable principal component pursuit with nonnegative constraint, J. Inequal. Appl., 148 (2015) · Zbl 1323.90067
[419] Mao, J.; Zhang, Z., A local convex method for rank-sparsity factorization, Pattern Recogn. Lett., 71, 31-37 (2016)
[420] He, H.; Han, D., A distributed Douglas-Rachford splitting method for multi-block convex minimization problems, Adv. Comput. Math., 42, 27-53 (2016) · Zbl 1332.90198
[421] Wang, J.; Song, W., An algorithm twisted from generalized ADMM for multi-block separable convex minimization models, J. Comput. Appl. Math. (2016)
[428] Kim, J.; He, Y.; Park, H., Algorithms for nonnegative matrix and tensor factorizations: A unified view based on block coordinate descent framework, J. Global Optim., 58, 2, 285-319 (2014) · Zbl 1321.90129
[429] Bian, X., Sparse and Low-rank Modeling on High Dimensional Data: A Geometric Perspective (2014), North Carolina State University: North Carolina State University USA, (Ph.D. thesis)
[430] Shu, X., Advanced Imaging via Multiplexed Sensing and Compressive Sensing (2013), University of Illinois at Urbana-Champaign, (Ph.D. thesis)
[432] Zhou, Z.; Jin, Z., 2DPCA-based motion detection framework with subspace update of background, IET Comput. Vision (2016)
[433] Boykov, Y.; Veksler, O.; Zabih, R., Fast approximate energy minimization via graph cuts, IEEE Trans. PAMI, 23, 11, 1222-1239 (2001)
[434] Kolmogorov, V.; Zabih, R., What energy functions can be minimized via grapgh cuts?, IEEE Trans. PAMI, 26, 2, 147-159 (2004)
[435] Golub, G.; Loan, C. Van, Matrix Computation (1989), Johns Hopkins University Press
[437] Kwak, N., Principal component analysis based on \(l_1\)-norm maximization, IEEE Trans. Pattern Anal. Mach. Intell., 30, 1672-1680 (2008)
[438] Sobral, A.; Bouwmans, T.; Zahzah, E., LRSLibrary: Low-Rank and Sparse tools for Background Modeling and Subtraction in Videos, (Handbook on Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing, vol. 1 (2016), CRC Press), (Chapter 18)
[440] He, R.; Tan, T.; Wang, L., Recovery of corrupted low-rank matrix by implicit regularizers, IEEE Trans. Pattern Anal. Mach. Intell. (2011)
[443] Shah, M.; Deng, J.; Woodford, B., Video background modeling: recent approaches, issues and our proposed techniques, Mach. Vis. Appl., 25, 5, 1105-1119 (2014)
[444] Narayana, M.; Learned-Miller, E., Background subtraction-separating the modeling and the inference, Mach. Vis. Appl. (2017)
[445] Javed, S.; Bouwmans, T.; Jung, S., Background subtraction based on minimum spanning tree low-rank learning, ACM Symp. Appl. Comput. (2017), submitted for publication
[450] Sun, Y.; Tao, X.; Li, Y.; Lu, J., Robust 2D principal component analysis: A structured sparsity regularized approach, IEEE Trans. Image Process., 2515-2526 (2015) · Zbl 1408.94624
[451] Yang, J.; Zhang, D.; Frangi, A.; Yang, J., Two-dimensional PCA: A new approach to appearance based face representation and recognition, IEEE Trans. Pattern Anal. Mach. Intell., 26, 131-137 (2004)
[453] Ye, J.; Janardan, R.; Li, Q., Two-dimensional linear discriminant analysis, Adv. Neural Inf. Process. Syst., 354-363 (2004)
[454] Zhang, D.; Zhou, Z., \((2 D)^2\) PCA:2-Directional 2-Dimensional PCA for efficient face representation and recognition, Neurocomputing, 39, 224-231 (2005)
[455] Ye, J., Generalized low rank approximations of matrices, Mach. Learn., 61, 167-191 (2005) · Zbl 1087.65043
[456] Ye, J., Generalized low rank approximations of matrices revisited, IEEE Trans. Neural Netw., 21, 621-632 (2010)
[457] Shi, J.; Yang, W.; Zheng, X., Robust generalized low rank approximations of matrices, PLoS One (2015)
[458] Shimada, A.; Nonaka, Y.; Nagahara, H.; Taniguchi, R., Video background modeling: Recent approaches, issues and our solutions, Mach. Vis. Appl., 25, 5, 1121-1131 (2014)
[459] Tan, H.; Cheng, B.; Feng, J.; Feng, G.; Wang, W.; Zhang, Y., Low-n-rank tensor recovery based on multi-linear augmented Lagrange multiplier method, Neurocomputing (2013)
[462] Li, L.; Wang, P.; Hu, Q.; Cai, S., Efficient background modeling based on sparse representation and outlier iterative removal, IEEE Trans. Circuits Syst. Video Technol. (2014)
[464] Zhang, Z.; Yan, S.; Zhao, M.; Li, F., Bilinear low-rank coding framework and extension for robust image recovery and feature representation, Knowl.-Based Syst. (2015)
[471] Javed, S.; Bouwmans, T.; Jung, S., SBMI-LTD: Stationary Background Model Initialization based on Low-rank Tensor Decomposition, ACM Symp. Appl. Comput. (2017), submitted for publication
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