tracking vehicles using multiple detections from a

1

Examensarbete 30 hpJuni 2015

Tracking Vehicles using Multiple Detections from a Monocular Camera

Viktor Bäck

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Tracking Vehicles using Multiple Detections from aMonocular Camera

Viktor Bäck

This thesis concerns image based tracking of vehicles using a monocular camera. Aclassifier is used to detect and classify objects in the images from the camera. Foreach detected object the classifier outputs several classifications, each including aconfidence value. The objective of this thesis is to investigate how these classificationsand confidence values can be used in a single target tracking framework in the bestpossible way. This is achieved by evaluating several tracking methods that utilize theclassifications and confidence values in different ways. The relationship between theconfidence values and the accuracy of the corresponding classifications is alsoinvestigated.

The methods are evaluated using data from real-world scenarios. It is found thatclassifications with high confidence values are more accurate on average than thosewith low confidence values. The differences in the average performance for theconsidered methods are found to be small.

Image based tracking of vehicles is a key component in active safety systems invehicles. Such systems can warn the driver or automatically brake the vehicle if acollision is about to happen, thereby preventing accidents.

ISSN: 1401-5757, UPTEC F15 055Examinator: Tomas NybergÄmnesgranskare: Thomas SchönHandledare: Daniel Ankelhed, Niklas Ollesson

Sammanfattning

Att köra bil är bland det farligaste en genomsnittlig person gör. Förarens säkerhetberor både på hennes egen och andras förmåga att vara uppmärksam i trafiken.Detta medför problem eftersom människors förmåga att upptäcka och reagerakorrekt i farliga situationer av sin natur är begränsad. Så länge det är enmänniskasom kör bilen kommer dessa problem att kvarstå. Passiva säkerhetssystem, såsom bilbälten och krockkuddar, kan lindra skadorna till följd av en kollision. Föratt förhindra att olyckan inträ↵ar krävs dock så kallade aktiva säkerhetssytem,vilket är vad detta arbete handlar om.

På samma sätt som en människa använder sina sinnen kan en bil använda sen-sorer för att få information om omgivningen. Sådana sensorer kan till exempelvara en eller flera kameror, eller radar. Informationen från dessa sensorer bearbe-tas därefter stegvis. Först tas information fram om vart intressanta objekt så sombilar och fotgängare befinner sig, samt vilken hastighet dem rör sig med. Därefterkan man utifrån denna information bedöma om en kollision är påväg att inträf-fa. Om så är fallet kan olika strategier för att undvika kollisionen tillämpas. Tillexempel så kan föraren varnas med en ljudsignal, eller så kan bilen automatisktaktivera bromsarna.

I det här arbetet undersöks olika metoder för att bestämma position och has-tighet för bilar. Den enda sensoren som används av en kamera. En klassifice-rare används för att detektera och klassificera objekt i bilderna från kameran.För varje detekterat objekt ger klassificeraren ifrån sig ett konfidensvärde. Måletmed det här arbetet är att undersöka hur dessa klassificeringar och konfidensvär-den på bästa möjliga sätt kan användas för målföljning av bilar. Detta uppnåsgenom att utvärdera målföljningsmetoder som använder klassificeringarna ochkonfidensvärdena på olika sätt. Förhållandet mellan konfidensvärdena och nog-grannheten av motsvarande klassificeringar undersöks även.

Metoderna utvärderas med hjälp av data från verkliga scenarion. Utvärdering-en visar att klassificeringar med hög konfidens i genomsnitt är noggrannare änklassificeringar med låg konfidens. Skillnaden i den genomsnittliga prestandanför metoderna visas vara liten.

i

Acknowledgments

First of all I would like to thank my supervisors Daniel Ankelhed and NiklasOllesson at Autoliv Electronics AB, as well as my subject reviewer at UppsalaUniversity, Thomas Schön. You always pointed my in the right direction when Igot o↵ track.

Special thanks to Karl Granström and Gustaf Hendeby for taking your timeand answering my questions. Your ideas and advise have been truly helpful.

Also, thank you Peter Hall and Jacob Roll for the opportunity to do my thesisat Autoliv. It has been really great to get to know Autoliv as a company, and allthe wonderful people working there.

I would also like to thank David Molin, who was doing his thesis at Autoliv atthe same time as me, for all the interesting conversations and for accompanyingme on the climbing walls of Hangaren.

Last but definitely not least, I thank my family for their unfaltering support.

Linköping, June 2015

Viktor Bäck

v

Contents

Notation ix

1 Introduction 1

1.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Target Tracking 9

2.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.1 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . 10

2.2 Data Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Nearest Neighbor Association . . . . . . . . . . . . . . . . . 122.2.3 Probabilistic Data Association . . . . . . . . . . . . . . . . . 13

2.2.3.1 Association Probability . . . . . . . . . . . . . . . 132.2.3.2 State Update . . . . . . . . . . . . . . . . . . . . . 14

2.2.4 M/N Initiation and Deletion of Tracks . . . . . . . . . . . . 15

3 Methods 17

3.1 Measurement Group Partitioning and Association . . . . . . . . . . 173.2 Clustering Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Average Cluster . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Simple Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.3 Weighted Sum Cluster . . . . . . . . . . . . . . . . . . . . . 203.2.4 Nonlinear Weighted Sum Cluster . . . . . . . . . . . . . . . 203.2.5 Regression Cluster . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Probabilistic Data Association . . . . . . . . . . . . . . . . . . . . . 23

4 Error Analysis 25

vii

viii Contents

4.1 Track-Marking Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Error Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.2 5-Largest Tracking Error . . . . . . . . . . . . . . . . . . . . 264.2.3 Clustering Error . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Overtaking Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Confidence Value Error Analysis . . . . . . . . . . . . . . . . . . . . 27

5 Results 29

5.1 Method Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Method Evaluation in Overtaking Scenarios . . . . . . . . . . . . . 295.3 Confidence Value Evaluation . . . . . . . . . . . . . . . . . . . . . . 31

6 Conclusions 33

6.1 Overall Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 336.1.1 Simple Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . 336.1.2 Nonlinear Weighted Sum Cluster . . . . . . . . . . . . . . . 346.1.3 Weighted Sum Cluster . . . . . . . . . . . . . . . . . . . . . 346.1.4 Average Cluster . . . . . . . . . . . . . . . . . . . . . . . . . 346.1.5 Probabilistic Data Association . . . . . . . . . . . . . . . . . 346.1.6 Regression Cluster . . . . . . . . . . . . . . . . . . . . . . . 34

6.2 Performance in Overtaking Scenarios . . . . . . . . . . . . . . . . . 346.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7 Future Work 49

Bibliography 51

Notation

Classification, measurement and region of interest (ROI) are words that will beused somewhat interchangeably in this thesis. Although, in actuality a classifica-tion is transformed into a measurement, which corresponds to a ROI than can bedrawn as a rectangle on the image. Also, image and frame both refer to a singleimage from the camera mounted on the ego vehicle.

