Fast ℓ _{1}minimization algorithm for robust background subtraction
 Huaxin Xiao^{1}Email author,
 Yu Liu^{1} and
 Maojun Zhang^{1}
DOI: 10.1186/s1364001601505
© The Author(s) 2016
Received: 29 June 2016
Accepted: 30 November 2016
Published: 12 December 2016
Abstract
This paper proposes an approximative ℓ _{1}minimization algorithm with computationally efficient strategies to achieve realtime performance of sparse modelbased background subtraction. We use the conventional solutions of the ℓ _{1}minimization as a preprocessing step and convert the iterative optimization into simple linear addition and multiplication operations. We then implement a novel background subtraction method that compares the distribution of sparse coefficients between the current frame and the background model. The background model is formulated as a linear and sparse combination of atoms in a prelearned dictionary. The influence of dynamic background diminishes after the process of sparse projection, which enhances the robustness of the implementation. The results of qualitative and quantitative evaluations demonstrate the higher efficiency and effectiveness of the proposed approach compared with those of other competing methods.
Keywords
Approximative ℓ _{1}minimization Background subtraction Sparsity representation1 Introduction
Foreground or motion detection is a problem involving the segmentation of moving objects from a given image sequence or video surveillance. Because of its fundamental and pivotal role in the field of advanced computer vision, such as tracking, event analytics, and behavior recognition, foreground segmentation has drawn considerable attention over the past decades [1]. Generally, background subtraction (BGS) is an effective and efficient technique for addressing the issue of foreground segmentation. In this technique, some strategies are employed to establish or estimate a background model, and then the current frame is compared with the background model to segment the foreground objects. However, the scene typically includes other periodical or irregular motion (e.g., shaking trees and flowing water) arising from the nature of the captured video, which challenges the feasibility of BGS [2].
Various methods have been proposed to deal with the BGS problem, such as the statistical models: Gaussian mixture model (GMM) [3]. Framebased methods consider spatial configurations as a significant cue for background modeling, such as eigenbackground model [4]. In addition, a number of popular approaches have been developed that are not restricted to the above categories, such as artificial neural networks like selforganizing background subtraction (SOBS) [5] and local feature descriptors [2]. All of the abovementioned approaches and algorithms can be categorized as classic BGS methods that make overly restrictive assumptions on the background model.
In this paper, we propose a sparsebased BGS strategy that can be distinguished from the above classic methods owing to looser model assumptions. We employ a dictionary learning algorithm to train bases, which formulates the background modeling step as a sparse representation problem. The current image frame is then projected over this trained dictionary to obtain a corresponding coefficient. Different scene contents have different coefficients, reflecting the fact that the foreground does not lie on the same bases or subspaces spanned by the background. This condition is helpful in identifying changes in the scene by comparing the spanned coefficients. Given that dynamic texture and statistical noise are typically distributed through the entire space anisotropically, their influence on an actual signal will be obviously weakened after application of the sparse projection process. This characteristic enhances the robustness of the proposed method to corrupted signals and noisy scenes.
On the other hand, existing ℓ _{1}minimization (ℓ _{1}min) or sparse coding algorithms are not sufficiently fast for realtime implementation of BGS. Inspired by the theory of data separation of sparse representations [6], we simplify the ℓ _{1}min process and apply it as a preprocessing step. In the proposed approximative ℓ _{1}min algorithm, the test/observed signal is separated into a number of basic atoms. For each atom, the sparse coefficient is calculated by an existing ℓ _{1}min algorithm, which obtains a number of sparse coefficient vectors equivalent to the total number of atoms. The sparse coefficient of the atom is defined as the children sparse vector in this paper. We assume that any observed/test data can be linearly represented by these atoms. Consequently, the sparse coefficient of any test/observed signal can also be regarded as a linear combination of the children sparse vectors. Therefore, the ℓ _{1}min process is simplified into addition and multiplication operations.

1. A novel formulation of BGS is proposed. The proposed method regards the distribution of sparse coefficients rather than the sparse error as the criterion of foreground detection, where the existing sparsebased BGS directly utilizes the frames of scenes [8, 9] or learned frames [10] to construct the dictionary. A twostage sparse projection processing is employed to obtain precise detection results even with dynamic scenes.

