PhD Thesis Defense Presentation: Robust Low-rank and Sparse Decomposition for Moving Object Detection - From Matrices to Tensors

Robust Low-rank and Sparse Decomposition for Moving
Object Detection
From Matrices to Tensors
Andrews Cordolino Sobral
L3I/MIA, Universit´e de La Rochelle
Ph.D. European Label
Supervisors: El-hadi Zahzah (L3I) and Thierry Bouwmans (MIA)
May 11, 2017 - La Rochelle, France
Andrews Cordolino Sobral (L3I/MIA) Universit´e de La Rochelle 1

Table of contents
Thesis outline
Introduction to intelligent video surveillance.
Background subtraction process / Challenges.
Framework for low-rank and sparse decomposition / Application for
background/foreground separation.
Contributions:
A uniﬁed model for low-rank and sparse decomposition.
Matrix/tensor completion methodology for background model
initialization.
Double constraint RPCA for robust foreground detection in maritime
scenes.
Tensor-based models for handling multidimensional streaming data.
Collaborative external research contributions.
Conclusions and future perspectives.

Video surveillance cameras are everywhere
* From IBM Intelligent Video Analytics.
s

...and an intelligent video surveillance system is needed
* From BOSCH Intelligent Video Analysis.

Intrusion detection
Vehicle monitoring
People counting
Abandoned object detection
* From Aventura Intelligent Video Analytics.

Understanding an intelligent video surveillance framework
!
Video
Pre-processing
Object
Detection
!
Example of automatic
incident detection
Location of the incident
Intrusion
detection
Abandoned
objects
Tracking and
counting of
people, cars, etc.
Anomaly
detection
Traffic
surveillanceRoad traffic data
collection
Incident report
Human expert
Object Tracking
Activity
Recognition

Our focus
!
Video
Pre-processing
Object
Detection
!
Example of automatic
incident detection
Location of the incident
Intrusion
detection
Abandoned
objects
Tracking and
counting of
people, cars, etc.
Anomaly
detection
Traffic
surveillanceRoad traffic data
collection
Incident report
Human expert
Object Tracking
Activity
Recognition

CCTV systems
The most commonly used equip-
ments are:
Stationary cameras
Pan-Tilt-Zoom (PTZ) cameras
* From Sky NEWS: British Are World’s Most Watched People.

Background subtraction (BS) process
Model
initialization
Frames
Model update
Background
model
Foreground
detection

BS 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.
Play

BS 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 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])
* [Sobral, 2013] https://github.com/andrewssobral/bgslibrary.

BS 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 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.

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

Note
This assumption holds a particular association to the problem of

Note
This assumption holds a particular association to the problem of
A L
= +
S
The process of background/foreground separation can be regarded as a
matrix separation problem.

Robust Principal Component Analysis (RPCA)
This deﬁnition is also known as Robust Principal Component Analysis
(RPCA), and can be 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).

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).

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.

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...

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

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?

learned model?
Solvers: How to design an eﬃcient optimization algorithm that is faster and more
scalable? online or oﬄine?

learned model?
Multidimensionality: How to represent the input data?

learned model?
Multidimensionality: How to represent the input data?
...and without forgetting the BS constraints!
In summary
Designing an eﬃcient RPCA algorithm for background/foreground separation need to
take into account the BS challenges and the mathematical issues of RPCA.

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]*.
* Google Scholar: https://scholar.google.fr/citations?user=nRQi4O8AAAAJ

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
Some key issues and challenges remain, such as handling complex/dynamic background
scenarios and performing in a incremental / real-time manner.

The uniﬁcation
#1 contribution

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.

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}.

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).

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.

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.

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.

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!

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

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].

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

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].

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.

General overview of the DLSM framework

Focus of the #2 contribution

Background model initialization
#2 contribution

#2 contribution
Is matrix completion (or even tensor completion) robust to the problem
of background model initialization?
Model
initialization
Frames
Model update
Background
model
Foreground
detection
Let’s see!

