AUTOMATIC ESTIMATION OF MULTIPLE MOTION FIELDS

USING OBJECT TRAJECTORIES AND OPTICAL FLOW

Manya V. Afonso, Jorge S. Marques and Jacinto C. Nascimento

Instituto de Sistemas e Rob´otica, Instituto Superior T´ecnico, Lisbon, Portugal

Keywords:

Vector ﬁelds, Video segmentation, Optical ﬂow, Trajectories.

Abstract:

Multiple motion ﬁelds are an efﬁcient way of summarising the movement of objects in a scene and allow an

automatic classiﬁcation of objects activities in the scene. However, their estimation relies on some kind of

supervised learning e.g., using manually edited trajectories. This paper proposes an automatic method for the

estimation of multiple motion ﬁelds. The proposed algorithm detects multiple moving objects and their veloc-

ities in a video sequence using optical ﬂow. This leads to a sequence of centroids and corresponding velocity

vectors. A matching algorithm is then applied to group the centroids into trajectories, each of them describing

the movement of an object in the scene. The paper shows that motion ﬁelds can be reliably estimated from the

detected trajectories leading to a fully automatic procedure for the estimation of multiple motion ﬁelds.

1 INTRODUCTION

Video surveillance systems aim to detect and track

moving objects (e.g., pedestrians), and to character-

ize their behaviors in the scene. Surveillance sys-

tems should be able to learn typical behaviors from

video data in an unsupervised fashion, without us-

ing speciﬁc knowledge about the actions performed

by humans in the monitored environment. Such sys-

tems typically involve the following steps: (a) de-

tection of moving objects; (b) extraction of informa-

tive features (e.g. position, motion, shape); (c) track-

ing of the extracted features; and (d) classiﬁcation of

the observed behavior based on the extracted features

(Turaga et al., 2008).

In outdoor applications, the object trajectories

play an important role since they allow the system

to characterize typical behaviors and discriminate ab-

normal ones. One way of modeling trajectories is

the recently proposed approach using multiple mo-

tion ﬁelds, each representing a speciﬁc type of mo-

tion(Nascimento et al., 2009). The estimation of the

motion ﬁelds is done automatically once we have a

set of trajectories containing all typical activities in

the scene. This model was applied with success to

several problems. However, the training trajectories

were hand edited to compensate for object detection

and tracking errors.

This paper aims to overcome this difﬁculty. It

proposes an automatic object tracking algorithm and

its application to the estimation of multiple motion

ﬁelds. The goal is to automate the estimation of mul-

tiple motion ﬁelds from the video stream.

The paper is organized as follows. Section 2, de-

scribes related work in this area. Section 3, presents

the proposed automatic trajectory extraction algo-

rithm. Section 4, describes the comparison between

the detected trajectories and manually extracted ones,

used as ground truth. Experimental results with real

data are presented in Section 5.

2 RELATED WORK

2.1 Segmentation

Segmentation is a key step in most video surveillance

systems. It aims to detect objects of interest in the

video stream, using their visual and motion proper-

ties. It plays a key role since it reduces the amount

of information to be processed by higher processing

levels and locates the position of the targets. There is

a large spectrum of segmentation methods that can be

broadly classiﬁed into the following classes: (i) statis-

tical approaches, (ii) non-statistical approaches, and

(iii) spatio-temporal approaches.

The methods of the ﬁrst class adopt statistical

models for the background image, e.g., each pixel of

the background is modeled as a Gaussian distribution

457

Afonso M., Marques J. and Nascimento J. (2012).

AUTOMATIC ESTIMATION OF MULTIPLE MOTION FIELDS USING OBJECT TRAJECTORIES AND OPTICAL FLOW.

In Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods, pages 457-462

DOI: 10.5220/0003783404570462

 SciTePress

(Wren et al., 1997) or a mixture of Gaussians (Stauf-

fer et al., 2000; McKenna and Gong, 1999). Also,

minimization of Gaussian differences has been used

(Ohta, 2001). Dynamic belief network (Koller et al.,

1994) is another type of statistical method. This class

of works also comprises a combination of frame dif-

ferences and statistical background models (Collins

et al., 1999).

Non-statistical based approaches adopt determin-

istic descriptions of the background image. The

simplest approach is called background subtraction

method, in which, a pixel-wise difference between

the current frame and the background model is per-

formed. Foreground/backgorund pixels are deter-

mined if the resulting difference is above/under a

pre-determined threshold, respectively (Gonzalez and

Woods, 2002). Other methods consider an admis-

sible range for each pixel intensity, maximum rate

of change in consecutive images or the median of

largest inter-frames absolute difference (Haritaoglu

et al., 2000).

