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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

Thierry Bouwmans, Andrews Sobral, Sajid Javed, Soon Ki Jung, El-Hadi Zahzah

arXiv:1511.01245v3cs.CV

TL;DR

Rigorous large-scale quantitative evaluation of robust subspace methods for background/foreground separation has been limited. The paper reviews these formulations under a unified DLAM view and reports that fundamental DLAM models outperform MOG and KDE.

  • Problem

    Existing evaluations of robust subspace methods lacked rigorous quantitative testing on large-scale datasets with accurate ground truth and balanced real-world challenges.

    Source: §1.3 · p. 47

  • Method

    The paper reviews six robust subspace formulations under a unified DLAM framework and evaluates their suitability for background/foreground separation, including incremental and real-time implementation.

    Source: §11 · p. 116

  • Results

    Fundamental DLAM models outperform MOG and KDE, while methods designed specifically for background/foreground separation offer performance largely comparable to SubSENSE.

    Source: §11 · p. 117

  • Takeaways & Limitations

    DLAM offers a suitable framework for background/foreground separation, with robustness to illumination changes and dynamic backgrounds improvable through structured constraints.

    Source: §11 · p. 117

    A unified review of sparse-plus-additive decompositions (DSAM) is explicitly outside the paper’s scope.

    Source: §1.2.10 · p. 45

Abstract

from arXiv · show

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 in 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. 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.

1 Introduction

This section presents a unified survey of robust low-rank-plus-additive formulations for background/foreground separation, covering their organization, modeling choices, optimization methods, and solvers. It frames this review as a basis for comparative evaluation of methods in the application.

1 Introduction

  • Foreground detection separates moving objects from static background; RPCA models video as low-rank background plus sparse foreground, motivating the broader DLAM framework.The paper describes classical PCA-based background modeling and contrasts its limitations with robust PCA based on low-rank-plus-sparse decomposition.Source: §1 · pp. 2–3

1.1 Problem Formulations based on Decomposition into Low-rank plus Additive Matrices: A Preliminary Overview

  • The preliminary overview classifies implicit or explicit low-rank-plus-additive formulations as RPCA, RNMF, RSR, RST, RMC, and RLRM.Source: §1.1 · p. 4

1.1.1 Robust Principal Component Analysis (RPCA)

  • PCP [401][55] convexly separates low-rank background and sparse foreground and can perfectly recover them under minimal assumptions, but batch computation, lost frame structure, noise, missing entries, and localized objects limit deployment.The review describes variants targeting entry-wise noise, quantization error, block-sparse foreground, and background variation across low-dimensional subspaces.Source: §1.1.1 · pp. 4–5

1.1.2 Robust Non-negative Matrix Factorization (RNMF)

  • RNMF methods address nonnegative data and outliers: MahNMF has qualitative performance similar to IALM, RobustXray outperforms IALM in noise, and RANMF handles grouped outliers with controllable factor denseness.MahNMF uses Manhattan distance, while RobustXray uses l1 loss or Bregman-divergence approximations.Source: §1.1.2 · p. 8

1.1.3 Robust Matrix Completion (RMC)

  • RMC addresses partial observations with robust losses and factorizations; FRMC compensates global motion, while MAMC weights background reliability against slow objects and camouflage, with RMAMC adding noise robustness.Source: §1.1.3 · pp. 8–9

1.1.4 Robust Subpace Recovery (RSR)

  • RSR spans bi-sparse RoSuRe for camera jitter, efficient ROSL/ROSL+ recovery with quadratic/linear complexity and IALM-matched accuracy, and ROC-PCA for orthogonal outliers.Source: §1.1.4 · pp. 9–10

1.1.5 Robust Subspace Tracking (RST)

  • RST supports online processing of streaming frames: GRASTA tracks changing subspaces, t-GRASTA is four times faster with half the memory, and pROST improves on GRASTA for multimodal backgrounds.The review also describes structured-sparsity GOSUS for real-time updates and recursive-least-squares tracking for dynamic backgrounds.Source: §1.1.5 · p. 10

1.1.6 Robust Low Rank Minimization (RLRM)

  • RLRM includes DECOLOR’s joint outlier-support and background learning and LRMF-MoG’s broad noise modeling, while DLAM spans implicit or explicit two- or three-term decompositions and convex or nonconvex optimization.Source: §1.1.6 · pp. 11–12
  • For robust low-rank minimization, proxy losses approximate original losses to make problems solvable, and solvers may use strongly convex approximations to design efficient algorithms.Source: §1.1.6 · p. 12

1.2.1 Notations

  • The paper standardizes notation by using A for observations, L for the low-rank matrix, S for the residual or sparse matrix, and E for noise; m and n denote A’s columns and rows.For background/foreground separation, m is the pixels per frame and n is the frames in a sequence.Source: §1.2.1 · p. 19

1.2.2 Norms

  • The norm overview includes mixed l2,1 norms for spatially homogeneous fitting and column outliers, and the nuclear norm as the sum of singular values.Source: §1.2.2 · pp. 19–20

1.2.3 Loss Functions and Regularization Functions

  • The l1 loss is robust to outliers but may be unstable, non-unique, over-penalizing, and unable to handle collinearity, whereas the l2 loss is stable and yields one solution but is sensitive to outliers and missing data.Source: §1.2.3 · p. 21

1.2.4 Definition and Classifications of Outliers

  • Outliers include element-wise and row- or column-wise corruption, as well as missing data; PCP may handle observation-space outliers but fail on orthogonal-complement outliers.Missing data can arise from acquisition problems or moving objects in background initialization.Source: §1.2.4 · p. 24

1.2.5 Decomposition into Low-rank plus Additive Matrices

  • The reviewed formulations are unified as Decomposition into Low-rank plus Additive Matrices (DLAM).Source: §1.2.5 · p. 24

1.2.6 Decomposition Problem

  • The general decomposition is M_k = M_1 + M_2 + M_3 = L + S + E.Source: §1.2.6 · p. 24
  • Implicit decompositions use A ≈ L, explicit decompositions use A = L + S, and stable explicit decompositions use A = L + S + E to separate outliers from noise.In robust formulations, S may be a residual or a sparse matrix, depending on the framework.Source: §1.2.6 · pp. 25–26