All variables that represent vectors, matrices or sets are written in bold font.Vectors are by default column vectors.

A table with the most frequently used abbreviations in this thesis are pre-sented below.

Abbreviations

Abbreviation Explanation

5-LTE 5-Largest Tracking ErrorAC Average ClusterCE Clustering ErrorEKF Extended Kalman FilterGNN Global Nearest NeighbourKF Kalman FilterNN Nearest NeighbourPDA Probabilistic Data AssociationPDF Probability Density FunctionRC Regression ClusterROI Region of InterestSC Simple ClusterSO Spatial OverlapSSM State Space ModelTO Temporal OverlapTE Tracking ErrorWSC Weighted Sum Cluster

ix

1Introduction

Active safety systems in vehicles is a field that has gained more and more atten-tion in recent years as a result of embedded electronic systems becoming cheaperand more powerful. It is estimated that more than 90% of all vehicle accidentsare caused by human error [1]. The potential of such systems to prevent injuriesand save lives is therefor immense.

Systems are currently in use that are able to warn the driver if a threateningsituation occurs, or brake the car using autonomous emergency braking (AEB) inorder to prevent accidents. The possibility of fully autonomous cars sharing theroad with human drivers has also gained a lot of attention in media lately, not atleast due to the Google Self-Driving Car project [2].

The European New Car Assessment Programme (Euro NCAP) is a programmethat assesses the safety of vehicle models. A rating is assigned to a vehicle modelbased on how it performs in tests that simulate real-world accident scenarios.In 2014 AEB was included in the Euro NCAP rating system [3]. This acts as acatalyst which increases the demand for active safety systems.

1.1 Previous Work

Target tracking is a challenging problem that has occupied the research commu-nity for several decades. It is also a highly relevant problem, with applicationssuch as active safety system for vehicles [4], air tra�c control [5] and ballisticmissile surveillance [6]. The challenges come from the fact that sensors, as wellas models, are always inaccurate to some degree. In target tracking applicationsthis inaccuracy typically results in (i) measurements not originating from a target,known as clutter (ii) measurements originating from another target (iii) lack ofmeasurement due to occlusion (iv) probability of detection less than unity.

Several methods that handle these problems have been considered. A category

1

2 1 Introduction

of commonly used non-probabilistic methods are based on data association, i.e.explicit association of measurements to tracks. One simple but common methodis the Nearest Neighbour (NN) method which associates the closest gated mea-surement to the target [7]. The Global Nearest Neighbour (GNN) method is a gen-eralization of NN which handles association of measurements to multiple targetsby solving a optimal assignment problem [7]. The Probabilistic Data Association(PDA) filter, first presented in [8], handles single-target tracking in a clutteredenvironment by considering all events where at most one of the measurementsin the target gate is target originated. A multi-target extension of PDA called theJoint Probabilistic Data Association (JPDA) filter is presented in [9]. Track initia-tion and deletion is added to this framework in the Integrated Probabilistic DataAssociation (IPDA) filter [10], and its multi-target version the Joint IntegratedProbabilistic Data Association (JIPDA) filter [11]. In [12] the Multi-HypothesisTracking (MHT) method is presented, which considers the data association prob-lem over time by considering all measurement to track association events. TheMHT method su↵ers from combinatorial growth of the number of associationevents, and is known to be more computationally demanding and more compli-cated to implement than e.g. JPDA, which limits its use in real-time systems.

Multi-detection versions of PDA, JPDA and IPDA have recently been pre-sented in [13], [14], [15], respectively. A common application for these methodsare the over-the-horizon radar (OTHR) problem [16].

Another category of methods uses random finite sets (RFS) in order to for-mulate and solve the problem in a Bayesian setting. Explicit data associationis thereby avoided. Examples of such methods are the Probability HypothesisDensity (PHD) filter [17] and the Bernoulli filter [18]. Various extension of thesemethods exist that handles multi-target tracking, initialization/deletion of tracks,etc.

1.2 Problem Formulation

The system setup is as follows. A monocular camera is mounted in the frontof the ego-vehicle and is directed in the vehicle’s forward direction. Images fromthe camera are processed in real-time by a detection algorithm, which determinesclass and pixel location of objects in the image, as well as a confidence value foreach detected object. The confidence value gives a measure of the certainty of theclassification. The output from the detection stage is then used as measurementsin the tracking stage, according to the tracking-by-detection paradigm. The de-tection is thus done independently of the tracking. The images from the camerahave a resolution of 1024x592 pixels.

The objective of this thesis is to investigate how the classifications and the cor-responding confidence values can be used in a single target tracking frameworkin the best possible way. This is achieved by investigating di↵erent clusteringmethods (Section 3.2) and a modified version of the Probabilistic Data Associa-tion (PDA) method (Section 3.3). The relationship between the confidence valuesand the accuracy of the corresponding classifications is also investigated (Section

1.3 Limitations 3

4.4).The tracking is done using a model which describes the target motion in 3-

dimensional world coordinates. The tracking is evaluated in world coordinates(see Chapter 4) using manually marked reference tracks in the image plane. Asimple state transition model is used in order to make it easier to investigate thebest use of the classifications. Indeed, using a simple model allows one to isolatethe contribution from the classifications, without having to consider contribu-tions from e.g. optical flow, etc. This is done in the hope that the analysis is stillvalid for more advanced models.

1.3 Limitations

To limit the scope of this thesis, the following assumptions are made.

1. Single target tracking

2. Preceding tra�c on highway roads

3. Same model tuning can be used for all implemented methods

Only single target tracking methods are considered. Measurement associationproblems that arise due to the presence of multiple targets are handled in a waythat is independent of the used tracking method, as described in Section 3.1. Thisis done so that the focus can be on investigating how the single target tracking canbe improved. In order to limit the e↵ect of measurement association problemsdue to multiple targets, di�cult scenarios such as tra�c in cities are avoided.Hence only data from highway tra�c is considered.

It is also assumed that the same model (as given in Section 1.4) can be usedfor all considered methods.

1.4 Model

For each image frame from the camera, the detection algorithm outputs severalclassifications for each detected object in the image. Each classification containsinformation about the class of the object, a region of interest (ROI) which containsinformation about the pixel location of its corners, as well as an estimate of thecertainty of the classification,

P(object is of class T | true detection of object). (1.1)

The height of the ROI is not detected. Instead it is derived from a constant width-to-height ratio which is dependent on the object class. An example of a set ofclassifications for a car is shown in Figure 1.1. The confidence value is shownin the upper left corner of each ROI. Note that the ROI of each classification isassumed to be right-angled.