The third class is based on spatio-temporal seg-

mentation of the video signal. These methods try to

detect moving regions taking into account not only the

temporal evolution of the pixel intensities and color

but also their spatial properties. Segmentation is per-

formed in a 2D+T region of image-time space, con-

sidering the temporal evolution of neighboring pix-

els. This can be done in several ways, e.g. by using

spatio-temporal entropy (Ma and Zhang, 2001), or the

3D structure tensor deﬁned from the pixels spatial and

temporal derivatives (Souvenir et al., 2005).

2.2 Optical Flow

Optical ﬂow is a measure of the apparent motion of

all image pixels from one frame to the next. With

some exceptions, it is an estimate of the motion ﬁeld

(Barron et al., 1994), (Simoncelli, 1993). Denoting

the image intensity at position (x, y) and time t by

I(x, y,t), the optical ﬂow equation is

(∇

I)(x, y,t)u+ (∇

I)(x, y,t)v+ (∇

I)(x, y,t) = 0,

(1)

where x, y indicates a pixel location, and ∇

I, ∇

I, and

∇

I respectively denote the image gradient along the

x, y and t directions. This equation is not enough to

obtain the velocity vector (u, v) associated to each im-

age point since it has an inﬁnite number of solutions.

This is known as the aperture problem. Additional

constraints like smoothness have to be used.

There are several methods for computing the

optical ﬂow. Gradient based methods that im-

pose smoothness constraints on the ﬁeld of veloc-

ity vectors, the Horn and Schunck method (Horn

and Schunck, 1981) that uses a global smoothness

constraint or the Lucas-Kanade method (Lucas and

Kanade, 1981) that assumes the velocity is locally

constant combining local constraints over local re-

gions. These methods are considered more accurate

than the ones based on template matching, but they

can only be used if the displacements is small (Bar-

ron et al., 1994).

There are several ways to deal with large displace-

ments. One approach is region-matching based meth-

ods which do not actually solve (1), but try to ﬁnd the

most likely position for an image region in the next

frame. Yet, another method is the Bayesian multi-

scale coarse to ﬁne algorithm (Simoncelli, 1993). A

coarse to ﬁne warping scheme involving two nested

ﬁxed point iterations for energy minimization func-

tional was proposed in (Brox et al., 2004) and a vari-

ant of this method (Sand and Teller, 2008), wherein

video motion is represented by a set of particles and

particle trajectories yielding the displacements as well

as trajectories representing their motion.

2.3 Multiple Motion Field Model

Multiple motion ﬁelds were recently proposed as a

tool to summarize objects motion in a scene and

to statistically characterize their trajectories (Nasci-

mento et al., 2009). Each trajectory is assumed to be

a sequence of segments, each of them being generated

by one of the motion ﬁelds, the so-called active ﬁeld.

Model switching is performed based on a probabilis-

tic mechanism whose parameters depend on the posi-

tion of the object in the image. This model is ﬂexible

enough to represent a wide variety of trajectories and

allows the representation of space-varying behaviors

as required.

Each trajectory is a length-n sequence of positions

x = (x

, ..., x

) with x

∈ R

. Each position represents

the centroid of the object in the image. We assume

each trajectory is the output of a switched dynamical

system

= x

t−1

+ T

t−1

) + w

, (2)

where k

∈ {1, . . . , M} is the label of the active ﬁeld

at time t; {T

, . . . , T

} are M vector ﬁelds which de-

scribe typical motion patterns and (w

, . . . , w

) are in-

dependent samples of a zero-mean Gaussian random

vector with identity covariance.

The label sequence k = (k

, ..., k

) is assumed to

be a Markov chain of order one with space-dependent

transition probabilities

Pr{k

= j|k

t−1

= i, x

t−1

} = b

t−1

) (3)

Matrix B(x

t−1

) = (b

t−1

)) is a space-varying

stochastic matrix. The model parameters (motion

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

458

ﬁelds, noise variances, switching matrix ﬁeld) can be

retrieved from observed object trajectories using the

Expectation-Maximization (EM) algorithm.

3 PROPOSED APPROACH

In this section, we describe the segmentation and fea-

ture extraction processes. We ﬁrst perform back-

ground subtraction to detect the active regions in a

frame, and then estimate the velocity ﬁelds at the cen-

troid of each active region by computing the optical

ﬂow using the template matching algorithm. The re-

sult of these steps is a sequence of vectors containing

the spatial coordinates of the centroids of the active

regions, over the entire set of frames, and also the cor-

responding velocity ﬁelds.