1.2.7 Minimization problem

  • The minimization objectives constrain low rank and sparsity with rank and l0 losses or surrogates; these problems may be NP-hard or nonconvex, with difficult convergence guarantees.The RPCA formulation relaxes rank and l0 to the nuclear and l1 norms to obtain a tractable convex problem.Source: §1.2.7 · pp. 27–28

1.2.8 Algorithms for solving the optimization problem

  • Solvers fall into regularization-based and statistical-inference approaches, and their efficiency depends on iteration and SVD costs.Examples include SVT, APG, and ALM; approximate SVD options include partial, linear-time, limited-memory, and Block Lanczos with Warm Start methods.Source: §1.2.8 · pp. 28–29

1.2.9 Adequacy for the background/foreground separation

  • PCP assumes complete observations with sparse, sufficiently random corruption and no noise, conditions that may fail in applications with missing entries or impulsive and Gaussian noise.Variants such as SPCP, QPCP, and BPCP address entry-wise noise, quantization error, or outliers in entire columns.Source: §1.2.9 · p. 34
  • Background/foreground methods model spatial coherence with structured sparsity or penalties and temporal coherence with optical flow or motion saliency, sometimes combining both.Strategies include graph- and depth-based regularization, confidence or weighting maps, and transformations that align frames; computing motion parameters can be costly in RASL, prompting a cheaper method by Ebadi and Izquierdo.Source: §1.2.9 · pp. 35–36
  • Incremental algorithms update low-rank and additive matrices from prior estimates and new observations, while real-time PCP and SPCP implementations use submatrix computations or GPUs.The review cites incremental methods across RPCA, subspace tracking, and RNMF frameworks, and identifies real-time implementations for PCP and SPCP.Source: §1.2.9 · p. 37
  • For noisy video, ARF denoising followed by OR-PCA outperforms MOG, PBAS, and Codebook models combined with ARF; VBSE handles blurred and noisy sequences without input parameters, while MAGRPCA outperforms RMAMC and GoDec in cluttered bootstrapping scenes.The OR-PCA with ARF results are reported on i-LIDS and CDnet 2014; the MAGRPCA comparison concerns scenes where heavy foreground objects occlude the background.Source: §1.2.9 · p. 38
  • Camera-motion methods use image or low-rank transformations, preprocessing compensation, or convolutional sparse representations; ARPCA-BS [90][87][91] and DSPSS [88] address additional scene variations.The review describes ARPCA-BS as more robust to varying foreground sizes, illumination changes, and dynamic backgrounds than RPCA-PCP and RPCA-LBD [129], and reports that DSPSS scores better on ChangeDetection.net than SemiSoftGoDec, GSRPCA-LSD, and SPGFL.Source: §1.2.9 · pp. 38–39
  • Dynamic-background and intermittent-motion methods use MRFs, saliency-guided updates, motion masks, or graph regularization; gs-incPCP suppresses ghosts using background estimates from short and long frame windows.For ghost suppression, gs-incPCP uses two simultaneous background estimates based on N1 and N2 previous frames, with N1 ≪ N2; DSPSS uses prior knowledge of outlier distribution to accelerate convergence.Source: §1.2.9 · pp. 40–41
  • For background initialization, MAGRPCA, SLMC, and FRMC address many requirements; MAGRPCA and incPCP-TI are table-listed broad-coverage leaders.For background/foreground separation, the review also identifies SMD-RPCA and SRPCA among methods addressing many challenges.Source: §1.2.9 · p. 41, Table 14 · p. 42

1.2.10 Sparse decompositions

  • Sparse dictionary learning represents background images using few learned atoms, while sparse regression casts foreground detection as sparse-corruption or outlier estimation.Dictionary-learning variants differ in dictionary-learning algorithms, decomposition terms, norms, and solvers; examples include K-SVD [75][73][74][344][165][318], matching pursuit, Lasso [344], and online methods [448].Source: §1.2.10 · pp. 43–44
  • Compressive sensing enables foreground detection from lower-dimensional measurements without reconstructing the background or foreground first, and structured sparsity can encode spatial and temporal grouping.The review describes convex and greedy solvers for sparse recovery and cites Dynamic Group Sparsity [404][305][392], Spatio-Temporal Group Sparsity, and adaptive measurement selection in ARCS.Source: §1.2.10 · pp. 44–45

1.2.11 Mixed decompositions

  • RPCA-CS [395] combines low-rank and sparse recovery from compressive measurements using an underdetermined linear operator.Its stated formulation minimizes measurement reconstruction error subject to rank and sparsity constraints.Source: §1.2.11 · p. 45
  • Mixed formulations include SpaRCS and accelerated, support-informed, split-variable, low-latency, and online variants [395][207][470][184][185][418], plus compressive-sensing dictionary approaches [186], but their study is out of scope.The dictionary approaches use learned sparse representations for moving-object detection; one uses spatial-temporal image patches and random projection.Source: §1.2.11 · pp. 45–46

1.3 Motivations and Contributions

  • The survey targets novices, experts, and reviewers, addressing a literature with more than 390 background/foreground-separation papers since 2009 and spatial, temporal, incremental, and real-time demands.The authors identify the absence of rigorous quantitative evaluation on large-scale datasets with accurate ground truth and balanced real-world challenges as a major gap.Source: §1.3 · p. 47
  • The survey presents a unified view of low-rank-plus-additive decompositions and reviews methods across RPCA, RNMF, RMC, RSR, subspace tracking, and robust low-rank minimization.For each method, it examines solvers and incremental or real-time versions for foreground detection, as well as advantages and drawbacks under dynamic backgrounds or illumination changes.Source: §1.3 · p. 48
  • The survey evaluates decomposition methods on the BMC 2012 dataset using its quantitative framework for a fair, broad comparison.The paper presents this as a systematic evaluation and comparative analysis of methods for background/foreground separation.Source: §1.3 · pp. 48–49

2 Robust Principal Component Analysis · 2.1 RPCA via Principal Component Pursuit

This section reviews Principal Component Pursuit (PCP) for separating low-rank background from sparse foreground outliers, covering batch solvers, incremental variants, real-time implementations, and regularization-parameter selection. It also surveys block-sparse RPCA and modified PCP formulations that alter approximations, use prior information, or adapt the solver.