4 1 Introduction

Figure 1.1: Classifications with corresponding confidence values (shown inthe upper left corner of each ROI) for a car.

The classifications are then transformed and used as measurements y in thetracking stage according to

y =

0BBBBBBBBBB@

xHCP [pixels]width [pixels]ybottom [pixels]size [meters]

1CCCCCCCCCCA, (1.2)

where xHCP is the horizontal center pixel, width is the width in pixels, and ybottomis the y-coordinate of the bottom edge of the ROI. The component size is an as-sumed width of the vehicle in world coordinates and is constant for each vehicleclass. This prior knowledge about objects based on their class allows one to esti-mate the distance to the objects. Since size does not contain any new information,it can be considered an artificial measurement.

As a model for the target dynamics we consider a constant velocity model,where the acceleration is modeled as normally distributed noise. With measure-ment signal as in (1.2), the model can be formulated using a state space model(SSM) as

xk+1 = f (xk , uk) + wk (1.3a)yk = h(xk) + vk , (1.3b)

where wk ⇠ N (0, Qk) and vk ⇠ N (0, Rk) are process noise and measurementnoise, respectively, which are assumed to be normally distributedwith covariance

1.4 Model 5

x

y

z

forward direction

Figure 1.2: Sketch of the ego-vehicle and the xyz-coordinate system. Theorigin of the coordinate system coincides with the location of the camera,which is placed on the windscreen. The forward direction of the vehicle isindicated by an arrow.

matrices Qk and Rk . The input vector uk contains information about the ego-vehicle motion:

uk = vego✓ego

!, (1.4)

where vego and ✓ego are the velocity and and yaw rate of the ego-vehicle, respec-tively. The state vector, which contains information about a target, has the follow-ing components

xk =

0BBBBBBBBBBBBBBBBBBB@

wxvxyvyz

1CCCCCCCCCCCCCCCCCCCA

, (1.5)

where w is the width of the vehicle in world coordinates, vx and vy are the velocityin the x- respectively y-direction in the xyz-coordinate system that is attached tothe camera in the ego-vehicle, as given in Figure 1.2. The point (x, y, z), whichis also measured in world coordinates, is located on the lower center part of theROI that frames the backside of the target vehicle (see Figure 1.3).

The measurement model (1.3b) models the fisheye e↵ect of the camera andtransforms world coordinates (x, y, z) to image coordinates (xp, yp) according tothe pinhole camera model, which is given by

xpyp

!=

1x

�fxy�fyz

!, (1.6)

where fx and fy are the focal points in the x- and y-direction, respectively.

6 1 Introduction

1.5 Dataset

All considered tracking methods are implemented and evaluated in Chapter 5using data that have been recorded in real-world scenarios. The data includesclassifications, motion data of the ego-vehicle and images from the camera. Themethods are then evaluated by comparing estimated tracks with reference tracks,known as markings, that represent the true position of objects in the image plane.Each marking contains the class of the object and an ROI in pixel coordinates (seeFigure 1.4),

yM =

0BBBBBB@

xHCP [pixels]width [pixels]ybottom [pixels]

1CCCCCCA , (1.7)

with notation as in (1.2). The markings have been obtained by manually drawingROIs for objects in the recorded image frames, which is done with almost pixelprecision.

1.6 Outline

An outline of the chapters is given below.

• Chapter 2 gives an introduction to single tracking.

• Chapter 3 presents all methods that are considered in this thesis.

• Chapter 4 gives a detailed description of how the methods are evaluated.

• Chapter 5 presents method evaluation results.

• Chapter 6 presents comments on the results from the method evaluation.

• Chapter 7 presents thoughts on future work.

1.6 Outline 7

Figure 1.3: ROI corresponding to the state vector xk shown in blue, wherethe point (x, y, z) is marked with a white dot.

Figure 1.4: Car with a marking shown in red.

2Target Tracking

Target tracking amounts to estimating the state of one or many targets over timegivenmeasurements. In a typical scenario measurements arrive sequentially overtime, and whenever a newmeasurement is available the state estimate is updatedaccordingly.

The di↵erent stages of a typical target tracking algorithm are described indetail in the following sections.

2.1 Filtering

In the filtering step the state estimate is updated based on the previous stateestimate and newmeasurements. This is achieved by using a model of the system,such as the state space model (SSM)

xk+1 = f (xk , uk) + wk (2.1a)yk = h(xk) + vk , (2.1b)

where wk ⇠ N (0, Qk) and vk ⇠ N (0, Rk) are process noise and measurementnoise, respectively, which we assume are normally distributed. The state transi-tion function f is in general a nonlinear function of the previous state xk and,possibly, an input signal uk . It should capture the dynamics of the system, andmakes it possible to predict future states based on a previous state. The observa-tion function h is also in general a nonlinear function of the state estimate xk andcontains information about how the state is measured.

2.1.1 Kalman Filter

The filtering problem amounts to estimating p(xk | y1:k), i.e. the probability den-sity function of the state at time k given all measurements up to time k. In the

9

10 2 Target Tracking

case when f and h are linear, the SSM (2.1) can be written as

xk+1 = Fkxk + Bkuk + wk (2.2a)yk = Hkxk + Dkuk + vk , (2.2b)

where Fk , Bk , Hk and Dk are constant matrices. Unlike (2.1), an exact solutionto the filtering problem exists for (2.2) and is given by the normal distributionN (xk|k , Pk|k), with parameters given by the recursive equations

xk|k = xk|k�1 + Kk yk , (2.3a)Pk|k = (I � KkHk)Pk|k�1, (2.3b)yk = yk � Hk xk|k�1, (2.3c)

Kk = Pk|k�1HTk S�1k , (2.3d)

Sk = HkPk|k�1HTk + Rk . (2.3e)

The equations (2.3) are collectively known as the Kalman filter (KF). State predic-tions can then be calculated from the filtered state estimate using the dynamicsof the model according to

xk+1|k = Fk+1xk|k + Bk+1uk+1, (2.4a)

Pk+1|k = Fk+1Pk|kFTk+1 +Qk+1. (2.4b)

2.1.2 Extended Kalman Filter

If the state transition model f or the observation model h in (2.1) are nonlinear,then there exist no closed form solution to the filtering problem. Hence one hasto rely on methods that solve the problem approximately. Several such methodshave been considered, e.g the unscented Kalman filter (UKF) and particle filters(PF). In this thesis one of the most common approaches is considered; the socalled extended Kalman filter (EKF). The EKF utilizes a quite intuitive approach;instead of using Hk in the update equations (2.3), use the Jacobian

JH, k =@h@x

��xk|k�1

. (2.5)

Similarly, Fk+1 is replaced by the Jacobian of f in the prediction equation (2.4b),

JF, k+1 =@f@x

��xk|k , uk

. (2.6)

When calculating the innovation yk and the state prediction xk+1|k , however, thenonlinear functions h and f should be used;

yk = yk � h(xk|k�1, uk), (2.7a)xk+1|k = f (xk|k , uk+1). (2.7b)

The EKF is ad-hoc in the sense that there is no guarantee that it converges to thecorrect solution. That being said, it has been empirically shown to work well formany systems with mild nonlinearities.

