Anomaly Detection in Crowded Scenes Using Log-Euclidean Covariance

Matrix

Efsun Sefa Sezer and Ahmet Burak Can

Department of Computer Engineering, Hacettepe University, Beytepe, Ankara, Turkey

Keywords:

Anomaly Detection, Video Surveillance, Log-Euclidean Covariance Matrices, One-class SVM.

Abstract:

In this paper, we propose an approach for anomaly detection in crowded scenes. For this purpose, two impor-

tant types of features that encode motion and appearance cues are combined with the help of covariance matrix.

Covariance matrices are symmetric positive deﬁnite (SPD) matrices which lie in the Riemannian manifold and

are not suitable for Euclidean operations. To make covariance matrices suitable for use in the Euclidean space,

they are converted to log-Euclidean covariance matrices (LECM) by using log-Euclidean framework. Then

LECM features created in two different ways are used with one-class SVM to detect abnormal events. Exper-

iments carried out on an anomaly detection benchmark dataset and comparison made with previous studies

show that successful results are obtained.

1 INTRODUCTION

In recent years, abnormal crowd behavior analysis

has become a popular topic of research in computer

vision. Due to increasing security concerns, secu-

rity cameras are used in many areas such as airports,

metro stations, shopping malls and hospitals. This

has led to the amount of video data being acquired.

Processing these data manually is very hard and time

consuming. The visual attention module of the human

brain (Wang et al., 2017) is limited and thus, human

attention shows a great decline after a certain period

of time. This is a serious problem in manual anal-

ysis of large amounts of data. Therefore, intelligent

surveillance systems have a vital role to play. These

systems reduce the need for human power and enable

to obtain meaningful information from large amount

of video data. The main purposes of intelligent video

surveillance systems are to analyze videos effectively,

distinguish between normal and abnormal conditions

and alert security personnel about abnormal events.

Although various methods are used to design intelli-

gent surveillance systems, general approach is mod-

eling normal events and identifying abnormal events

that do not ﬁt into the model. The reasons for re-

searcher to prefer mentioned approach are that the

anomaly deﬁnition varies according to the content,

namely, situations considered abnormal for a particu-

lar scene may be considered normal in another scene

and the difﬁculties in ﬁnding the abnormal training

samples.

In this work, we propose an efﬁcient approach to

detect anomalies in videos. For that, log-Euclidean

covariance matrices are used with one-class SVM

classiﬁcation method. Covariance matrices are cre-

ated with appearance and motion cues. For appear-

ance cues, gradient-based features are chosen. For

motion cues, optical ﬂow-based features are used.

Unlike traditional methods, which utilize gradient-

based or optical ﬂow-based features for motion rep-

resentation, two important types of features that en-

code motion and appearance cues are combined with

the help of covariance matrix. Covariance matri-

ces are symmetric positive deﬁnite (SPD) matrices

which lie in the Riemannian manifold and are not

suitable for traditional Euclidean operations. Most of

the computer vision algorithms are developed for data

points located in Euclidean space. For this reason,

covariance matrices are mapped to Euclidean space

by utilizing log-Euclidean framework (Arsigny et al.,

2007). The model building process, which is the ﬁrst

step in the detection of abnormal situations, is per-

formed by using features obtained from normal events

and one-class SVM (OCSVM). In the detection pro-

cess, dissimilar events meaning that do not ﬁt the

model are marked as abnormal. Figure 1 shows the

overview of our approach. We evaluate our approach

on UMN (umn, 2006) anomaly detection benchmark

dataset. Experiments reveal that successful results are

obtained and the proposed method detects abnormal

events as soon as they occur. We organize the rest of

Sezer, E. and Can, A.

Anomaly Detection in Crowded Scenes Using Log-Euclidean Covariance Matrix.