2. A novel ℓ _{1}min algorithm is proposed for realtime BGS implementation. The existing ℓ _{1}min algorithms are computationally expensive for the proposed BGS framework. We therefore convert the iterative processing of an existing ℓ _{1}min algorithm into simple addition and multiplication operations, with minimal sacrifice to the accuracy.
2 Related work
2.1 ℓ _{1}min algorithms
where ∥α∥_{1} represents the sparse constraint and λ is a scalar weight.
In [11], P _{ λ } was regarded as a LASSO problem and solved by least angle regression [12]. Numerous methods have been subsequently proposed to solve the unconstrained problem P _{ λ }, such as the coordinatewise descent method [13], fixedpoint method [14], and Bregman iterative algorithm [15]. Presently, ℓ _{1}min algorithms for sparse model or CS have achieved remarkable breakthroughs with respect to recovered results and computational efficiency. However, these algorithms are not sufficiently fast for realtime implementation of BGS because optimization is conducted in an iterative manner. Hence, the motivation of the present study is the development of a specialized ℓ _{1}min algorithm for realtime sparsebased BGS.
2.2 Sparsebased BGS
Sparsebased BGS avoids modeling of the background with parametric or nonparametric models, which provides a substantial advantage. The only assumption made on the background is that any variation in its appearance can be captured by the sparse error. Cevher et al. [7] regarded BGS as a sparse approximation problem and obtained a lowdimensional compressed representation of the background. Huang et al. [8] added a prior of group sparsity clustering as a new constraint in the process of sparse recovery and extended CS theory to manage dynamic background scenes efficiently. However, the balance between the signal sparsity prior and group sparsity prior required control by parametric tuning. Sivalingam et al. [9] regarded the foreground as the ℓ _{1}min of the difference between the current frame and the estimated background model. Zhao et al. [10] proposed a robust dictionary learning algorithm that prunes the foreground objects out as outliers at the training step. Xue et al. [16] cast foreground detection as a fused Lasso problem with a fused sparsity constraint. Later, Xiao et al. [17] extended the assumptions of CS for BGS [7] by adding an assumption that the projection of the noise over the dictionary is irregular and random.
2.3 Low rankbased BGS
The lowrank model based BGS assumes that the background of a scene can be captured by a lowrank matrix while the foreground can be regarded as a sparse error [18]. Qiu and Vaswani [19] proposed a realtime principal components pursuit (PCP) algorithm to recover the low matrix. Subsequently, robust PCA (RPCA) [20] was proposed to pursue the lowrank representation by an iterative optimization approach. Cui et al. [21] utilized lowrank decomposition to obtain the background motion and group sparsity [8] by which the foreground was constrained. The DECOLOR [22] method incorporates the Markov random field prior to restrict the foreground model and domain transformations to address a moving background. A simple and fast incremental PCP (incPCP) [23] is proposed for video background modeling. In a most recent work [24], the authors estimated a dense motion field to facilitate the process of matrix restoration.
Subspace tracking also plays an important role in low rankbased BGS. He et al. [25] proposed an online subspace estimation algorithm GRASTA to separate the foreground and background in subsampled video. Seidel et al. [26] replaced the ℓ _{1}norm in RPCA with a smoothed ℓ _{ p }norm and presented a robust online subspace tracking algorithm based on alternating minimization on manifolds. Xu et al. [27] formulated the online estimation procedure as an approximate optimization process on a Grassmannian.
3 Proposed method
3.1 Proposed approximative ℓ _{1}min algorithm
where the projection γ _{ i } of y over e _{ i } is the pixel value of y at site i in the problem of image or video processing.
where β _{ i } is the sparse coefficient of e _{ i } and is defined as the children sparse vector. In this paper, we solve the problem \(P_{\lambda }^{\mathbf {e}}\) with the Bregman iterative algorithm [15]. For the same size signals, Eq. (4) only need to be solved one time.
For a given problem or application, once the size of the processing signal is decided, e _{ i } is also known. Then, we can presolve the children sparse vector β _{ i } in Eq. (4) by an existing ℓ _{1}min algorithm. The sparse solution α of a new signal y can be rapidly estimated by Eq. (5) where the weights γ _{ i } is the value of y at site i. The iterative process in existing ℓ _{1}min algorithms is replaced by simple addition and multiplication operations.
