Presentation summary: * Moving object detection by background modeling and subtraction. * Solved and unsolved challenges. * Framework for low-rank and sparse decomposition. * Some applications of RPCA on: * * Background modeling and foreground separation. * * Very dynamic background. * * Multidimensional and streaming data. * LRSLibrary1 + demo.

1. Robust Low-rank and Sparse Decomposition for Moving
Object Detection
Andrews Cordolino Sobral
Ph.D. on Computer Vision and Machine Learning
@ University of La Rochelle, France (European doctorate label)
Senior AI Research Engineer
@ ActiveEon, Paris office, France
1

2. Summary
Moving object detection by background modeling and subtraction.
Solved and unsolved challenges.
Framework for low-rank and sparse decomposition.
Some applications of RPCA on:
Background modeling and foreground separation.
Very dynamic background.
Multidimensional and streaming data.
LRSLibrary1
+ demo.
1
https://github.com/andrewssobral/lrslibrary
2

3. Background modeling and subtraction (BMS) process
Model
initialization
Frames
Model update
Background
model
Foreground
detection
3

4. BMS challenges
“Solved” and “unsolved” issues:
Baseline
Shadow
Bad weather
Thermal
Dynamic background
Camera jitter
Intermittent object motion
Turbulence
Low framerate
Night scenes
PTZ cameras
* Pierre-Marc Jodoin. Motion Detection: Unsolved Issues
and [Potential] Solutions. Scene Background Modeling and
Initialization (SBMI), ICIAP, 2015.
4
Play

5. BMS methods
A large number of algorithms have been proposed for background
subtraction over the last few years [Sobral and Vacavant, 2014],
[Bouwmans, 2014], [Xu et al., 2016]:
Traditional methods (several implementations available in BGSLibrary*):
Basic methods (i.e. [Cucchiara et al., 2001])
Statistical methods (i.e. [Stauffer and Grimson, 1999])
Non-parametric methods (i.e. [Elgammal et al., 2000])
Fuzzy based methods (i.e. [Baf et al., 2008])
Neural and neuro-fuzzy methods (i.e. [Maddalena and Petrosino, 2012])
Decomposition into low-rank + sparse components
Introduced in [Cand`es et al., 2011]. In general, the decomposition is done
by matrix and tensor methods. Our focus
* [Sobral, 2013] https://github.com/andrewssobral/bgslibrary.
5

6. Decomposition into low-rank + sparse components
This framework considers that the data (matrix A) to be processed satisfy
two important assumptions:
The inliers (latent structure) are drawn from a single (or a union of)
low-dimensional subspace(s) (matrix L)
The corruptions are sparse (matrix S)
A L
= +
S
6

7. Decomposition into low-rank + sparse components
Note
This assumption holds a particular association to the problem of
background/foreground separation.
A L
= +
S
The process of background/foreground separation can be regarded as a
matrix separation problem.
7

8. Robust Principal Component Analysis (RPCA)
This definition is also known as Robust Principal Component Analysis
(RPCA), and it is formulated as follows:
minimize
L,S
rank(L) + card(S),
subject to A = L + S,
(1)
where rank(L) represents the rank of L and card(S) denotes the
number of non-zero entries of S.
The low-rank minimization concerning L offers a suitable framework for
background modeling due to the high correlation between frames.
However, the above equation yields a highly non-convex optimization
problem (NP-hard).
8

9. RPCA via Principal Component Pursuit (PCP)
[Cand`es et al., 2011] showed that L and S can be recovered by solving a
convex optimization problem, named as Principal Component Pursuit
(PCP).
The card(.) is replaced with the 1-norm and the rank(.) with the nuclear
norm* ||.||∗, yielding the following convex surrogate:
minimize
L,S
||L||∗ + λ||S||1,
subject to A = L + S,
(2)
where λ > 0 is a trade-off parameter between the sparse and the
low-rank regularization.
The minimization of ||L||∗ enforces low-rankness in L, while the minimization of ||S||1
maximize the sparsity in S.
* Sum of singular values.
9

