A Fixed Size Assumption Based Data Association Method for

Coalescing Objects Tracking using a Laser Scanner Sensor.

Pawel Kmioteka,b and Yassine Ruicheka

(a) Systems and Transportation Laboratory

University of Technology of Belfort-Montbeliard - France

pawel.kmiotek,yassine.ruichek@utbm.fr

(b) Department of Computer Science

AGH University of Science and Technology - Krakow, Poland

pawel.kmiotek@agh.edu.pl

Abstract—Data association is one of the crucial parts in thereliable objects’ tracking system. In this paper, a methodology fordata association using an Oriented Bounding Box based objectrepresentation is presented. A laser scanner sensor is used forobjects perception. A data association algorithm for coalescingobjects is described. The algorithm is based on the NearestNeighbours principle enriched by an Inter-Rays uncertaintyparadigm and a Fixed Size assumption, which are introducedto improve the tracking process in terms of objects size andcenter position estimation. Experimental results are presented todemonstrate the effectiveness of the proposed technique.

I. INTRODUCTION

The work presented in this paper is a part of a project

launched in the Systems and Transportation Laboratory of the

University of Technology of Belfort-Montbeliard. The project

is focused on the study of the concept of intelligent vehicles

and their integration in the city of the future. The aim is to

develop a vehicle having the ability to navigate autonomously

in various urban environments. The research developments are

based on an experimental platform consisting of an electrical

vehicle with an automatic control, equipped with several

sensors and communication interfaces. To reach the objective,

the first primary task is to develop a perception system

for detecting, localising and tracking objects in this type of

environments. In this paper, the emphasis is put on tracking

of compact dynamic objects using a laser range finder (LRF).

Representation of dynamic objects is crucial for tracking

and trajectory planning. In the literature concerning tracking,

points with elliptical uncertainty are used for representing

objects position[1][2]. This representation is good enough

for obstacle detection, collision warning or driving assistance

systems in well structured environments like highways [2][3].

In the urban areas, however, the objects movements are less

predictable. Thus, for the task of autonomous navigation in

demanding urban areas, these representation methods are not

sufficient. Oriented Bounding Box (OBB) [4][5][6] provides

a good approximation of the size, shape and orientation angle

of dynamic objects, with a good data compression ratio.

In this paper, the OBB based model with an Inter-Rays

(IR) uncertainty paradigm and a Fixed Size (FS) assumption

is used to represent dynamic objects. The IR uncertainty and

FS assumption are introduced to increase the tracking system

reliability by better object’s size and centre position estimation.

Data association is an important part of multiple-objects

tracking. The raw data points clustering is the basic stage of

the objects separation. In the literature, fixed threshold [7][5],

or adaptive threshold [8][9], are used.

We use raw data points clustering with tract-to-cluster corre-

lation to achieve the preliminary data association and to detect

the three following possible situations: new object appearance,

separate object tracking, coalescing objects tracking. Each of

the detected situation is treated separately in terms of data

association. In this paper, emphasis is put on coalescing objects

data association and tracking. In the literature, there are many

variants of the Nearest-Neighbour (NN), Probabilistic Data

Association (PDA), Joint PDA (JPDA) [1] algorithms, used

to track coalescing objects. The drawback of these methods is

that they do not take into account the size of the objects and it’s

uncertainty, caused by the interaction context objects/sensor.

The proposed data association method uses NN principle

enriched by an association constraint, which is based on

track size. The track size is computed by a size estimation

methodology founded on the IR uncertainty paradigm and

objects’ FS assumption.

The Extended Kalman Filter (EKF) with Discrete White

Noise Acceleration Model (DWNA) [1] and ego odometry is

used for objects tracking.

The paper is organized as follows. Section II presents

the OBB representation for dynamic objects, with the IR

uncertainty paradigm and the FS assumption. The data asso-

ciation method is described in section III. The tracking model

is briefly explained in the section IV. Before concluding,

experimental results are presented in section V.