An important question remains concerning the numerical distance between the sparse solution of an existing ℓ _{1}min algorithm and the proposed algorithm. The distance is, in fact, acceptable for many applications that demand a compositive result (e.g., foreground detection or recognition), but not for applications that expect the highest quality result possible (e.g., image deblurring or denoising). If tolerable in a specific application, the proposed ℓ _{1}min algorithm can be used as an acceleration engine, which can dramatically improve the computational efficiency. The numerical error between the solution of an existing ℓ _{1}min and the proposed algorithm and the computational burden will be discussed in detail in Section 4.1.
3.2 Proposed sparsebased BGS
According to the sparse coefficients, we can pick up the patches that contain the foreground object. The selected patches of subsampled images correspond the same position of the original frames. For eliminating the inaccurate results caused by image patches, a secondstage of patch refinement is applied to the region determined in the first stage to obtain the final foreground detection.
3.2.1 Background model
where α _{ i } is the sparse coefficient and D is a prelearned and overcompleted dictionary.
Compared with traditional methods of obtaining bases such as wavelet and PCA, overcompleted dictionary learning does not emphasize the orthogonality of bases. Thus, its representation of the signal has better adaptability and flexibility. In this paper, the dictionary D is prelearned by the algorithm in [29] with a natural image training set. This paper constructs the training set with some images that contains nature scenes. The images for foreground detection do not include dictionary training set. The training images are separated as the same size as the patches P ^{ i }. We set the regularization parameter in [29] as 1.2/K where K×K is the size of P ^{ i }. In this paper, D is global and suitable for arbitrary scenes, which indicates that, once D is learned, it can be employed for any testing dataset.
where \(\gamma _{j}^{'i}\) are the projection coefficients of the current frame patch P ^{ i } over the basis e _{ j }.
3.2.2 Firststage foreground detection
where i represents the ith patch of I ^{′} and Δ _{1}(i) and Δ _{2}(i) are the differences in the distributions and values of the sparse coefficients between the current patch D α ^{′} and the background model D α in Eq. (9). Due to adoption of identity basis vectors as basis functions e _{ j }, \({\gamma _{j}^{i}}\) equals to the pixel value of the ith patch at site j.
Given that the distributions and values of the sparse coefficients reflect which subspace is expanded by the test frame, we can use these parameters to determine whether a monitored scene has moving content. Specifically, an unchanging image content tends to have identical distributions and corresponding values. In contrast, if a foreground object enters the scene and changes the content, it generates distinct distributions and values for the sparse coefficients.
where μ _{1} and μ _{2} are the unitary parameters that determine the respective weights of Δ _{1}(i) and Δ _{2}(i). Because the ℓ _{1}norm, or least absolute deviation, can better represent the distribution of the sparse coefficient and ensure a more distinguishable difference, μ _{1} is set to a relatively large value (0.60–0.75) as the dominant weight, while μ _{2} is smaller (0.25–0.40).
3.2.3 Secondstage foreground detection
where neighbor(Δ) defines a neighborhood patch of the current sliding window, as shown by the black square on the righthand side of Fig. 4.
Equation (12) enhances the effect of segmentation because the question of whether a pixel belongs to a foreground object depends not only on its own intensity but also on the intensities of its neighborhood regions. As shown in Fig. 3 d, patchwise refinement based on firststage detection achieves far more precise results, where the resulting foreground outlines show good agreement with the ground truth results shown in Fig. 3 b.
3.2.4 Background update
where α _{ i } and α i′ are the sparse coefficients of background model \({P_{B}^{i}}\) and current image patch P ^{ i }, respectively, and ρ∈[0.2,0.5] is the learning rate.
4 Experimental results and discussion
To evaluate the performance of the proposed method, the experimental study was divided into two parts: one part tested the proposed approximative ℓ _{1}min algorithm and the other part tested the proposed BGS method. All experiments are performed using MATLAB on a laptop with a 2.50GHz Intel Core i74710MQ processor and 16 GB of memory.