Background model (BM) initialization
The main challenge is to obtain a first background model when video
frames contain foreground objects.
Classification of background model initialization methods according to [Bouwmans et al., 2017].
The approaches presented here are in red.
Type of methods Related works
Temporal Statistics Mean, Color Median, MoG [Stauffer and Grimson, 1999], BE-AAPSA [Ramirez-Alonso et al., 2017]
Subintervals of Stable Intensity IMBS-MT [Bloisi et al., 2016], LaBGen [Laugraud et al., 2016]
Model Completion RSL2011 [Reddy et al., 2011]
Optimal Labeling Photomontage [Agarwala et al., 2004]
Subspace Estimation Eigen [Oliver et al., 2000], RSL [De La Torre and Black, 2003], RPCA [Candès et al., 2011]
Missing Data Reconstruction Matrix Completion [Sobral et al., 2015a], Tensor Completion [Sobral and Zahzah, 2016]
Neural Networks SC-SOBS [Maddalena and Petrosino, 2012], BEWiS [De Gregorio and Giordano, 2015]

Proposed approach
Low-rank
Reconstruction
original size reduced size
moving pixels
filled with zeros
Motion Detection
Frame Selection
+
background model
input images
Proposed approach to background model initialization.

Joint motion detection and frame selection
Frames
0 50 100 150 200 250 300
Differencebetween
consecutiveframes
0
0.2
0.4
0.6
0.8
1
Frame Selection
normalized vector derivative vector selected frames
Illustration of the frame selection operation.

Joint motion detection and frame selection
Number of selected frames after the frame selection process.
# Sequence Frames Selected Reduction τ
1 Board 228 64 71.93% 0.125
2 Candela m1.10 350 84 76.00% 0.100
3 CAVIAR1 610 88 85.57% 0.100
4 CAVIAR2 460 83 81.96% 0.125
5 CaVignal 258 65 74.81% 0.125
6 Foliage 394 68 82.74% 0.600
7 Hall&Monitor 296 94 68.24% 0.075
8 HighwayI 440 59 86.59% 0.100
9 HighwayII 500 49 90.20% 0.075
10 HumanBody2 740 86 88.38% 0.050
11 IBMtest2 90 33 63.33% 0.100
12 People&Foliage 341 55 83.87% 0.100
13 Snellen 321 70 78.19% 0.125
14 Toscana 6 6 0.00% -

Low-rank reconstruction process
10 20 30 40 50 60 70 80 90
1
2
3
4
5
6
7
8
× 104
10 20 30 40 50 60 70 80 90
1
2
3
4
5
6
7
8
× 104
10 20 30 40 50 60 70 80 90
1
2
3
4
5
6
7
8
× 104
10 20 30 40 50 60 70 80 90
1
2
3
4
5
6
7
8
× 104
TensorrepresentationMatrixrepresentation
Selected frames Moving regions (black pixels) Selected frames with missing entries
Tensor completion process
Low-rank reconstruction
Matrix completion process

Evaluated 13 matrix completion algorithms
List of matrix completion algorithms evaluated for BM initialization.
Type Method Main techniques Author(s)
RM
IALM Augmented Lagrangian [Lin et al., 2010]
RMAMR Augmented Lagrangian [Ye et al., 2015]
MF
SVP Hard thresholding [Jain et al., 2010]
OptSpace Grassmannian [Keshavan et al., 2010]
MC-NMF Non-negative factors [Xu et al., 2012]
LMaFit Alternating [Wen et al., 2012]
ScGrassMC Grassmannian [Ngo and Saad, 2012]
LRGeomCG Riemannian [Vandereycken, 2013]
GROUSE Online algorithm [Balzano and Wright, 2013]
OR1MP Matching pursuit [Wang et al., 2015]
GoDec Randomized [Zhou and Tao, 2011]
SSGoDec Randomized [Zhou and Tao, 2011]
GreGoDec Randomized [Zhou and Tao, 2013]
RM - Rank Minimization
MF - Matrix Factorization

