algorithms and for tra jectory - COREGokul, Sanjna, Sandra, Deepthi, Annelie, Ra-neesh, Jagadeep,...

Thesis for the Degree of Do tor of Philosophy

Data asso iation algorithms and

metri design for traje tory

estimation

Abu Sajana Rahmathullah

Department of Signals and Systems

Chalmers University of Te hnology

Göteborg, Sweden

June 2016

Data asso iation algorithms and metri design for traje tory estimation

Abu Sajana Rahmathullah

ISBN 978-91-7597-400-2

© Abu Sajana Rahmathullah, 2016.

Doktorsavhandlingar vid Chalmers tekniska högskola

Ny serie nr 4081

ISSN 0346-718X

Department of Signals and Systems

Chalmers University of Te hnology

SE�412 96 Göteborg

Sweden

Telephone: +46 (0)31 � 772 1000

Typeset by the author using L

A

T

E

X.

Chalmers Reproservi e

Göteborg, Sweden 2016

To everyone

i

Abstra t

This thesis is on erned with traje tory estimation, whi h �nds appli ations

in various �elds su h as automotive safety and air tra� surveillan e. More

spe i� ally, the thesis fo uses on the data asso iation part of the problem,

for single and multiple targets, and on performan e metri s.

Data asso iation for single-traje tory estimation is typi ally performed

using Gaussian mixture smoothing. To limit omplexity, pruning or merg-

ing approximations are used. In this thesis, we propose systemati ways to

perform a ombination of merging and pruning for two smoothing strate-

gies: forward-ba kward smoothing (FBS) and two-�lter smoothing (TFS).

We present novel solutions to the ba kward smoothing step of FBS and a

likelihood approximation, alled smoothed posterior pruning, for the ba k-

ward �ltering in TFS.

For data asso iation in multi-traje tory estimation, we propose two it-

erative solutions based on expe tation maximization (EM). The appli ation

of EM enables us to independently address the data asso iation problems

at di�erent time instants, in ea h iteration. In the �rst solution, the best

data asso iation is estimated at ea h time instant using 2-D assignment,

and given the best asso iation, the states of the individual traje tories are

immediately omputed using Gaussian smoothing. In the se ond solution,

we average the states of the individual traje tories over the data asso ia-

tion distribution, whi h in turn is approximated using loopy belief propaga-

tion. Using simulations, we show that both solutions provide good trade-o�s

between a ura y and omputation time ompared to multiple hypothesis

tra king.

For evaluating the performan e of traje tory estimation, we propose two

metri s that behave in an intuitive manner, apturing the relevant features

in target tra king. First, the generalized optimal sub-pattern assignment

metri omputes the distan e between �nite sets of states, and addresses

properties su h as lo alization errors and missed and false targets, whi h

are all relevant to target estimation. The se ond metri omputes the dis-

tan e between sets of traje tories and onsiders the temporal dimension of

traje tories. We re�ne the on epts of tra k swit hes, whi h allow a traje -

iii

Abstra t

tory from one set to be paired with multiple traje tories in the other set

a ross time, while penalizing it for these multiple assignments in an intuitive

manner. We also present a lower bound for the metri that is remarkably

a urate while being omputable in polynomial time.

Keywords: Traje tory estimation, data asso iation, metri s, Gaussian

mixtures, smoothing, expe tation maximization

iv

A knowledgments

This thesis would not have been possible without the support from many.

I would like to start by thanking Prof. Mats Viberg and Prof. Tomas M K-

elvey for giving me the opportunity to work with the Signal Pro essing

group at Chalmers.

I would like to express my immense gratitude to my supervisor Lennart

Svensson. You have been a great guide to me in several aspe ts, from

produ ing this thesis to writing to skiing. I really appre iate your honest,

onstru tive and elaborate feedba ks. I have learnt a lot from our te hni al

dis ussions and from the way you an onne t seemingly di�erent problems.

Thanks to your warm and friendly personality that made working with you

su h a pleasure.

Many thanks to my o�supervisor Daniel Svensson for his support over

the years. I really appre iate that you have been there to help me with

guidan e and feedba ks. I have also enjoyed all the dis ussions I have had

with you.

Thanks also to my o�supervisor Ángel F. Gar ía-Fernández for the

exhilarating dis ussions, prompt feedba ks and for being a great host during

my two months stay in Australia.

