Robust tensor subspace learning for anomaly detection

ORIGINAL ARTICLE

Robust tensor subspace learning for anomaly detection

Jie Li • Guan Han • Jing Wen • Xinbo Gao

Received: 28 November 2010 / Accepted: 1 March 2011 / Published online: 12 March 2011

� Springer-Verlag 2011

Abstract Background modeling plays an important role

in many applications of computer vision such as anomaly

detection and visual tracking. Most existing algorithms for

learning appearance model are vector-based methods

without maintaining the 2D spatial structure information of

objects in an image. To this end, a robust tensor subspace

learning algorithm is developed for background modeling

which can capture the appearance changes through adap-

tively updating the tensor subspace. In the tensor frame-

work, the spatial structure information is maintained and

utilized for feature extraction of objects. Then by incor-

porating the robust scheme, we can weight individual pixel

of an image to reduce the influence of outliers on back-

ground modeling. Furthermore an incremental algorithm

for the robust tensor subspace learning is proposed to adapt

to the variation of appearance model. The experimental

results illustrate the effectiveness of the proposed robust

learning algorithm for anomaly detection.

Keywords Background modeling � Tensor subspace �Robust learning � Incremental learning � Anomaly detection

1 Introduction

Anomaly detection in video surveillance by using station-

ary cameras to monitor an environment of interest has

gained much more attention to public security, due to the

increasing societal threats from terrorists and crime, which

can be accomplished by learning background model rep-

resenting normal state and identifying image regions that

anomalous with respect to that background model. In this

paper, we focus on detecting drastic changes of the ‘‘nor-

mal’’ background model. It is suitable for a wide variety of

scenarios, such as outdoor parking lots, airport lounge,

market hall entrance and etc. However, the main challenge

of background representation can be attributed to handling

the appearance variation of the scene over time. Illumina-

tion changes and camera shaking are regarded as extrinsic

appearance variation, whereas intrinsic changes are resul-

ted from object motion and pose variation in the scene [5].

Therefore, robust modeling such appearance variation is of

great importance to anomaly detection.

In recent years, many works have been done in back-

ground modeling. Oliver et al. [4] firstly proposed eigen-

background modeling by preforming principal component

analysis (PCA) method. The background model can be

represented by the mean image and linear combination of

the first p significant eigenvectors. However, the traditional

PCA is sensitive to outliers, which can be absorbed into the

background model. To enhance the robustness of PCA

model, Xu and Yuille [14] introduced a binary variable to

consider an entire contaminated image as an outlier and

discard it. Torre and Black [9] described a method of robust

subspace learning based on robust M-estimation, which is

capable of constructing weight for each pixel in every

image. However, these methods are performed in batch

mode, thus requiring all training images to be given in

advance. The model has to be recomputed from scratch if

the subspace is updated sequentially with new images [11].

Another drawback is intensive computation because the

optimization problem has to be solved iteratively. To tackle

these problems, several methods have been developed for

online learning. Li et al. [3] proposed an incremental and

J. Li � G. Han � J. Wen � X. Gao (&)

Video and Image Processing System Lab,

School of Electronic Engineering,

Xidian University, Xi’an 710071, China

e-mail: [email protected]; [email protected]

123

Int. J. Mach. Learn. & Cyber. (2011) 2:89–98

DOI 10.1007/s13042-011-0017-0

robust subspace learning algorithm. Moreover, a weighted

incremental PCA algorithm for subspace learning is pre-

sented [6], which enables assigning arbitrary temporal and

spatial weights. These aforementioned methods for back-

ground modeling represent an image as a vector in high-

dimensional space, ignoring the underlying local geometric

structure information. Consequently, model representation

with image-as-matrix learning methods has attracted much

more attention in recent years. Multilinear algebra is uti-

lized to make analysis of image ensembles resulting from

the confluence of different factors related to scene structure,

illumination, and viewpoint [13]. He et al. [1] presented a

new algorithm for image representation taking multilinear

algebra and differential geometry into consideration, called

tensor subspace analysis (TSA). Sun et al. [7] developed the

dynamic and streaming tensor analysis to solve high order

tensor problems. Li et al. [2] employed a tensor subspace

learning for visual tracking. Tao et al. [8] proposed a

Bayesian tensor analysis method and applied it to 3D face

modeling. However, these tensor-based algorithms share an

open problem that they do not introduce a robust analysis

scheme for subspace learning, as observations may contain

outliers in real world.

