Optimizing Interaction Force for Global Anomaly Detection in Crowded Scenes

R. Raghavendra1 Alessio Del Bue1 Marco Cristani 1,2 Vittorio Murino 1,2

Istituto Italiano di Tecnologia (IIT), Italy1

Dipartimento di Informatica, University of Verona, Italy2

Abstract

This paper presents a novel method for global anomaly

detection in crowded scenes. The proposed method intro-

duces the Particle Swarm Optimization (PSO) method as a

robust algorithm for optimizing the interaction force com-

puted using the Social Force Model (SFM). The main ob-

jective of the proposed method is to drift the population of

particles towards the areas of the main image motion. Such

displacement is driven by the PSO fitness function aimed at

minimizing the interaction force, so as to model the most

diffused and typical crowd behavior. Experiments are ex-

tensively conducted on public available datasets, namely,

UMN and PETS 2009, and also on a challenging dataset

of videos taken from Internet. The experimental results re-

vealed that the proposed scheme outperforms all the avail-

able state-of-the-art algorithms for global anomaly detec-

tion.

1. Introduction

Crowd analysis has become a popular topic in computer

vision applications that involves the analysis of human be-

havior from a surveillance context. This is because conven-

tional methods designed for surveillance applications fail

drastically for the following reasons: (1) overlapping be-

tween individual subjects; (2) random variations in the den-

sity of people over time; (3) low resolution videos with tem-

poral variations of the scene background.

The main objective of crowd behavior analysis involves

not only the modeling of people mass dynamics but also to

detect or even predict possible abnormal behaviors in the

scene. In particular for surveillance scenarios, this task is

of paramount importance since early detection, or even pre-

diction, may reduce the possible dangerous consequences

of a threatening event, or may alert a human operator for

inspecting more carefully the ongoing situation.

Anomaly detection in crowded scenes can be classified

into two types, namely [6]: (1) local abnormal event, indi-

cating that behavior of a local image area is different from

that of its neighbors in spatio-temporal terms; (2) global ab-

normal event, corresponding to an anomaly of the whole

scene. In this work, we address the latter issue, by con-

sidering its application to more realistic scenario than the

standard acted scenes in the available public datasets, such

as detecting anomalies in prison scenes.

Several works have been proposed for global anomaly

detection in crowded scenes [11, 13, 14, 10], where the ma-

jority are based on particle advection schemes. In this case,

a rectangular grid of particles is typically placed on each

frame and advected using the underlying motion. These

schemes are quite useful since they do not need the seg-

mentation, detection or tracking of individuals that are very

challenging tasks in crowd scenes. The first work using par-

ticle advection schemes for crowd behavior analysis was in-

troduced in [13]. Here, the particle flow is computed by

moving a grid of particles using the fourth-order Runge-

Kutta-Fehlberg algorithm [9] along with the bilinear inter-

polation of the optical flow field. This method is further ex-

tended in [14] using chaotic invariants capable of analyzing

both coherent and incoherent scenes. In [10], streaklines are

introduced and integrated with a particle advection scheme

capable of incorporating the spatial change in the particle

flow. In [11] the social force model (SFM) [7] is exploited

to detect abnormal events. After the superposition of a fixed

grid of particles on each frame, the SFM is used to esti-

mate the interaction force under the belief that it can por-

tray significant information to describe (abnormal) crowd

behavior. So, after estimating the so-called force flow, a

bag of words method [3] and a Latent Dirichlet Allocation

(LDA) [4] are employed to discriminate between normal

and abnormal frames, while localizing the abnormal areas

as those showing the highest force magnitude. In addition

to the above mentioned particle advection schemes, there

also exist a few model-based approaches facing the same

problem. In [12], 3D bricks are extracted and used as a de-

scriptor to represent the scene, and followed by a dynamic

programming algorithm aimed at detecting the anomaly. In

[6], a sparse reconstruction cost is proposed to detect the

presence of anomalies in crowded scenes. Here, the normal

dictionary to measure the abnormality of the test sample is

constructed using local spatio-temporal patches. Further, to

reduce the size of the dictionary, a new selection method is

proposed based on sparsity consistency constraints.