4.1 Performance of the proposed approximative ℓ _{1}min algorithm
In the first experiment, we compared the performance of solving the problem P _{1} or P _{ λ } by eight ℓ _{1}min algorithms including gradient projection for sparse reconstruction (GPSR) [31], SPGL1Lasso [32], orthogonal matching pursuit (OMP) [33], subspace pursuit (SP) [34], DGS [8], the Bregman iterative algorithm [15], l1ls [35], and the proposed approximative ℓ _{1}min algorithm.
4.2 Performance of the proposed BGS algorithm
This section evaluates the performance of the proposed BGS method and is divided into two parts: qualitative and quantitative evaluation. All tested videos are 160×128. The dictionary sizes in the twostage foreground detection are 8×8 pixels with 256 atoms in the first stage and 3×3 pixels with 256 atoms in the second stage. We qualitatively and quantitatively compare the proposed method with classic BGS algorithms including SOBS [5], ViBe [36], and SuBS [2], as well as the sparse and lowrank model of Xiao et al. [17], DECOLOR [22], MAMR [24], RePROCS [37], and GOSUS [27]. For all algorithms, we adjusted parameters to obtain what appeared to be optimal results on the tested dataset.
4.2.1 Qualitative evaluation
Movement in captured scenes can be divided into two parts. One part represents the foreground, which is an independent object that has no relationship to the scene. The other part is periodical or irregular, such as rain, snow, waves, and moving trees, and should be classified as the background based on its relevance to the scene. Therefore, an ability to distinguish the two types of movement becomes an important criterion for motion detection. In this section, we conduct experiments on realimage sequences from the I2R dataset [30] and CDnet dataset [38].
SuBS [2] can handle the dynamic background well and generate robust detection results. Due to the postprocess in SuBS, the results seem to be overly smooth. Similarly, DECOLOR [22] method has the same problem because the single regularized parameter cannot adequately distinguish the lowrank part (background) from the sparse error part (foreground). The Fountain and Fountain02 sequences present another form of nonstationary background. The results of SOBS [5] and the proposed method manage these conditions well. However, the floating water leads to falsepositive results of Vibe [36], MAMR [24], and RePROCS [37]. Weather variations such as rain and snow, which can be regarded as an irregular background motion, are also a challenge for BGS. The Snow fall and Skating datasets reflect this situation. However, the lowrank model GOSUS [27] cannot detect the left person in Skating due to the falling snow. The proposed method effectively eliminates the influence of the dynamic textures, and accurately detect the foreground. More discussion about the models comparison is shown in the following section.
4.2.2 Quantitative evaluation
Here, tp is the number of pixels correctly classified as the foreground, whereas tp+fn and tp+fp are the number of pixels detected as foreground pixels by the ground truth and the proposed method, respectively. Therefore, Recall and Precision denote the percentage of detected true positives as compared to the total number of true positives in the ground truth and the total number of detected pixels in the proposed method. Because Recall and Precision conflict to each other, we employ the Fmeasure as the primary metric in the quantitative evaluation.
The quantitative Fmeasure metric (%) of the compared BGS methods on CDnet [38] datasets
Dataset  Classic methods  Lowrank methods  Sparse methods  

SOBS  ViBe  SuBS  DECO  MAMR  GOSUS  RePROCS  Xiao  Proposed  
Office  96.63  90.32  97.02  95.34  85.23  91.54  90.87  89.61  94.31 
PETS2006  85.26  84.32  85.12  79.13  77.63  78.21  79.33  75.16  86.16 
Fountain01  11.21  6.05  15.63  2.71  6.35  7.55  8.36  7.15  8.61 
Fountain02  85.81  63.38  84.69  75.36  77.65  70.23  67.93  78.38  83.44 
Parking  36.68  45.33  72.76  34.61  58.03  30.84  40.37  59.71  75.31 
Sofa  62.18  61.97  62.69  50.31  63.46  51.24  47.87  67.49  69.63 
Cubicle  72.05  79.65  79.33  77.67  69.38  71.23  69.34  70.91  73.64 
Copy Machine  57.21  81.71  89.74  78.18  70.68  79.22  74.32  69.61  75.61 