In this paper, in order to automatically learn and update

a low dimensional subspace representation of background

model in tensor space, a robust tensor subspace learning

(RTSL) is proposed. Firstly, an image is represented as a

second order tensor, or a matrix. Therefore, the relationship

between the row vectors of the matrix as well as that

between the column vectors could be captured for model

representation [1, 12]. Secondly, in combination with

robust analysis, a weighted tensor model is built to

decrease the influence of outliers on background modeling.

Thirdly, an incremental algorithm is developed to adap-

tively reflect the appearance changes of the scene by

updating the mean and covariance when a new observation

arrives, which enables efficient estimation of the tensor

subspace for real-time performance. Finally we apply the

proposed algorithm to video anomaly detection with sim-

ilarity measurement between the reference and current

background model.

The rest of this paper is organized as follows. The

details of the presented robust tensor subspace learning

algorithm are described in Sect. 2. Application of anomaly

detection is introduced in Sect. 3. Experimental results and

analysis are presented in Sect. 4. Finally we give a con-

clusion about this paper and our future work.

2 Robust tensor subspace learning

Multilinear algebra or the algebra of high-order tensors is

usually used to make analysis of multifactor structure of

image ensembles [11, 13]. In this section, we first briefly

introduce mathematical notation and operations for tensors.

Next we describe a weighted tensor representation for new

observations, weighting individual pixel with different

weights. Finally we propose an incremental learning

algorithm for tensor subspace to efficiently update the

eigenspace.

2.1 Tensor analysis

A tensor is a higher generalization of a vector (first order

tensor) and a matrix (second order tensor) [7]. Tensors

define multilinear operators over a set of vector spaces. In

this paper, we denote scalars by lower case letters (e.g., a),

vectors by bold lower case letters (e.g., a), matrices by bold

upper-case letters (e.g., A), and higher-order tensors by

Euclid math upper-case letters (e.g., A).

We denote an N-order tensor as A 2 RI1�I2�...�IN ; where

the dimensionality of the nth mode (or dimension) of A is

In (1 B n B N). Elements of A are denoted as

Aði1; i2; . . .; iNÞ or ai1i2;...;iN The mode-n matrix unfolding of

an N-order tensor A is vectors in RIn ; obtained by keeping

index n fixed and varying the other indices. Hence the

mode-n matrix unfolding A(n) is in RIn�

Qd 6¼n

Id

� �

.

The mode-n product of a tensor A 2 RI1�I2�...�IN and a

matrix U 2 RJn�In is denoted by A�n U; and defined as

ðA �n UÞði1; . . .; in�1; jn; inþ1; . . .iNÞ

¼XIn

in¼1

Aði1; i2; . . .; iNÞUðjn; inÞ: ð1Þ

The result is a tensor in RI1�...In�1�Jn�Inþ1�...IN . In general,

a tensor A can multiply a sequence of matrices

UnNn¼1 2 R

Jn�In�� as A�1 U1 � . . .�N UN 2 RJ1�J2�...�JN ;

which can be written as AQN

i¼1

�iUi. An N-order tensor

can also be decomposed as the mode-n product of the

orthogonal space spanned by UiNi¼1

�� ; which is defined as

A ¼ BYN

i¼1

�iUi; ð2Þ

where B is the core tensor defined as

B ¼ AYN

i¼1

�iUTi : ð3Þ

2.2 Weighting tensor representation

In general, the robust problem is that the positions of

outliers are unknown beforehand. However, it is aware that

observations may contain outliers, which are not consistent

90 Int. J. Mach. Learn. & Cyber. (2011) 2:89–98

123

with current background model [6]. The reconstruction

error of inliers should be smaller than that of outliers,

which makes outliers detection easier. This can be

accomplished by projecting new observations into the

current subspace with robust scheme. Therefore, weights

are constructed to control the influence of each pixel of an

image.