2.1.1 Principal Component Pursuit

  • PCP exactly recovers the low-rank and sparse matrices under the stated minimal assumptions.For background modeling, the low-rank component represents a subspace that can change over time, while sparse outliers represent moving foreground; the foreground mask is obtained by thresholding the sparse component. Separation depends on the low-rank and sparsity assumptions and on optimization convergence, and an excessively high or low background rank can respectively absorb moving objects or miss dynamic-background multimodality.Source: §2.1.1 · pp. 50, 52

2.1.2 Algorithms for solving PCP

  • IALM is at least five times faster than APG, while ADMM reduces computation cost relative to ALM by minimizing L and S serially.The review describes interior-point methods as poorly scalable for large matrices, and notes that IALM uses significantly fewer partial SVDs than EALM while converging practically as fast.Source: §2.1.2 · pp. 53–54
  • For small target rank, l1-filtering solves PCP in O(r^2(m+n)) time and is described as a truly linear-cost, parallelizable method for very large data.The review also notes that other fast-solver approaches may retain high complexity or sacrifice convergence guarantees.Source: §2.1.2 · p. 55
  • AltProj has complexity O(r^2mnlog(1/ε)), runs around 19 times faster than IALM, and visually produces more accurate background extraction.The review also reports that IADM outperformed SVT, APG, IALM, and ADM with less computation time on the I2R dataset.Source: §2.1.2 · pp. 54, 56

2.1.3 Algorithms for incremental PCP

  • ReProCS-cPCA can exactly recover the support of S_t with high probability and bounds reconstruction errors for both S_t and L_t by a small, time-invariant value.Batch PCP is unsuitable for direct incremental use when sparse support is correlated or when principal directions change over time, which can cause rank growth; ReProCS itself also struggles with large outlier supports.Source: §2.1.3 · pp. 56–57

2.1.4 Methods for real time implementation of PCP

  • GPU CAQR achieved 30× speedup over CPU Intel MKL, and a split low-rank recovery scheme accelerated ALM by more than 365 times.Pope et al. combine SVD reduction, seeding from the previous frame block, and partitioning; they report PCP can suit real-time surveillance separation on corresponding hardware.Source: §2.1.4 · pp. 58–59

2.1.5 Methods for finding the optimal PCP solution

  • MDL-based model selection chooses λ by measuring how well a low-rank model captures data regularity, and experiments found the best λ differed from Candes et al.'s theoretical value [45].The approach generates candidate decompositions across decreasing λ values using warm-restarted ALM, rather than selecting λ through cross-validation on downstream application results.Source: §2.1.5 · p. 59
  • Block-sparse RPCA outperforms original PCP and ReProCS, while Biao and Lin’s block-based approach is more robust on I2R than single Gaussian, MOG, and KDE.The first variant determines λ using motion in a second-pass RPCA; the second uses block affiliation to the moving-object class.Source: §2.1.5 · p. 60

2.1.6 Modified-PCP

  • Fixed-rank PCP includes an incremental algorithm, while Bao et al.’s inductive version reduces the cost of processing new samples and partial subspace knowledge permits recovery under weaker incoherence assumptions when accurate.The partial-subspace method was successfully applied to backgrounds whose image subspace changes gradually over time.Source: §2.1.6 · pp. 60–62
  • Yuan et al.’s nuclear-norm-free model has a median-based closed-form background estimate requiring no iterations, and is reported to extract background more accurately than original PCP.The median is taken at each pixel across all video frames, making computation significantly cheaper than iterative numerical schemes.Source: §2.1.6 · p. 60
  • Capped-norm PCP uses thresholds to control recovery precision and outperforms original PCP and SPCP on I2R.Its capped nuclear and l1 norms approximate rank and sparsity; the formulation can be solved with a Difference of Convex functions method or a greedy approach.Source: §2.1.6 · pp. 60–61
  • Stable p,q-PCP outperforms original SPCP and SPCP solved via NSA [13] on I2R, using a PIRA solver based on ADMM.The nonconvex Schatten-p and lq penalties strengthen low-rank and sparsity; Shao et al.’s related variant uses an Lq-seminorm and ALM or APG solvers with root-finder strategies.Source: §2.1.6 · p. 62
  • SVD-free Rank-1 RPCA [91] achieves more than double the speed-up for the same performance target as its SVD counterpart and handles camera movement and illumination changes [90].Rank-1 soft constraint is also applied to edge images for moving-object detection under global illumination changes.Source: §2.1.6 · p. 63
  • NSMP and WNSMP show more robustness on I2R than RPCA solved by IALM and GoDec.They use nuclear- and spectral-norm minimization; WNSMP weights singular values, and its nonconvex weighted-nuclear-norm formulation has a closed-form solution.Source: §2.1.6 · pp. 63–64
  • Welsch M-estimation outperforms Huber and l1-l2 estimators on I2R, and the M-estimator-based model is more robust to outliers than mean square error.The framework alternately recovers low-rank structure and corrupted errors using proximity operators of implicit regularizers.Source: §2.1.6 · p. 64

2.2 RPCA via Stable Principal Component Pursuit

Because PCP assumes exactly low-rank and exactly sparse components, it does not directly accommodate entry-wise noise common in real applications; stable PCP extends the formulation to noisy observations. The section reviews SPCP models, solvers, real-time implementations, and modifications.

2.2.1 Stable Principal Component Pursuit

  • SPCP models observations as low-rank, sparse, and entry-wise noise components, with the noise Frobenius norm bounded by δ, and relaxes PCP for recovery.Zhou et al. proposed the model to provide stable and accurate recovery in the presence of entry-wise noise.Source: §2.2 · p. 64, §2.2.1 · p. 65

2.2.2 Algorithms for solving SPCP

  • NSA works with the fully nonsmooth SPCP objective and is reported as the fastest among the considered optimal-gradient, alternating-linearization, and alternating-direction methods, substantially outperforming ASALM [13].ASALM and its variants converge to an optimum but ASALM iterations are too slow for real-time use and have unknown complexity; PSPG uses significantly fewer partial SVDs and less CPU time than NSA and ASALM.Source: §2.2.2 · p. 66