10. RPCA limitations
However, the RPCA via PCP has some limitations:
Low-rank component = exactly low-rank.
Sparse component = exactly sparse.
The input matrix is considered as the sum of a true low-rank matrix plus a true sparse
matrix.
That’s not all...
10

11. RPCA challenges (outliers)
In real applications the observations are often corrupted by noise, and
missing data can occurs.
11

12. RPCA challenges (design)
Moreover, designing a RPCA algorithm needs to address some of the
following questions:
Decomposition: Decompose the input data into one, two, or more terms.
Convexity, norms and constraints: Is there a suitable norm or constraint for each
term? Use a convex surrogate norm or not?
Loss function and regularization: Is there a suitable loss function that is globally
continuous and differentiable? Is there a suitable regularization to improve the
learned model?
Solvers: How to design an efficient optimization algorithm that is faster and more
scalable? online or offline?
Multidimensionality: How to represent the input data?
...and taking into the BMS constraints!
In summary
Designing an efficient RPCA algorithm for background/foreground separation need to
take into account the BMS challenges and the mathematical issues of RPCA.
12

13. RPCA methods
A large number of approaches for robust low-rank and sparse modeling have been
proposed in the last few years ([Zhou et al., 2014], [Lin, 2016],
[Davenport and Romberg, 2016], and [Bouwmans et al., 2016]).
2010–2011 2011–2012 2012–2013 2013–2014 2014–2015 2015–2016
200
400
600
800
1,000
1,200
1,400
# of citations of [Cand`es et al., 2011].
In [Bouwmans et al., 2016], more than 300 papers addressed the problem of
background/foreground separation.
Some key issues and challenges remain, such as handling complex/dynamic background
scenarios and performing in a incremental / real-time manner.
13

14. RPCA applications
14

15. Decomposition into Low-rank and Sparse Matrices (DLSM)
A unified model is proposed to represent the state-of-the-art methods in
a more general framework, named DLSM (Decomposition into Low-rank
and Sparse Matrices) [Bouwmans, Sobral et al., 2016].
The DLSM framework categorizes the matrix separation problem into
three main approaches: implicit, explicit and stable.
and it is formulated as follows:
A =
Y
y=1
Ky (3)
where, in most of the cases, Y ∈ {1, 2, 3}.
15

16. Implicit approaches (Y = 1)
The first matrix K1 is the best low-rank approximation (e.g. K1 = L) of
the matrix A, where A ≈ L.
This is an “implicit decomposition” due to the fact that we have any
constraint with respect to the foreground objects.
The residual matrix S (sparse or not) is recovered by S = A − L.
e.g. Low-Rank Approximation (LRA).
16

17. Low-Rank Approximation (LRA)
LRA is formulated as:
minimize
L
f (A − L),
subject to rank(L) = r,
(4)
where f (.) denotes a loss function (i.e. ||.||2
F ) and r (1 ≤ r < rank(A)) is the desired rank.
)]kF(. . . vec)1F(vec= [A
kF. . .1FframeskSequence of background modelskSequence of
i
Tviσiu=1i
r
=rA
(rank-1 approximation)1A
Input matrix (full rank) Low-rank approximation
A closed form solution can be estimated by computing the “truncated” Singular Value
Decomposition (SVD) of A.
17

18. Limitations of LRA
LRA is formulated as:
minimize
L
f (A − L),
subject to rank(L) = r,
(4)
where f (.) denotes a loss function and r (1 ≤ r < rank(A)) represents the desired rank.
18

19. Affine rank minimization
In many applications, we need to recover a minimal rank matrix subject to
some problem-specific constraints, often characterized as an affine set.
This affine rank minimization problem is defined as follows:
minimize
L
rank(L),
subject to A(L) = b,
(5)
where A : Rm×n → Rp denotes a linear mapping and b ∈ Rp represents a
vector of observations of size p.
19