Evaluated 10 tensor completion algorithms algorithms
List of tensor completion algorithms evaluated for BM initialization.
Type Method Main techniques Author(s)
CP
NCPC Non-negative factors [Xu and Yin, 2013]
BCPF Bayesian CP Factorization [Zhao et al., 2015]
TenALS Alternating [Jain and Oh, 2014]
SPC Smooth PARAFAC [Yokota et al., 2016]
TD
HoRPCA-IALM Augmented Lagrangian [Goldfarb and Qin, 2014]
FaLRTC Trace norm [Liu et al., 2013b]
geomCG Riemannian [Kressner et al., 2013]
TMac Alternating [Xu et al., 2015b]
t-SVD Fourier domain [Zhang et al., 2014]
t-TNN Nuclear norm [Hu et al., 2015]
CP - CANDECOMP/PARAFAC decomposition.
TD - Tucker decomposition / HOSVD / N-mode SVD.

Dataset
Scene Background Initialization (SBI) dataset
The SBI dataset1 [Maddalena and Petrosino, 2015] contains 14 image sequences and their
corresponding ground truth backgrounds.
1
http://sbmi2015.na.icar.cnr.it/SBIdataset.html

Qualitative results (top-5 algorithms)
Frame
Ground truth
LRGeomCG
LMaFit
RMAMR
MC-NMF
TMac

Quantitative results
Summary of the top-1 best algorithms for each scene.
Scenes Top-1 MC Top-1 TC Scene Top-1
Board IALM TMac M IALM
Candela m1.10 LRGeomCG SPC T SPC
CAVIAR1 LMaFit TMac M LMaFit
CAVIAR2 LRGeomCG TMac M LRGeomCG
CaVignal LRGeomCG TMac M LRGeomCG
Foliage GROUSE TMac M LRGeomCG
HallAndMonitor LRGeomCG t-TNN T t-TNN
HighwayI RMAMR TMac M RMAMR
HighwayII IALM TMac M IALM
HumanBody2 LRGeomCG TMac M LRGeomCG
IBMtest2 LMaFit TMac M LMaFit
PeopleAndFoliage LRGeomCG TMac M LRGeomCG
Snellen LRGeomCG TMac M LRGeomCG
Toscana LRGeomCG SPC M LRGeomCG
M Matrix-based completion.
T Tensor-based completion.

Comparison with the state-of-the art
Comparison with the state-of-the art methods [Maddalena and Petrosino, 2015]. The best
scores are in bold, and the top-1 matrix and tensor completion algorithms are highlighted in red
and blue, respectively.
Method AGE pEPs pCEPs MS-SSIM PSNR CQM
Mean 14.1944 22.5150 18.4428 0.8737 25.6980 43.5839
Color Median 10.3744 13.4008 10.5571 0.8533 28.0044 42.4746
MOG2 14.3579 4.0847 2.8080 0.8935 25.9576 38.1916
KNN 20.6968 7.5118 4.5180 0.7595 18.4701 26.3836
BE-AAPSA 11.4846 12.5518 10.0605 0.9247 27.8024 41.8124
WS2006 5.2885 3.5335 1.2118 0.9349 28.8791 39.6334
IMBS-MT 4.2092 3.8819 2.2602 0.9598 33.4090 44.9362
LaBGen 2.9945 1.3972 0.9246 0.9764 35.2028 47.2947
RSL2011 5.8228 5.3511 4.0186 0.9172 29.9272 40.5713
Photomontage 5.8238 4.6952 3.7274 0.9334 31.8573 43.9038
LRGeomCG 8.7644 14.1305 11.0810 0.9302 28.9596 45.5625
TMac 8.8685 14.3577 11.2884 0.9284 28.7507 45.4125
SC-SOBS 1 3.5023 4.1508 2.2295 0.9765 35.2723 50.1138
SC-SOBS 2 4.6049 4.7435 2.5370 0.9645 32.2024 45.7614
BEWIS 3.8665 2.4286 1.4238 0.9675 32.0143 44.3728
http://sbmi2015.na.icar.cnr.it/SBIdataset.html

