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Comparison of Matrix Completion Algorithmsfor Background Initialization in Videos
Andrews Sobral1,2(B), Thierry Bouwmans2, and El-hadi Zahzah1
1 Lab. L3I, Universite de La Rochelle, 17000 La Rochelle, France2 Lab. MIA, Universite de La Rochelle, 17000 La Rochelle, France
Abstract. Background model initialization is commonly the first stepof the background subtraction process. In practice, several challengesappear and perturb this process such as dynamic background, boot-strapping, illumination changes, noise image, etc. In this context, thiswork aims to investigate the background model initialization as a matrixcompletion problem. Thus, we consider the image sequence (or video)as a partially observed matrix. First, a simple joint motion-detectionand frame-selection operation is done. The redundant frames are elimi-nated, and the moving regions are represented by zeros in our observationmatrix. The second stage involves evaluating nine popular matrix com-pletion algorithms with the Scene Background Initialization (SBI) dataset, and analyze them with respect to the background model challenges.The experimental results show the good performance of LRGeomCG [17]method over its direct competitors.
Keywords: Matrix completion · Background modeling · Backgroundinitialization
1 Introduction
Background subtraction (BS) is an important step in many computer vision sys-tems to detect moving objects. This basic operation consists of separating themoving objects called “foreground” from the static information called “back-ground” [2,16]. The BS is commonly used in video surveillance applications todetect persons, vehicles, animals, etc., before operating more complex processesfor intrusion detection, tracking, people counting, etc. Typically the BS processincludes the following steps: a) background model initialization, b) backgroundmodel maintenance and c) foreground detection. With a focus on the step (a), theBS initialization consists in creating a background model. In a simple way, thiscan be done by setting manually a static image that represents the background.The main reason is that it is often assumed that initialization can be achievedby exploiting some clean frames at the beginning of the sequence. Naturally,this assumption is rarely encountered in real-life scenarios, because of contin-uous clutter presence. In addition, this procedure presents several limitations,c© Springer International Publishing Switzerland 2015V. Murino et al. (Eds.): ICIAP 2015 Workshops, LNCS 9281, pp. 510–518, 2015.DOI: 10.1007/978-3-319-23222-5 62
Comparison of Matrix Completion Algorithms 511
because it needs a fixed camera with constant illumination, and the backgroundneeds to be static (commonly in indoor environments), and having no movingobject in the first frames. In practice, several challenges appear and perturb thisprocess such as noise acquisition, bootstrapping, dynamic factors, etc [11].
The main challenge is to obtain a first background model when more thanhalf of the video frames contain foreground objects. Some authors suggest theinitialization of the background model by the arithmetic mean [9] (or weightedmean) of the pixels between successive images. Practically, some algorithms are:(1) batch ones using N training frames (consecutive or not), (2) incrementalwith known N or (3) progressive ones with unknown N as the process gen-erates partial backgrounds and continues until a complete background image isobtained. Furthermore, initialization algorithms depend on the number of modesand the complexity of their background models. However, BS initialization hasalso been achieved by many other methodologies [2,11]. We can cite for examplethe computation of eigen values and eigen vectors [15], and the recent researchon subspace estimation by sparse representation and rank minimization [3]. Thebackground model is recovered by the low-rank subspace that can graduallychange over time, while the moving foreground objects constitute the correlatedsparse outliers.
In this paper, the initialization of the background model is addressed asa matrix completion problem. The matrix completion aims at recovering alow rank matrix from partial observations of its entries. The image sequence(or video) is represented as a partially observed real-valued matrix. Figure 1shows the proposed framework. First, a simple joint motion-detection and frame-selection operation is done. The redundant frames are eliminated, and the mov-ing regions are represented with zeros in our observation matrix. This operationis described in the Section 2. The second stage involves evaluating nine popularmatrix completion algorithms with the Scene Background Initialization (SBI)data set [12] (see Section 3). This enables to analyze them with respect to thebackground model challenges. Finally, in Sections 4 and 5, the experimentalresults are shown as well as conclusions.