In this paper, we propose a new scheme for global

anomaly detection by performing particle advection us-

ing Partical Swarm Optimization (PSO) and Social Force

Model (SFM). The proposed process starts from random

positioned particles over each frame, assuming that the in-

teraction force (obtained using SFM) is discriminant for

characterizing crowd behavior. However, unlike previous

algorithms, this force is not straightforwardly utilized in a

classification framework to detect the anomalies [11], but it

is further processed to augment its discriminant properties.

After the force flow estimation, we apply a minimization

process aimed at optimizing the position of the particles at

each frame, so that particles converge naturally towards the

significant moving areas in the scene, and in particular to-

wards the parts which likely show a higher magnitude of

interaction force. Then, we estimate the evolution of the

interaction force frame by frame in order to identify the oc-

currence of the global anomaly in a crowded scenario.

There are several characteristic features which differen-

tiate our approach with respect to other related works in the

literature [11, 13, 14, 10]. First, particles are spread ran-

domly over the image and can move in a continuous way

according to an optimization criterion. Second, we use PSO

[8] for particle advection which considers not only the in-

dividual particle motion, but also the global motion of the

particles as a whole, i.e. it “simulates” in some way social

interactions among the particles. In addition, this model

has proved to outperform the other approaches considered

the state of the art for global anomaly detection when tested

on a set of public video datasets and of real videos retrieved

in the Internet.

The rest of the paper is organized as follows. Section

2 discusses the proposed algorithm by first introducing the

basic notions of PSO and SFM. Section 3 presents the re-

sults on public available databases as well as on a set of

real videos taken from Internet. Finally, Section 4 draws

the conclusions and sketches future work.

2. The proposed algorithm

Figure 1 shows the various stages followed in the design

of the proposed framework for global anomaly detection.

The proposed scheme consists of two main components: (1)

particle advection using PSO-SFM, and (2) global anomaly

detection.

2.1. Particle advection using PSO­SFM

This section describes our proposed particle advection

using PSO-SFM. In earlier attempts [11, 13], the particle

advection is carried out by placing a rectangular grid of

particles over each video frame. Then, the velocity for

each particle is calculated using fourth-order Runge-Kutta-

Fehlberg algorithm [9] along with the bilinear interpolation

of the optical flow field. In general, a drawback of this

approach is that it assumes that a crowd follows a fluid-

dynamical model which is too restrictive when modeling

masses of people. The elements of the crowd may also

move with unpredictable trajectories that will result in an

unstructured flow. Moreover, the use of a rectangular grid

for particles is a coarse approximation with respect to the

continuous evolution of the social force. To overcome these

drawbacks, we propose a novel particle advection using

PSO aiming at modeling the crowd behavior. Before pre-

senting the detailed description about our proposed scheme,

we first provide brief introduction on PSO and SFM in the

following subsections.

Particle Swarm Optimization Particle Swarm Opti-

mization is a stochastic, iterative, population-based opti-

mization technique aimed at finding a solution to an op-

timization problem in a search space [8]. The main ob-

jective of PSO is to optimize a given criterion function

called fitness function f . PSO is initialized with a pop-

ulation, namely a swarm, of N-dimensional particles dis-

tributed randomly over the search space: each particle is so

considered as a point in this N-dimensional space and the

optimization process manages to move the particles accord-

ing to the evaluation of the fitness function in an iterative

way. More specifically, at each iteration, each particle is

updated according to best values called pbesti that is de-

pending on the i− th particle and gbest that is independent

from the specific particle i.e. gbest is valid for the whole

swarm. The pbesti value represents the best position as-

sociated with the best (i.e., minimum or maximum) fitness

value of particle i obtained at each iteration, and having fit-

ness value f(pbesti). The gbest value represents the best

position among all the particles in the swarm, i.e. the po-

sition of the particle assuming the minimum or maximum

value when evaluated by the fitness function. The rate of

the position change (velocity) for the particle i is called vi,

which is updated according to the following equations [8]:

vnewi = IW · vold

i + C1 · rand1 · (pbesti − xoldi )