Remarks
The ﬁrst four best ranked algorithms (headed by LRGeomCG) are based
on the matrix completion approach.
SBI dataset is based on RGB color images – may not be multidimensional enough for
the power of tensor completion methods.
Tensor-based approaches has been highlighted only on two scenes: Candela m1.10 by
SPC and HallAndMonitor by t-TNN.
Related publications:
(SBMI/ICIAP, 2015, [Sobral et al., 2015a])
(PRL, 2016, [Sobral and Zahzah, 2016]).
MATLAB codes: https://github.com/andrewssobral/mctc4bmi.

Focus of the #3 contribution

Dealing with very dynamic background
#3 contribution

Context
The development of automatic video surveillance applications for
maritime environment is a very diﬃcult task due to the complexity of the
scenes: moving water, waves, etc.
The motion of the objects of interest (i.e. ships or boats) can be mixed with the
dynamic behavior of the background (non-regular patterns).

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.

Proposed method
Combine some ideas of [Oreifej et al., 2013] and [Ye et al., 2015].
The weighting matrix proposed by [Ye et al., 2015] can be used as a
shape constraint (or region constraint),
While the conﬁdence map proposed by [Oreifej et al., 2013]
reinforces the pixels belonging from the moving objects.

Proposed method
Combine some ideas of [Oreifej et al., 2013] and [Ye et al., 2015].
The weighting matrix proposed by [Ye et al., 2015] can be used as a
shape constraint (or region constraint),
While the confidence map proposed by [Oreifej et al., 2013]
reinforces the pixels belonging from the moving objects.
Moreover,
Instead of using dense optical flow (temporal descriptor) as a
preliminary step, we suggest to use a saliency detector (spatial
descriptor).
We call our approach as SCM-RPCA (Shape and Confidence Map-based
RPCA).

Why a spatial descriptor?
In some cases:
The object of interest can move very slowly (e.g. long distance boats).
The background can be very dynamic (e.g. boats in the sea).
Optical ﬂow may not be suﬃcient to ensure the object detection.

Why a spatial descriptor?
In some cases:
The object of interest can move very slowly (e.g. long distance boats).
The background can be very dynamic (e.g. boats in the sea).
Optical flow may not be sufficient to ensure the object detection.
Moreover,
The dense optical flow computation requests high computational cost, while
computing the saliency map is commonly faster.
Here, the BMS2 method proposed by [Zhang and Sclaroff, 2014] was
selected, due to its speed performance and accuracy results.
2
http://cs-people.bu.edu/jmzhang/BMS/BMS.html

Block diagram of the proposed approach
(a) Input image (b) Saliency detection
(c) Object confidence map
(d) Shape constraint
(e) Foreground mask
RPCA
The double constraints (conﬁdence map and shape) can be built from two diﬀerent
types of source, but here we focus only on spatial saliency maps.

Comparison of the SCM-RPCA and related works.
Author(s) Minimization
Single constraint
[Oreifej et al., 2013] minimize
L,S,E
||L||∗+λ1||Π(S)||1+λ2||E||2
F
subject to A = L + S + E
[Ye et al., 2015] minimize
L,S,E
||L||∗+λ1||S||1+λ2||E||2
F
subject to W ◦ A = W ◦ (L + S + E)
Double constraint
SCM-RPCA (proposed)
[Sobral et al., 2015b]
minimize
L,S,E
||L||∗+λ1||Π(S)||1+λ2||E||2
F
subject to A = L + W ◦ S + E
W weighting matrix / shape constraint (binary case)
Π(.) conﬁdence map

Datasets
UCSD
[Mahadevan and Vasconcelos, 2010]
MarDT
[Bloisi et al., 2013]
The UCSDa and MarDTb datasets consist of 18 and 28 video sequences, respectively, both
acquired from stationary and moving cameras.
a
http://www.svcl.ucsd.edu/projects/background_subtraction/ucsdbgsub_dataset.htm
b
http://www.dis.uniroma1.it/~labrococo/MAR/index.htm
Four sequences from UCSD and three sequences from MarDT were selected, and all sequences
come from stationary cameras.