Throughout the paper, we use the following notations. Scalars are denotedby lowercase letters, e.g., x; vectors are denoted by lowercase boldface letters,e.g., x; matrices by uppercase boldface, e.g., X. In this paper, only real-valueddata are considered.
2 Joint Motion Detection and Frame Selection
In order to reduce the number of redundant frames, a simple joint motion detec-tion and frame selection operation is applied. First, the color images are con-verted into its gray-scale representation. So, let a sequence of N gray-scale images(frames) I0 . . . IN captured from a static camera, that is, I ∈ R
m×n where mand n denote the frame resolution (rows by columns). The difference betweentwo consecutive frames (motion detection step) is calculated by:
Dt =√
(It − It−1)2 | t= 1,...,N , (1)
512 A. Sobral et al.
Matrix Comple�on
original size reduced size
moving pixels filled with zeros
Mo�on Detec�onFrame Selec�on
+
background model
input images
Fig. 1. Block diagram of the proposed approach. Given an input image, a joint motiondetection and frame selection operation is applied. Next, a matrix completion algorithmtries to recover the background model from the partially observed matrix. In this paper,the processes described here are conducted in a batch manner.
where Dt ∈ Rm×n denotes the matrix of pixel-wise L2-norm differences from
frame t − 1 to frame t. Next, the sum of all elements of Dt, for t = 1, . . . , N , isstored in a vector d ∈ R
N whose t-th element is given by:
dt =m∑
i=1
n∑
j=1
Dt(i, j), (2)
where Dt(i, j) is the matrix element located in the row i ∈ [1, . . . , m] and columnj ∈ [1, . . . , n]. Then, the vector d is normalized between 0 and 1 by:
d =dt − dmin
dmax − dmin| t= 1,...,N , (3)
where dmin and dmax denote the minimum value and the maximum value of thevector d. The frame selection step is done by calculating the derivative of d by:
d′ =d
dtd, (4)
Next, the vector d′ is also normalized by Equation 3 and represented by d′.
Finally, the index of the more relevant frames are given by thresholding d′:
(5)y =
{1 if |d′ − μ′| > τ
0 otherwise,
where μ′ denotes the mean value of the vector d′, and τ ∈ [0, . . . , 1] controls the
threshold operator. In this paper, R ≤ N represent the set of all frames wherey = 1, and the parameter τ was chosen experimentally for each scene: τ = 0.025for HallAndMonitor, τ = 0.05 for HighwayII, τ = 0.10 for HighwayI, and τ =0.15 to all other scenes. Figure 3 illustrates our frame selection operation, in thisexample, with τ = 0.025, only 92 relevant frames are selected from a total of296 frames (68, 92% of reduction). In the next section, the matrix completionprocess is described.
Comparison of Matrix Completion Algorithms 513
Frames0 50 100 150 200 250 300
Diff
eren
ce b
etw
een
cons
ecut
ive
fram
es
0
0.2
0.4
0.6
0.8
1Frame Selection
vector d normalized derivative of vector d selected frames
Fig. 2. Illustration of frame selection operation. The normalized vector (in blue) showsthe difference between two consecutive frames. The derivative vector draw how muchthe normalized vector changes (in red), and then it is thresholded and the frames areselected (in orange).