3.1 Active Region Detection

We represent a frame at time t with M rows and N

columns by a matrix I

∈ R

M× N

. Given a sequence

of K frames { I

,t = 1, ..., K}, we estimate the back-

ground image B ∈ R

M× N

by taking a subset of frames

and applying a median ﬁlter to the evolution of each

pixel in time. We segment each frame I

, by subtract-

ing the background and thresholding the difference

image with a predeﬁned positive value λ, to produce

a binary image J

∈ { 0, 1}

M× N

with the value at pixel

m, n,

(m, n) =



1, if |I

(m, n) − B(m, n)| > λ,

0, otherwise.

(4)

For multiple moving objects, this requires ﬁnding the

connected pixels above this threshold. We do not as-

sume beforehand, the number of moving objects. We

therefore apply a clustering algorithm on this binary

image J

to ﬁnd connected regions (assuming 8 neigh-

bors), each cluster corresponding to a moving object.

In practice, because of the empirical nature of the

threshold value λ, there may be false active regions

(i.e. the threshold being exceeded where there is no

motion) or the active region corresponding to a single

moving object may get split into two or more clus-

ters. We solve the second problem by dilating the bi-

nary image J

before clustering, and solve the prob-

lem of false active regions by discarding the clusters

with small area. The number of active regions de-

tected at the frame t,will be denoted by n

3.1.1 Optical Flow using Template matching

Region based matching approaches for computing the

optical ﬂow deﬁne the velocity vector of an object

moving across successive frames as the vector of dis-

placements (i, j) that produces the best ﬁt between

image regions at different times. The best ﬁt means

that a distance measure between a region in a frame,

, and its possible location in the next frame, I

t+1

, is

minimum for the vector (i, j), or a similarity measure

such as cross-correlation is maximum. These meth-

ods are intuitively simple and relatively easy to im-

plement, and the computational load is low since it

has to be computed only for the active regions.

Let the bounding box of the k − th active region

be I

∈ Z

. Then assuming suitable boundary condi-

tions, the velocity vector is computed by solving the

optimization problem,

, v

) = arg min

(i, j)∈(−d

)

E(i, j), (5)

E(i, j) =

∑

(m,n)∈I

(m, n)−I

t+1

(m+i, n+ j))

+α(|i|+| j|).

(6)

where d

is the maximum displacement, and α > 0

is the regularization parameter that controls the rela-

tive weight of the penalty term. Since the entire block

is assumed to be displaced by the vector (u

, v

), we

consider this the optical ﬂow at the centroid of the k

region.

Figure 1 illustrates the optical ﬂow between two

frames at times t and t + 1, for a man walking. The

velocity vector is shown superimposed over the frame

at time t and its binary image, at the centroid of the

active region corresponding to the man.

Figure 1: Optical ﬂow between 2 frames. (a),(b) grayscale

frames at times t and t + 1, (c), (d) active regions corre-

sponding to the moving person in the two frames.

3.2 Motion Correspondence

At the end of the optical ﬂow step, we have a set of

centroid locations {(x

, y

), i = 1, ..., n

} and their

respective motion vectors {(u

, v

), i = 1, ..., n

}. We

therefore need to apply a region association algorithm

to integrate the centroids in successive frames into tra-

jectories. This differs from the approach presented in

(Nascimento et al., 2010), where the pre-processing

consists of two steps: active region detection using

the Lehigh Omnidirectional Tracking System (LOTS)

algorithm followed by region association. The ap-

proach herein proposed is threefold regarding the pre-

processing in [Nascimento et al., 2010]: (1) it com-

putes directly the vector ﬁelds, (2) it reduces drasti-

cally the errors that may arise in the region associa-

tion mechanism, and (3) consequently avoids manual

AUTOMATIC ESTIMATION OF MULTIPLE MOTION FIELDS USING OBJECT TRAJECTORIES AND OPTICAL

FLOW

459

corrections to obtain the trajectories. In this paper,

we directly apply a motion correspondencealgorithm,

namely, the GOA tracker (Veenman et al., 2001) on

the sequence of centroid positions.

Let the vector x

= [x

, y

]

denote the position

vector of the centroid of active region k in frame t. For

the centroids of two regions i and j in two successive

frames t and t + 1, the association cost is

(i, j) = kx

t+1

− (x

+ v

, (7)

where v

= [u

, v

]

is the velocity vector at the cen-

troid of active region k in frame t.