+C2 · rand2 · (gbest − xoldi ); (1)

xnewi = xold

i + vnewi , (2)

where IW is the inertia weight, whose value should be tuned

to provide a good balance between global and local explo-

rations, and it may result in fewer iterations on average for

finding near optimal soluction. The scalar values C1 and

C2 are acceleration parameters used to drive each particle

towards pbesti and gbest. Low values of C1 and C2 allow

Particle Advection Global Anomaly Detection

VideoParticle

Advection --

PSOSFM

Change in Magnitude of

Interaction Force

Moving

Average

Filtering

Threshold

Particle Advection

Using PSO-SFMGlobal Anomaly Detection

Normal

Anomaly

Figure 1. Proposed framework for global anomaly detection

the particles to roam far from target regions, while high val-

ues result in abrupt movements towards the target regions.

rand1 and rand2 are random numbers between 0 and 1.

Finally, xoldi and xnew

i are the current and updated particle

positions, respectively, and the same applies for the devia-

tion voldi and vnew

i .

Social Force Model The SFM [7] provides a mathemat-

ical formalization to describe the movement of each indi-

vidual in a crowd on the basis of its interaction with the

environment and other obstacles. The SFM can be written

as:

mi

dWi

dt= mi

(

Wpi − Wi

τi

)

+ Fint, (3)

where mi denotes the mass of the individual, Wi indicates

its actual velocity which varies given the presence of obsta-

cles in the scene, τi is a relaxing parameter, Fint indicates

the interaction force experienced by the individual which is

defined as the sum of attraction and repulsive forces, and

Wpi is the desired velocity of the individual.

PSO and SFM constitute the basic elements used to de-

velop our model for crowd behavior analysis which will be

described in the following section.

2.2. Our approach

First, given a video sequence, the PSO begins with a ran-

dom initialization of the particles in the first frame. From

such initial stage, we obtain a first guess of pbesti for each

particle i and the global gbest. The particles are defined by

their 2-D value corresponding to the pixel coordinates in the

frames. At each iteration, the pbesti value is updated only

if the present position of the particle is better than the previ-

ous position according to fitness function evaluated on the

model interaction force. Finally, the gbest is updated with

the position obtained from the best pbesti after reaching the

maximum number of iterations or if the desired fitness value

is achieved. We then use the final particle positions as the

initial guess in the next frame and the same iterative process

is repeated until the end of the video sequence. Therefore,

the movement of the particles is updated according to the

fitness function which drives the particle towards the areas

of minimum interaction force.

Fitness function The fitness function aims at capturing

the best interaction force exhibited by each movement in

the crowded scene. Each particle is evaluated according to

its interaction force calculated using SFM and optical flow

[5]. The Optical Flow (OF) is actually a suitable candidate

to substitute the pedestrian velocities in the SFM model.

In order to use the OF jointly with SFM, we first define

the intensity of the optical flow computed at a given position

in the image for the particle i as:

Wi = Oavg(xnewi ), (4)

where Oavg(xnewi ) indicates the average OF at the particle

coordinates xnewi . The average is computed over L previous

frames. Then, the desired velocity of the particle Wpi is

defined as:

Wpi = O(xnew

i ), (5)

where O(xnewi ) represents the OF intensity of the particle

i, whose coordinates are estimated using equation (1). In

fact, this OF value is an average value computed in a small

spatial neighborhood to avoid numerical instabilities of the

OF. Finally, we calculate the interaction force Fint using

equation 3 as follows:

Fint(xnewi ) = mi ·

dWi

dt−

mi

τi

(W pi − Wi) , (6)

where the velocity derivative is approximated as the differ-

ence of the OF at the current frame t and t − 1, that isdWi

dt= [O(xnew

i )|t − O(xnewi )|t−1]. As observed from

equation (3), the interaction force allows an individual to

change its movement from the desired path to the actual

one. This process is in some way mimicked by the parti-

cles which are driven by the OF towards the image areas of

larger motion. In this way, the more regular the pedestri-

ans’ motion, the less the interaction force, since the people

motion flow varies smoothly. So, in a normal crowded sce-

nario the interaction force is expected to stabilize at a certain

(low) value complying the typical motion flow of the mass

of people. It is then reasonable to define a fitness function

aimed at minimizing the interaction force, and moving par-

ticles towards these sinks of small interaction force, thereby

allowing particles to simulate a “normal” situation of the

crowd.

Algorithm 1: Particle Advection using SFM-PSO

Input: Oavg and O // Average optical flow and opti-

cal flow computed for the whole set of video frames

Initialization: {vi}k

i=1, {xi}

k

i=1// Initial particle ve-

locity and position and k is the number of particles.

pbesti // Initial local best position

gbest // Initial global best position

for Frame = 1 : No. of Frames do

while Iter ≤ maxiter and Fint(gbest) <

BestF it do

for i = 1 : k do

Calculate Fint using Equation (6)

if Fint(xnewi ) < Fint(pbesti) then

pbesti = xnewi

end if

end for // Particles loop

Set gbest = argmini {pbesti}Estimate new positions for particles using equa-

tion (1) and (2)

end while

Output: Optimized interaction forces and

new particles’ positions.

end for // Frames loop

Hence, we can write our fitness function as:

FitX = mini

{Fint (xnewi )} (7)

where, xi denotes the i − th particle. The proposed PSO-

SFM scheme is summarized in Algorithm 1 where maxiter

indicates the maximum number of iterations employed and

BestF it indicates the desired fitness value which is set to

0 in our case.

As an example, Figure 2 (a) - (d) shows the computed

interaction force with the proposed particle advection using

PSO-SFM for both normal (Figure 2 (a)-(b)) and anomaly

video frames (Figure 2 (c)-(d)). In the figures, we map

on the image plane the magnitude of the interaction forces

assigned to every particle. As observed in Figure 2, the

presence of the high magnitude interaction force over time

Normal

(a)(a)

(b)

Abnormal

(c)(c)

(d)

Figure 2. Illustration of the proposed scheme. (a) input nor-

mal frame. (b) Interaction force corresponding to (a). (c) Input

anomaly frame. (d) Interaction force corresponding to (c).

can provide useful information about the existence of an

anomaly. Thus, in this case, we expect the particles to have

high magnitude interaction force overall.

2.3. Global anomaly detection

Our framework allows to formulate the detection of

global anomalies as the detection of the changes in the in-

teraction force magnitude. This process is valid with the

proposed particle advection scheme since the presence of

global abnormality can be distinguished by the presence of

high magnitude of the interaction force assumed by the par-

ticles (see Figure 2). Since all the available test videos con-

tains a certain amount of frames in which normal behavior

is assumed, we take advantage of this information in the

comparison process, like all the other previous algorithms

[11]. In practice, we carry out the following steps to decide

whether a given frame contains an anomaly or not:

1. First, compute the sum of the interaction forces of a

reference frame Fr. This reference frame(s) represents

a normal behavior scene in the given video sequence.

Actually, all the public datasets considered have an ini-

tial (variable, but at least one frame) set of frames rep-

resenting a normal behavior which can be used for ref-

erence. Thus, we obtain Fr as follows:

Fr =k

∑

i=1

Fint(xnewi )|r (8)

2. Compute the sum of interaction forces corresponding

to all particles in the current frame Ft as:

Ft =k

∑

i=1

Fint(xnewi )|t (9)

3. Compute the change in the magnitude force at each

frame t as:

Ct = |Ft − Fr| (10)

4. Repeat steps 2 - 3 for all frames to obtain the profile

(values of Ct for all the video frames) corresponding

to the change of the force magnitude.