DOI: 10.5220/0006618402790286

In Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 4: VISAPP, pages

279-286

ISBN: 978-989-758-290-5

279

Figure 1: Overview of the proposed approach. First, motion and appearance features are extracted. Then they are combined

with the help of covariance matrix. Log-Euclidean framework is employed to use the covariance matrices in the Euclidean

space. Anomaly detection is performed using log-Euclidean covariance features with OCSVM.

this paper as follows: Previous works are expressed in

Section 2. LECM features and log-euclidean frame-

work are described in Section 3. Anomaly detection

with OCSVM is explained in Section 4. Experimen-

tal results, comparison with previous approaches are

given in Section 5. Finally, we conclude this paper in

Section 6.

2 RELATED WORKS

Anomaly detection is the problem of ﬁnding situa-

tions that do not conform to the expected behavior

and it is an important topic that has been explored in

different application areas. Various approaches have

been proposed for anomaly detection. They can be

divided into two main categories: trajectory analysis

and motion analysis.

In trajectory-based approaches (Piciarelli et al.,

2008; Marsden et al., 2016; Fu et al., 2005), model-

ing crowd behaviors requires the separation and track-

ing of each objects in the scene. Object detection

and tracking cannot be done sufﬁciently in crowded

scenes due to the variable structure of crowded envi-

ronments. This makes the applicability of trajectory-

based approaches to crowded scenes considerably dif-

ﬁcult.

In motion-based approach, as opposed to

trajectory-based approaches, behavioral analysis is

carried out without object tracking. Thus, they per-

form well in crowded scenes with difﬁcult problems

such as closure and noise. For example, (Mehran

et al., 2009) use social force model to identify and

localize abnormal behaviors in crowded scenes. So-

cial force model is a method of mathematical crowd

behavior modeling based on Newton principles. (Lee

et al., 2013) propose motion inﬂuence matrix for

anomaly detection. In this study, anomaly detection

is performed according to the value of the motion

inﬂuence matrix which is high for abnormal events

and low for normal events. (Shi et al., 2010) calculate

the motion vectors between two consecutive frames

using phase correlation. Then normal events are mod-

eled using STCOG (spatial-temporal co-occurrence

Gaussian Mixture Models) and events that do not

ﬁt into the model are marked as abnormal. (Wang

and Snoussi, 2015) extract the histogram of the

optical ﬂow orientation (HOFO) features for motion

representation. Using the HOFO features with

kernel principal component analysis and one-class

SVM, they obtain favorable results. (Colque et al.,

2017) use spatio-temporal feature descriptor called

HOFM (Histograms of Optical Flow Orientation and

Magnitude) to determine the anomalies. HOFM is

generated by the direction and magnitude information

of the optical ﬂow. Since the process of obtaining

magnitude and direction information does not require

complex operations, this work is suitable for use

in real-time systems. When anomaly detection

is conducted using only motion information, it is

difﬁcult to identify the anomalies originating from

the size and appearance of the object. Considering

this situation, (Reddy et al., 2011) use appearance

information in addition to motion information. They

model features separately for efﬁcient computation.

Classiﬁcation is done using up to two classiﬁers. In

the ﬁrst stage, velocity information is checked. If the

anomaly is not detected, it is passed to the second step

where anomaly detection is performed using size and

texture information. (Mahadevan et al., 2010) model

crowd behavior using Mixture of Dynamic Texture

(MDT). MDT represents motion and appearance

cues together. However, it has high computational

complexity. (Ryan et al., 2011) use textures of optical

ﬂow with Gaussian Mixture Model (GMM). (Zhang

et al., 2016) deal with abnormal event detection in

two parts, appearance and motion. For abnormal

situations caused by appearance such as unusual

objects, unexpected appearance, strange positions,

unidentiﬁed objects, they use spatio-temporal gradi-

ent features with Support Vector Data Description

(SVDD). For motion anomaly, statistical histogram

is used. In the ﬁnal part, the results obtained from the

motion and appearance detection are combined.