3 Matrix Completion
As explained previously, the matrix completion aims to recover a low rank matrixfrom partial observations of its entries. Considering the general form of low rankmatrix completion, the optimization problem is to find a matrix L ∈ R
n1×n2
with minimum rank that best approximates the matrix A ∈ Rn1×n2. Candes
and Recht [6] show that this problem can be formulated as:
minimize rank(A),subject to PΩ(A) = PΩ(L),
(6)
where rank(A) is equal to the rank of the matrix A, and PΩ denotes the samplingoperator restricted to the elements of Ω (set of observed entries), i.e., PΩ(A) hasthe same values as A for the entries in Ω and zero values for the entries outsideΩ. Later, Candes and Recht [6] propose to replace the rank(.) function withthe nuclear norm ||A||∗=
∑ri=1 σi where σ1, σ2, ..., σr are the singular values of
A and r is the rank of A. The nuclear norm make the problem tractable andCandes and Recht [6] have proved theoretically that the solution can be exactlyrecovered with a high probability. In addition, Cai et. al [4] propose an algorithmbased on soft singular value thresholding (SVT) to solve this convex relaxationproblem. However, in real world application the observed entries may be noisy.In order to make the Equation 6 robust to noise, Candes and Plan [5] proposea stable matrix completion approach. The equality constraint is replaced by||PΩ(A − L)||F≤ ε, where ||.||F denotes the Frobenious norm and ε is an upperbound on the noise level. Recently, several matrix completion algorithms havebeen proposed to deal with this challenge, and a complete review can be foundin [21].
In this paper, we address the background model initialization as a matrixcompletion problem. Once frame selection process is done, the moving regions
514 A. Sobral et al.
Fig. 3. Illustration of the matrix completion process. From the left to the right: a) theselected frames in vectorized form (our observation matrix), b) the moving regions arerepresented by non-observed entries (black pixels), c) the moving regions filled withzeros (modified version of the observation matrix), and d) the recovered matrix afterthe matrix completion process.
of the R selected frames are determined by:
(7)Mk(i, j) =
{1 if 0.5(Dk(i, j))2 > β
0 otherwise
where k ∈ R, and β is the thresholding parameter (in this paper, β = 1e−3 forall experiments). Next, the moving regions of each selected frame are filled withzeros by Ik ◦ Mk, where Mk denotes the complement of Mk, and ◦ denotes theelement-wise multiplication of two matrices. For color images, each channel isprocessed individually, then they are vectorized into a partially observed real-valued matrix A = [vec(I1) . . . vec(Ik)], where A ∈ R
n1×n2, n1 = (m × n), andn2 = k. Figure 3 illustrates our matrix completion process. It can be seen that thepartially observed matrix can be recovered successfully even with the presence ofmany missing entries. So, let L the recovered matrix from the matrix completionprocess, the background model is estimated by calculating the average value ofeach row, resulting in a vector l ∈ R
n1×1, and then reshaped into a matrixB ∈ R
m×n.
4 Experimental Results
In order to evaluate the proposed approach, nine matrix completion algorithmshave been selected, and they are listed in Table 1. The algorithms were groupedin two categories, as well as its main techniques (following the same definitionof Zhou et al. [21]).
In this paper, the Scene Background Initialization (SBI) data set was cho-sen for the background initialization task. The data set contains seven imagesequences and corresponding ground truth backgrounds. It provides also MAT-LAB scripts for evaluating background initialization results in terms of eightmetrics1. Figure 4 show the visual results for the top three best matrix comple-tion algorithms, and Table 2 reports the quantitative results of each algorithm1 Please, refer to http://sbmi2015.na.icar.cnr.it/ for a complete description of each
metric.
Comparison of Matrix Completion Algorithms 515
Table 1. List of low-rank matrix completion algorithms evaluated in this paper.
Category Method Main techniques Reference
Rank MinimizationIALM Augmented Lagrangian [10, Linetal.(2010)]RMAMR Augmented Lagrangian [20, Yeetal.(2015)]
Matrix Factorization
SVP Hard thresholding [13, Mekaetal.(2009)]OptSpace Grassmannian [8, Keshavanetal.(2010)]LMaFit Alternating [19, Wenetal.(2012)]ScGrassMC Grassmannian [14, NgoandSaad(2012)]LRGeomCG Riemannian [17, Vandereycken(2013)]GROUSE Online algorithm [1, Balzanoetal.(2013)]OR1MP Matching pursuit [18, Wangetal.(2015)]
Fig. 4. Visual comparison for the background model initialization. From top to bottom:1) example of input frame, 2) background model ground truth, and background modelresults for the top 3 best ranked MC algorithms: 3) LRGeomCG, 4) LMaFit, and 5)RMAMR.
over the data set2. The algorithms are ranked as follow: 1) for each algorithmwe calculate its rank position for each metric, we call it as metric rank (i.e.