Park  59.70  69.53  58.69  75.81  70.96  72.33  66.98  70.14  74.73 
Dining Room  71.73  75.49  70.36  82.47  78.33  76.48  70.38  72.30  84.67 
Snow fall  67.01  82.49  85.03  83.46  82.36  81.84  76.70  73.41  85.27 
Skating  76.33  73.67  80.25  83.81  85.68  79.83  75.93  78.64  82.11 
Tram Crossroad  74.18  85.64  71.36  74.62  76.69  75.65  67.33  71.49  75.29 
Turnpike  78.98  90.64  89.67  88.37  85.44  85.45  79.43  79.61  86.82 
Winter Street  51.04  30.58  55.56  66.13  49.64  38.45  33.20  57.91  60.34 
Tram Station  71.32  70.69  72.91  69.54  70.03  72.98  68.26  61.41  76.17 
Turbulence0  2.64  5.36  8.65  38.34  35.67  37.94  33.62  29.34  40.34 
Turbulence3  74.96  65.48  80.28  77.67  81.36  68.29  59.33  79.56  87.64 
Average  63.05  64.57  69.98  68.53  68.03  64.96  61.64  66.21  73.33 
Average frames per second (FPS) of each method
Dataset  Classic methods  Lowrank methods  Sparse methods  

SOBS  ViBe  SuBS  DECO  MAMR  GOSUS  RePROCS  Xiao  Proposed  
FPS  28.5  31.9  48.7  2.3  3.6  0.54  0.78  2.6  29.3 
Platform  C++  C++  C++  MATLAB  MATLAB  MATLAB  MATLAB  MATLAB  MATLAB 
It is noted that the proposed BGS method obtained the best average Fmeasure compared to all other methods while SuBS [2] ranks second. Compared to the proposed method, SuBS [2] is sensitive to the Turbulence dataset due to the flow distortion. Besides, DECOLOR [22] has a good performance on Fmeasure while the frames per second (fps) processed by DECOLOR [22] (MATLAB implementation) is only 2.3. The proposed method can achieve 29.3 fps while this number of MAMR (MATLAB implementation) is about 3.6. This accelerated processing speed is possible because the proposed method replaces an iterative optimization by linear addition and multiplication operations. For the baseline category (Office and PETS2006 datasets), the performances of all methods considered are acceptable. For the Fountain01 dataset, all the methods failed because the fountain movement exceeds the background updating capabilities of the methods. In contrast, the movement of Fountain02 is smooth and continuous, and SOBS [5] and SuBS [2] both perform well. The proposed method demonstrates competitive results for the thermal and turbulence categories (Park, dining room, turbulence0 and turbulence3 datasets). This is because the datasets of these two categories present distinct irregular fluctuations similar to noise that cannot be formulated by a mathematical expression. The proposed method employs sparsity over a prelearned dictionary that can restrain this condition. The fps performance of lowrank methods such as RePROCS [37] and GOSUS is poor. This is because that the iterative pursuit of lowrank matrix or sparse matrix is timeconsuming. The proposed approximative ℓ _{1}min algorithm avoid the iterative process and employ the power of sparse representation.
5 Conclusions
Sparse and lowrank model based BGS applications and methods have received considerable attention. However, the iterative optimization process used to obtain sparse or lowrank solutions is computationally expensive. This paper proposed the approximative ℓ _{1}min algorithm to provide a level of computational efficiency unobtainable by previous sparse model based approaches. Moreover, the proposed approach employed the sparsity rather than the sparse error to detect the foreground, which has been proven effective and robust to dynamic and corrupted scenes.
However, this work is at a preliminary stage. For example, how the signal should be separated into basic atoms e _{ i } remains an open question, even though a satisfactory result can be obtained in separating the signal using the simplest method, as demonstrated in Eq. (3) by this work. Another future work is to measure the numerical differences of the sparse solution between the proposed ℓ _{1}min method and existing ℓ _{1}min algorithms. The difference is acceptable for motion detection, but this does not ensure it can be used for other applications. Thus, mathematically defining this difference is required to determine the potential of the proposed algorithm.
Declarations
Acknowledgements
This research was partially supported by National Natural Science Foundation (NSFC) of China under project No. 61403403 and No. 61402491.