5000

0 100 200 300 400 500 6000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Number of frames

Pr

(P

rofile

)

(a)(a)

2500

0 100 200 300 400 500 6000

500

1000

1500

2000

2500

Number of frames

SP

(S

mo

oth

en

pro

file

)

(b)(b)

Figure 3. Profile (a) Before Smoothing (b) After Smoothing

As an example, Figure 3(a) shows the profile obtained

from a sequence of the UMN dataset after following

the above mentioned steps 1 - 4.

5. Finally, we use the moving average filter to smooth out

the short term fluctuations that are present in the ob-

tained profile at the previous step, so to get a smoothed

profile Cst (see Figure 3(b)). Each frame is then classi-

fied as either normal or abnormal according to a thresh-

old as follows:

Lt =

{

Abnormal if Cst > th

Normal otherwise

where Cst represents the smoothed profile, th repre-

sents a threshold value, and Lt holds the final detection

result of the given video sequence.

3. Experiments and discussion

To validate the performance of the proposed approach

for global anomaly detection, we conducted an exten-

sive set of experiments on four different datasets such

as UMN [2], PETS 2009 [1], UCF [11], and prison riot

dataset (collected from the web).

In the following experiments, all the video frames are

resized to a fixed width of 200 × 200 pixels. For the

particle advection scheme, the particle density (i.e., the

number of particles) is kept constant at 25% of num-

ber of pixels, and number of iterations is fixed to 100.

To detect the changes of the interaction force magni-

tude, we use the first frame as the reference frame.

This is because, in all the following datasets, the ini-

tial (roughly) 40% of the video frames represents the

normal behavior which is then followed by the abnor-

mal behavioral frames. Finally, the performance is val-

idated by plotting the ROC curves obtained over all

possible values of the threshold th.

3.1. UMN Dataset

The UMN dataset consists of eleven video sequences

acquired in three different crowded scenarios including

both indoor and outdoor scenes. All these sequences

exhibit a escape panic scenario and hence, they start

with the normal behavior frames followed by the ab-

normal activity. Figure 4 illustrates the results of the

proposed scheme obtained on the UMN dataset. Fig-

ure 4(a) shows the normal and abnormal crowd be-

havior frames from the dataset, and Figure 4(b) indi-

cates the corresponding interaction force obtained us-

ing the proposed PSO-SFM based particle advection.

From this figure, it can be observed that the presence

of high magnitude of the majority of the particles in-

teraction force is an evidence that an abnormal frame

has occurred. Figure 4(c) shows the detection results

of the normal and abnormal frames from the dataset

using step 5 of the proposed global anomaly detection

algorithm. Figure 5 shows the detection results ob-

Normal

(a)

(b)

(c)

N

Abnormal

(a)

(b)

(c)

A

Figure 4. Computed Force Flow obtained on example sequence.

(a) Input frame. (b) Force field. (c) Detection (N indicates Normal

and A indicates Abnormal frame)

tained on two different sequences of the UMN dataset

showing that at abnormal frames always correspond a

higher magnitude of interaction force in the particles.

Figure 6 and 7 show the performance of the pro-

posed scheme on three different scenes of UMN and

of the whole dataset, respectively. The quantitative re-

sults shown in Table 1 indicate the best performance

of the proposed scheme over available state-of-the-art

methods.

(a)

(b(b

(c

(d

(a)

(b)

N

N

A

(b)

(c)

(d)

N

A

Figure 5. Results of the proposed scheme on other sequences of

the UMN dataset. (a) Normal behavior in scene 2 with its cor-

responding interaction force and detection.(b) Abnormal behavior

in scene 2 with its corresponding interaction force and detection.

(c) Normal behavior in scene 3 with its corresponding interaction

force and detection. (d) Abnormal behavior in scene 3 with its

corresponding interaction force and detection.