II. OBJECT REPRESENTATION

A. OBB based model for object representation

Urban environments are characterised by limited spaces

available for navigation and there are little objects movement

constraints. In these conditions, geometrical representation of

dynamic objects is necessary. Oriented bounding box (OBB)

is a way of representing objects geometry with sufficient

approximation for the means of navigation.

ICVES 2009

978-1-4244-5441-9/09/$26.00 ©2009 IEEE 62

Fig. 1. Inter-Rays uncertainty paradigm.

The OBB based representation is described by two vectors z

(1) and σ2z (2). The first one represents the OBB geometry and

includes the centre coordinations cx, cy, the orientation angle θ

and the size dx, dy. The second vector represents uncertainties

on the components of the vector z.

z = [cx, cy, θ, dx, dy]T (1)

σ2z = [σ2

cx, σ2cy, σ

2θ , σ2

dx, σ2dy]T (2)

To construct the OBB based measurement, a specific method

is used. The OBB construction method consists of the four

following main steps. The first step is to find a contour of

the tracked objects using a convex-hull technique [10]. In

the second step, a method based on Rotating Calipers (RC)

technique [11] is used to construct an OBB, which is best

aligned to the object’s contour. The third step consists of

the uncertainty computation. Finally, the forth step concerns

the application of the IR uncertainty paradigm and the FS

assumption. In this paper, we will focus on the forth step. The

previous steps are described in details in [4].

B. Inter-Rays uncertainty

An important aspect of OBB extraction is the fact that the

raw data points representing the extremities of the extracted

OBB do not coincide with the real object’s extremities (see

Figure 1).

In the Figure 1, minX , minY , maxX , maxY are respec-

tively the minimum x coordinate, the minimum y coordinate,

the maximum x coordinate and the maximum y coordinate

of the extracted OBB. The line Lr (respectively Lr + n)

is crossing the point maxY (respectively minX) and is

perpendicular to the OBB side to which maxY (respectively

minX) belongs. The Inter-Rays (IR) real object’s extremities

position estimation and their variances are added to the OBB’s

size and OBB’s size uncertainty. The real object’s extremi-

ties are situated between the raw data points delimiting the

OBB (maxY , minX) and the points Pr and Pr + n. Pr

(respecitvely Pr+n) is the intersection point between the ray

r (respecitvely r + n) with the line Lr (respectively Lr + n).

Considering the OBB’s local X axis, the real object’s

extremity position is uniformly distributed with the mean

µIRx, which is equal to the half of the IR line segment length

dIRx. The IR line segment is defined by the point maxY and

Pr. To fulfil Kalman Filter assumption, the distribution of the

real object’s extremity position is approximated by a normal

distribution with the mean µIRx, and the variance σ2IRx, which

is set to dIRx

6 . The Inter-Rays values z[µIRx] and z[σ2IRx] are

used in each iteration of the tracking algorithm to correct the

size of the OBB measurement [4]. The correction equations

are expressed as follows:

z[dx] = zperc[dx] + z[µIRx] (3)

z[σ2dx] = zperc[σ

2dx] + z[σ2

IRx] (4)

where zperc is the percepred measurement, z is the corrected

measurement used for tracking.

The same process is applied for the OBB’s local Y axis.

C. Fixed Size assumption

The idea of the fixed size (FS) assumption is based on the

fact that, in general cases, objects’ size does not change during

the tracking. However, due to the LRF’s limited resolution and

change of the relative distance and orientation of the observed

object, measurements of the object’s size vary in time. The

principle of the FS assumption is that the track representing

the tracked object cannot reduce it’s size. The FS algorithm

takes place in each iteration of the tracking after the track

prediction and measurements extraction.

For the following algorithm description, we consider the

local OBB’s X axis. The same process is applied to the local

OBB’s Y axis.

Having the percepted OBB measurement with the IR line

segment length zperc[dIRx], we obtain the corrected IR line

segment length z[dIRx] associated with the OBB measure-

ment:

z[dIRx] = min(zperc[dIRx], xk−1[dIRx]) (5)

where xk−1[dIRx] is the IR line segment length associated

with the track at time k-1. The quantity z[dIRx] is then

memorised in the track xk:

xk[dIRx] = z[dIRx] (6)

After using the equation (3) and (4), the next step consists

of the measurement’s size correction by using the following

equation:

z[dx] = max(z[dx], xk[dx]) (7)

where xk[dx] is the track predicted size at the time k.