2.2 Data Association 11

2.2 Data Association

If more than one target is being tracked or clutter is present, one faces the prob-lem of how the measurements should be assigned to the di↵erent targets, andhow they should be used to update the state of each target. Data associationmethods present a solution to this problem.

2.2.1 Gating

The first step in data association methods often consists of gating the measure-ments, which means that for each target we only consider measurements thatare su�ciently ’close’ based on the current estimate of the targets state and somedistance norm. This reduces computational cost, and is motivated by the obser-vation that measurements ’far away’ from a target are unlikely to have originatedfrom that target. Several di↵erent gating strategies exist which consider di↵erentdistance norms. One of the most commonly used is ellipsoidal gating, where ameasurement yk is considered to be inside the gate if

d2k := (yk � yk)T S�1k (yk � yk) �G (2.8)

is fulfilled, where �G > 0 is a parameter that determines the size of the gate,yk is the predicted measurement and Sk is the covariance of the innovation, ascalculated in (2.3) (see Figure 2.1a). The volume of the ellipsoidal gating regionis

VG = CM

p|Sk |GM/2, (2.9)

whereM is the number of elements in yk , and CM is given in terms of the Gammafunction � as

CM =⇡M/2

�⇣M2 + 1

⌘ =

8>>>>>>>><>>>>>>>>:

⇡M/2⇣M2

⌘!, M even

2M+1⇣M+12

⌘!⇡(M�12 )

(M + 1)!M odd.

. (2.10)

The distance norm d2k , known as the Mahalanobis distance, of the innovationtakes into account the uncertainty of the predicted measurement, resulting ina larger gating region if the uncertainty is large. If the state and measurementmodels are accurate, then d2k is approximately �2 distributed with M degrees offreedom. Hence it is common to choose a value of �G that corresponds to the, say,99th percentile of the �2 distribution, which means that approximately 99% ofthe measurements are expected to fall into the gate. This often gives a reasonabletrade-o↵ between having a gate which is too large and thus resulting in a highcomputational burden, and not gating all measurements that originate from thetarget.

Gating alone does not necessarily solve all data association problems. Con-sider the following scenarios. (i) Suppose that more than one measurement falls


d2k = �G

y1k

y2k

y3k

yk

(a)

d2k = �G

y1k

y2k

y3k

y1k

y2k d2k = �G

(b)

Figure 2.1: (a) Predicted measurement yk and an ellipsoidal gating regiongiven by d2k �G. Measurements y1k and y2k are inside the gate, while y3kis not. (b) Two overlapping gating regions corresponding to two di↵erenttracks, with predicted measurement y1k and y2k , respectively. Measurementy2k is inside both gates.

into a target gate; it could be a measurement from another target or it could beclutter. In this case it is not clear how the target state should be updated usingthe gated measurements. (ii) Another problematic situation is shown in Figure2.1b, where a measurement is in the gate of two di↵erent target gates.

In the following sections some common data associationmethods that addressproblem (i) are presented. However, since the emphasis in this thesis is on single-target tracking, rather than multi-target tracking, methods that handle problem(ii) are not considered. Instead, measurement association problems due to multi-ple targets are handled as described in Section 3.1.

2.2.2 Nearest Neighbor Association

A simple data association method that is common in radar applications is NearestNeighbour (NN) data association, where the distance to each measurement insidethe gate is calculated according to (2.8). The measurement with the smallestdistance value, i.e. which is closest to the predicted measurement, is then used toupdate the state of the target. The other measurements inside the gate are simplyneglected. This resolves the association problem in Figure 2.1a.

There exist global nearest neighbor association (GNN) methods that deal withthe problem in Figure 2.1b by associating measurements with tracks in such away that the total cost of all associations are minimized, thus obtaining an opti-mal solution to the association problem.


2.2.3 Probabilistic Data Association

The probabilistic data association (PDA) method estimates the posterior statePDF by considering all possible measurement association events. In each asso-ciation event it is assumed that at most one measurement is target originated andthe other measurements are clutter. A derivation (which is inspired by [15]) ofthe PDA method now follows.

Suppose that N measurements {y}k := {yjk}Nj=1 are inside the gate of a trackat time k, and let Yk := {{y}t | t k} be the set of all measurements up to time k.Furthermore, let Aj be the event that measurement yj is target originated, and A0be the event where all measurement are clutter. Let A be the set of all mutuallyexclusive and exhaustive events Aj .

The Probabilistic Data Association (PDA) filter, first presented in [8], solvesthe measurement association problem by marginalizing the posterior state PDFwith respect to all possible association events,

p(xk |Yk) =X

Aj2Ap(xk , Aj |Yk)

=X

Aj2Ap(xk |Aj , Yk)P(Aj |Yk), (2.11)

where P(Aj |Yk) is the probability of the association event Aj given measurementsup to time k.

2.2.3.1 Association Probability

The association probability P(Aj |Yk) can be expressed in terms of a likelihoodand a prior conditioned on the previous measurements according to

P(Aj |Yk) = ⌘p({y}k |Aj , Yk�1)P(Aj |Yk�1), (2.12)

where all terms that do not depend on Aj are included in the normalization con-stant ⌘. The association Aj at time k is assumed to be independent of previousmeasurements, i.e. P(Aj |Yk�1) = P(Aj ). Also, the prior P(Aj ) is assumed to beuniformly distributed, which means that it can be included in the normalizationconstant.

For j � 1 the likelihood in (2.12) is proportional to

p({y}k |Aj , Yk�1) / PDPG�N�1⇤j

k , (2.13)

where PD is the probability of detection, and PG is the probability that a detectedmeasurement is inside the track gate. The clutter density is assumed to Poissondistributed with parameter (see [7])

� =NVG

, (2.14)


where VG is the volume of the track gate (2.9). The spatial likelihood of the truemeasurement ⇤j

k can be obtained by calculating the corresponding innovation

and evaluating the innovation PDF N (yjk � h(xk) | 0, Sk), which is given by

⇤jk =

ed2j /2

PG(2⇡)M/2p|S|k

, (2.15)

where M is the dimension of the measurement vector yk and dj is the Maha-lanobis distance (2.8). The innovation PDF is restricted to the gating region byincluding the gating probability PG in (2.15).