Figure 6. ROC Curves of abnormal behavior detection on different

scenes in UMN dataset

3.2. Prison Riot Dataset

In order to evaluate the proposed method on real ap-

plications, we collected a real video set from websites

such as YouTube and ThoughtEquity.com. The col-

lected video dataset comprises of 7 sequences that rep-

resent riots in prisons that are captured with different

angles, resolutions, background and include abnormal-

ity like fighting with each other, clashing, etc.. All

the collected sequences start with the normal behavior

which is then followed by a sequence of abnormal be-

havioral frames. Figure 8 shows the interaction force

obtained on the frames from this dataset. Figure 9 il-

Figure 7. ROC performance on UMN dataset

Table 1. Performance of proposed scheme on UMN dataset

Method Area Under ROC

Optical Flow [11] 0.84Social Force [11] 0.96

Chaotic Invariants [14] 0.99NN [6] 0.93

Sparse Reconstruction (Scene 1) [6] 0.995Sparse Reconstruction (Scene 2) [6] 0.975Sparse Reconstruction (Scene 3) [6] 0.964

Sparse Reconstruction (Full Dataset) [6] 0.978Proposed Scheme (Scene 1) 0.9961Proposed Scheme (Scene 2) 0.9932Proposed Scheme (Scene 3) 0.9991

Proposed Scheme (Full Dataset) 0.9961

Table 2. Performance of the proposed scheme on the Prison dataset

Method Area Under ROC

Optical Flow 0.5801Proposed Scheme (Full Dataset) 0.8903

lustrates the performance of the proposed method on

some frames of the different sequences in this dataset.

The ROC curves in Figure 10 demonstrates that the

proposed method outperforms optical flow methods in

distinguishing the abnormal sequences from the nor-

mal ones. The quantitative results of this comparison

are reported in Table 2.

3.3. Results on PETS 2009 Dataset

This section describes the result obtained on PETS

2009 ’S3’ dataset. This dataset is different from the

other dataset used in this paper, in the sense that abnor-

mality begins smoothly and this makes the detection

more challenging because of the gradual transaction

(a)

(b)

(c)

(d)

N

N

A

N

A

Figure 8. Results of the proposed scheme on the prison dataset.

(a) Normal behavior frame and its corresponding interaction force

and detection result on sequence 1. (b) Abnormal behavior frame

and its corresponding interaction force and detection result on se-

quence 1. (c) Normal behavior frame and its corresponding in-

teraction force and detection result on sequence 2. (d) Abnormal

behavior frame and its corresponding interaction force and detec-

tion result on sequence 2.

Figure 9. ROC curve of abnormal behavior detection in the differ-

ent sequences of the Prison dataset

Table 3. Performance of proposed scheme on PETS 2009 dataset

Method Area Under ROC

Optical Flow Scene 1 0.8834Proposed Scheme Scene 1 0.9414

Optical Flow Scene 2 0.9801Proposed Scheme Scene 2 0.9914

from normal to abnormal activity. Figure 11 shows the

interaction force estimated using the proposed scheme

on PETS 2009 and Figure 12 shows the corresponding

Figure 10. ROC curves showing the comparison of proposed

scheme over optical flow method on the Prison dataset

(a)

(b)(b)

(c)

(d)

A

N

N

A

N

Figure 11. Samples frames from PETS 2009 dataset. (a) Input

frame from S3 (14-16). (b) The corresponding interaction force.

(c) Input frames from S3 (14-33. (d) The corresponding interac-

tion force.

Figure 12. The ROC curves of abnormal behavior detection in the

PETS 2009 database.

Table 4. Performance of proposed scheme on UCF dataset

Method Area Under ROC

Optical Flow 0.884Proposed Scheme (Full Dataset) 0.986

ROC curve. Table 3 shows the quantitative results of

the comparison, illustrating that the proposed scheme

outperforms the optical flow method.