20. Matrix Completion (MC)
In many applications, we need to recover a minimal rank matrix subject to
some problem-specific constraints, often characterized as an affine set.
This affine rank minimization problem is defined as follows:
minimize
L
rank(L),
subject to A(L) = b,
(5)
where A : Rm×n → Rp denotes a linear mapping and b ∈ Rp represents a
vector of observations of size p.
A special case of problem (5) is the matrix completion problem:
minimize
L
rank(L),
subject to PΩ(L) = PΩ(A),
(6)
where PΩ(.) denotes a sampling operator restricted to the elements of Ω
(set of observed entries). Let’s take an example!
20

21. MC for Background Model Estimation
Conceptual illustration
A ).(ΩPSampling operator )A(ΩP
,)A(ΩP) =L(ΩPsubject to
,∗||L||
L
minimize
L
Application to background model estimation
A ).(ΩPSampling operator )A(ΩP
,)A(ΩP) =L(ΩPsubject to
,∗||L||
L
minimize
L
21

22. Explicit approaches (Y = 2)
The matrices K1 = L and K2 = S are usually assumed to be the low-rank
and sparse representation of the data, where A ≈ L + S.
This is an “explicit decomposition” due to the fact that we have two
constraints:
the first one enforcing a low-rank structure over the matrix L, and
the second one enforcing a sparse structure over the matrix S.
Explicit approaches usually work better for the problem of
background/foreground separation in comparison to the implicit methods.
e.g. Robust Principal Component Analysis (RPCA) proposed by [Cand`es et al., 2011].
22

23. Background/foreground separation with RPCA via PCP
Components
Video Low-rank Sparse Foreground
Background model Moving objects Classification
Demo
23

24. Stable approaches (Y = 3)
The matrices K1 = L, K2 = S and K3 = E are usually assumed to be the
low-rank, sparse and noise components, respectively, where
A ≈ L + S + E.
This decomposition is called “stable decomposition” as it separates the
sparse components in S and the noise in E.
In the case of background/foreground separation, the noise matrix E can
also represent some dynamic properties of the background.
e.g. Stable Principal Component Pursuit (Stable PCP) proposed by Zhou et
al. [Zhou et al., 2010].
24

25. PCP vs Stable PCP
Input video RPCA via PCP RPCA via Stable PCP
Visual comparison of foreground segmentation between PCP and Stable PCP for
dynamic background.
25

26. Stable PCP for dynamic background scenes
Stable PCP try to deal with this problem under the term where the
multi-modality of the background (i.e. waves) can be considered as
noise component (E).
Some authors used an additional constraint to improve the
background/foreground separation:
[Oreifej et al., 2013] used a turbulence model driven by dense optical flow to
enforce an additional constraint on the rank minimization.
[Ye et al., 2015] proposed a robust motion-assisted matrix restoration (RMAMR)
where a dense motion field given by optical flow is mapped into a weighting matrix.
[Sobral et al., 2015b] presented a double-constrained RPCA (SCM-RPCA), where
the sparse component is constrained by shape and confidence maps both extracted
from spatial saliency maps.
26

27. SCM-RPCA on very dynamic background
Input Background Foreground
minimize
L,S,E
||L||∗+λ1||Π(S)||1+λ2||E||2
F
subject to A = L + W ◦ S + E
[Sobral et al., 2015b]
27

28. SCM-RPCA over MarDT dataset
Input frame Saliency map Sparse component Foreground mask Ground truthLow-rank component
For the MarDT scenes, the temporal median of the saliency maps was subtracted, due
to the high saliency from the buildings around the river.
Related publication: (IEEE AVSS, 2015, [Sobral et al., 2015b]).
MATLAB code: https://sites.google.com/site/scmrpca.
28

29. Dealing with multidimensional and
streaming data
29

30. Multispectral data
Usually a multispectral video consists of a sequence of multispectral
images sensed from contiguous spectral bands.
Each multispectral image can be represented as a three-dimensional data
cube, or tensor.
Processing a sequence of multispectral images with hundreds of bands
can be computationally expensive.
30