The process of identifying unusual events in com-

plex scenes requires use of high dimensional features

VISAPP 2018 - International Conference on Computer Vision Theory and Applications

280

(Sabokrou et al., 2015). Training with these fea-

tures is very difﬁcult and leads to problems such as

a decrease in the prediction power of the model. To

overcome these limitations, sparse methods have been

proposed. For instance, (Cong et al., 2011) propose

Multi-scale Histogram of Optical Flow (MHOF) to

detect abnormal events. MHOF is formed by combin-

ing two optical ﬂow histograms according to a certain

threshold value and it represents motion in more de-

tail than the standard optical ﬂow histogram. They in-

troduce the sparse reconstruction cost (SRC) over the

normal dictionary to distinguish anomalies from nor-

mal ones. (Huo et al., 2012) use MHOF features with

the multi-instance learning method. Unlike (Cong

et al., 2011), MHOF features are extracted from only

moving pixels. Different from previous studies, (Pen-

nisi et al., 2016) propose a real-time method based

on segmentation and without the need for a training

phase. They use entropy and TOV (Temporal Occu-

pancy Variation) to identify abnormal events.

3 LOG-EUCLIDEAN

COVARIANCE MATRIX (LECM)

When previous studies are examined, it is seen that

the optical ﬂow feature which represents motion

of the crowd is frequently used in determining the

anomalies (Colque et al., 2017; Cong et al., 2011;

Wang and Snoussi, 2015). In this way, it is possible

to detect irregularities in the direction of movement

and speed. As is known, in application such as object

recognition, object tracking, action recognition, ab-

normal event detection successful result are obtained

by combining motion and appearance cues (Shotton

et al., 2006; Sanin et al., 2013; Zhang et al., 2016) . In

this study, both approaches are followed by creating

two different forms of covariance matrix. In the ﬁrst

form of the covariance matrix, optical ﬂow-based and

gradient-based features are used together. In the sec-

ond form, only optical ﬂow-based features are used.

For optical ﬂow estimation, Horn-Schunk (Horn and

Schunck, 1981) method is used.

Covariance matrix is introduced by (Tuzel et al.,

2006) to computer vision community for pedestrian

detection, object recognition. Later, they have been

successfully applied many areas such as object track-

ing, face recognition, action recognition (Sanin et al.,

2013; Guo et al., 2013). Let I(x,y,t) denote a video

sequence and F = { f

} be the feature vectors. Then

the covariance matrix is deﬁned as:

N − 1

∑

k=1

( f

− µ)( f

− µ)

(1)

where N is the size of the feature set and µ is the mean

of the feature vectors. The use of covariance matrix

has many advantages:

• It is a simple and effective way of integrating var-

ious features.

• It is a low dimensional descriptor. Dimension of

covariance feature is independent of the size of the

region where it is computed.

While the advantages listed above make the covari-

ance matrix based approaches attractive, it is an im-

portant problem that the covariance matrices are de-

ﬁned in the Riemannian manifold and not suitable for

the Euclidean operations. In order to make the co-

variance matrices suitable for Euclidean operations,

the log-Euclidean metric (Arsigny et al., 2007) is pro-

posed.

3.1 Log-Euclidean Framework on

Symmetric Positive Deﬁnite

Matrices

Covariance matrix is a symmetric positive deﬁnite

(SPD) matrix and SPD matrices do not lie in a vector

space. In order to make covariance matrices suitable

for use in the Euclidean space, log-Euclidean frame-

work is employed. According to this framework, co-

variance matrices are mapped to the Euclidean space

by using the matrix logarithm operation. The log-

covariance matrix estimation is performed as follows.

Let SPD(n) and S(n) denote the space of n × n real

SPD matrices and n × n real symmetric matrices, re-

spectively. The eigen-decomposition of a covariance

matrix S ∈ S(n) is S = UΛU

, where U is an or-

thonormal matrix and Λ = Diag(λ

,...,λ

) is a diago-

nal matrix that contains the eigenvalues λ

of S. If S is

positive deﬁnite matrix, S ∈ SPD(n), then λ

> 0 for

i = 1,...,n. Using eigen-decomposition, the exponen-

tial of a S ∈ S(n) can be calculated as follows:

exp(S) = U.Diag(exp(λ

),...,exp(λ

)).U

(2)

Logarithm of S ∈ SPD(n) is the following form:

log(S) = U.Diag(log(λ

),...,log(λ

)).U

(3)

Because the covariance matrix is a symmetric matrix,

half-vectorization is performed and ﬁnal representa-

tion contains

n(n+1)

values.