2.2.3 Methods for real time implementation of SPCP

  • DFC divides matrix factorization into parallel subproblems and combines their solutions, enabling near-linear to superlinear speedups in practice with high-probability recovery guarantees.DFC-PROJ and DFC-NYS use different sampling schemes; ensemble variants reduce recovery error at negligible added parallel-running cost.Source: §2.2.3 · pp. 66–67

2.2.4 Modified-SPCP

  • GreBsmo makes bilateral-factorization SPCP 30–100 times faster than most existing algorithms, such as IALM.Its l1 regularization enables soft-thresholding when updating the sparse component.Source: §2.2.4 · p. 67
  • Modified-SPCP includes nuclear-norm-free models, which are reported to extract background more accurately than original SPCP and more robustly under frame-wise blur.The unblurred model has one tuning parameter and requires iterative solution; the blurred extension models blur with an operator H [438].Source: §2.2.4 · p. 68
  • Other modifications include online under-complete dictionary SPCP, a robust-Huber variational formulation [11], and a three-term turbulence decomposition that outperforms MoG, Kernel Density Estimation, and PCA.The dictionary method uses alternating minimization and fast trainable encoders; the variational approach uses a smooth convex penalty, while the three-term model represents turbulence as dense Gaussian error [11].Source: §2.2.4 · pp. 68–69

2.3 RPCA via Quantization based Principal Component Pursuit

Quantization-based PCP (QPCP) accommodates pixel quantization by allowing the low-rank and sparse components to approximate observations within the quantization precision. Becker et al. solve the resulting convex cone problem using TFOCS with smoothing and an optimal first-order method.

2.3 RPCA via Quantization based Principal Component Pursuit

  • Becker et al. allow L + S to agree with observations A up to a maximum quantization error of 0.5 per pixel, rather than requiring exact equality.The formulation models real pixel values in [0,255] through quantized observations taking values from 0 to 255.Source: §2.3 · p. 69
  • QPCP replaces the equality constraint L + S = A with ∥A − L − S∥_l∞ ≤ 0.5, so the l∞-norm captures pixel quantization error.Source: §2.3 · p. 69
  • Becker et al. solve QPCP by formulating the convex cone problem and its dual in TFOCS, then applying smoothing and an optimal first-order method.The approach also allows the problem to be solved in compressed sensing.Source: §2.3 · pp. 69–70

2.4 RPCA via Block based Principal Component Pursuit

Tang and Nehorai proposed block-based principal component pursuit (BPCP), which decomposes A into a low-rank matrix and a block-sparse matrix. The section also describes its column-outlier treatment, background/foreground limitations, and ALM solver.

2.4 RPCA via Block based Principal Component Pursuit

  • BPCP decomposes A = L+S, enforcing low-rankness on L and block sparsity on S.Its optimization uses the nuclear norm and the l2,1-norm; the latter is the l1-norm of the vector of column l2-norms. The added κ(1 − λ)||L||l2,1 term ensures exact zero columns in L for outliers, which are assumed to have zero columns in L to eliminate ambiguity.Source: §2.4 · p. 70
  • The l2,1 penalty on L enforces exact zero columns corresponding to outliers, with those columns assumed zero to eliminate ambiguity.The l2,1-norm is the l1-norm of the vector formed from the column l2-norms.Source: §2.4 · p. 70
  • RPCA-LBD performed better than RPCA-PCP in background/foreground separation, but its assumption that S is mostly zero by columns is unsuitable when foreground appears in every frame.The review argues that block sparsity should instead be imposed on frame pixels, as in RPCA-BS [90][87][91].Source: §2.4 · p. 70
  • Tang and Nehorai designed an efficient augmented Lagrangian method (ALM) algorithm to solve the convex BPCP problem.The algorithm decomposes A into low-rank and block-sparse matrices using the l2,1-norm and the additional κ(1 − λ)||L||l2,1 term.Source: §2.4 · p. 70

2.5 RPCA via Local Principal Component Pursuit … 2.9 RPCA via Log-sum Heuristic Recovery

These sections adapt RPCA to union-of-subspaces data, column outliers, controlled sparsity, identifiability constraints, and dense-error settings. Reported advantages include better, faster performance than two baselines for SpaCtrl and denser-error removal than PCP for LHR.

2.5 RPCA via Local Principal Component Pursuit

  • LPCP replaces explicit low-rank modeling with sparse self-representation for union-of-subspaces data; l2,1 regularizes rows of U, while l1 sparsifies S [399].Wohlberg et al. solve the problem with Split Bregman, alternating five subproblems: two linear-system problems and three shrinkage problems.Source: §2.5 · p. 71

2.6 RPCA via Outlier Pursuit

  • Outlier Pursuit targets corruption of entire columns and recovers the uncorrupted matrix’s column space rather than the exact matrix; Stable Outlier Pursuit extends it to noisy data.Xu et al. use Singular Value Thresholding to solve both formulations.Source: §2.6 · pp. 71–72

2.7 RPCA with Sparsity Control

  • SpaCtrl controls outlier sparsity with row-wise l2 regularization; its incremental EWLS estimator uses the full data history and exponentially discounts past data when β < 1 [259].The discounting enables operation in nonstationary backgrounds.Source: §2.7 · p. 72
  • With λ = 9.69 × 10−4, RPCA-SpaCtrl performs better than naive PCA and RSL [371] with less computation time.Source: §2.7 · p. 73

2.8 RPCA with Sparse Corruption

  • RPCA-SpaCorr imposes joint conditions that limit row- and column-wise density in S and sparsity in L’s singular vectors to ensure component identifiability.Its formulations also bound ||L||l∞ or ||A − S||l∞ by b [155].Source: §2.8 · p. 73

2.9 RPCA via Log-sum Heuristic Recovery

  • LHR replaces the nuclear-norm formulation with an l1 heuristic and then uses a log-sum term to represent signal sparsity.The log-sum model is nonconvex, but a convex upper bound can be defined.Source: §2.9 · p. 74
  • LHR removes denser errors from corrupted matrices than PCP, addressing settings where high rank or dense corruption may limit convex approaches.Its log-sum model is nonconvex; Deng et al. solve it with majorization-minimization, alternating between constructing and minimizing a convex upper bound.Source: §2.9 · pp. 73–74

2.10 RPCA via Iteratively Reweighted Least Squares Minimization · 2.11 RPCA via Stochastic Optimization · 2.12 Bayesian Robust Principal Component Analysis

These sections review RPCA variants using iteratively reweighted least squares, online stochastic optimization, and Bayesian inference, describing their decomposition strategies, modeling choices, and reported performance.