As described earlier, an image is actually a matrix,

which can be considered as a second order tensor. The

residual error of a new observation X is defined by

R ¼ X �XY2

i¼1

�i UiUTi

� �; ð4Þ

where XQ2

i¼1

�i UiUTi

� �is the approximation of X , so that

the residual error is minimized by

minXI1

i1¼1

XI2

i2¼1

r2i1i2; ð5Þ

where ri1i2 is the element of R. Through the robust M-

estimation method [3], the minimization problem can be

solved by a robust function q(r):

minXI1

i1¼1

XI2

i2¼1

q ri1i2ð Þ: ð6Þ

Differentiating (6) with respect to u, the element of Ui,

we obtain

XI1

i1¼1

XI2

i2¼1

/ ri1i2ð Þdri1i2

du¼ 0; ð7Þ

where /(x) = dq(x)/dx. A weight function can be defined

as w(x) = /(x)/x, and (7) can be written as

XI1

i1¼1

XI2

i2¼1

w ri1i2ð Þri1i2

ori1i2

ou¼ 0: ð8Þ

Then (8) can be regarded as a new problem:

minXI1

i1¼1

XI2

i2¼1

w ri1i2ð Þr2i1i2: ð9Þ

It is crucial to note that weights of each pixel can be

determined with respect to the residual error. We choose

the robust function as

qðxÞ ¼ k2

2log 1þ x

k

� �2� �

; ð10Þ

where k is a scale parameter that controls the convexity of

the function, then we can obtain the weight function

wðxÞ ¼ 1

1þ x=kð Þ2: ð11Þ

The parameter k controls the sharpness of the robust

function and determines what residual errors are treated as

outliers [6]. Now we introduce a method to estimate the

robust parameters.

First a new observation X can be unfolded along mode-n

(n = 1, 2) to matrix Xn. Then we define rj(n) as the standard

deviation of the jth column vector of Xn. Supposing that the

current subspace model is a prototype model, we can make

an approximate evaluation of rj(n) with

rjðnÞ ¼ maxp

i¼1

ffiffiffiffiffiffiffiffikiðnÞ

quijðnÞ��

��; ð12Þ

where ki(n) and uij(n) are the ith eigenvalue of the current

energy matrix and the ith row jth column element of the

current projection matrix in mode-n, respectively. Then k

can be defined as

kjðnÞ ¼ brjðnÞ; ð13Þ

where b is a fixed robust coefficient. b can be set to a high

value (e.g., b = 10) for fast model updating, but at the risk

of accepting outliers into the model. Actually there is no

fixed value for b, it is application-dependent and has to be

chosen empirically.

When new observations arrive, we can obtain the

weights by computing w(rij), and construct weight matrix

in mode-n. Considering that outliers should not be absor-

bed into the background model as much as possible, we

express the weight as

W ¼ minðW1;W2Þ; ð14Þ

where W1, W2 is the weight matrix in mode-1 and mode-2.

The weight matrix characterizes the variation between the

vectors in each mode. Then the weighted tensor can be

defined as

x�ij ¼ffiffiffiffiffiffiwijp

xij: ð15Þ

It is obvious that the weights control the contribution of

new observations to the model ranging from 0 to 1. If

wij = 1, it means that the corresponding pixel is fully

reliable. On the contrary, if wij = 0, it means that the

corresponding pixel is irrelevant to the model.

2.3 Incremental updating for tensor subspace

In this section, we propose an efficient method for incre-

mental subspace learning as new observations arrive by

updating the mean and covariance. It is desirable to focus

more on new images and less on previous observations.

Therefore a forgetting factor a is incorporated to control

the updating rate of tensor subspace [7].