I want to thank my ollaborator/Masters thesis student Raghavendra

Selvan for all the valuable dis ussions we have had. Thanks for helping me

with the papers for this thesis despite short noti e.

I would also like to thank Prof. Ba-Ngu Vo for allowing me to visit his

group at Curtin University. My sin ere thanks to Karl, Samuel and Erik

for their valuable time in proofreading this thesis. Thanks to all my urrent

and former olleagues at Chalmers�Maryam, Lars, Ay a, Anders, Malin,

Astrid, Naga, Keerthi, Sathya, Srikar, Rahul�who have made the work

pla e su h a relaxed environment. I have enjoyed my �ve years at Chalmers

that would not have been possible without you all.

Thanks to all my friends who add a lot of fun to my life outside Chalmers.

Thanks go to Ninva, Dipti, Gokul, Sanjna, Sandra, Deepthi, Annelie, Ra-

neesh, Jagadeep, Aravindh and Kavitha.

I annot thank enough my family�Zerina, Rahmathullah, Ziana, Asan

v

A knowledgments

and Hussain�who have always been an immense support for me. Thanks

for the un onditional love you have for me. Last but not the least, thanks

to Ri kard for his love and support. Thanks for being there during the ups

and downs of my life.

vi

Publi ations

This thesis is based on the following publi ations:

Paper I

A.S. Rahmathullah, L. Svensson and D. Svensson, �Merging-based forward-

ba kward smoothing on Gaussian mixtures�, In Pro . 17th International

Conferen e on Information Fusion (FUSION), Salaman a, Spain, July 2014.

Paper II

A.S. Rahmathullah, L. Svensson and D. Svensson, �Two-�lter Gaussian mix-

ture smoothing with posterior pruning�, In Pro . 17th International Con-

feren e on Information Fusion (FUSION), Salaman a, Spain, July 2014.

Paper III

A.S. Rahmathullah, R. Selvan and L. Svensson, �A bat h algorithm for es-

timating traje tories of point targets using expe tation maximization�, A -

epted for publi ation in IEEE Transa tions on Signal Pro essing, April 2016.

Paper IV

R. Selvan, A.S. Rahmathullah and L. Svensson, �Expe tation maximization

with hidden state variables for multi-target tra king�, To be submitted to

IEEE Transa tions on Signal Pro essing.

Paper V

A.S. Rahmathullah, Á.F. Gar ía-Fernández and L. Svensson, �Generalized

optimal sub-pattern assignment metri �, Submitted to IEEE Transa tions

on Signal Pro essing, Jan 2016.

Paper VI

A.S. Rahmathullah, Á.F. Gar ía-Fernández and L. Svensson, �A metri on

the spa e of �nite sets of traje tories for evaluation of multi-target tra k-

ing algorithms�, Submitted to IEEE Transa tions on Signal Pro essing,

May 2016.

vii

Contents

Abstra t iii

A knowledgments v

Publi ations vii

Contents ix

I Introdu tory hapters xv

1 Introdu tion 1

2 Models, obje tives and on eptual solution 5

2.1 State spa e representation . . . . . . . . . . . . . . . . . . . 5

2.2 Problem statement and on eptual solution . . . . . . . . . . 6

2.2.1 Predi tion and �ltering . . . . . . . . . . . . . . . . . 6

2.2.2 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Single traje tory estimation 11

3.1 Linear and Gaussian �ltering and smoothing . . . . . . . . . 11

3.1.1 Kalman �ltering . . . . . . . . . . . . . . . . . . . . . 11

3.1.2 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Non-linear models . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Gaussian �lters . . . . . . . . . . . . . . . . . . . . . 14

3.2.2 Parti le �lters . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Gaussian mixture �ltering and smoothing . . . . . . . . . . . 16

3.3.1 Optimal solution . . . . . . . . . . . . . . . . . . . . 17

3.4 Gaussian mixture redu tion . . . . . . . . . . . . . . . . . . 19

3.4.1 Pruning . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4.2 Merging . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4.3 Choi e of GMR . . . . . . . . . . . . . . . . . . . . . 20

ix

Contents

3.4.4 GMR for FBS and TFS . . . . . . . . . . . . . . . . 20

4 Multi�traje tory estimation 23

4.1 Data asso iation . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Tra king algorithms . . . . . . . . . . . . . . . . . . . . . . . 24