Authors’ contributions
HX carried out the main part of this manuscript. YL participated in the design of the approximative ℓ _{1}min algorithm. MZ participated in the discussion. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 T Bouwmans, Traditional and recent approaches in background modeling for foreground detection: An overview. Comput. Sci. Rev. 11:, 31–66 (2014).View ArticleMATHGoogle Scholar
 P StCharles, G Bilodeau, R Bergevin, Subsense: a universal change detection method with local adaptive sensitivity. IEEE Trans. Image Process.24(1), 359–373 (2015).MathSciNetView ArticleGoogle Scholar
 C Stauffer, WEL Grimson, in Proceedings of the IEEE Comput. Vis. Pattern Recognit. (CVPR). Adaptive background mixture models for realtime tracking (IEEEFt. Collins, 1999), pp. 246–252.Google Scholar
 NM Oliver, B Rosario, AP Pentland, A Bayesian computer vision system for modeling human interactions. IEEE Trans. Pattern Anal. Mach. Intell.22(8), 831–843 (2000).View ArticleGoogle Scholar
 L Maddalena, A Petrosino, A selforganizing approach to background subtraction for visual surveillance applications. IEEE Trans. Image Process. 17(7), 1168–1177 (2008).MathSciNetView ArticleGoogle Scholar
 YC Eldar, G Kutyniok (eds.), Compressed Sensing: Theory and Applications (Cambridge University Press, Cambridge CB2 8RU, 2012).
 V Cevher, A Sankaranarayanan, MF Duarte, D Reddy, RG Baraniuk, R Chellappa, in Proceedings of the European Conf. Comput. Vis. (ECCV). Compressive sensing for background subtraction (SpringerMarseille, 2008), pp. 155–168.Google Scholar
 J Huang, X Huang, D Metaxas, in Proceedings of the IEEE Int. Conf. Comput. Vis. (ICCV). Learning with dynamic group sparsity (IEEEKyoto, 2009), pp. 64–71.Google Scholar
 R Sivalingam, D Alden, B Michael, M Roland, V Morellas, N Papanikolopoulos, in Proceedings of the IEEE Int. Conf. Rob. Autom. (ICRA). Dictionary learning for robust background modeling (IEEEShanghai, 2011), pp. 4234–4239.Google Scholar
 C Zao, X Wang, WK Cham, Background subtraction via robust dictionary learning. EURASIP J. Image Video Process, 1–12 (2011).
 M Osborne, B Presnell, B Turlanch, A new approach to variable selection in least squares problems. IMA J. Numer. Anal. 20(3), 389–404 (2000).MathSciNetView ArticleMATHGoogle Scholar
 B Efron, T Hastie, I Johnstone, R Tibshirani, Least angle regression. Ann. Stat. 32(2), 407–499 (2004).MathSciNetView ArticleMATHGoogle Scholar
 J Friedman, T Hastie, R Tibshirani, Pathwise coordinate optimization. Ann. Appl. Stat. 1(2), 302–332 (2007).MathSciNetView ArticleMATHGoogle Scholar
 E Hale, W Yin, Y Zhang, A fixedpoint continuation method for ℓ _{1} regularized minimization with applications to compressed sensing. CAAM TR0707, Rice University. 43:, 1–44 (2007).Google Scholar
 W Yin, S Osher, D Goldfarb, J Darbon, Bregman iterative algorithms for compressed sensing and related problems. SIMA J. Imag. Sci. 1(1), 143–168 (2008).MathSciNetView ArticleMATHGoogle Scholar
 G Xue, L Song, J Sun, Foreground estimation based on linear regression model with fused sparsity on outliers. IEEE Trans. Circ. Syst. Video Technol. 23(8), 1346–1357 (2014).View ArticleGoogle Scholar
 H Xiao, Y Liu, S Tan, J Duan, M Zhang, A noisy videos background subtraction algorithm based on dictionary learning. KSII Trans. Internet Inf. Syst. 8(6), 1946–1963 (2014).View ArticleGoogle Scholar
 T Bouwmans, E Zahzah, Robust PCA via principal component pursuit: a review for a comparative evaluation in video surveillance. Comp. Vision Image Underst. 122:, 22–34 (2014).View ArticleGoogle Scholar
 C Qiu, N Vaswani, in Proceedings of the IEEE Communication, Control, and Computing. Realtime robust principal components’ pursuit (IEEETamil Nadu, 2010), pp. 591–598.Google Scholar