For j = 0 the likelihood is proportional to

p({y}k |Aj , Yk�1) / (1 � PDPG)�N . (2.16)

The normalization constant ⌘ can now be obtained by marginalizing (2.12)with respect to the association events Aj , using (2.13) and (2.16). This yields thefollowing expressions for the association probability,

P(Aj |Yk) =

8>>>>>>>><>>>>>>>>:

(1 � PDPG)�(1 � PDPG)� + PDPG

PNj=1 ⇤

jk

j = 0

PDPG⇤jk

(1 � PDPG)� + PDPGPN

j=1 ⇤jk

j � 1.

(2.17)

2.2.3.2 State Update

A weighted sum of the innovations is calculated as

yk =NX

j=1

P(Aj |Yk)yjk , (2.18)

where yjk is the innovation corresponding to measurement yjk . The sum of PDFs

(2.11) is now approximated using a single Gaussian distribution N (xk |xk|k , Pk|k),where

xk|k = xk|k�1 + Kk yk , (2.19)

and

Pk|k = P0k|k + dPk , (2.20)

where

P0k|k = P(A0|Yk)Pk|k�1 + (1 � P(A0|Yk))P⇤k|k (2.21)

dPk = Kk

26666664

NX

j=1

P(Aj |Yk)yjk(y

jk)

T � yk yTk

37777775K

Tk (2.22)

P⇤k|k = (I � KkHk)Pk|k�1. (2.23)


The Kalman gain Kk and the state prediction are calculated as in the standardEKF filter presented in Section 2.1.

2.2.4 M/N Initiation and Deletion of Tracks

If a measurement is not associated to an existing track, it could either be clutter orbe a measurement from a new target. For this reason a tentative track is createdwhich is evaluated to make sure that it really is a new target. If the tentativetrack receives su�ciently many measurements over a period of time, it will beconsidered a confirmed track, otherwise it will be deleted.

A common track initiation procedure is M/N initiation, which is now ex-plained. In order for a tentative track to be confirmed, it first needs to receivemeasurements on N1 consecutive time steps. Then, the track must receive mea-surements on M2 out of the next N2 time steps. If these conditions are fulfilled,the track is considered confirmed.

When a confirmed track stops receiving new measurements, a strategy whichdeletes the track is employed. One simple strategy is to delete the track if it hasnot received a new measurement for ND consecutive time steps.

3Methods

This chapter presents a detailed description of all considered methods.

3.1 Measurement Group Partitioning and Association

As the objective of this thesis is not to evaluate the multi-target tracking perfor-mance of di↵erent methods, but rather their single target tracking performance,all implemented methods utilize an algorithm which forms groups of measure-ments, and associates at most one such group to each target. Each group of mea-surements should then correspond to an object in the image. While this is notnecessarily always true, it resolves the association problem due to adjacent ob-jects in the image in a consistent manner, which is independent of the used track-ing method. The measurement group partitioning and association is described indetail below.

In order to successfully use a single target tracking algorithm in situationswhere vehicles are located close to each other in the image, groups of measure-ments are formed and assigned to tracks as explained in Algorithm 1. The over-lap in Algorithm 1 is calculated using the spatial overlap norm

SO =area (A1 \ A2)area (A1 [ A2)

, (3.1)

where A1 and A2 are the two measurement ROIs (see Figure 3.1). An example ofa measurement group partitioning is shown in Figure 3.2.

Each track is then associated with at most one measurement group by choos-ing the closest group inside the track gate. The distance to the measurementgroup is defined as the Mahalanobis distance (2.8) to the measurement in thegroup with the highest confidence value. This association is done so that a mea-surement group can be associated to at most one track, thus reducing the risk

17

18 3 Methods

Algorithm 1Measurement Group Partitioning

1: measurements initially have no id2: count 0 . Number of measurement groups3: while 9measurements with no id do

4: count count+15: find measurement m1 with highest confidence and no id6: assign id count to m1 and all measurements that overlaps with m17: end while

8: for each measurement m2 with more than one id do

9: for each id i of m2 do

10: find measurement m3 with highest confidence and id i11: calculate overlap of m2 and m312: end for

13: assign m2 the id i that corresponds to the largest overlap on line 1114: end for

A1

A2

A1 \ A2 :

A1

A2

A1 [ A2 :

Figure 3.1: Intersection of two ROIs (left), and union (right).

that two tracks are tracking the same target by sharing the same measurementgroup. It should be noted though that in the case when several tracks competeover the same measurement group, the group is associated to one of the tracksmore or less at random. A more optimal approach could be used, e.g. where thegroup is associated with the closest track. However, this has not been considered,since empirical results suggest that this situation is rare and should not a↵ect theevaluation of the considered methods.

3.2 Clustering Methods

After the measurements have been partitioned into measurement groups, one ap-proach is to cluster eachmeasurement group into one single measurement, whichis then filtered using a standard EKF. These methods are referred to as clusteringmethods in this thesis. All considered clustering methods are presented below.

3.2 Clustering Methods 19

0.95

0.8

0.7

0.9

(a)

0.95

0.8

0.7

0.9

id = 1

id = 1

id = 1

(b)

0.95

0.8

0.7

0.9

id = 1

id = 1

id = 1, 2

id = 2

(c)

0.95

0.8

0.7

0.9

id = 1

id = 1

id = 1

id = 2

(d)

Figure 3.2: (a) 4 measurements with corresponding confidence values. (b)The id 1 is assigned to the measurement with confidence value 0.95 and toall measurements that it overlaps. (c) The id 2 is assigned to the measure-ment with confidence value 0.9 and to all measurements that it overlaps.(d) The measurement with confidence value 0.7 that was assigned two id’sis now assigned id 1, since the spatial overlap with the measurement withconfidence value 0.95 is greater than that with confidence value 0.9.

20 3 Methods

3.2.1 Average Cluster

In Average Cluster (AC) the mean value of the measurements is calculated andused as output. Hence the confidence values are not used.

3.2.2 Simple Cluster

In Simple Cluster (SC) the measurement with the highest confidence value isused as output, thereby discarding the other measurements. This method is sen-sitive to outliers and tends to produce a noisy output. Also, useful informationmight be lost when discarding measurements.

3.2.3 Weighted Sum Cluster

In Weighted Sum Cluster (WSC) a weighted sum of the measurements is calcu-lated for each measurement group according to

y =1

Pj p

j

X

j

pjyj , (3.2)

where yj are the measurements in the measurement group, with correspondingconfidence values pj . The WSC is less sensitive to outliers than SC since it usesmore measurements.

In e↵ect it assumes that the confidence values gives an indication of the accu-racy of the measurements. Note that a large group of low confidence measure-ments can give the same contribution to the output as a smaller group of highconfidence measurements.

Variations of this method are also considered, where only the m measure-ments with the highest confidence values are used in (3.2). Note that if m = 1,then WSC is reduced to SC.