3.2 LECM-1 Feature

In this section, we explain the ﬁrst form of the pro-

posed covariance descriptor. When the previous stud-

ies are reviewed, it is commonly seen that gradi-

ent and optical ﬂow-based features are used together

Anomaly Detection in Crowded Scenes Using Log-Euclidean Covariance Matrix

281

(Zhang et al., 2016; Zhu et al., 2016). These two fea-

tures are complementary to each other and give in-

formation about appearance and motion, respectively.

When they are used together, they produce good re-

sults. The feature vector f

(x,y,t) which is extracted

from (x,y,t) pixel position is the following form:

(x,y,t) = [x,y,t,g,o]

(4)

where

g =

|,|I

+ I

(5)

o =



u,v,

∂u

∂t

∂v

∂t



∂u

∂x

∂v

∂y





∂v

∂x

−

∂u

∂y



(6)

g and o represent appearance and motion cues. The

ﬁrst four gradient-based features in (5) denote the ﬁrst

and second order intensity gradients at pixel location

(x,y,t) and the last term is the gradient magnitude.

The optical ﬂow-based features in (6) denote the hor-

izontal and vertical components of the ﬂow vector

(u,v), the ﬁrst order derivatives of the horizontal and

vertical components of the optical ﬂow with respect to

time (∂u/∂t, ∂v/∂t). The last two optical ﬂow based

features are the spatial divergence and vorticity of the

ﬂow ﬁeld (Ali and Shah, 2010).

3.3 LECM-2 Feature

The second form of covariance matrix is created by

using only optical ﬂow-based features. The feature

vector f

(x,y,t) which is extracted from (x,y,t) pixel

position is the following form:

(x,y,t) = [x,y,t,o]

(7)

where o =



u,v,

∂u

∂t

∂v

∂t



∂u

∂x

∂v

∂y





∂v

∂x

−

∂u

∂y



,Gten,Sten



(8)

The optical ﬂow-based features in (8) denote the hor-

izontal and vertical components of the ﬂow vector

(u,v), the ﬁrst order derivatives of the horizontal and

vertical components of the optical ﬂow with respect

to time (∂u/∂t, ∂v/∂t) and the spatial divergence and

vorticity of the ﬂow ﬁeld. Gten, Sten are tensor invari-

ants which remain unchanged no matter which coor-

dinate system they are referenced in (Ali and Shah,

2010). Gten,Sten are derived from gradient tensor of

optical ﬂow and the rate of strain tensor. The gradient

tensor of optical ﬂow ∇u(x,y,t) is a 2×2 dimensional

matrix and deﬁned as:

∇u(x,y,t) =







∂u

∂x

∂u

∂y

∂v

∂x

∂v

∂y







(9)

The rate of strain tensor S(x, y,t) is deﬁned as follows:

S(x,y,t) =

(∇u(x,y,t) + ∇

u(x,y,t)) (10)

Gten and Sten are deﬁned using ∇u(x,y,t) and

S(x,y,t) as follows:

Gten(x, y,t) =

(tr

(∇u(x,y,t) − tr(∇

u(x,y,t)))

(11)

Sten(x,y,t) =

(tr

(S(x,y,t)− tr(S

(x,y,t))) (12)

where tr(.) represents the trace operation.

4 DETECTION OF ANOMALOUS

EVENTS

Abnormal event detection is a daunting task due to its

context-dependent nature. This means that, an event

considered abnormal in one scenario may be consid-

ered normal in another scenario. In automatic surveil-

lance systems, abnormal event detection is performed

by modeling expected patterns in a given dataset and

ﬁnding patterns that do not conform to expected be-

havior. The expected behaviors are modeled using

normal samples.