Let

be the size n

t+1

× n

matrix whose entries

are computed using the above cost function. In gen-

eral, the number of centroids in successive frames is

not the same, that is, n

t+1

6= n

. A centroid in frame t

may have a correspondingpoint in framet+1 belong-

ing to its trajectory or it may be the last point belong-

ing to that particular trajectory. Likewise, a centroid

in frame t + 1 may belong to a trajectory having an

associated point in frame t, or it could be the starting

point of a new trajectory. To be able to account for

the “birth” or “death” of trajectories, we pad the ma-

trix

with entries equal to γ to form a square cost

matrix C

of size (n

+ n

t+1

) × (n

+ n

t+1



, γ1

t+1

×n

t+1

)

γ1

×n

)

, γ1

×n

t+1

)



, (8)

where 1 stands for a matrix whose entries are all 1,

γ > 0 is the cost of starting a new trajectory or end-

ing an existing one. A column of this matrix corre-

sponds to a centroid in frame t and a row corresponds

to a centroid in frame t + 1. The Hungarian match-

ing algorithm (Kuhn, 1955) is then used to solve the

matching problem by minimizing the assignment cost

C =

t+1

∑

i, j=1

, (9)

where b

∈ {0, 1} and have the constraints

t+1

∑

i=1

= 1, ∀ j ;

t+1

∑

j=1

= 1, ∀ i . (10)

4 COMPARISON WITH GROUND

TRUTH

Once we have computed a trajectory of an object in

a given video sequence as described in the previous

section, we would like to compare it with the trajec-

tory for the same object from the set of previously

computed trajectories which we consider the ground

truth. We would like to do this automatically, without

the user having to manually locate the sequence of

frames corresponding to the trajectory. This has the

possible problems that the trajectory and its counter-

part from the ground truth may not exactly coincide in

terms of starting and ending frames, may have some

missed detections in some frames, or a single trajec-

tory may have multiple trajectories in the ground truth

corresponding to different segments (e.g. an object

moving in loops).

We denote an automatically detected trajectory

by {x}

t∈T

, deﬁned over a subset of frames T =

, . . . , t

}. Similarly, a trajectory i from the ground

truth set is denoted as {x

gt,i

}

t∈T

, with T

being the

set of relevant frames. If there are some frames in

common between T and T

, we deﬁne the set of over-

lapping frames, T

OL,i

= T ∩ T

, and compute the mean

square error between the two trajectories at the times

corresponding to the overlapping frames

E(i) =

OL,i

∑

t∈T

OL,i

− x

gt,i

. (11)

We then ﬁnd all trajectories in the ground truth for

which the MSE is within a limit of κ times the mini-

mum value, E

min

, {i : E(i) ≤ κE

min

}. From this set,

the trajectories which have no overlapping frames are

the ones corresponding to different continuous seg-

ments of the trajectory {x}

5 EXPERIMENTAL RESULTS

We illustrate our proposed approach on a set of video

sequences of a university campus (Instituto Superior

T´ecnico, Lisbon) recorded with a single static cam-

era. The total number of frames was 69596, acquired

at a rate of 25 frames per second. For our compu-

tation of optical ﬂow, the sequence was subsampled

at a rate of 5 frames. The frame size is 432× 540.

There were 7 pedestrian activities of interest in this

sequence, which are listed in Table 1.

Figure 2 shows two trajectories corresponding to

people walking, detected over a segment of 2000

frames, and their respective velocity vectors superim-

posed. The velocities were scaled by a factor of 10

and subsampled by a factor of 8 for display.

Figure 3 shows the two trajectories from Figure 2

and the corresponding ground truth trajectories, re-

trieved by the library search. The trajectory in Fig-

ure 3(a) corresponds to a single trajectory from the

ground truth library. The other trajectory of a person

leaving the building and entering it again, which ap-

pears as a loop in Figure 3(b) has two non-overlapping

matches in the ground truth (corresponding to activi-

ties “entering” and “leaving”), which are 70 frames

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

460

apart. Therefore, for the estimation of the motion

ﬁelds using the EM algorithm, we split this trajectory

into two segments corresponding to the two ground

truth trajectories for these activities.

Figure 2: Motion ﬁelds computed using optical ﬂow super-

imposed over the corresponding trajectories.

Estimated

Ground Truth

(a)

Estimated

Ground Truth

(b)

Figure 3: Extracted trajectories using the proposed ap-

proach and their corresponding ground truth trajectories.