After correcting the percepted measurement’s size, the

measurement’s center must be appropriately translated. The

updating of the center position is achieved as follows.

Firstly, the visibility factor V Fx is computed for the OBB’s

local X axis:

V Fx =max(βf

minX , βfmaxX)

βfminX + β

fmaxX

(8)

63

Fig. 2. Visibility factor associated to the OBB’s local X axis.

where βminX and βmaxX correspond respectively to the an-

gles between OBB’s sides minXside and maxXside normals

and their radius vectors (see Figure 2). f is a parameter, which

is set to 4.

In the second step, the direction factor DFx associated

to the OBB’s local X axis is computed using the following

equation:

DFx =

{

1, if βmaxX > βminX (9a)

−1, if βmaxX < βminX (9b)

In the last step, the difference between the percepted size

zperc[dx] and the corrected size z[dx] is calculated:

∆dx = z[dx] − zperc[dx] (10)

Finally, the measurement’s center translation is expressed

as follows:

z[cx] = z[cx] + 12V Fx · DFx · ∆dx

z[σ2cx] = 1

2z[σ2dx]

(11)

III. DATA ASSOCIATION

One of the most important tasks of autonomous navigation

in urban areas is tracking of dynamic objects. Data association,

which is closely related to the objects representation and

sensory data, is a crucial part of the tracking process. In

this section, a data association methodology suitable for the

OBB representation and laser scanner data is proposed. The

emphasis is put on association of raw data points of coalescing

objects. Since geometrical features are taken into account, only

objects being previously recognised as separated ones can be

tracked correctly.

The data association algorithm is composed of the following

stages : preliminary association (raw data points clustering),

tracks to clusters correlation and raw data points to track

association. The preliminary association is based on distance

thresholding. Points belong to the same cluster if the Euclidean

distance between them is below a certain threshold. Each

cluster is represented by an Axis Aligned Bounding Box

(AABB). The raw data points clustering uses two thresholds:

Track prediction

Raw data points to tracks

prediction distances computation

0 gate 1 gate 2 or more gates

New track measurement

creation (clustering)

Intermediate

measurement

extraction

New Track

Points set

One Track

Points sets

Ambiguous

Points set

Intermediate

measurements

Ambiguous

points

association

Measurements creation

Track state

update

Data

Process

Measurements

Measurements

Decission

Raw data points classification

Fig. 3. Schema of NN+FS data association algorithm.

a general threshold and a neighbouring points one. Applied to

the points produced by neighbouring rays, the neighbouring

threshold is greater than the general one, which is used for all

non neighbour points.

After raw data points are clustered into an AABB, track to

cluster correlation takes place. The track is correlated with the

cluster if the track’s OBB intersects with the cluster’s AABB.

If the track do not intersect any cluster, the track is correlated

with the closest cluster. There are three possible outputs

of track to cluster correlation. A cluster can be correlated

with zero, one, two or more tracks. These cases represent

respectively the following situations: appearance of a new

object, tracking of separated object, and multi-object tracking.

Basing on the results of the previous step, raw data points

to track association follows. In this stage, raw data points,

positively associated with a track, create a measurement. Each

measurement is in the OBB format (see (1) and (2)).

In the first situation (appearance of a new object), all the

points are used to create the measurement. In the second one,

Mahalanobis distance based gating is used to associate raw

data points with a track. Not associated points are processed

to create new tracks. In the last case, a method based on the

Nearest-Neighbour principle is used. In this paper, two variants

of the method are proposed.

The first variant is a pure NN algorithm applied to raw data

points and tracks in interest. For each pair of raw data point

and a track correlated with the cluster, Mahalanobis distance

is calculated. The raw data point validated by tracks’ gates is

assigned to the closest track.