31. Limitations of matrix-based approaches
Matrix-based low-rank and sparse decomposition methods work only on a
single dimension and consider the input frame as a vector.
Multidimensional data for efficient analysis can not be considered.
The local spatial information is lost and erroneous foreground regions
can be obtained.
Some authors used a tensor representation to solve this
problem [Li et al., 2008, Hu et al., 2011, Tran et al., 2012,
Tan et al., 2013, Sobral et al., 2014, Sobral et al., 2015c].
31

32. Tensor decomposition and factorization
Tensor decompositions have been widely studied and applied to many
real-world problems [Kolda and Bader, 2009].
They were used to design low-rank approximation algorithms for
multidimensional arrays taking full advantage of the multi-dimensional
structures of the data.
Three widely-used models for low rank decomposition on tensors are:
Tucker/Tucker3 decomposition.
CANDECOMP/PARAFAC (CP) decomposition.
Tensor Robust PCA decomposition.
32

33. Tucker vs CP decomposition
Tucker decomposition
CP decomposition
33

34. RPCA on tensors
Some authors extended the Robust PCA framework for matrices to the
multilinear case [Goldfarb and Qin, 2014, Lu et al., 2016].
Tensor Robust PCA decomposition
The RPCA for matrices was reformulated into its “tensorized” version. For
an N-order tensor X, it can be decomposed as:
X = L + S + E, (7)
where L, S and E represent the low-rank, sparse and noise tensors.
34

35. Extending a matrix-based RPCA to tensors
Stochastic RPCA on matrices [Feng et al., 2013]:
minimize
W,H,S
1
2
||X − WHT
− S||2
F +
λ1
2
(||W||2
F +||H||2
F ) + λ2||S||1,
subject to L = WHT
.
(8)
Extension for tensors (OSTD) [Sobral et al., 2015c]:
minimize
W,H,S
1
2
N
i=1
||X[i]
− Wi HT
i − S[i]
||2
F +
λ1
2
(||Wi ||2
F +||Hi ||2
F ) + λ2||S[i]
||1,
subject to L[i]
= Wi HT
i .
(9)
X[n]
: n-mode matricization of tensor X.
Xi : ith matrix.
35

36. Comparison
OSTD was compared with 3 other ones:
CP-ALS [Kolda and Bader, 2009]
HORPCA [Goldfarb and Qin, 2014]
BRTF [Zhao et al., 2016]
CP-ALS, HORPCA, and BRTF are based on batch optimization strategy.
A total of 100 frames at time were used to reduce the computational cost
for the whole video sequence (fourth-order tensor).
OSTD processes each multispectral image or RGB image per time
instance.
The algorithms were evaluated on MVS dataset [Benezeth et al., 2014], the first dataset for
background subtraction on multispectral video sequences.
36

37. Qualitative results I
RGB image ground truth proposed approach BRTF HORPCA CP-ALS
Visual comparison of background subtraction results over three scenes of
the MVS dataset.
37

38. Qualitative results II
Video GT OSTD
38

39. Quantitative results
MVS dataset: Comparison of average F-measure score in (%) with other approaches.
Methods 1st
2nd
3rd
4th
5th
Avg
CP-ALS
RGB 58.69 RGB 71.25 RGB 51.32 RGB 60.21 RGB 49.35 RGB 58.16
MSB 71.61 MSB 83.50 MSB 68.54 MSB 78.63 MSB 66.97 MSB 73.85
HORPCA
RGB 63.23 RGB 78.52 RGB 55.69 RGB 67.56 RGB 58.80 RGB 64.76
MSB 80.65 MSB 84.79 MSB 68.12 MSB 77.56 MSB 74.47 MSB 77.11
BRTF
RGB 68.56 RGB 79.21 RGB 63.56 RGB 73.22 RGB 62.51 RGB 70.32
MSB 85.30 MSB 89.63 MSB 68.11 MSB 84.65 MSB 77.91 MSB 82.76
OSTD
RGB 78.63 RGB 85.96 RGB 79.56 RGB 76.32 RGB 71.23 RGB 76.69
MSB 93.65 MSB 95.17 MSB 90.64 MSB 89.29 MSB 92.66 MSB 92.28
39