In this work, we use one-class SVM for building

normal models. The main reasons for preferring one-

class classiﬁcation methods in the detection of abnor-

mal events are that there are a wide range of abnormal

events and difﬁculties in collecting samples of these

cases. For that purpose, (Sch

olkopf et al., 2001) pro-

pose a method that adapts the classical SVM method-

ology to the one class classiﬁcation problem. In one-

class SVM, ﬁrstly the distribution of normal data is

determined. Classiﬁcation is done according to the

presence or absence of the test data in this distribu-

tion. Let x

,...,x

be training examples belonging

to one class X and Φ : X → H. H is a feature space

and Φ is a kernel map that transforms training sam-

ples to another space. The process of separating the

normal samples from the others using the kernel is

achieved by solving the following quadratic program-

ming problem :

min

||w||

∑

i=1

− ρ (13)

subject to

(w.Φ(x

)) ≥ ρ − ξ

i = 1, 2, ...,l ξ

≥ 0 (14)

where w is a vector deﬁning the hyper-plane,, v is reg-

ularization parameter, l is the number of training sam-

ples, ξ

is the slack variable and ρ the distance to the

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282

Figure 2: Example frames from the UMN dataset. Top line: normal events. Bottom line: abnormal events.

origin in feature space.

The decision function is deﬁnes as:

f (x) = sign((w.Φ(x)) − ρ) (15)

The function in (15) will produce a positive value for

the samples in the training set.

5 EXPERIMENTAL RESULTS

In this section, ﬁrstly, we provide information about

the abnormal crowd behavior dataset UMN and eval-

uation metrics. Then qualitative and quantitative re-

sults are given.

5.1 Dataset

The UMN dataset (umn, 2006) is used to measure the

performance of the proposed method. It contains 11

video and 7739 frames with a 320 × 240 resolution.

Videos are captured in 1 indoor and 2 outdoor scenes.

Each video starts with normal behavior and ends with

an abnormal behavior of escape. Example scenes are

shown in Figure 2.

5.2 Evaluation Metric

In order to conduct a quantitative analysis on the pro-

posed method, ROC (Receiver Operating Character-

istics) curve and Area Under Curvature (AUC) are

used. For the ROC curve and the area under the curve

(AUC), the true positive rate (TPR) and the false pos-

itive rate (FPR) should be determined. TPR and FPR

values are calculated using the false positive (FP), true

positive (TP), false negatives (FN) and true negatives

(TN).

T PR =

T P

T P + FN

(16)

FPR =

FP + T N

(17)

where TP denotes the correctly detected abnor-

mal events, FN denotes incorrectly detected nor-

mal events, FP denotes incorrectly detected abnormal

events, TN denotes correctly detected normal events.

5.3 Results

This section contains the results of the proposed

methods and comparison with previous studies. ROC

curves for LECM-1 and LECM-2 features are pre-

sented in Figure 5 and Figure 6, respectively. Re-

sults show that LECM-1 feature produces better re-

sults than LECM-2. Both approaches produce a lower

AUC value in the second scene than the other scenes.

This is due to the fact that the Scene-2 is dim light

indoor scene and there are changes in lighting con-

ditions. These problems adversely affect feature ex-

traction stage, and thus the system performance is

reduced. Figure 3 and Figure 4 show the qualita-

tive results for LECM-1 and LECM-2. It is observed

that LECM-1 makes the transition between abnormal

and normal events better than LECM-2. Especially

in the second scene, LECM-2 features cannot cor-

rectly detect the end of abnormal events. The rea-

son is that LECM-1 is constructed by using features

that are complementary to each other. The results also

Anomaly Detection in Crowded Scenes Using Log-Euclidean Covariance Matrix

283

Figure 3: The qualitative results of the abnormal behavior

detection using LECM-1. Each row shows detected ab-

normal events in three different videos. The ground truth

bar and the proposed approach bar show the labels of each

frame for that video. In that bars, green color indicates nor-

mal events and red color indicates abnormal events.

show that combining motion and appearance cues im-

proves detection accuracy and can also reduce false

alarms. In systems designed to detect anomalies, it

is important to deﬁne the moments of transition from

normal events to abnormal events correctly. In this

sense, LECM-1 features give alarms with a high de-

gree of accuracy from the moment when abnormal

conditions have begun to be seen.