3.2.4 Nonlinear Weighted Sum Cluster

In Nonlinear Weighted Sum Cluster (NWSC) a weighted sum of the measure-ments is calculated for each measurement group according to

y =1

Pj p

j

X

j

pjyj , (3.3)

where yj are the measurements in the measurement group and pj are trans-formed confidence values given by

pj =

8>>><>>>:

0.5pj

0.85pj < 0.85

c1E(pj ) + c2 pj � 0.85.(3.4)

3.2 Clustering Methods 21

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p

p

Figure 3.3: Transformed confidence values (blue line) plotted against confi-dence values. The black dashed line is a guide for the eye.

An explanation of (3.4) now follows. In Chapter 5 the mean error, E(p), for alarge number of measurements is calculated and plotted against their correspond-ing confidence values p (see Figure 5.1b). By doing the transformation (3.4), pj

should contribute to the sum (3.3) in a way that corresponds to the shape of E(p).This can be seen by noting that E(p) is approximately linear for p < 0.85. It alsoholds that

E(0.85) ⇡ max E(p) �min E(p)2

. (3.5)

This results in (3.4), where c1 and c2 are constants such that pj attains the values0.5 and 1 at pj = 0.85 and and pj = 1, respectively. The transformation (3.4) isvisualized in Figure 3.3.

3.2.5 Regression Cluster

In Regression Cluster (RC) a function g(y, ✓) is fitted to the measurements in aleast squares sense by solving the optimization problem

min✓

X

j

⇣pj � g( ˆyj , ✓)

⌘2+

X

i

ri (✓)2, (3.6)

where ✓ = (✓1, ✓2, . . . , ✓np )T is a vector with np parameters and ri (✓) are quadratic

penalty terms which impose constraints on ✓. The measurement vector in (3.6)

22 3 Methods

is given in the alternative form

ˆyj =

0BBBBBBBBBB@

xjleft

xjright

yjbottom

1CCCCCCCCCCA, (3.7)

where xjleft and x

jright are the x-coordinate of the left and right edge of the ROI of

measurement j , respectively. We consider a Gaussian shaped function as regres-sion function,

g(y, ✓) = ✓1 exp⇣�(y � µ(✓))T⌃(✓)�1(yj � µ(✓))

⌘, (3.8)

where

µ(✓) =

0BBBBBB@

✓2✓3✓4

1CCCCCCA (3.9)

⌃(✓) =

0BBBBBB@

✓5 0 00 ✓6 00 0 ✓7

1CCCCCCA , (3.10)

with constraints

0 ✓1 1 (3.11)

minj

xjleft ✓2 max

jxjleft (3.12)

minj

xjright ✓3 max

jxjright (3.13)

minj

yjbottom ✓4 max

jyjbottom (3.14)

0 ✓5, ✓6, ✓7 104. (3.15)

Hence the number of parameters np = 7.Note that the confidence values of the measurements can not be seen as re-

alizations of a PDF. For this reason the normalization constant of the Gaussiandistribution is replaced by the parameter ✓1 in (3.8). The Levenberg-Marquardtalgorithm [19], which is a nonlinear least squares method, is used to find a localminimum of (3.6). When a solution is found the vector (3.9) is used as outputfrom RC.

A requirement for the RC method to work is that su�ciently many measure-ments are available when solving (3.6), otherwise the Levenberg-Marquardt al-gorithm will result in a system of equations that is badly conditioned. For thisreason the WSC method is used if the number of measurements are less than 8.

3.3 Probabilistic Data Association 23

3.3 Probabilistic Data Association

A modified version of the PDA method in Section 2.2.3 is presented here. As it issuspected that measurements with high confidence are more accurate, the confi-dence values are included in the modified method. This is done by multiplyingthe likelihood (2.13), where yj is the target originated measurement, with thecorresponding confidence value pj ,

p({y}k , {p}k |Aj , Yk�1) / PDPG�N�1⇤j

kpj , (3.16)

where {p}k is the set of all confidence values at time k. This results in a slightlydi↵erent expression for the association probability

P(Aj |Yk , {p}k) =

8>>>>>>>><>>>>>>>>:

(1 � PDPG)�(1 � PDPG)� + PDPG

PNj=1 ⇤

jkp

jj = 0

PDPG⇤jkp

j

(1 � PDPG)� + PDPGPN

j=1 ⇤jkp

jj � 1.

(3.17)

The state update and prediction is identical to the standard PDA given in Section2.2.3.

4Error Analysis

This chapter presents the error norms that are used in the method evaluation inChapter 5. The errors are calculated for estimated tracks using reference tracks,known as markings (see Section 1.5). All estimated tracks, however, do not have acorresponding marking, so there is a need to pair up tracks with markings beforethe errors can be calculated. This procedure is presented in detail below.

4.1 Track-Marking Pairs

In order to calculate the error for an estimated track, there is a need to determineif a marking exists that corresponds to the tracked object. It is also necessary todetermine the time interval that they both have in common. This is achieved byintroducing the concepts of spatial overlap and temporal overlap (see [20]). Thespatial overlap of a confirmed track and a marking for a single image frame isgiven by

SO =area (T \M)area (T [M)

, (4.1)

where T and M are the ROIs of a track and a marking, respectively (as defined in(3.1)). The temporal overlap is defined as

TO = overlap in frame span. (4.2)

A track and a marking are then paired up if the following criteria are fulfilled:

SO � 0.2 TO � 18 frames, (4.3)

where SO is the average spatial overlap during the time interval given by thetemporal overlap. The criteria (4.3) has empirically been shown to result in goodtrack-marking pairs.

25

26 4 Error Analysis

4.2 Error Norms

The following sections present error norms that are used in the method evalua-tion in Chapter 5. Suppose that a marking yM , as given in (1.7), is available for agiven track-marking pair at a given frame. The di↵erence yM � y in image coordi-nates is then transformed to a di↵erence P (yM �y) in world coordinates using thetransformation (1.6). This is done under the assumption that the x�componentof the marking is the same as that of the estimated track. The vector y can ei-ther be a measurement, the output from the clustering, or obtained using themeasurement model (1.3b). The error is then calculated in world coordinates as

E(y) :=��P (yM � y)

��2 , (4.4)

where k · k2 denotes the 3-dimensional Euclidean norm. Note that the artificialmeasurement size, as given in (1.2), is not included when calculating the error(4.4).

4.2.1 Tracking Error

The tracking error (TE) is presented in Algorithm 2.

Algorithm 2 Tracking Error1: a empty array2: for each track-marking pair p do

3: for each frame in the temporal overlap of p do

4: calculate E(y), where y is obtained from the measurement model (1.3b)5: concatenate the error calculated on row 4 to a6: end for

7: end for

8: result mean value of a . tracking error

4.2.2 5-Largest Tracking Error

The 5-largest tracking error (5-LTE) is presented in Algorithm 3. The 5-LTE givesa measure of the worst performance of a method.