2.10 RPCA via Iteratively Reweighted Least Squares Minimization

  • Guyon et al. use IRLS with a weighted background mask and an l2,1 mixed norm to encourage spatially homogeneous fitting.Their decomposition adds a local penalty to increase coherence of the sparse moving-object component; related variants add spatial or spatio-temporal constraints,.Source: §2.10 · pp. 74–75

2.11 RPCA via Stochastic Optimization

  • OR-PCA updates a low-dimensional basis and coefficients one frame at a time, enabling real-time processing without batch optimization.Modified OR-PCA adds Laplacian and Gaussian preprocessing to reduce memory and computation, but can perform weakly in highly variable scenes; continuous MRF constraints [174] improve results, especially for dynamic backgrounds, with further variants addressing feature selection, depth, noise, and camera jitter.Source: §2.11 · pp. 75–76
  • INMF-based OR-PCA outperforms original OR-PCA, RPCA-PCP, and DECOLOR in experiments.It uses nonnegative constraints, structured sparsity, and frame weights to balance ghost avoidance against adaptation to background changes.Source: §2.11 · p. 76

2.12 Bayesian Robust Principal Component Analysis

  • Bayesian RPCA models low-rank and sparse components probabilistically; Ding et al. use MCMC, while Babacan et al. [18] use the more computationally efficient VB approach.The reviewed approaches include beta-Bernoulli priors, automatic relevance determination, and a generative model that represents data noise as a mixture of Gaussians.Source: §2.12 · p. 76

2.12.1 Bayesian Robust PCA

  • BRPCA learns low-rank and sparse components with beta-Bernoulli priors and models sparse-component dependence across time and space.MCMC and VB have approximately the same iteration complexity; VB may reach a local optimum, while Ding et al. report that MCMC works effectively in practice.Source: §2.12.1 · pp. 78–79

2.12.2 Variational Bayesian Robust Principal Component Analysis

  • VBRPCA [18] uses mean-field variational Bayes to approximate posterior distributions by alternating updates that minimize Kullback-Leibler divergence.Its low-rank component is factorized, with Gaussian priors on factor columns that can reduce the estimated rank during inference.Source: §2.12.2 · pp. 80–81

2.12.3 Factorized Variational Bayesian RPCA (FVBRPCA)

  • FVBRPCA performs slightly better than RPCA solved via IALM, VBRPCA [18], and GoDec.Its variational approach approximates the posterior with a factorized distribution and estimates model variables from their expectations.Source: §2.12.3 · p. 82

2.12.4 Bayesian RPCA with MoG noise(MoG-BRPCA)

  • MoG-BRPCA models data noise as a mixture of Gaussians, enabling it to fit a wide range of noise types and combinations.The passage identifies Laplacian, Gaussian, and sparse noise as examples.Source: §2.12.4 · p. 82

2.13 Approximated RPCA

This section describes GoDec’s alternating low-rank-plus-sparse approximation and Semi-Soft GoDec’s soft-thresholding variant, including their computational and incremental extensions.

2.13.1 ”Go Decomposition” (GoDec)

  • GoDec alternates low-rank and sparse approximations to decompose a matrix, using hard thresholding and bilateral random projection to reduce computation.Although its subproblems have nonconvex constraints, global solutions exist; the low-rank update uses singular-value hard thresholding, and the sparse update uses entry-wise hard thresholding. Bilateral Random Projection replaces the SVD in the low-rank update.Source: §2.13.1 · pp. 82–83

2.13.2 Semi-Soft GoDec

  • Semi-Soft GoDec soft-thresholds sparse entries, automatically determines k via threshold τ, and achieves four-times-faster background modeling with the same or lower error.Chen et al. extended it as incremental low-rank and sparse decomposition (ILRSD), which processes new matrix columns with a given low-rank structure for video coding in H.264/AVC and HEVC.Source: §2.13.2 · p. 83

2.14 Sparse Additive Matrix Factorization (SAMF) · 2.15 Variational Bayesian Sparse Estimator (VBSE)

SAMF extends RPCA to multiple sparse-noise structures, including row- and column-wise sparsity, while VBSE applies a known linear measurement transformation to outlier-corrupted data. On CAVIAR, reported experiments found sSAMF and VBSE outperforming their respective comparison methods.

2.14 Sparse Additive Matrix Factorization (SAMF)

  • SAMF extends RPCA to handle row-wise and column-wise sparse noise, beyond RPCA’s element-wise sparse and low-rank terms [268][45].Source: §2.14 · p. 83
  • For background/foreground separation, SAMF’s low-rank and element-wise sparse terms represent static background and moving foreground; sSAMF adds a segment-wise term from precomputed over-segmentation [268].The segment-wise formulation relies on the assumption that pixels with similar image intensities tend to belong to the same object.Source: §2.14 · p. 83
  • On CAVIAR, segmentation-based sSAMF outperformed PCP via IALM, corresponding to LE-SAMF [268][45].The passage reports no quantitative scores.Source: §2.14 · p. 84
  • SAMF reduces a partial problem to standard matrix factorization, then uses mean update for variational Bayesian approximation, yielding a global optimum for a large parameter subset at each step [268].Source: §2.14 · p. 84

2.15 Variational Bayesian Sparse Estimator (VBSE)

  • VBSE uses a known measurement matrix to transform outlier-corrupted data and estimate outlier amplitudes, enabling separation in blurred and noisy video sequences [66][67].Because multiplication by a wide matrix compresses the sparse term, the transformed outliers are not necessarily sparse.Source: §2.15 · p. 84
  • VBSE models column sparsity in U and V with Gaussian priors and Gamma hyperpriors, sparse-term amplitudes with independent zero-mean Gaussian precisions, and dense noise with zero-mean Gaussian priors [67].The method combines these stages in a hierarchical Bayesian model and uses approximate Bayesian inference.Source: §2.15 · p. 84
  • On CAVIAR, VBSE outperformed PCP solved via APG and PCA solved via IALM [67][45].The passage reports no quantitative scores.Source: §2.15 · p. 84

3 Robust Non-negative Matrix Factorization

Robust NMF methods use nonnegative low-rank factorizations to separate background and foreground, with different assumptions and losses targeting heavy-tailed noise, separable data, or grouped outliers. The reviewed methods also differ in their optimization strategies and reported I2R comparisons.