Evaluated algorithms
The SCM-RPCA was compared with its direct competitors:
PCP [Cand`es et al., 2011].
Stable PCP [Aravkin et al., 2014].
3WD [Oreifej et al., 2013]
RMAMR [Ye et al., 2015].
PCP and stable PCP are not constrained, while 3WD and RMAMR are
single-constrained RPCA.
Here, 3WD and RMAMR used saliency maps (instead of optical ﬂow) as
input constraint.

Visual comparison over UCSD dataset
Input frame Saliency maps from BMS SCM-RPCA 3WD RMAMRGround truth
surfersboatsbirds

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.

Visual results
Input Background Foreground

Quantitative results on UCSD dataset
Quantitative results on four videos of UCSD Background Subtraction Dataset.
Birds Surfers Boats Ocean Rank
Re Pr F1 Re Pr F1 Re Pr F1 Re Pr F1 Avg.F1
PCP 0.842 0.094 0.170 0.754 0.075 0.137 0.814 0.100 0.178 0.748 0.115 0.200 0.171
Lag-SPCP-QN 0.413 0.322 0.362 0.244 0.282 0.261 0.405 0.215 0.281 0.484 0.313 0.380 0.321
RMAMR 0.823 0.229 0.358 0.775 0.248 0.376 0.816 0.230 0.359 0.777 0.175 0.286 0.345
3WD 0.586 0.604 0.595 0.538 0.405 0.462 0.673 0.473 0.556 0.563 0.337 0.422 0.509
SCM-RPCA 0.573 0.638 0.604 0.518 0.565 0.541 0.663 0.550 0.602 0.457 0.544 0.497 0.561
The SCM-RPCA outperformed the previous methods with the highest F-measure
average over the selected video sequences.

Computational cost evaluation on UCSD dataset
Computational cost evaluation over four videos of UCSD Background Subtraction Dataset.
Birds Surfers Boats Ocean
(242 × 156 × 71) (344 × 224 × 41) (344 × 224 × 31) (316×196×176)
Iter Time∗
Iter Time∗
Iter Time∗
Iter Time∗
PCP +
100 27.29 +
100 21.19 +
100 18.47 +
100 110.53
Lag-SPCP-QN 29 10.12 53 16.27 39 10.01 18 29.49
RMAMR 34 10.63 35 13.09 33 11.44 35 44.22
3WD 30 4.53 26 4.28 31 4.06 42 29.96
SCM-RPCA 29 4.59 25 4.37 27 3.82 43 33.02
(width × height × length) denotes the frame resolution and the number of processed frames.
∗
Time for matrix decomposition (in seconds). Does not include the time to compute the input constraint (saliency maps).
+
Iteration limit 100 reached.
The algorithms are 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.

Remarks
The experimental results of the SCM-RPCA indicate a better enhancement
of the object foreground mask when compared with its direct competitors.
The combination with conﬁdence map and shape constraint can reduce
the amount of false positive pixels.
The SCM-RPCA algorithm has a slightly change in the number of
iterations and computation time compared to the original 3WD.
Related publication: (IEEE AVSS, 2015, [Sobral et al., 2015b]).
MATLAB codes: https://sites.google.com/site/scmrpca/.

Dealing with multidimensional and
streaming data
#4 contribution

Context
Most of background subtraction algorithms were designed for mono (i.e.
graylevel) or trichromatic cameras (i.e. RGB) within the visible spectrum
or near infrared part (NIR).
Recent advances in multispectral imaging technologies give the possibility
to record multispectral videos for video surveillance
applications [Benezeth et al., 2014].