 E Candès, X Li, Y Ma, J Wright, Robust principal component analysis?J. ACM. 58(3), 1–37 (2011).MathSciNetView ArticleMATHGoogle Scholar
 X Cui, J Huang, S Zhang, D Metaxas, in Proceedings of the European Conf. Comput. Vis. (ECCV). Background subtraction using low rank and group sparsity constraints (SpringerFirenze, 2012), pp. 612–625.Google Scholar
 X Zhou, C Yang, W Yu, Moving object detection by detecting contiguous outliers in the lowrank representation. IEEE Trans. Pattern Anal. Mach. Intell. 35(3), 597–610 (2013).View ArticleGoogle Scholar
 P Rodríguez, B Wohlberg, in Proceedings of the IEEE Image Processing. A Matlab implementation of a fast incremental principal component pursuit algorithm for video background modeling (IEEEParis, 2014), pp. 3414–3416.Google Scholar
 X Ye, J Yang, X Sun, K Li, C Hou, Y Wang, Foregroundbackground separation from video clips via motionassisted matrix restoration. IEEE Trans. Circ. Syst. Video Technol. 25(11), 1721–1734 (2015).View ArticleGoogle Scholar
 J He, L Balzano, A Szlam, in Proceedings of the IEEE Comput. Vis. Pattern Recognit. (CVPR). Incremental gradient on the Grassmannian for online foreground and background separation in subsampled video (IEEEBoston, 2012), pp. 1568–1575.Google Scholar
 F Seidel, C Hage, M Kleinsteuber, pROST—a smoothed Lpnorm robust online subspace tracking method for realtime background subtraction in video. Mach. Vis. Appl. 122:, 1–13 (2013).Google Scholar
 J Xu, V Ithapu, L Mukherjee, JM Rehg, V Singh, in Proceedings of the IEEE Int. Conf. Comput. Vis. (ICCV). Gosus: Grassmannian online subspace updates with structuredsparsity (IEEESydney, 2013), pp. 3376–3383.Google Scholar
 D Donoho, X Huo, Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory. 47(7), 2845–2862 (2001).MathSciNetView ArticleMATHGoogle Scholar
 J Mairal, F Bach, J Ponce, G Sapiro, Online learning for matrix factorization and sparse coding. J. Mach. Learn. Res. 11:, 19–60 (2010).MathSciNetMATHGoogle Scholar
 L Li, W Huang, IYH Gu, Q Tian, Statistical modeling of complex backgrounds for foreground object detection. IEEE Trans. Image Process. 13(11), 1459–1472 (2004).View ArticleGoogle Scholar
 M Figueiredo, R Nowak, S Wright, Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Sign. Process. 1(4), 586–597 (2007).View ArticleGoogle Scholar
 E Berg, M Friedlander, Sparse optimization with leastsquares constraints. SIAM J. Optim. 21(4), 1201–1229 (2011).MathSciNetView ArticleMATHGoogle Scholar
 J Tropp, A Gilbert, Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory. 53(12), 4655–4666 (2007).MathSciNetView ArticleMATHGoogle Scholar
 D Wei, M Olgica, Subspace pursuit for compressive sensing signal reconstruction. IEEE Trans. Inf. Theory. 55(5), 2230–2249 (2009).MathSciNetView ArticleGoogle Scholar
 SJ Kim, K Koh, M Lustig, S Boyd, D Gorinevsky, An interiorpoint method for largescale l1regularized least square. IEEE J. Sel. Top. Sign. Process. 1(4), 606–617 (2007).View ArticleGoogle Scholar
 O Barnich, MV Droogenbroeck, Vibe: A universal background subtraction algorithm for video sequences. IEEE Trans. Image Process. 20(6), 1709–1724 (2011).MathSciNetView ArticleGoogle Scholar
 H Guo, N Vaswani, C Qiu, in Proceedings of IEEE Global Signal and Information Processing. Practical ReProcs for separating sparse and lowdimensional signal sequences from their sum—part 2 (IEEEAtlanta, 2014), pp. 369–373.Google Scholar
 N Goyette, P Jodoin, F Porikli, J Konrad, P Ishwar, in Proceedings of the IEEE Comput. Vis. Pattern Recognit. Workshops (CVPRW). Changedetection.net: a new change detection benchmark dataset (IEEEBoston, 2012), pp. 1–8.Google Scholar