3.1 Manhattan Non-negative Matrix Factorization (MahNMF)

  • MahNMF minimizes Manhattan distance to model heavy-tailed Laplacian noise in a nonnegative matrix.Its RRI solver uses closed-form factor updates but is not scalable to large matrices, while Nesterov smoothing alternates factor updates and improves approximation accuracy iteratively.Source: §3.1 · p. 85
  • On I2R, MahNMF gives visual results similar to PCP via IALM and GoDec.MahNMF estimates a nonnegative low-rank part and a sparse residual, and is designed to handle outliers and heavy-tailed noise; results are reported on I2R.Source: §3.1 · p. 85

3.2 Near-separable Non-negative Matrix Factorization (NS-NMF)

  • Xray finds anchor columns incrementally under separability, with at most r parallelizable iterations and no required normalization or preprocessing.RobustXray extends this approach to l1 loss for background separation, using a separable background model in which frame-to-frame pixel variability is represented by a small set of observed pixels.Source: §3.2 · p. 86
  • On I2R, RobustXray outperforms robust NMF local search, robust low-rank local search, and PCP via IALM.The comparison is reported on the I2R dataset.Source: §3.2 · p. 86

3.3 Robust Asymmetric Non-negative Matrix Factorization (RANMF)

  • RANMF targets grouped outliers and controls factor-matrix density through asymmetric constraints, yielding a well-conditioned basis.Its 3D total-variation term supports denoising and foreground-mask segmentation, and SRAM is the proposed solver.Source: §3.3 · p. 87
  • On I2R, RANMF outperforms PCP via IALM and DECOLOR.The comparison is reported for the I2R dataset.Source: §3.3 · p. 87

4 Robust Matrix Completion

Robust matrix completion (RMC) separates low-rank and sparse components when the locations of missing entries are known; unlike RPCA, it treats the support of missing entries as given and can be viewed as supervised learning.

4.1 lσ-norm loss function (RMC-lσ)

  • RMC-lσ slightly outperforms PCP via IALM in background-detail recovery and runs more than 3 times faster, but neither method recovers missing foreground entries.The passage notes that PCP via IALM may add some background details to the foreground.Source: §4.1 · p. 88

4.2 Robust Bilateral Factorization (RMC-RBF)

  • RBF [327] produces visual results similar to RPCA via IALM and GRASTA, while running more than 2 times faster than IALM and more than 3 times faster than GRASTA.It uses two smaller factor matrices and ADMM to handle incomplete and grossly corrupted measurements.Source: §4.2 · pp. 88–89

4.3 Matrix Factorization (RMC-MF)

  • RMC-MF is more accurate than convex RMC and runs more than 7 times faster than convex RMC, more than 4 times faster than GRASTA, and more than 2 times faster than UNN-BF; both RMC variants slightly outperform GRASTA and UNN-BF.The scalable non-convex model factorizes the low-rank matrix into two smaller matrices, and its solver performs SVD on smaller matrices than the convex model.Source: §4.3 · pp. 89–90

4.4 Factorized Robust Matrix Completion (FRMC)

  • Online FRMC completes recovery 7 to 9 times faster than GRASTA with comparable separation quality; the algorithm was developed in both batch and online modes.Its nuclear-norm surrogate supports large matrix-completion problems, and its subproblems are solved with spectral projected gradient iterations.Source: §4.4 · p. 90

4.5 Motion-Assisted Matrix Completion (MAMC)

  • RMAMC [434] outperforms RPCA via IALM on several datasets.MAMC uses dense motion fields to weight likely background pixels, suppress slowly moving objects, and improve recovery of background that appears in only a few frames; RMAMC adds robustness to noise.Source: §4.5 · pp. 90–91

5 Robust Subspace Recovery

Robust Subspace Recovery (RSR) covers robust decompositions beyond RPCA and RNMF. The section describes bi-sparsity, orthonormal-subspace learning, and orthogonal-complement PCA approaches to handling corruption and outliers.

5.1 Robust Subspace Recovery via Bi-Sparsity (RoSuRe)

  • RoSuRe [26] recovers a union of low-dimensional subspaces under sparse corruption using a bi-sparsity model with sparse coefficients and errors.Its minimization imposes A = L + S and L = LW4, with sparsity penalties on W4 and S.Source: §5.1 · p. 91
  • RoSuRe separates foreground from background in static- and moving-camera experiments [26][27][25], while periodic nonzeros in W4 can reveal camera-position groups with similar backgrounds.After rows and columns of W4 are permuted by camera position, its block-diagonal structure groups images into subspaces.Source: §5.1 · pp. 91–92

5.2 Robust Orthonomal Subspace Learning (ROSL)

  • ROSL [342] uses group-sparse coefficients under orthonormal bases, with rank bounded by the number of nonzero coefficient rows, and replaces nuclear-norm minimization with a rank heuristic.Its formulation minimizes coefficient row sparsity and sparse errors; the l1 reformulation uses a nonconvex objective.Source: §5.2 · p. 92
  • On I2R, ROSL’s recovery accuracy and efficiency are slightly better than PCP solved via EALM, IALM, Random Projection, and LMaFit, and it is more than 10 times faster than RPCA solved via IALM.The reported comparison is for ROSL, not its accelerated ROSL+ variant.Source: §5.2 · p. 92

5.3 Robust Orthogonal Complement Principal Component Analysis (ROC-PCA)

  • ROC-PCA targets orthogonal outliers by projecting data onto the orthogonal complement of the principal subspace and decomposing it into mean, outlier, and noise components.It uses a generalized M-estimator and a fast alternating algorithm combining Stiefel-manifold optimization with iterative nonlinear thresholding.Source: §5.3 · p. 93

6 Robust Subspace Tracking

Robust subspace tracking processes streaming observations incrementally, discarding them after processing rather than storing the full input. The reviewed methods use varied outlier, alignment, sparsity, and adaptation strategies for online background/foreground separation.