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.

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].

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.
Two widely-used models for low rank decomposition on tensors are:
Tucker/Tucker3 decomposition.
CANDECOMP/PARAFAC (CP) decomposition.

Tucker vs CP decomposition
Tucker decomposition
CP decomposition

RPCA on tensors
Recently, 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.

Proposed approach
Most of tensor subspace learning approaches has the following drawbacks:
Apply matrix SVD into the unfolded matrices (computationally
expensive, especially for large matrices).
Work in a batch manner (not suitable for streaming multispectral
video sequences).
In order to overcome these limitations, we extended the online stochastic
RPCA proposed by [Feng et al., 2013] for tensors.
A stochastic optimization is applied on each mode of the tensor.
The low-dimensional subspace is updated iteratively followed by
processing of one video frame per time instance.

Comparison
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 (proposed approach) [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.

Dataset
MVS (Multispectral Video Sequences) dataset [Benezeth et al., 2014]
The proposed method was evaluated on MVS dataset. This is the ﬁrst dataset on MVSa
available for research community in background subtraction.
a
http://ilt.u-bourgogne.fr/benezeth/projects/ICRA2014/
The MVS dataset contains a set of 5 video sequences with 7 multispectral bands (6 visible
spectra and 1 NIR spectrum). Each sequence presents a well known BS challenge, such as color
saturation and dynamic background.

The proposed approach 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.
Due to this limitation, they were applied for each 100 frames at time
(reducing the computational cost) of the whole video sequence
(fourth-order tensor).
The proposed method processes each multispectral image or RGB image
per time instance.

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.

Qualitative results II
Video GT Proposed

Qualitative results III
Input
Low-rank
Sparse
Mask
RGB VS-1 VS-2 VS-3 VS-4 VS-5 VS-6 NIR
Visual results of the proposed method on each RGB and multispectral
band.

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
BRTF
Proposed

Computational time
Computational time for the ﬁrst 100 frames varying the image resolution.
Size HORPCA CP-ALS BRTF Proposed
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.

Remarks
Experimental results show that the proposed methodology outperforms the
other considered approaches.
We have achieved almost real time processing, since one video frame is
processed at time.
Related publications:
(ICIAR, 2014, [Sobral et al., 2014]).
(IEEE ICCV Workshop on RSL-CV, 2015, [Sobral et al., 2015c]).
MATLAB codes: https://github.com/andrewssobral/ostd.

Collaborative external research
contribution

Collaborative research with Jordi Gonzalez at CVC
(Barcelona, Spain)
Evaluation of subspace clustering algorithms to the problem of
human action recognition from 3D skeletal data (work in progress).
Robust subspace clustering of human activities through skeletal data.
Diﬀerently from previous approaches, subspace clustering methods consider
the inliers are drawn from the union of low-dimensional subspaces.

Building the actions representation matrix
Construction of the action representation matrix.
Temporal modeling procedure applied in the skeletal representation to deal with rate variations,
temporal misalignment, and noise.
* From [Vemulapalli et al., 2014].

Setup
Datasets for human action recognition from 3D skeletal data.
Dataset # of actions # of subjects # of sequences
UTKinect-Action [Xia et al., 2012]3 10 10 199
Florence3D-Action [Seidenari et al., 2013]4 9 10 215
Skeletal representations:
AJP (Absolute Joint Positions).
RJP (Relative Joint Positions).
JAQ (Joint Angles Quaternions).
SE3AP (SE3 Lie Algebra with Absolute Pairs) [Vemulapalli et al., 2014].
SE3RP (SE3 Lie Algebra with Relative Pairs) [Vemulapalli et al., 2014].
3
http://cvrc.ece.utexas.edu/KinectDatasets/HOJ3D.html
4
http://www.micc.unifi.it/vim/datasets/3dactions/