2 Full experimental evaluation and related source code can be found in the mainwebsite: https://sites.google.com/site/mc4bmi/
516 A. Sobral et al.
Table
2.
Quanti
tati
ve
resu
lts
over
SB
Idata
set,
and
the
glo
balra
nk
for
each
matr
ixco
mple
tion
met
hod.T
he
bold
met
ric
valu
essh
owth
ebes
tsc
ore
for
each
met
ric.
For
each
scen
e,th
ere
sult
sare
ord
ered
by
the
rank
colu
mn.
HallAndM
onitor
Method
AG
E?
EP
s?
pEP
s?
CEP
s?
pC
EP
S?
MSSSIM
?P
SN
R?
CQ
M?
Scene
Rank?
LRG
eom
CG
2.0
550
190
0.0
022
00.0
000
0.9
938
37.9
811
46.3
255
1RM
AM
R2.0
499
190
0.0
022
00.0
000
0.9
938
37.9
755
46.3
222
2LM
aFit
2.0
583
194
0.0
023
00.0
000
0.9
938
37.8
487
46.2
893
3ScG
rass
MC
2.0
693
193
0.0
023
00.0
000
0.9
937
37.9
046
46.2
838
4G
RO
USE
2.2
201
191
0.0
023
00.0
000
0.9
923
37.5
319
45.8
794
5O
R1M
P2.2
025
374
0.0
044
00.0
000
0.9
926
34.9
021
44.8
283
6IA
LM
3.6
143
2336
0.0
277
1190
0.0
141
0.9
627
30.5
907
40.9
249
7SVP
4.1
486
3230
0.0
382
1574
0.0
186
0.9
474
28.2
921
41.4
398
8O
ptS
pace
6.7
051
5950
0.0
704
2694
0.0
319
0.9
299
25.1
834
37.6
430
9
Hig
hwayI
Method
AG
E?
EP
s?
pEP
s?
CEP
s?
pC
EP
S?
MSSSIM
?P
SN
R?
CQ
M?
Scene
Rank?
LRG
eom
CG
2.7
715
192
0.0
025
16
0.0
002
0.9
769
35.8
950
58.6
193
1RM
AM
R2.7
601
193
0.0
025
16
0.0
002
0.9
769
35.8
899
58.6
283
1LM
aFit
2.7
781
196
0.0
026
16
0.0
002
0.9
770
35.8
636
58.6
193
3ScG
rass
MC
3.2
306
1088
0.0
142
593
0.0
077
0.9
714
33.4
838
58.5
694
4G
RO
USE
6.1
324
621
0.0
081
138
0.0
018
0.9
614
30.3
643
56.2
861
5O
R1M
P3.9
587
1202
0.0
157
691
0.0
090
0.9
637
32.4
145
57.8
051
6IA
LM
6.4
223
1836
0.0
239
837
0.0
109
0.9
507
29.5
142
57.9
214
7SVP
6.9
160
5694
0.0
741
2530
0.0
329
0.9
095
27.6
621
53.1
073
8O
ptS
pace
14.7
067
19754
0.2
572
13930
0.1
814
0.7
957
22.8
624
43.8
142
9
Hig
hwayII
Method
AG
E?
EP
s?
pEP
s?
CEP
s?
pC
EP
S?
MSSSIM
?P
SN
R?
CQ
M?
Scene
Rank?