Assuming that the weights on the previous model and

the current observation are a and 1 - a, respectively, we

can update the mean as

Int. J. Mach. Learn. & Cyber. (2011) 2:89–98 91

123

�X0 ¼ a �X þ ð1� aÞX ; ð16Þ

where �X and X are the current mean and the new

observation, respectively. Then the new covariance matrix

of the nth mode is updated sequentially without storing the

previous data, as

C0n ¼ aCn þ ð1� aÞXnXTn ; ð17Þ

where Xn is the mode-n matrix unfolding of the mean-

normalized tensor. The current mode-n covariance matrix

Cn can be presented as

Cn ¼ UnSnUTn ; ð18Þ

where Un and Sn are the previous projection matrix and

energy matrix, respectively. We can obtain the new pro-

jection matrix U0n and energy matrix S0n by applying the

singular value decomposition (SVD) to C0n. The number of

basis is calculated by keeping the energy ratio

S0n

F

�Xnk k2

Flarger than 0.92 to preserve 92% of the

energy for the reconstruction. The pseudo code of the

robust tensor subspace learning algorithm is listed in

Table 1.

3 Anomaly detection algorithm based on robust tensor

subspace learning

3.1 Overview of anomaly detection

Anomaly detection is an active research topic in video

surveillance. With a stationary camera, the background

model of the scene is relatively static or unchanged. If a

moving object stays in the scene, it will be incrementally

learnt in the reconstruction to be part of background; if a

stationary object in the background moves, it will incre-

mentally disappear from the reconstruction and be looked

as a foreground object. Both the scenarios lead to the

background changes, i.e., anomaly.

The framework of anomaly detection in this paper

includes two modules: robust background modeling and

model detection. At the first module, background model is

learned online by the proposed algorithm as new obser-

vations arrive. At the second module, reconstruction

background images are compared with the reference model

to reflect the background changes and detect anomaly. The

architecture of the proposed anomaly detection framework

is shown in Fig. 1.

3.2 Background modeling based on RTSL

Our goal is to apply the robust tensor subspace learning

algorithm to represent the background model of the scene.

Given a training set X ¼ fX 1;X 2; . . .;XNgin RI1�I2 and

the initial projection matrix Unj2n¼1 constrained by

||Un|| = I. The mean of the training set can be formulated

as

�X ¼ 1

N

XN

i¼1

X i: ð19Þ

Table 1 The robust tensor subspace learning algorithm

Input

New observation X 2 RI1�I2

Previous projection matrices U 2n¼1

�� 2 R

Jn�In

Previous energy matrices S 2n¼1

�� 2 R

Jn�Jn

Previous mean �X 2 RI1�I2

Output

New projection matrices U 2n¼1 2 R�� J

0n�In

New energy matrices Sn2n¼1

�� 2 R

J0n�J

0n

New mean �X 2 RI1�I2

Algorithm

Compute the residual error R ¼ X � XQN

i¼1

�iUiUTi

� �

For n = 1 to 2

Compute maximum standard deviation

rjðnÞ ¼ maxp

i¼1

ffiffiffiffiffiffiffiffikiðnÞ

puijðnÞ��

��

Estimate the parameter kjðnÞ ¼ brjðnÞ, where b is a constant

Weight matrix Wn, where wijðnÞ ¼ 1

1þ rij=kijðnÞð Þ2

End

Construct the weights for tensor W = min (W1, W2)

Weighted tensor X�, where X�ij ¼ffiffiffiffiffiffiwijp

xij

Update mean �X0 ¼ a �X þ ð1� aÞX�

For n = 1 to 2

Mode-n unfold X� as Xn 2 RIn�ðQ

d 6¼nIdÞ

Construct the covariance matrix Cn ¼ UnSnUTn

Update covariance matrix C0n ¼ aCn þ ð1� aÞXnXTn

Decomposition ½U0n;S0n� ¼ SVD(C0nÞCompute new projection matrices and energy matrices Un and Sn

End

Tensor subspace learning

Weighting and updating

Training setNew

observations

DetectionSimilarity

measurementReconstruction

Robust background modeling Model detection

Fig. 1 The architecture of the anomaly detection framework


123

With the mode-n (n = 1, 2) matrix unfolding Xi(n) of

each image sample X i, the mode-n covariance matrix can

be denoted by

Cn ¼1

N

XN

i¼1

XiðnÞXTiðnÞ: ð20Þ

Then the projection matrix Un and the energy matrix Sn

can be obtained by performing SVD on Cn [7].

Subsequently we can weight the raw observations and

update the mean and covariance incrementally by using

(4)–(18). With decomposing C0n in (17), we can obtain the

updated projection matrix. The reconstruction can be

computed by projecting it into the current tensor subspace

as (2) and (3).