We validate our method by computing the statis-

tics of the root mean square error (RMSE) between

the trajectories obtained using our method and the

ground truth trajectories from (Nascimento et al.,

2010). We present the mean and variance of the er-

ror for each class of activities and overall values, in

Table 1.

Table 1: Statistics of the Error in the Trajectories.

Trajectory class Error Mean Error Standard

(pixels) Deviation (pixels)

entering 4.03 0.99

leaving 3.80 1.07

walking along 2.46 0.35

crossing park up 4.12 1.29

crossing park down 5.33 1.80

passing through cars 4.49 0.87

browsing 3.23 0.82

Overall 4.31 1.59

We also validate the trajectories by estimating

the motion ﬁelds from the model (2) with parame-

ters estimated by the EM method (Nascimento et al.,

2009). It is not possible to directly compare the vec-

tor ﬁelds obtained from both sets of trajectories since

the ﬁeld estimates depend on the initialization of the

EM method. We therefore, evaluated each set of vec-

tor ﬁelds by measuring its ability to predict the target

position at the next time instant. For this dynamic

model, the predictor and prediction error are given by

= x

t−1

+ T

t−1

), (12)

= x

− x

t−1

− T

t−1

). (13)

In practice, we do not know the active ﬁeld k

, and

therefore select the error with the smallest norm (ideal

switching). We can now deﬁne an SNR measure

SNR = 10log



∑

t=2

− x

t−1

∑

t=2

min

− x

t−1

− T



(14)

The same number of trajectories per activity was

used to train the model for both the ground truth and

our extracted trajectories, with a total of 69 trajecto-

ries. Figure 4 shows both sets of trajectories after ap-

plying a projective transformation (homography) be-

tween the image and a plane parallel to the ground.

This is done to achieve viewpoint invariance, by pro-

jecting all image measurements onto a view orthogo-

nal to the ground plane (the so-called birds eye view).

(a) (b)

Figure 4: Trajectories from (a) the ground truth and (b) ob-

tained using the proposed approach, superimposed on the

starting frame, after applying a homography.

Figure 5 shows the motion ﬁelds estimated us-

ing the EM algorithm, trained with trajectories corre-

sponding to a single activity, with one model. Figures

5(a) and 5(b) show the motion ﬁelds for the activity

“entering” using the trajectories from the ground truth

set, and those obtained using the proposed approach,

respectively. Similarly, Figures 5(a) and 5(b) present

the corresponding motion ﬁelds for the activity “pass-

ing through cars”.

(a) (b) (c) (d)

Figure 5: Motion ﬁelds for the activities “Entering” and

“Passing through cars”, estimated from (a),(c) the GT tra-

jectories, and (b),(d) extracted trajectories.

Table 2 shows the best SNRs obtained for the

AUTOMATIC ESTIMATION OF MULTIPLE MOTION FIELDS USING OBJECT TRAJECTORIES AND OPTICAL

FLOW

461

Table 2: SNR between the trajectory and predicted trajectory from the estimated motion ﬁelds.

No. of models 1 2 3 4 5 6 7 8

Ground truth 1.37 4.57 4.09 6.23 5.62 5.4 5.48 3.58

Proposed method 1.15 4.42 4.13 6.02 5.52 3.45 5.05 2.96

ground truth trajectories (corresponding to all activi-

ties) and those obtained using the proposed approach

for different numbers of ﬁelds estimated by the EM

algorithm. It can be seen that the maximum SNR ob-

tained from the extracted trajectories is close to that

obtained from the ground truth data.

6 CONCLUSIONS

We have proposed a method for automatically com-

puting the trajectories and velocity ﬁelds of multi-

ple moving objects in a video sequence, using opti-

cal ﬂow. The trajectories obtained were found to be

close to the manually edited ground truth trajectories,

for a large set of activities occurring in the video se-

quences. The motion ﬁelds estimated from these tra-

jectories using the EM method led to an SNR close

to that obtained with the ground truth trajectories.

Hence the proposed method allows fully automatic

extraction of multiple motion ﬁelds. Current and fu-

ture work includes extending the method to denser en-

vironments such as crowds of moving people.

ACKNOWLEDGEMENTS

This work was supported by Fundac¸

ao para a

encia e Tecnologia (FCT), Portuguese Ministry

of Science and Higher Education, under projects

PTDC/EEA-CRO/098550/2008 and PEst-OE/EEI/

LA0009/ 2011.

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