6.1 Grassmannian Subspace Tracking (GRASTA)

  • On I2R, GRASTA [144] tracks non-stationary subspaces using an l1 cost for incomplete, outlier-corrupted data and is reported more robust than IALM-RPCA and ReProCS, with less time required.It models foreground pixels as sparse outliers and solves an alternating Grassmannian optimization problem.Source: §6.1 · p. 94

6.2 Transformed Grassmannian Subspace Tracking (t-GRASTA)

  • For sequences with simulated camera jitters, t-GRASTA [147] outperforms RASL and GRASTA, runs four times faster than state-of-the-art algorithms, and requires half their memory.It jointly estimates a low-rank subspace, sparse occlusions or foreground, and image transformations, with batch and online modes.Source: §6.2 · p. 95

6.3 Grassmannian Adaptive Stochastic Gradient with l2,1-norm (GASG21)

  • GASG21 uses stochastic gradient descent for l2,1-norm minimization, fitting inliers while handling column outliers; experiments report a slight improvement over OP with less computation time.The l2,1 norm suits column-outlier corruption because outliers have large residuals while inliers have small residuals.Source: §6.3 · pp. 95–96

6.4 Lp-norm Robust Online Subspace Tracking (pROST)

  • pROST [319] uses weighted smoothed Lp-quasi-norm penalties and spatio-temporal pixel-label dependencies, outperforming GRASTA on multi-modal backgrounds and coping with more outliers and higher-rank subspaces.It applies to fully or incompletely observed data and can operate in real time; previously foreground-labeled pixels receive smaller weights to reduce their influence on background maintenance.Source: §6.4 · pp. 96–97

6.5 Grassmannian Online Subspace Updates with Structured-sparsity (GOSUS)

  • GOSUS uses group sparsity and block-wise ADMM for online subspace updates, slightly outperforming IALM-RPCA, GRASTA [143], and BRMF on Wallflower and I2R.Its mixed l2/l1 penalty encourages similarly weighted pixels within selected groups while promoting sparsity across groups.Source: §6.5 · p. 97

6.6 Fast Adaptive Robust Subspace Tracking (FARST)

  • FARST combines recursive least-squares basis updates with l1 foreground extraction and outperforms GRASTA and PRMF for dynamic backgrounds.Unlike GRASTA, which adapts slowly to background change, FARST is described as having optimal global convergence and fast adaptation to background dynamics.Source: §6.6 · pp. 97–98

7 Robust Low Rank Minimization

The section surveys robust low-rank minimization and factorization methods that model outliers or noise, and reports comparisons on background/foreground datasets for several methods.

7.1 LRM with contiguous outliers detection (DECOLOR)

  • On I2R, DECOLOR outperforms PCP solved via IALM, the mean model, and MOG, but its computation time is prohibitive.It models contiguous foreground support with a Markov random field and alternates convex background optimization using SOFT-IMPUTE with graph-cut support estimation.Source: §7.1 · pp. 99–100

7.2 LRM with Direct Robust Matrix Factorization (DRMF)

  • On I2R, experimental results show DRMF outperforms SVD, PCP solved via IALM, and SPCP; Xiong et al. provide no DRMF-R background/foreground separation results.DRMF excludes outliers from the low-rank approximation when their number is not too large; DRMF-R counts outlier patterns using the structured l2,0-norm.Source: §7.2 · p. 100, §7.3 · p. 101

7.4 Probabilistic Robust Matrix Factorization (PRMF)

  • On I2R, PRMF gives visual results similar to PCP solved via IALM, GoDec, and BRPCA with less computation time.It uses an l1-norm loss and l2 regularizer, with an expectation-maximization algorithm exploiting a hierarchical Laplace-distribution representation.Source: §7.4 · p. 102

7.5 Bayesian Robust Matrix Factorization (BRMF)

  • On SABS, MBRMF outperforms PCP solved via IALM, BRPCA, VBRPCA, DECOLOR, and PRMF.BRMF adds learned mean vectors and precision matrices, a Laplace-mixture noise model, and a Markov extension that assumes outliers have spatial or temporal proximity.Source: §7.5 · pp. 102–103

7.6 Practical Low-Rank Matrix Factorization (PLRMF)

  • PLRMF results are reported on Structure from Motion, not background/foreground separation.Its formulation combines a robust l1-norm measurement with a nuclear-norm regularizer and constrains U to be column-orthogonal.Source: §7.6 · p. 103

7.7 Low Rank Matrix Factorization with MoG noise (LRMF-MOG)

  • On I2R, LRMF-MOG outperforms SVD, RSL, PCP solved via IALM, CWM, and PCA-l1.Its mixture-of-Gaussians noise model is presented as able to represent a wider range of noise distributions than purely Gaussian or Laplacian models.Source: §7.7 · p. 104

7.8 Unifying Nuclear Norm and Bilinear Factorization

  • On I2R, UNNBF outperforms PCP solved via IALM, GRASTA, and PRMF with less computation time.The method unifies bilinear factorization and nuclear-norm regularization; an augmented Lagrange multiplier algorithm solves its model.Source: §7.8 · p. 104

7.9 Robust Rank Factorization (RRF)

  • On I2R, LOIRE outperforms PCP solved via IALM and SemiSoft GoDec in computation efficiency with similar visual accuracy.LOIRE models observations as A = BX + S + E, with sparse outliers and dense Gaussian noise, and is solved using an Alternative Matrix Descent Algorithm.Source: §7.9 · p. 105

8 Experimental Evaluation

The evaluation compares 32 low-rank-plus-additive-matrix algorithms on BMC synthetic and real videos using ground-truth-based metrics; OR-PCA with MRF achieves the best F-Measure on both, while most RPCA algorithms outperform adaptive MOG.