Selected subspace clustering algorithms for evaluation on skeletal action datasets.
Representation Method Author(s)
low-rank
LRR [Liu et al., 2013a]
LRSC [Vidal and Favaro, 2014]
sparse
SSC [Elhamifar and Vidal, 2009]
RSSC [Xu et al., 2015a]
LS3C [Patel et al., 2013]

Preliminary results (work in progress)
Performance comparison with state-of-the-art methods.
Author(s) Approach Recognition rate
UTKinect-Action dataset
[Xia et al., 2012] Histograms of 3D joints 90.92%
[Zhu et al., 2013] Random forests 87.90%
[Vemulapalli et al., 2014] Points in a Lie Group 97.08%
proposed LRSC + AJP or RSSC + RJP 95.10%
Florence3D-Action dataset
[Seidenari et al., 2013] Multi-Part Bag-of-Poses 82.00%
[Cippitelli et al., 2016] Key poses 82.10%
[Vemulapalli et al., 2014] Points in a Lie Group 90.88%
proposed RSSC + AJP 79.00%

Conclusions

Hierarchical overview of the DLSM framework

Summary and contributions
The thesis presented here has provided the following contributions:
A uniﬁed model for low-rank and sparse decomposition.
Matrix/tensor completion methodology for background model
initialization.
Double-constrained version of RPCA for robust foreground detection
in dynamic background.
Tensor-based methods for background/foreground separation in
multidimensional streaming data.
A collaborative work in conjunction with CVC/UAB that enabled the
European Label of this thesis, and a publication project.
Finally, a new library, named LRSLibrary, that provides a collection of
low-rank and sparse decomposition algorithms.

Future perspectives
Matrix/tensor completion methodology
More robust approach for frame-selection.
Evaluation of incremental and real-time approaches.

Future perspectives
SCM-RPCA
How diﬀerent sources can improve the foreground segmentation.
Development of an incremental version for streaming applications.

Future perspectives
SCM-RPCA
How diﬀerent sources can improve the foreground segmentation.
Development of an incremental version for streaming applications.
Tensor-based methods
Consider the recent advances on randomized
RPCA [Erichson et al., 2016].
Implementation C/C++ with GPU support for high scalability.

Publications I
The thesis has led to the following publications5:
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),
20166.
Journal papers (4)
2017 - Sobral, Andrews; Gong, Wenjuan; Gonzalez, Jordi; Bouwmans, Thierry; Zahzah,
El-hadi. “Robust Subspace Clustering of Human Activities from 3D Skeletal Data”, (in
progress).
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].

Publications II
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].
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].
Books (1)
2017 - Bouwmans, Thierry; Sobral, Andrews; Zahzah, El-hadi. Handbook on
“Background Subtraction for Moving Object Detection: Theory and Practices”, (in
progress)7.
Book chapters (2)
2017 - Sobral, Andrews; Bouwmans, Thierry; Zahzah, El-hadi. “Robust Tensor Models”.
Chapter in the handbook “Background Subtraction for Moving Object Detection: Theory
and Practices”, (in progress).

Publications III
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].
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].

Publications IV
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].
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].
5
The reader can refer to https://scholar.google.fr/citations?user=0Nm0uHcAAAAJ for an updated list of publications
and their citations.
6
http://rfia2016.iut-auvergne.com/index.php/autres-evenements/
detection-d-objets-mobiles-par-soustraction-de-fond
7
https://sites.google.com/site/foregrounddetection/

LRSLibrary
A new library, named LRSLibrary [Sobral et al., 2016]a
that 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.
It has been fundamental for all the experiments conducted in the thesis.