3.4. UCF dataset

Finally, the effectiveness of the proposed algorithm is

also evaluated on the UCF dataset [11] composed by

12 video sequences representing normal and abnormal

scenes collected from the web. Also in this case, Fig-

ure 13 demonstrates that the proposed scheme outper-

forms the optical flow procedure, and this is further

corroborated by the quantitative results reported in Ta-

ble 4.

Figure 13. The ROC curves of abnormal behavior detection in the

UCF dataset.

Thus, from the above illustrated experiments, it is well

justified that, the proposed algorithm outperforms the

available state-of-the-art methods on realistic datasets

like UCF and Prison Riots, other than UMN and PETS

2009 benchmark datasets.

4. Conclusions

We introduced a new particle advection scheme todetect global abnormal activity from crowded videoscenes using PSO and SFM. We demonstrated the ca-pability of the method in capturing the crowd dynam-ics by performing the optimization of the interactionforce. The main advantages of the proposed methodis that it does not require the tracking of individualsnor to perform a learning stage. This further justifiesthe applicability of our proposed scheme for real timeapplications. Finally, results have also indicated that

our method is efficient, robust and highly performingin detecting the global abnormal activities on very dif-ferent types of crowded scenes.

References

[1] Pets 2009 dataset. http://ftp.cs.rdg.ac.uk/

PETS2009/. 5

[2] Umn dataset. mha.cs.umn.edu/movies/

crowd-activity-all.avi. 5

[3] K. Barnard, P. Duygulu, N. de Freitas, F. Forsyth,

D. Blei, and M. Jordan. Matching words and

pictures. Journal of Machine Learning Research,

3(1):11071135, 2003. 1

[4] D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent dirich-

let allocation. Journal of Machine Learning Research,

34(1):993 1022, 1981. 1

[5] T. Brox, A. Bruhn, N. Papenberg, and J.Weickert. High

accuracy optical flow estimation based on a theory for

warping. In European Conference on Computer Vision

(ECCV), 2004. 3

[6] Y. Cong, J. Yuan, and J. Liu. Sparse reconstruction

cost for abnormal event detection. In IEEE Conference

on Computer Vision and Pattern Recognition (CVPR),

2011. 1, 6

[7] D. Helbing and P. Molnar. Social force model for

pedestrian dynamics. Physical Review E, 51(4):42 –

82, 1995. 1, 3

[8] J. Kennedy and R. Eberhart. Particle swarm optimiza-

tion. In IEEE International Conference on Neural Net-

works, Perth, Australia, pages 1942–1948, 1995. 2

[9] F. Lekien and J. Marsden. Tricubic interpolation in

three dimensions, journal of numerical methods and

engineering. Journal of Numerical Methods and En-

gineering, 63(3):455 – 471, 2005. 1, 2

[10] R. Mehran, B. Moore, and M. Shah. A streakline rep-

resentation of flow in crowded scenes. In European

Conference on Computer Vision (ECCV), 2010. 1, 2

[11] R. Mehran, A. Oyama, and M. Shah. Abnormal crowd

behavior detection using social force model. In Pro-

ceedings of IEEE Conference on Computer Vision and

Pattern Recognition, pages 935 – 942, 2009. 1, 2, 4, 5,

6, 8

[12] O.Boiman and M.Irani. Detecting irregularities in im-

ages and in video. International Journal of Computer

Vision (IJCV), 74(1):17 31, 2007. 1

[13] S.Ali and M. Shah. A lagrangian particle dynamics ap-

proach for crowd flow segmentation and stability anal-

ysis. In Proceedings of IEEE Conference on Computer

Vision and Pattern Recognition, pages 1 – 6, 2007. 1, 2

[14] S. Wu, B. Moore, and M. Shah. Chaotic invariants of

lagrangian particle trajectories for anomaly detection in

crowded scenes. In Proceedings of IEEE Conference

on Computer Vision and Pattern Recognition (CVPR),

2010. 1, 2, 6