From the above analysis, the model can be learnt effi-

ciently through our method, which is crucial for anomaly

detection at the next module.

3.3 Anomaly detection based on background modeling

The initial background model can be represented as the

normal state or reference model. Whether or not an

anomaly occurs in video can be classified through a simi-

larity measurement between the reference model and the

current reconstruction.

The task of similarity measurement can be separated

into three components: luminance, contrast and structure

[10], denoted as

M ¼ ½lðx; yÞ�g � ½cðx; yÞ�h � ½sðx; yÞ�c; ð21Þ

where g[ 0, h[ 0 and c[ 0 are parameters used to adjust

the importance of the three components. l(x, y), c(x, y) and

s(x, y) are luminance, contrast and structure comparison

functions respectively. M is the value of similarity

measurement. The higher the M value is, the more

similar the two models are. The mean intensity of the

reference model X and the reconstruction Y are lx, ly

estimated by

lx ¼1

N

XN

i¼1

xt; ð22Þ

where N is the number of pixel. The standard deviation is

utilized as an estimate of contrast, given by

rx ¼1

N � 1

XN

i¼1

xi � lxð Þ2 !1=2

: ð23Þ

Then we define the three comparison functions

respectively by

lðx; yÞ ¼2lxly þ C1

l2x þ l2

y þ C1

ð24Þ

cðx; yÞ ¼ 2rxry þ C2

r2x þ r2

y þ C2

ð25Þ

sðx; yÞ ¼ rxy þ C3

rxry þ C3

ð26Þ

where C1, C2 and C3 = C2/2 are non-negative constant,

avoiding instability when the denominator is very close to

zero. rxy can be estimated as

rxy ¼1

N � 1

XN

i¼1

ðxi � liÞðyi � lyÞ: ð27Þ

In particular, we set g = h = c = 1, and substitute (22)-

(27) into (21) to yield

M ¼2lxly þ C1

� �2rxy þ C2

� �

l2x þ l2

y þ C1

� �r2

x þ r2y þ C2

� �: ð28Þ

In this way, consecutive reconstruction background

models are measured with the reference model. The

criterion for anomaly detection is defined as

reconstruction 2 anomaly if ðMn �Mnþ1Þ[ threshold

normal otherwise

�

ð29Þ

We denote Mn as the measurement of the nth frame, if

the difference between Mn and Mn?1 is less than a

threshold, it means the scene is normal, otherwise we

consider that the current background is varying, and

anomaly occurs in the scene. Then the background model

incrementally updates to build a new model. When a stable

background model has been learnt, anomaly ends, and the

reference model should be updated by the new background

model.

4 Experimental results and analysis

In this section, several experiments are carried out to

evaluate the performance of the proposed algorithm for

anomaly detection. Our experiments demonstrate the con-

tribution of the proposed RTSL by comparing with the

incremental and robust subspace learning algorithm [3],

referred as IRSL. Two video sequences are used in the

experiments which recorded by stationary cameras. These

videos are captured outdoor without drastic illumination

changes. Each video consists of 160 9 120 pixel gray scale

images. In video 1, a bicycle enters in a scene and has a

stop for a while, and then it leaves the scene. There are also

a few of walking persons and swaying trees in the scene.

Video 2 contains a person placing a suitcase on the ground

and leaving. Following this the first person comes back and

stands beside his suitcase. Then a second person arrives


123

and talks with the first person. The first person leaves the

scene without taking his luggage. The second person takes

the first person’s suitcase and leaves the scene.

In the first experiment, the forgetting factor a, the robust

coefficient b and the threshold for RTSL are chosen as

0.95, 5, and 0.03, respectively. For IRSL, the PCA

dimension p = 20, the update rate a = 0.95, and b = 5.

Figures 2 and 3 show the anomaly detection results by

RTSL and IRSL, respectively. Our proposed algorithm is

able to effectively capture the background changes and

detect the two anomalies when a bicycle stays in and leaves

from the scene, and successfully filter out outliers caused

by some walking people and swaying trees. The anomalous

regions of the representative frames (792, 1031) are high-

lighted by white boxes. In contrast, the IRSL cannot per-

form well. There is a false detection in the 371th frame of

video 1. Due to a slight camera shaking, the vector-based

method fails to build the background model robustly,

which can be further illustrated in Fig. 4. The sharp

declines in the value of similarity measurement correspond

to anomalies, which are highlighted by black boxes. The

false detection is shown in Fig. 4b.