[Agarwala et al., 2004] Agarwala, A., Dontcheva, M., Agrawala, M., Drucker, S., Colburn, A., Curless, B., Salesin, D., and
Cohen, M. (2004). Interactive digital photomontage. In ACM SIGGRAPH, SIGGRAPH ’04, pages 294–302, New York, NY,
USA. ACM.
[Aravkin et al., 2014] Aravkin, A. Y., Becker, S., Cevher, V., and Olsen, P. (2014). A variational approach to stable principal
component pursuit. The Conference on Uncertainty in Artificial Intelligence.
[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.
[Balzano and Wright, 2013] Balzano, L. and Wright, S. J. (2013). On GROUSE and incremental SVD. In IEEE International
Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).
[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).
[Bloisi et al., 2013] Bloisi, D. D., Iocchi, L., and Pennisi, A. (2013). Mar - maritime activity recognition dataset.
[Bloisi et al., 2016] Bloisi, D. D., Pennisi, A., and Iocchi, L. (2016). Parallel multi-modal background modeling. Pattern
Recognition Letters, pages –.
[Bouwmans, 2014] Bouwmans, T. (2014). Traditional and recent approaches in background modeling for foreground detection:
An overview. In Computer Science Review.
[Bouwmans et al., 2017] Bouwmans, T., Maddalena, L., and Petrosino, A. (2017). Scene background initialization: a
taxonomy. Pattern Recognition Letters.
[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.
[Candès et al., 2011] Candès, E. J., Li, X., Ma, Y., and Wright, J. (2011). Robust Principal Component Analysis? Journal of
the ACM.
[Chang et al., 2015] Chang, X., Nie, F., Ma, Z., Yang, Y., and Zhou, X. (2015). A convex formulation for spectral shrunk
clustering. In AAAI Conference on Artificial Intelligence.
[Cippitelli et al., 2016] Cippitelli, E., Gasparrini, S., Gambi, E., and Spinsante, S. (2016). A human activity recognition system
using skeleton data from rgbd sensors. Journal of Computational Intelligence and Neuroscience.Andrews Cordolino Sobral (L3I/MIA) Université de La Rochelle 114

[Agarwala et al., 2004] Agarwala, A., Dontcheva, M., Agrawala, M., Drucker, S., Colburn, A., Curless, B., Salesin, D., and
Cohen, M. (2004). Interactive digital photomontage. In ACM SIGGRAPH, SIGGRAPH ’04, pages 294–302, New York, NY,
USA. ACM.
[Aravkin et al., 2014] Aravkin, A. Y., Becker, S., Cevher, V., and Olsen, P. (2014). A variational approach to stable principal
component pursuit. The Conference on Uncertainty in Artificial Intelligence.
[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.
[Balzano and Wright, 2013] Balzano, L. and Wright, S. J. (2013). On GROUSE and incremental SVD. In IEEE International
Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).
[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).
[Bloisi et al., 2013] Bloisi, D. D., Iocchi, L., and Pennisi, A. (2013). Mar - maritime activity recognition dataset.
[Bloisi et al., 2016] Bloisi, D. D., Pennisi, A., and Iocchi, L. (2016). Parallel multi-modal background modeling. Pattern
Recognition Letters, pages –.
[Bouwmans, 2014] Bouwmans, T. (2014). Traditional and recent approaches in background modeling for foreground detection:
An overview. In Computer Science Review.
[Bouwmans et al., 2017] Bouwmans, T., Maddalena, L., and Petrosino, A. (2017). Scene background initialization: a
taxonomy. Pattern Recognition Letters.
[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.
[Candès et al., 2011] Candès, E. J., Li, X., Ma, Y., and Wright, J. (2011). Robust Principal Component Analysis? Journal of
the ACM.
[Chang et al., 2015] Chang, X., Nie, F., Ma, Z., Yang, Y., and Zhou, X. (2015). A convex formulation for spectral shrunk
clustering. In AAAI Conference on Artificial Intelligence.
[Cippitelli et al., 2016] Cippitelli, E., Gasparrini, S., Gambi, E., and Spinsante, S. (2016). A human activity recognition system
using skeleton data from rgbd sensors. Journal of Computational Intelligence and Neuroscience.Andrews Cordolino Sobral (L3I/MIA) Université de La Rochelle 115

PhD Thesis Defense Presentation: Robust Low-rank and Sparse Decomposition for Moving Object Detection - From Matrices to Tensors

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