40. Computational time
Computational time for the first 100 frames varying the image resolution.
Size HORPCA CP-ALS BRTF OSTD
160 × 120 00:01:35 00:00:40 00:00:22 00:00:04
320 × 240 00:04:56 00:02:09 00:03:50 00:00:12
The algorithms were implemented in MATLAB running on a laptop
computer with Windows 7 Professional 64 bits, 2.7 GHz Core i7-3740QM
processor and 32Gb of RAM.
OSTD achieved almost real time processing, since one video frame is
processed at time.
Related publications:
(IEEE ICCV Workshop on RSL-CV, 2015, [Sobral et al., 2015c]).
MATLAB code: https://github.com/andrewssobral/ostd.
40

41. LRSLibrary
41

42. LRSLibrary
LRSLibrary [Sobral et al., 2016]a
provides a collection of low-rank and sparse
decomposition algorithms in MATLAB.
a
https://github.com/andrewssobral/lrslibrary
The LRSLibrary was designed for background/foreground separation in videos, and it
contains a total of 104 matrix-based and tensor-based algorithms.
42

43. 43

44. Publications I
This presentation was based on the following publications2:
Talks (1)
2016 - Sobral, Andrews. “Recent advances on low-rank and sparse decomposition for
moving object detection.”. Workshop/atelier: Enjeux dans la d´etection d’objets mobiles
par soustraction de fond. Reconnaissance de Formes et Intelligence Artificielle (RFIA),
20163.
Journal papers (3)
2016 - Sobral, Andrews; Zahzah, El-hadi. “Matrix and Tensor Completion Algorithms for
Background Model Initialization: A Comparative Evaluation”, In the Special Issue on
Scene Background Modeling and Initialization (SBMI), Pattern Recognition Letters
(PRL), 2016. [Sobral and Zahzah, 2016].
2016 - Gong, Wenjuan; Zhang, Xuena; Gonzalez, Jordi; Sobral, Andrews; Bouwmans,
Thierry; Tu, Changhe; Zahzah, El-hadi. “Human Pose Estimation from Monocular
Images: A Comprehensive Survey”, Sensors, 2016. [Gong et al., 2016].
44

45. Publications II
2016 - Bouwmans, Thierry; Sobral, Andrews; Javed, Sajid; Ki Jung, Soon; Zahzah,
El-Hadi. “Decomposition into Low-rank plus Additive Matrices for
Background/Foreground Separation: A Review for a Comparative Evaluation with a
Large-Scale Dataset”, Computer Science Review, 2016. [Bouwmans et al., 2016].
Book chapters (1)
2015 - Sobral, Andrews; Bouwmans, Thierry; Zahzah, El-hadi. “LRSLibrary: Low-Rank
and Sparse tools for Background Modeling and Subtraction in Videos”. Chapter in the
handbook “Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image
and Video Processing”, CRC Press, Taylor and Francis Group, 2015. [Sobral et al., 2016].
Conferences (7)
2015 - Sobral, Andrews; Javed, Sajid; Ki Jung, Soon; Bouwmans, Thierry; Zahzah,
El-hadi. “Online Stochastic Tensor Decomposition for Background Subtraction in
Multispectral Video Sequences”. ICCV Workshop on Robust Subspace Learning and
Computer Vision (RSL-CV), Santiago, Chile, December, 2015. [Sobral et al., 2015c].
45