In the second variant (NN+FS), the Fixed Size assumption

is used to improve the NN association algorithm. Figure 3

shows the schema of the NN+FS variant. In the first stage

of the algorithm, raw data points are classified into three

classes: New Track Points (NTP), One Track Points (OTP),

Ambiguous Points (AP). The classification process is based on

the relation between raw data points and the gates of the tracks

in the cluster. The first class represents the points which are

outside all the gates. The second class consists of sets of points

which are inside only one track (one set per one track). The

last class consists of a set with points which are inside more

64

Fig. 4. Classification of raw data points using three classes.

than one gate (see Figure 4). The size of the gate is related

to the sum of the track prediction size and track prediction

position convariances by applying the following gate rule:

Td2 ≥ d2(ij) = νT

(ij)S−1ν(ij) (12)

where ν(ij) is the vector defined from the ith raw data point to

the jth track’s prediction OBB, S−1 is the inverse of the track

prediction covariance matrix. A value of the gate’s threshold

Td2 is taken from χ2 distribution with two degrees of freedom

and expresses the number of sigmas of the Normal distribution.

The points from the first class (NTP) are separated into

clusters (using the same clustering method as before in the

preliminary association). For each cluster, a new track is

created.

Each OTP class set associated with a track serves as a source

for obtaining intermediate measurement. In this process, the

basic OBB extraction method presented in [10] is used.

The aim of the last stage of the proposed NN+FS algorithm

is to tackle ambiguous points association using the AP class

set and intermediate measurements. Each ambiguous point

P(i) in this set has a list L(i) = {x(j)} of tracks to which

it may belong (point is inside the gate of each track of the

list), where i ∈ [1; N ] is a point number, N is the AP set

cardinality, x(j) is the jth track. For each pair composed by

a point P(i) and a track x(j) from the list L(i), the hypothesis

HP(i),x(j)that the ith point originate from the jth track

is tested. For this point-track pair (P(i), x(j)), we create a

temporary OBB z(ij)temp, constructed by including the point

in the track’s intermediate measurement. If the temporary OBB

size is not greater than the jth track prediction OBB size,

the point can be associated with the jth track. Otherwise,

the point is not associated with the track. It happens that

the point P(i) is associated with more that one track. In this

case, this point is associated with the jth track for which

the difference Diff(ij) = Diff(ij)[dx] · Diff(ij)[dy] is the

smallest, where Diff(ij)[dx] = |z(ij)temp[dx]−x(j)k[dx]| and

Diff(ij)[dy] = |z(ij)temp[dy] − x(j)k[dy]|.Tracks that have no raw data points associated stay valid

for the next iteration and its age is increased. The tracks that

exceed the maximum live span are deleted.

IV. TRACKING

The object’s state estimation is done by the means of

Extended Kalman Filter (EKF). All values of the track’s

state vector are expressed in the local ego-vehicle coordinate

system. Tracks are represented by the augmented OBB state

vector xk:

xk = [cx, cx, cy, cy, θ, θ, dx, dy]T (13)

Since tracking is done from dynamic platform, odometry

information is used to increase the tracking accuracy. State

change of the ego-vehicle is represented as differences of

position ∆x, ∆y and angle ∆γ between consecutive instants.

Thus, the input to the state transition equation is defined as:

uk = [∆x, ∆y, ∆γ] (14)

The Discrete White Noise Acceleration Model

(DWNA) [12] is used to describe objects kinematics

and process noise. Thus, taking into account the odometry

information, the track state transition is modelled as

follows [4]:

xk|k−1 = A(∆x, ∆y, ∆γ)F xk−1 + Buk + Gvk−1 (15)

where F is is the standard DWNA transition matrix, B

is the odometry-input model, G represents the noise gain

matrix, vk−1 is the process noise, defined with the Gaussian

distribution:

vk−1 = [cx, cy, θ, σdx, σdy], vk−1 ∼ N(0, Qk) (16)

where

Qk = Gvk−1GT (17)

with σdx and σdy are the process errors for OBB sizes dx and

dy respectively. The prediction covariance matrix is:

Pk|k−1 =∂A

∂x(xk−1)FPk−1

∂AT

∂x(xk−1)F

T + Qk (18)

where Pk−1 is the estimation covariance matrix.