We compare our approach with other methods

in Table 1. These methods are : SR (Cong et al.,

2011), MI (Lee et al., 2013), HF (Marsden et al.,

2016), MIDL (Huo et al., 2012), CMA (Zhang et al.,

2016), STCOG (Shi et al., 2010), FSCB (Pennisi

et al., 2016), HOFO SVM and HOFO PCA (Wang

and Snoussi, 2015), SF and OF (Mehran et al., 2009).

Figure 4: The qualitative results of the abnormal behavior

detection using LECM-2. Each row shows detected ab-

normal events in three different videos. The ground truth

bar and the proposed approach bar show the labels of each

frame for that video. In that bars, green color indicates nor-

mal events and red color indicates abnormal events.

LECM-1 outperforms SF, OF, STCOG, MIDL, HF

and is comparable to SR, MI, CMA, HOFO SVM

and HOFO PCA. It is important to note that we ob-

tained comparable or better results in comparison to

other methods using a very simple classiﬁcation tech-

nique. The computational cost of our work is lower

than SR and MIDL which are complex dictionary

learning-based approaches. Furthermore, unlike the

FSCB method, segmentation is not used in the pro-

posed approach. In crowded scenes, it is very dif-

ﬁcult to perform segmentation because there are too

many components to be analyzed and they have clo-

sure problems. When Table 1 is examined, it is seen

that the highest performance is achieved by HOFO

PCA. HOFO feature contains only motion informa-

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284

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

FPR

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

TPR

Scene-1 AUC = 0.9986

Scene-2 AUC = 0.9181

Scene-3 AUC = 0.9667

Figure 5: The ROCs for LECM-1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

FPR

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

TPR

Scene-1 AUC = 0.9857

Scene-2 AUC = 0.8790

Scene-3 AUC = 0.9468

Figure 6: The ROCs for LECM-2.

tion and are not discriminative enough for detecting

anomalies arising form object shapes and appearance.

In contrast to HOFO, LECM-1 feature contains both

appearance and motion cues and can provide success-

ful results when used with more advanced classiﬁca-

tion methods.

6 CONCLUSIONS

In this study, log-Euclidean covariance matrix is

formed in two different ways and used with OCSVM

effectively to detect abnormal events. As mentioned

before, it is difﬁcult to collect anomalous event in-

stances in the detection of abnormal events. For this

reason, OCSVM, which has been popular in recent

years, is preferred. In this respect, the method has

a simple and effective structure which is different

from the complicated works in the literature. Also,

our method is suitable for use in crowded scenes be-

cause object tracking, detection are not performed

and there is no need to set any threshold value dur-

ing anomaly detection. Experiments carried out on

Table 1: Performance comparison according to ROC curves

on the UMN dataset.

Approach Scene-1 Scene-2 Scene-3

SR 0.995 0.975 0.964

MI 0.995 0.853 0.98

HF 0.953 0.913 0.964

MIDL 0.8927 0.7541 0.9482

CMA 0.993 0.969 0.988

STCOG 0.9362 0.7759 0.9661

FSCB 09641 0.8764 0.9750

HOFO SVM 0.9845 0.9037 0.9815

HOFO PCA 0.9992 0.9880 0.9989

SF 0.96

OF 0.84

LECM-1 0.9986 0.9181 0.9667

LECM-2 0.9857 0.8790 0.9468

the UMN dataset indicate that the proposed method

provides satisfying results. The best results are ob-

tained by combining appearance and motion informa-

tion with the help of the covariance matrix. For future

work, we aim to make the proposed approach suitable

for use in scenes where local abnormal events are ob-

served.

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