4.2.3 Clustering Error

The clustering error (CE) is presented in Algorithm 4. The CE can only be calcu-lated for clustering methods, and not for e.g. PDA.

4.3 Overtaking Scenarios

In Section 5.2 all methods are evaluated in overtaking scenarios where the targetvehicle is seen from an angle (see Figure 4.1). This case is especially di�cult sincethe classifications tend to be non-symmetrically distributed around the target.

4.4 Confidence Value Error Analysis 27

Algorithm 3 5-Largest Tracking Error1: a empty array2: for each track-marking pair p do


4: calculate E(y), where y is obtained from the measurement model (1.3b)5: end for

6: concatenate the 5 largest errors calculated on row 4 to a7: end for

8: result mean value of a . 5-largest tracking error

Algorithm 4 Clustering Error1: a empty array2: for each track-marking pair p do


4: calculate E(y), where y is the output from the clustering5: concatenate the error calculated on row 4 to a6: end for

7: end for

8: result mean value of a . clustering error

The overtaking scenarios are selected using the following conditions,

x < 30m (4.5a)�10m < y < �1.7m or 1.7m < y < 10m, (4.5b)

where x and y are elements in the state vector xk (see Section 1.4).

4.4 Confidence Value Error Analysis

Most of the tracking methods in this thesis relies on the assumption that theconfidence values give an indication of the accuracy of the measurements. It istherefore of interest to investigate whether or not this assumption is true, and ifso, to what extent.

The dependence of the measurement accuracy on the confidence values areevaluated by computing the error according to the norm (4.4) for measurementsand then plotting the error against the corresponding confidence values. In orderto compute the error for a measurement of an object, a reference marking of theobject must be available. For each frame in the temporal overlap of each track-marking pair, as given in Section 3.1, the error is calculated for all measurementsin the measurement group that was assigned to the track at that frame. Thisallows one to plot, for example, the mean value and the standard deviation of theerror for di↵erent confidence value intervals, as is done in Section 5.3.

28 4 Error Analysis

Figure 4.1: Example of an overtaking scenario. The truck is located at(x, y, z) = (23, �3.1, �1.4).

5Results

In this chapter the methods presented in Chapter 3 are evaluated. All input sig-nals to the tracking stage have been recorded in real-world scenarios as describedin Section 1.5. The error of each method is calculated in world coordinates usingthe error norms presented in Chapter 4. An error analysis of the confidence val-ues is also given.

All methods are implemented usingM/N initiation (see Section 2.2.4) of tracks,with N1 = 3, M2 = 2 and N2 = 3. Tracks are deleted after ND = 4 consecutivemisses.

5.1 Method Evaluation

The methods are evaluated using data that consists of about 2 · 105 frames. Foreach method about 686 track-marking pairs were created, with an average lengthof about 305 frames.

The TE, 5-LTE and CE are shown in Table 5.1 for each method.

5.2 Method Evaluation in Overtaking Scenarios

The methods are evaluated using data that consists of about 2 · 105 frames. Onlytrack-marking pairs where the track satisfies (4.5) are considered. For eachmethodabout 253 track-marking pairs were created, with an average length of about 75frames.

The TE, 5-LTE and CE for overtaking scenarios are shown in Table 5.2 for eachmethod.

29

30 5 Results

Table 5.1: Estimated error in centimeters for all methods. The smallest andlargest mean values are shown in green and red, respectively.

TE 5-LTE CE

Method Mean Std Mean Std Mean Std

SC 21.84 11.02 38.36 14.46 24.43 12.05WSC, m = 2 21.27 10.74 37.05 14.39 22.55 11.25WSC, m = 3 21.04 10.62 36.58 14.26 21.84 10.94WSC, m = 4 20.93 10.55 36.39 14.19 21.52 10.81WSC, m = 5 20.89 10.56 36.35 14.22 21.37 10.79WSC, m = 6 20.91 10.58 36.42 14.22 21.32 10.79NWSC 21.06 10.59 36.47 14.21 21.19 10.77WSC 21.20 10.65 36.72 14.25 21.46 10.90AC 21.38 10.78 37.05 14.50 21.78 11.11PDA 22.58 11.30 37.91 14.80 - -RC 20.91 10.67 38.09 15.69 22.73 11.83

Table 5.2: Estimated error in centimeters for all methods in overtaking sce-narios. The smallest and largest mean values are shown in green and red,respectively.

TE 5-LTE CE

Method Mean Std Mean Std Mean Std

SC 17.43 10.69 28.78 14.17 21.10 12.57WSC, m = 2 16.96 10.22 26.98 13.02 19.10 11.55WSC, m = 3 16.94 10.09 26.68 12.45 18.46 11.11WSC, m = 4 16.96 10.15 26.67 12.61 18.19 10.96WSC, m = 5 17.19 10.27 26.85 12.59 18.25 10.99WSC, m = 6 17.37 10.37 27.11 12.64 18.32 11.00NWSC 17.81 10.38 27.24 12.35 18.32 10.90WSC 18.22 10.55 27.83 12.33 18.85 11.13AC 18.63 10.73 28.41 12.36 19.41 11.43PDA 19.34 11.76 28.91 14.00 - -RC 17.65 10.28 28.14 12.35 20.08 11.86

5.3 Confidence Value Evaluation 31

5.3 Confidence Value Evaluation

The measurement error (as given in Section 4.4) for about 3 · 104 measurementsis plotted against corresponding confidence values in Figure 5.1.

0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100err

or

[cm

]

confidence

(a)

0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

err

or

[cm

]

confidence

(b)

Figure 5.1: (a) Mean and standard deviation of measurement error in cen-timeters plotted against confidence (b) Mean of measurement error in cen-timeters plotted against confidence (zoomed-in version of Figure 5.1a).

6Conclusions

This chapter presents comments on the results in Chapter 5. The overall perfor-mance and performance in overtaking scenarios of the methods are discussed.The chapter ends with a summary of the conclusions.

6.1 Overall Performance

The TE, 5-LTE and CE for the di↵erent methods di↵er at most by a few centime-ters as can be seen in Table 5.1. Since these di↵erences are so small, one can notsay that one method is preferable to the others. There is a clear qualitative pat-tern however. As more measurements are included in WSC the TE and 5-LTEfirst decreases then increases, with WSC and m = 5 yielding the smallest errors.Since only using a few of the measurements with the highest confidence is bet-ter than using all of them, this indicates that the confidence values gives a validmeasure of the accuracy of the measurement. This is also clear from the measure-ment error plots in 5.1, where the error is smaller for measurements with higherconfidence.

Comments about the performance of each method is given below.

6.1.1 Simple Cluster

In SC only the measurement with the highest confidence is used. This resultsin a high sensitivity to outliers, i.e. measurements with low accuracy and highconfidence values. The presence of outliers is reflected by the high CE for SCin Table 5.1, and by the large variance of the error for the confidence values inFigure 5.1a.