46. Publications III
2015 - Javed, Sajid; Ho Oh, Seon; Sobral, Andrews; Bouwmans, Thierry; Ki Jung, Soon.
“Background Subtraction via Superpixel-based Online Matrix Decomposition with
Structured Foreground Constraints”. ICCV Workshop on Robust Subspace Learning and
Computer Vision (RSL-CV), Santiago, Chile, December, 2015. [Javed et al., 2015a].
2015 - Sobral, Andrews; Bouwmans, Thierry; Zahzah, El-hadi. ”Comparison of Matrix
Completion Algorithms for Background Initialization in Videos”. Scene Background
Modeling and Initialization (SBMI), Workshop in conjunction with ICIAP 2015, Genova,
Italy, September, 2015. [Sobral et al., 2015a].
2015 - Sobral, Andrews; Bouwmans, Thierry; Zahzah, El-hadi. “Double-constrained
RPCA based on Saliency Maps for Foreground Detection in Automated Maritime
Surveillance”. Identification and Surveillance for Border Control (ISBC), International
Workshop in conjunction with AVSS 2015, Karlsruhe, Germany, August,
2015. [Sobral et al., 2015b].
2015 - Javed, Sajid; Sobral, Andrews; Bouwmans, Thierry; Ki Jung, Soon. “OR-PCA
with Dynamic Feature Selection for Robust Background Subtraction”. In Proceedings of
the 30th ACM/SIGAPP Symposium on Applied Computing (ACM-SAC), Salamanca,
Spain, 2015. [Javed et al., 2015b].
2014 - Javed, Sajid; Ho Oh, Seon; Sobral, Andrews; Bouwmans, Thierry; Ki Jung, Soon.
“OR-PCA with MRF for Robust Foreground Detection in Highly Dynamic Backgrounds”.
In the 12th Asian Conference on Computer Vision (ACCV 2014), Singapore, November,
2014. [Javed et al., 2014].
46

47. Publications IV
2014 - Sobral, Andrews; Baker, Christopher G.; Bouwmans, Thierry; Zahzah, El-hadi.
“Incremental and Multi-feature Tensor Subspace Learning applied for Background
Modeling and Subtraction”. International Conference on Image Analysis and Recognition
(ICIAR’2014), Vilamoura, Algarve, Portugal, October, 2014. [Sobral et al., 2014].
2
The reader can refer to https://scholar.google.fr/citations?user=0Nm0uHcAAAAJ for an updated list of publications
and citations.
3
http://rfia2016.iut-auvergne.com/index.php/autres-evenements/
detection-d-objets-mobiles-par-soustraction-de-fond
47

48. [Baf et al., 2008] Baf, F. E., Bouwmans, T., and Vachon, B. (2008). Fuzzy integral for moving object detection. In IEEE
International Conference on Fuzzy Systems, pages 1729–1736.
[Benezeth et al., 2014] Benezeth, Y., Sidibe, D., and Thomas, J. B. (2014). Background subtraction with multispectral video
sequences. In International Conference on Robotics and Automation (ICRA).
[Bouwmans, 2014] Bouwmans, T. (2014). Traditional and recent approaches in background modeling for foreground detection:
An overview. In Computer Science Review.
[Bouwmans et al., 2016] Bouwmans, T., Sobral, A., Javed, S., Jung, S. K., and Zahzah, E. (2016). Decomposition into
low-rank plus additive matrices for background/foreground separation: A review for a comparative evaluation with a
large-scale dataset. Computer Science Review.
[Cand`es et al., 2011] Cand`es, E. J., Li, X., Ma, Y., and Wright, J. (2011). Robust Principal Component Analysis? Journal of
the ACM.
[Cucchiara et al., 2001] Cucchiara, R., Grana, C., Piccardi, M., and Prati, A. (2001). Detecting objects, shadows and ghosts in
video streams by exploiting color and motion information. In Proceedings 11th International Conference on Image Analysis
and Processing, pages 360–365.
[Davenport and Romberg, 2016] Davenport, M. A. and Romberg, J. (2016). An overview of low-rank matrix recovery from
incomplete observations. IEEE Journal of Selected Topics in Signal Processing, 10(4):608–622.
[Elgammal et al., 2000] Elgammal, A. M., Harwood, D., and Davis, L. S. (2000). Non-parametric model for background
subtraction. In Proceedings of the 6th European Conference on Computer Vision-Part II, ECCV ’00, pages 751–767, London,
UK, UK. Springer-Verlag.
[Feng et al., 2013] Feng, J., Xu, H., and Yan, S. (2013). Online robust PCA via stochastic optimization. In Advances in Neural
Information Processing Systems (NIPS).
[Goldfarb and Qin, 2014] Goldfarb, D. and Qin, Z. T. (2014). Robust low-rank tensor recovery: Models and algorithms. SIAM
Journal on Matrix Analysis and Applications.
[Gong et al., 2016] Gong, W., Zhang, X., Gonzalez, J., Sobral, A., Bouwmans, T., Tu, C., and Zahzah, E. (2016). Human pose
estimation from monocular images: A comprehensive survey. Sensors, 16(12).
[Hu et al., 2011] Hu, W., Li, X., Zhang, X., Shi, X., Maybank, S., and Zhang, Z. (2011). Incremental tensor subspace learning
and its applications to foreground segmentation and tracking. International Journal of Computer Vision (IJCV).
48