Fig. 2 The anomaly detection

result of RTSL on video 1. a the

original image, b the reference

image when anomaly occurs,

c the updated reference image

when anomaly ends


result of IRSL on video 1. a The




when anomaly ends


123

Fig. 4 The comparison of

similarity measurement on

video 1 with RTSL (a) and

IRSL (b)


result of RTSL on video 2.

a The original image, b the

reference image when anomaly

occurs, c the updated reference

image when anomaly ends


123

In the second experiment, the forgetting factor a, the

robust coefficient b and the threshold for RTSL are chosen

as 0.957, 6, and 0.03, respectively. For IRSL, the PCA

dimension p = 20, the update rate a = 0.95, and b = 3.

This experiment is for a comparison between RTSL and

IRSL in a more complex scenario. The anomaly detection

results with RTSL are demonstrated in Fig. 5, which is

capable of learning tensor subspace and performing well in

modeling background changes for anomaly detection. This

video actually includes five anomalies, which are suc-

cessfully detected by our proposed algorithm. These rep-

resentative frames (143, 334, 694, 1017, and 1129) are

shown in Fig. 5. As is shown in Fig. 6, the IRSL fails to

capture some of these anomalies. In the 457th frame, there

is a false detection. Furthermore, an anomaly is also

undetected in around the 1017th frame. Figure 7 shows

superior performance of RTSL over IRSL. It is observed

that the IRSL present a more fluctuant result, especially

when anomalies occur. The undetected anomaly is high-

lighted by a dotted box in Fig. 7b.

From the results in these experiments, it is note that the

proposed algorithm provides better performance than

IRSL. The IRSL is an image-as-vector method for learning

a low dimensional linear subspace representation of back-

ground model. Without the local geometric structure

information, the appearance model can be substantially

changed by some extrinsic and intrinsic variation which

could not be considered as an anomaly, making the


result of IRSL on video 2. a The




when anomaly ends


123

reconstruction far from the ‘‘real’’ model and causing false-

detected and undetected anomalies. On the contrary, The

RTSL can maintain the spatial structure information of the

2D appearance due to the tensor subspace analysis, capable

of adaptively reflecting the background variation. The

image-as-matrix method is more robust to outliers with a

strong disturbance-tolerable ability. From the similarity

measurement of the two methods, we can find the results of

RTSL show narrower fluctuation than that of IRSL, dem-

onstrating the reliability and stability of RTSL.

5 Conclusion and future work

We have developed a robust tensor subspace learning

algorithm for anomaly detection, which can incrementally

update the tensor subspace and robustly reflect the varia-

tion of background model over time. Most of subspace

learning methods regard an image as a vector in high

dimensional space with the ignorance of the geometry

relationship between pixels in an image. Compared with

these methods, the proposed algorithm is able to make full

use of the spatial structure information in the manner of

tensor representation, and capture the characteristics of

each mode to weight the new data with a robust approach

and filter out the outliers for background modeling. We

detect anomalies by measuring the similarity between

the reference model and the reconstruction frames. The

experimental results demonstrate the effectiveness of the

proposed algorithm.

It is noted that the proposed algorithm preforms well

under the condition of constant illumination. The tensor

formulation is calculated on the intensity which is sensitive

to drastic illumination changes. Reducing the effect of

illumination variations on appearance model to enhance the

robustness is required in our future work.

Acknowledgements We want to thank the helpful comments and

suggestions from the anonymous reviewers. This research was

supported partially by the National Natural Science Foundation of

China under Grant 60832005; by the Ph.D. Programs Foundation of

Ministry of Education of China under Grant 20090203110002; by the

Key Science and Technology Program of Shaanxi Province of China

under Grant. 2010K06-12; and by the Natural Science Basic Research

Plan in Shaanxi Province of China under Grant 2009JM8004.

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Robust tensor subspace learning for anomaly detection