33

34 6 Conclusions

6.1.2 Nonlinear Weighted Sum Cluster

NWSC yields a smaller error than WSC for each error norm. Measurementswith confidence values greater than 0.85 contribute more to the cluster outputin NWSC than in WSC. Thus, this indicates that measurements with higher con-fidence are more accurate than those with lower confidence, at least for measure-ments with confidence greater than 0.85.

6.1.3 Weighted Sum Cluster

WSC yields a smaller error than AC for each error norm. Measurements withhigher confidence values contribute more to the cluster output in WSC than inAC. Thus, this indicates that measurements with higher confidence are more ac-curate than those with lower confidence.

6.1.4 Average Cluster

The error for the AC method is relatively large compared to the other methods.This is most likely caused by situations such as that in Figure 6.1a, where themeasurements are non-symmetrically distributed around the target. The cluster-ing output for AC in this specific case is shown in Figure 6.1b, and the error forthe di↵erent methods is shown in Figure 6.2.

6.1.5 Probabilistic Data Association

The error for the PDAmethod is relatively large. The reason for this might be thatthe variance of the innovation PDF is too small, which causes the likelihood (3.16)to be small for measurements that are far away from the predicted measurement.This increases the sensitivity of PDA to persistent noise. An example where theerror for PDA is large is shown in Figure 6.3. The error for the di↵erent methodsfor this specific case is shown in Figure 6.4.

6.1.6 Regression Cluster

The 5-LTE and CE for the RC method is relatively large. The numerical methodthat is used to solve the optimization problem in RC sometimes converges to asolution which is very inaccurate, as is shown in Figure 6.5. The reason for thiscould be that the confidence values in some frames are distributed in a way thatdoes not resemble the regression function (3.8). The error for the methods forthis situation are shown in Figure 6.6.

6.2 Performance in Overtaking Scenarios

The method performance in overtaking scenarios agrees with the overall perfor-mance discussed in Section 6.1. However, note that the errors in Table 5.2 aresmaller than in Table 5.1. This is because the classifications are more accurate at

6.2 Performance in Overtaking Scenarios 35

(a)

(b)

Figure 6.1: Two images displaying data from the same frame. (a) Classifica-tions (cyan) with corresponding confidence values and a marking (red). (b)Clustering output for AC (magenta) and a marking (red).

36 6 Conclusions

0

5

10

15

20

25

30

35

40

SC

WSC

, m=2

WSC

, m=3

WSC

, m=4

WSC

, m=5

WSC

, m=6

NW

SCW

SC ACPD

ARC

err

or

[cm

]

(a)

500 510 520 530 540 550 560 570 5800

10

20

30

40

50

60

frames

err

or

[cm

]

SCWSC, m=3ACPDA

(b)

Figure 6.2: Estimated error for the situation in Figure 6.1. (a) TE for eachmethod. (b) TE in each frame for some of the methods.


(a)

(b)

Figure 6.3: Two images displaying data from the same frame. (a) Classifica-tions (cyan) with corresponding confidence values and a marking (red). (b)Estimated track for PDA (blue) and a marking (red).

38 6 Conclusions

0

5

10

15

20

25

30

35

40

SC

WSC

, m=2

WSC

, m=3

WSC

, m=4

WSC

, m=5

WSC

, m=6

NW

SCW

SC ACPD

ARC

err

or

[cm

]

(a)

560 580 600 620 640 6600

10

20

30

40

50

60

frames

err

or

[cm

]

WSC, m=3ACPDARC

(b)



(a)

(b)

Figure 6.5: (a) Classifications (cyan) with corresponding confidence valuesand a marking (red). (b) Clustering output for RC (magenta) and a marking(red).

40 6 Conclusions

0

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, m=2

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Figure 6.6: Estimated error for the situation in Figure 6.5. (a) TE for eachmethod (b) TE in each frame for some of the methods.

6.3 Summary 41

closer distance, and the average distance to the vehicles is smaller in overtakingscenarios than in general. In Table 5.2 it can be seen that WSC with 3 m 4yields the smallest TE and smallest 5-LTE. Interesting to note is that the di↵er-ence between the CE and the TE is more notable for these methods than for e.g.WSC. The TE is only slightly smaller than the CE for WSC, which could be a signthat the filter is not optimally tuned.

The performance of the methods in overtaking scenarios can be divided intothree categories as follows

1. All methods perform equally well, except possibly SCwhich performsworse

2. WSC with 3 m 4 performs better than the other methods

3. Neither 1 nor 2 applies

Examples where categories 1, 2 and 3 applies are shown in Figure 6.7, 6.9 and6.11, respectively. Corresponding error analyses are shown in Figure 6.8, 6.10and 6.12, respectively.

Category 1 seems to be most common, followed by category 2. It is thereforpossible that WSC with 3 m 4 would perform considerably better than theother methods if the occurrences of category 1 were excluded in the evaluation.

6.3 Summary

It has been shown that the confidence values can be used as a measure of theaccuracy of the measurements, with high confidence corresponding to a moreaccurate measurement. This has been shown by comparing measurements withmarkings, resulting in the plot in Figure 5.1 where the measurement error isplotted against corresponding confidence values. This has also been shown bythe improved performance of WSC with 3 m 5 compared to WSC and AC,as can be seen in Table 5.1 and Table 5.2. The TE and 5-LTE for PDA and RCare relatively large compared to the other methods. However, the di↵erences inerror for all considered methods are small. Thus the results do not show that onemethod is preferable to the others.

42 6 Conclusions

Figure 6.7: Classifications (cyan) with corresponding confidence values anda marking (red) for a category 1 overtaking scenario.

6.3 Summary 43

0

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44 6 Conclusions

Figure 6.9: Classifications (cyan) with corresponding confidence values anda marking (red) for a category 2 overtaking scenario.

6.3 Summary 45

0

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46 6 Conclusions

Figure 6.11: Classifications (cyan) with corresponding confidence valuesand a marking (red) for a category 3 overtaking scenario.

6.3 Summary 47

0

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7Future Work

This chapter presents comments on how the work in this theses can be continued.

• The PDAmethod could be improved if the measurement noise in the modelis increased. This would increase the variance of the innovation PDF, whichin turn would increase the association probability (3.17) for measurementsfar from the predicted measurement. This might reduce the sensitivity ofPDA to persistent noise, i.e. measurements with bad accuracy that occur onthe same location over several consecutive frames.

• The RC method could be improved if a global optimization algorithm isused to solve the optimization problem. This would, however, increase thecomputational time of the method.

• Persistent noise seems to be characteristic for the output from the useddetection algorithm. If this persistent noise is detected and removed, thetracking might improve. This should be done in the tracking stage sincetemporal information about the measurements is required.

49

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