The observation equation can be written as follows:

zk = Hxk|k−1 + wk (19)

where H is the observation model and wk , which represents

the measurement noise, is defined with a Gaussian distribution:

wk ∼ N(0, R)R = σ2zI5,5 (20)

where I5,5 is the identity matrix.

V. RESULTS

As a part of the ”intelligent vehicles and their integration

in the city of the future” project, a software platform is

developed to simulate the sensors and the multiple objects

tracking process. The simulator permits flexible changing of

all sensors parameters and mounting position. This allows to

test developed algorithms with different sensor configurations.

In the simulator laser range finder (LRF), LIDAR, stereovision

and odometry sensors are implemented.

For the test of the proposed algorithms, a Laser Range

Finder (LRF) is mounted in front of the vehicle. The step

65

Fig. 5. Objects configurations during the phase of ”multi-track dataassociation” (a) First scenario, (b) second scenario.

angle for the LRF is set to 1° with an angle range of 180°. In

the tests, the sensor range uncertainty σρ is set to 0.05m.

The proposed algorithm is evaluated using two scenarios,

with two tracked vehicles ( see Figure 5). The scenarios are

chosen to show the reliability of the proposed algorithm, which

stay stable even when two objects touch them selves.

In the first scenario, vehicles run towards each other by

travelling a symmetrical trajectory with respect to the Y axis of

the LRF reference. In the moment of vehicles frontal position,

the angles between the vehicles’ Y sides and the intersecting

LRF rays are close to the right angle. Thus, only the Y side

of the vehicles is seen by the LRF.

In the second scenario, the first vehicle runs towards the

second one, which does not move. When the two vehicles

become close to each other, the angles between the vehicles’

Y sides and the LRF rays are very small. Furthermore, the

X side of the second vehicle becomes occluded by the first

scenario.

In the second scenario, because of the vehicles orientation,

the LRF range uncertainty makes data association more diffi-

cult than in the first one.

We can see at the end of the two scenarios that the two

vehicles collide, and one vehicle pushes the other one (see

Figure 5). This part of the two scenario is considered to show

that NN+FS data association algorithm remains reliable even

in this extreme situation.

Three approaches are evaluated. The first is the pure NN

algorithm with the EKF based filtering. The second approach

is the pure NN algorithm with the EKF based filtering using

the IR uncertainty and the FS assumption. The third approach

is the NN+FS (NN enriched by track’s size information) with

the EKF based filtering using the IR uncertainty and the FS

assumption.

To evaluate the approaches, the vehicles’ size estimation

(Y side only - in the local tracks coordination system) is

compared. Size information gives the best insight into the

performance of the tested approaches, since badly associated

point directly influences track’s size. In the proposed scenarios

the Y side of the vehicles is visible during the phase of ”multi-

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

0 100 200 300 400 500 600 700 800 900

Object no.1 - Real Y side lenghtObject no.1 - Estimated Y side length

Object no.2 - Real Y side lenghtObject no.2 - Estimated Y side length

Fig. 6. First scenario - Evolution of the vehicles’ size (y side) using NNmethod without IR and FS.

3.6

3.8

4

4.2

4.4

4.6

4.8

5

0 100 200 300 400 500 600 700 800 900

Object no.1 - Real Y side lenghtObject no.1 - Estimated Y side length

Object no.2 - Real Y side lenghtObject no.2 - Estimated Y side length

Fig. 7. First scenario - Evolution of the vehicles’ size (y side) using NNmethod with IR and FS based tracking.

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

0 100 200 300 400 500 600 700 800 900

Object no.1 - Real Y side lenghtObject no.1 - Estimated Y side length

Object no.2 - Real Y side lenghtObject no.2 - Estimated Y side length

Fig. 8. First scenario - Evolution of the vehicles’ size (y side) using NN+FSmethod.

track data association” (see Figure 3), while the X side is

either not visible or occluded, and so cannot be used for

comparison.