LRG
eom
CG
2.6
840
268
0.0
035
40.0
001
0.9
919
35.7
076
46.1
997
1RM
AM
R2.7
003
271
0.0
035
50.0
001
0.9
919
35.6
695
46.2
002
2LM
aFit
2.6
919
275
0.0
036
70.0
001
0.9
919
35.5
752
46.0
673
3ScG
rass
MC
2.9
622
360
0.0
047
20.0
000
0.9
888
34.6
782
46.1
585
4IA
LM
4.9
261
306
0.0
040
20.0
000
0.9
830
31.5
964
46.1
296
5O
R1M
P3.2
510
843
0.0
110
102
0.0
013
0.9
888
32.2
682
42.0
740
6SVP
4.7
779
945
0.0
123
153
0.0
020
0.9
813
30.4
590
41.8
017
7G
RO
USE
4.3
955
1751
0.0
228
700
0.0
091
0.9
756
31.5
542
45.6
062
8O
ptS
pace
8.6
231
4722
0.0
615
1307
0.0
170
0.9
279
25.8
722
36.7
496
9
CaVig
nal
Method
AG
E?
EP
s?
pEP
s?
CEP
s?
pC
EP
S?
MSSSIM
?P
SN
R?
CQ
M?
Scene
Rank?
LM
aFit
11.9
504
3788
0.1
393
2700
0.0
993
0.9
027
24.3
417
39.8
279
1LRG
eom
CG
11.9
506
3789
0.1
393
2700
0.0
993
0.9
026
24.3
415
39.8
279
2RM
AM
R12.0
081
3817
0.1
403
2715
0.0
998
0.9
027
24.3
147
39.8
083
3G
RO
USE
12.8
057
3624
0.1
332
2205
0.0
811
0.8
846
23.2
489
38.8
799
4ScG
rass
MC
12.3
375
4084
0.1
501
2942
0.1
082
0.8
916
23.9
111
39.8
237
5IA
LM
12.2
618
4764
0.1
751
3531
0.1
298
0.8
779
23.7
957
40.6
135
6O
R1M
P12.5
266
4455
0.1
638
3356
0.1
234
0.8
855
23.7
925
39.7
716
7SVP
13.2
230
5628
0.2
069
3982
0.1
464
0.8
760
23.3
352
39.8
300
8O
ptS
pace
14.1
744
6176
0.2
271
4214
0.1
549
0.8
927
23.0
940
39.8
662
9
Foliage
Method
AG
E?
EP
s?
pEP
s?
CEP
s?
pC
EP
S?
MSSSIM
?P
SN
R?
CQ
M?
Scene
Rank?
GRO
USE
26.4
036
17840
0.6
194
13271
0.4
608
0.8
957
18.4
042
33.4
059
1LRG
eom
CG
26.4
006
18074
0.6
276
13459
0.4
673
0.8
970
18.4
522
33.2
733
2LM
aFit
26.4
093
18070
0.6
274
13442
0.4
667
0.8
970
18.4
490
33.2
663
2RM
AM
R26.5
144
18146
0.6
301
13548
0.4
704
0.8
965
18.4
160
33.2
392
4ScG
rass
MC
29.2
509
19189
0.6
663
15385
0.5
342
0.8
645
17.4
636
33.2
173
5O
R1M
P31.3
678
19094
0.6
630
15257
0.5
298
0.8
221
16.7
364
33.0
468
5O
ptS
pace
32.1
405
19156
0.6
651
15234
0.5
290
0.8
656
16.6
268
31.4
830
7IA
LM
31.6
036
19451
0.6
754
14986
0.5
203
0.7
473
16.8
016
31.0
114
8SVP
35.3
522
19469
0.6
760
16003
0.5
557
0.7
556
15.6
496
33.1
273
9
People
AndFoliage
Method
AG
E?
EP
s?
pEP
s?
CEP
s?
pC
EP
S?
MSSSIM
?P
SN
R?
CQ
M?
Scene
Rank?