8.1 Challenges in Video Surveillance

  • Video-surveillance background subtraction assumes a fixed camera, constant illumination, and a static background, but practical challenges include camera jitter, illumination changes, bootstrapping, dynamic backgrounds, and shadows.These conditions can introduce false detections or make foreground and background difficult to distinguish.Source: §8.1 · p. 106

8.2 Background Models Challenge Dataset

  • BMC combines 20 synthetic videos with frame-by-frame ground truth and nine long real videos designed to test reliability over time and under difficult conditions.Synthetic scenarios include fog, sun, and acquisition noise, with 10 videos for learning and 10 for evaluation; real videos last about one to four hours and may include vegetation, cast shadows, and sudden or long-term illumination changes.Source: §8.2 · p. 107

8.3 Performance Evaluation Metrics

  • The evaluation uses ground-truth-based measures derived from TP, TN, FP, and FN, including detection rate, specificity, error rates, precision, and F-Measure.The paper says good performance requires high detection rate without reducing precision, and F-Measure characterizes performance in Precision-Detection Rate space with a target close to one.Source: §8.3 · pp. 107–108

8.4 Experimental Results

  • The study evaluates 32 algorithms across decomposition categories using BMC Wizard and each algorithm’s original-paper parameters.Batch algorithms were initialized with 200 rather than 300 frames to reduce computation time; shared algorithms consequently had lower F-measures, but retained the same ranking and the earlier survey’s conclusions.Source: §8.4 · pp. 108–109

8.5.2 Real Videos

  • OR-PCA with MRF also leads real-video F-Measure, followed by OR-PCA without MRF, ROSL, PRMF, and DRMF.OR-PCA variants preserve their relative rank across synthetic and real videos; ROSL, PRMF, and DRMF are more robust on real video, while DECOLOR, RegL1-ALM, and GOSUS were not evaluated there due to computation time or an implementation problem. Most RPCA algorithms outperform adaptive MOG on both video types.Source: §8.5.2 · pp. 109–110

9 Implementation and Computational Cost

The section compares computational costs across algorithms, highlighting the savings from partial SVD and reporting that runtime rankings can change with image size.

9 Implementation and Computational Cost

  • CPU times were averaged across sequences and reported hh:mm:ss for 144×176 images with 200 training frames, matching prior I2R formats.All algorithms were implemented in Matlab using LRSLibrary; PROPACK partial SVD was used for IALM, LADMAP, LSADM, and LADM. Results for NSA [13] and PSPG came from their authors.Source: §9 · p. 111
  • Full-SVD EALM and ADM are very computationally expensive, while partial-SVD IALM, LADMAP, LSADM, and LADM significantly reduce computation on extremely low-rank problems.RPCA computational cost is mainly related to SVD, and only the largest few singular values are needed.Source: §9 · p. 111
  • For SPCP, PSPG [14] is most efficient, followed by NSA [13] and ASALM; GoDec is fastest in RPCA and makes large-video applications reachable in real time.Variational BRPCA is less computationally expensive than BRPCA.Source: §9 · p. 111
  • ROSL and ROSL+ meet time requirements under 15 seconds, like OR-PCA, while GOSUS is faster than GRASTA and pROST.DRMF is the best in the PRMF category.Source: §9 · pp. 111–112
  • Computation-time rankings vary with image size because algorithms differ in complexity and memory requirements, and several are suitable only for low-resolution video.A full visual ranking of matrix and tensor algorithms is provided at the LRSLibrary website.Source: §9 · p. 112

10 Experimental Results on I2R dataset and CD.net dataset

On I2R, RFDSA and MODSM outperform several robust decomposition and background-subtraction methods; on ChangeDetection.net, graph-regularized SRPCA outperforms incremental, spatially constrained, and background-subtraction methods.

10.1 Experimental Results on the I2R dataset

  • The I2R dataset comprises nine 176×144-pixel video sequences with dynamic backgrounds, illumination changes, and bootstrapping issues.The sequences include Curtain, Campus, Lobby, Shopping Mall, Airport, Restaurant, Water Surface, and Fountain.Source: §10.1 · pp. 113, 115

10.2 Experimental Results on the ChangeDetection.net dataset

  • ChangeDetection.net is a realistic, large-scale benchmark with nearly 90,000 frames across 31 sequences, six categories, and color and thermal IR modalities.The dataset was selected to cover a wide range of challenges.Source: §10.2 · p. 115

10.1 Experimental Results on the I2R dataset

  • On I2R, RFDSA and MODSM outperform RCPA-PCP via EALM, GRASTA, DECOLOR, MOG, and SOBS.The comparison groups results from different papers [283].Source: §10.1 · p. 115, Table 29 · p. 116

10.2 Experimental Results on the ChangeDetection.net dataset

  • On ChangeDetection.net, graph-regularized SRPCA with spatial and temporal constraints outperforms GoDec, GRASTA, DECOLOR, TVRPCA, MAMC, SLMC, and SOBS.The comparison groups results from different papers.Source: Table 30 · p. 116, §10.2 · p. 116

11 Conclusion

The review unifies robust low-rank-plus-additive formulations as DLAM and evaluates their use for background/foreground separation, including their optimization, incremental methods, and real-time potential. It concludes that DLAM is suitable for the task while identifying computational and spatial-temporal limitations and directions for future work.

11 Conclusion

  • The review covers RPCA, RNMF, RMC, RSR, RST, and RLRM, proposing DLAM as a unified view and evaluating solution methods, incremental algorithms, and real-time implementations.It reports comparative performance on the Background Models Challenge dataset.Source: §11 · pp. 116–117
  • DLAM is suitable for background/foreground separation; fundamental models outperform MOG and KDE, while specialized methods are comparable to SubSENSE and outperform SOBS, ViBe, and PBAS.Robustness to illumination changes and dynamic backgrounds can be improved with spatial and temporal constraints, including structured norms, group sparsity, dynamic tree sparsity, or MRFs.Source: §11 · p. 117
  • Original DLAM formulations often rely on batch algorithms whose computation time is too high for real-time requirements, prompting incremental and real-time work by Rodriguez et al. [309][307][311] and Vaswani et al. [297][121].Source: §11 · p. 117
  • Vectorizing images in original DLAM models loses spatial and temporal constraints, motivating MRFs, structured norms [329][361][177], or two-dimensional formulations that preserve spatial information with reduced computation.Examples of robust two-dimensional formulations include R2DRPCA [361] and RGLRAM.Source: §11 · p. 117
  • Future research includes less computational SVD for batch methods, DLAM that is both incremental and real-time, extensions to measurement-domain compressive sensing and tensor formulations, and efficient tensor incremental methods for separation and initialization.The review also notes tensor incremental background/foreground methods [298][169][350][270] and background initialization.Source: §11 · pp. 117–118

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