In the figures, one can see the real object Y side size (the

two tracked objects are identical in terms of size) and it’s

estimation for each vehicle.

Considering the pure NN algorithm (see Figures 6, 9),

the absence of the IR uncertainty leads to underestimated

object size. Furthermore, the absence of the FS assumption

do not guarantee that the object size decreases in time (what

is different with reality). The integration of the IR uncertainty

allows a better estimation of the objects size. The correct size

estimation is assured by using the FS assumption, despite

unfavourable position and/or orientation of the objects (see

Figures 7, 8, 10 and 11.

Figures 6 - 8 show the evolution of the vehicles’ size

estimation (y side only - in the local tracks coordination

system) in the first scenario. The ”multi-track data association”

phase starts after about 500 iterations. One can see that the first

approach (pure NN) fails, the objects’ size estimates get worse

with time. The points originally belonging to the track number

1 are associated with the track number 2. The second approach

66

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

0 50 100 150 200 250 300 350 400

Object no.1 - Real Y side lenghtObject no.1 - Estimated Y side length

Object no.2 - Real Y side lenghtObject no.2 - Estimated Y side length

Fig. 9. Second scenario - Evolution of the vehicles’ size (y side) using NNmethod without IR and FS.

3.5

4

4.5

5

5.5

6

6.5

7

0 50 100 150 200 250 300 350 400

Object no.1 - Real Y side lenghtObject no.1 - Estimated Y side length

Object no.2 - Real Y side lenghtObject no.2 - Estimated Y side length

Fig. 10. Second scenario - Evolution of the vehicles’ size (y side) using NNmethod with IR and FS based tracking.

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

0 50 100 150 200 250 300 350 400 450 500

Object no.1 - Real Y side lenghtObject no.1 - Estimated Y side length

Object no.2 - Real Y side lenghtObject no.2 - Estimated Y side length

Fig. 11. Second scenario - Evolution of the vehicles’ size (y side) usingNN+FS method.

(pure NN with IR and FS based tracking) stays stable for a

certain period, but finally it also fails. The result is similar to

the first approach one. Indeed, the points originally belonging

to the track number 2 are associated with the track number 1,

with the difference that the second track size stays constant

due to the FS assumption. Only the third approach (NN+FS)

stays stable and manages to correctly associate points to tracks.

Figures 9 - 11 show the evolution of the vehicles’ size

estimation (y side only - in the local tracks coordination sys-

tem) in the second scenario. The ”multi-track data association”

phase starts after about 300 iterations. In this scenario, the

first approach (pure NN) manages to give the correct data

association, this is due to the favourable raw data points

configuration (see Figure 9). However, one can see that the

sizes are greatly underestimated due to the absence of the IR

uncertainty and the FS assumption in the tracking.

The second approach (pure NN with IR and FS based track-

ing) fails and the technique performs as in the first scenario

(see Figure 10). The third approach (NN+FS) performs well

and all points are correctly associated (see Figure 11).

VI. CONCLUSIONS

A method for tracking coalescing objects using LRF sensory

data is presented. Based on an oriented bounding box (OBB)

representation, the method introduces two new concepts: an

Inter-Rays (IR) uncertainty paradigm, which depends on the

interaction context objects/sensor, and a fixed size (FS) as-

sumption, which supposes that the real objects size does note

change during the tracking process. These two concepts are

used in the main stages of the algorithm: OBB represen-

tation, data association and tracking. Considering the NN

pure algorithm, three objects tracking approaches (NN, NN

with IR and FS based tracking, NN with IR and FS based

data association and tracking) are evaluated to analyse the

contribution of the IR and FS concepts. The analysis is done

with different scenarios to take into account different objects

spacial configurations. The experimental results show that the

method using NN with IR and FS based data association and

tracking presents a good reliability. Many investigations are

in progress and concern the improvement of the Fixed Size

algorithm, the integration of occlusion constraints and the

probability based data association for coalescing objects.

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