LRG
eom
CG
38.7
467
64150
0.8
353
59210
0.7
710
0.8
505
15.1
638
27.6
167
1O
ptS
pace
41.8
200
60020
0.7
815
53183
0.6
925
0.7
535
14.1
746
25.9
930
2IA
LM
42.9
444
63161
0.8
224
58560
0.7
625
0.7
951
14.1
751
27.8
900
2G
RO
USE
40.3
918
63556
0.8
276
57097
0.7
435
0.8
130
14.6
485
26.5
857
4O
R1M
P42.8
580
62335
0.8
117
56809
0.7
397
0.7
861
14.0
424
26.7
015
4ScG
rass
MC
39.6
045
64076
0.8
343
58991
0.7
681
0.8
472
14.9
449
27.6
499
6LM
aFit
38.7
738
64155
0.8
354
59220
0.7
711
0.8
501
15.1
567
27.6
143
7RM
AM
R38.8
234
64189
0.8
358
59276
0.7
718
0.8
502
15.1
484
27.5
987
8SVP
45.0
733
63916
0.8
322
58803
0.7
657
0.7
644
13.6
320
27.0
416
9
Snellen
Method
AG
E?
EP
s?
pEP
s?
CEP
s?
pC
EP
S?
MSSSIM
?P
SN
R?
CQ
M?
Scene
Rank?
ScG
rass
MC
43.8
219
17838
0.8
602
16070
0.7
750
0.8
469
14.3
702
37.6
451
1LRG
eom
CG
41.8
853
18621
0.8
980
17379
0.8
381
0.8
807
14.8
166
37.4
253
2O
ptS
pace
49.2
605
16619
0.8
015
15292
0.7
375
0.7
419
12.8
053
29.4
492
2LM
aFit
41.8
688
18623
0.8
981
17387
0.8
385
0.8
809
14.8
203
37.4
234
4G
RO
USE
41.8
123
18629
0.8
984
17403
0.8
393
0.8
809
14.8
421
37.5
570
5IA
LM
46.0
978
18433
0.8
889
17084
0.8
239
0.8
330
14.0
292
37.1
551
6O
R1M
P50.4
572
18084
0.8
721
16677
0.8
043
0.7
504
13.1
602
36.2
566
7SVP
54.6
990
17649
0.8
511
16189
0.7
807
0.7
105
12.5
506
35.6
896
7RM
AM
R42.0
476
18625
0.8
982
17395
0.8
389
0.8
800
14.7
872
37.3
821
9
Glo
balra
nk
over
all
scenes
Method
Glo
balrank?
LRG
eom
CG
1LM
aFit
2RM
AM
R3
ScG
rass
MC
3G
RO
USE
5IA
LM
6O
R1M
P6
OptS
pace
8SVP
9
Comparison of Matrix Completion Algorithms 517
RMAMR have the first position for the AGE metric in the HallAndMonitorscene), next, 2) we sum the rank position value of each algorithm over the eightmetrics, and finally, 3) we calculate the rank position over the sum, and we callit as scene rank. For the Global Rank, first we sum the scene rank for each MCalgorithm, then we calculate its rank position over the sum. As we can see, theexperimental results show the good performance of LRGeomCG [17] methodover its direct competitors. Furthermore, in most cases the matrix completionalgorithms outperform the traditional approaches such as Mean [9], Median [7]and MoG [22] as can be seen in the full experimental evaluation available athttps://sites.google.com/site/mc4bmi/.
5 Conclusion
In this paper, we have evaluated nine recent matrix completion algorithms for thebackground initialization problem. Given a sequence of images, the key idea is toeliminate the redundant frames, and consider its moving regions as non-observedvalues. This approach results in a matrix completion problem, and the backgroundmodel can be recovered even with the presence of missing entries. The experimen-tal results on the SBI data set shows the comparative evaluation of these recentmethods, and highlights the good performance of LRGeomCG [17] method overits direct competitors. Finally, MC shows a nice potential for background model-ing initialization in video surveillance. Future research may concern to evaluateincremental and real-time approaches of matrix completion in streaming videos.
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