49. [Baf et al., 2008] Baf, F. E., Bouwmans, T., and Vachon, B. (2008). Fuzzy integral for moving object detection. In IEEE
International Conference on Fuzzy Systems, pages 1729–1736.
[Benezeth et al., 2014] Benezeth, Y., Sidibe, D., and Thomas, J. B. (2014). Background subtraction with multispectral video
sequences. In International Conference on Robotics and Automation (ICRA).
[Bouwmans, 2014] Bouwmans, T. (2014). Traditional and recent approaches in background modeling for foreground detection:
An overview. In Computer Science Review.
[Bouwmans et al., 2016] Bouwmans, T., Sobral, A., Javed, S., Jung, S. K., and Zahzah, E. (2016). Decomposition into
low-rank plus additive matrices for background/foreground separation: A review for a comparative evaluation with a
large-scale dataset. Computer Science Review.
[Cand`es et al., 2011] Cand`es, E. J., Li, X., Ma, Y., and Wright, J. (2011). Robust Principal Component Analysis? Journal of
the ACM.
[Cucchiara et al., 2001] Cucchiara, R., Grana, C., Piccardi, M., and Prati, A. (2001). Detecting objects, shadows and ghosts in
video streams by exploiting color and motion information. In Proceedings 11th International Conference on Image Analysis
and Processing, pages 360–365.
[Davenport and Romberg, 2016] Davenport, M. A. and Romberg, J. (2016). An overview of low-rank matrix recovery from
incomplete observations. IEEE Journal of Selected Topics in Signal Processing, 10(4):608–622.
[Elgammal et al., 2000] Elgammal, A. M., Harwood, D., and Davis, L. S. (2000). Non-parametric model for background
subtraction. In Proceedings of the 6th European Conference on Computer Vision-Part II, ECCV ’00, pages 751–767, London,
UK, UK. Springer-Verlag.
[Feng et al., 2013] Feng, J., Xu, H., and Yan, S. (2013). Online robust PCA via stochastic optimization. In Advances in Neural
Information Processing Systems (NIPS).
[Goldfarb and Qin, 2014] Goldfarb, D. and Qin, Z. T. (2014). Robust low-rank tensor recovery: Models and algorithms. SIAM
Journal on Matrix Analysis and Applications.
[Gong et al., 2016] Gong, W., Zhang, X., Gonzalez, J., Sobral, A., Bouwmans, T., Tu, C., and Zahzah, E. (2016). Human pose
estimation from monocular images: A comprehensive survey. Sensors, 16(12).
[Hu et al., 2011] Hu, W., Li, X., Zhang, X., Shi, X., Maybank, S., and Zhang, Z. (2011). Incremental tensor subspace learning
and its applications to foreground segmentation and tracking. International Journal of Computer Vision (IJCV).
49