4.2.1 Global nearest neighbour . . . . . . . . . . . . . . . . 25

4.2.2 Multiple hypothesis tra king . . . . . . . . . . . . . . 25

4.2.3 Joint probabilisti data asso iation . . . . . . . . . . 26

4.2.4 Probabilisti multiple hypothesis tra king . . . . . . 26

4.3 EM for data asso iation . . . . . . . . . . . . . . . . . . . . 26

5 Metri s 29

5.1 Need for a metri . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 Metri properties . . . . . . . . . . . . . . . . . . . . . . . . 30

5.3 Common metri s . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3.1 On ve tor spa es . . . . . . . . . . . . . . . . . . . . 31

5.3.2 On the spa e of �nite sets of ve tors . . . . . . . . . 32

5.3.3 On the spa e of �nite sets of traje tories . . . . . . . 33

6 Contributions and future work 37

6.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Referen es 43

II Appended papers 53

Paper I Merging�based forward�ba kward smoothing on Gaus-

sian mixtures 57

1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2 Problem formulation and idea . . . . . . . . . . . . . . . . . 59

2.1 Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3 Ba kground . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1 Forward-Ba kward Smoothing . . . . . . . . . . . . 62

3.2 Optimal Gaussian mixture FBS . . . . . . . . . . . . 62

3.3 Forward �lter based on pruning and merging approx-

imations . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Ba kward smoothing through merged nodes . . . . . . . . . 65

4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Smoothing on merged nodes . . . . . . . . . . . . . 69

5 Algorithm des ription . . . . . . . . . . . . . . . . . . . . . 70

x

Contents

6 Implementation and simulation Results . . . . . . . . . . . . 73

6.1 Simulation s enario . . . . . . . . . . . . . . . . . . . 73

6.2 Implementation details . . . . . . . . . . . . . . . . . 73

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A Weight al ulation . . . . . . . . . . . . . . . . . . . . . . . 76

Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Paper II Two��lter Gaussian mixture smoothing with poste-

rior pruning 81

1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2 Problem statement and idea . . . . . . . . . . . . . . . . . . 83

2.1 Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3 Ba kground . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.1 Two-�lter smoothing . . . . . . . . . . . . . . . . . . 84

4 Intragroup approximations of the ba kward likelihood . . . . 86

4.1 Stru ture of the ba kward likelihood . . . . . . . . . 86

4.2 Normalizability of the BL and intragroup approxima-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 Smoothed posterior-based pruning . . . . . . . . . . . . . . 90

5.1 Posterior-based pruning . . . . . . . . . . . . . . . . 90

6 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7 Implementation and simulation results . . . . . . . . . . . . 94

7.1 Simulation s enario . . . . . . . . . . . . . . . . . . . 94

7.2 Implementation details . . . . . . . . . . . . . . . . . 94

7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Paper III A bat h algorithm for estimating traje tories of

point targets using expe tation maximization 103

1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

2 Problem formulation and ba kground . . . . . . . . . . . . . 106

2.1 Nomen lature . . . . . . . . . . . . . . . . . . . . . . 106

2.2 Problem formulation . . . . . . . . . . . . . . . . . . 107

2.3 Ba kground . . . . . . . . . . . . . . . . . . . . . . . 108

3 EM for target tra king . . . . . . . . . . . . . . . . . . . . . 110

3.1 Expe tation maximization for MAP estimation . . . 110

3.2 Derivation of the EM algorithm for tra king multiple

point targets . . . . . . . . . . . . . . . . . . . . . . 112

4 Marginal data asso iation probabilities al ulation using Loopy

belief propagation . . . . . . . . . . . . . . . . . . . . . . . . 115

xi

Contents

4.1 Loopy belief propagation algorithm . . . . . . . . . . 116

5 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6 Comparison with PMHT and JPDA . . . . . . . . . . . . . . 119

6.1 Comparison with PMHT . . . . . . . . . . . . . . . . 120

6.2 Comparison with JPDA . . . . . . . . . . . . . . . . 122

7 Simulations and results . . . . . . . . . . . . . . . . . . . . . 125

7.1 Simulation s enarios . . . . . . . . . . . . . . . . . . 125

7.2 Handling sensitivity to initialization . . . . . . . . . . 127

7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8 Con lusions and future work . . . . . . . . . . . . . . . . . . 132

A Derivation of the EM algorithm for the Gaussian ase . . . . 133

B Smoothing with the RTS smoother . . . . . . . . . . . . . . 134

Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Paper IV Expe tation maximization with hidden state vari-

ables for multi�target tra king 141

1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 141


2.1 Problem formulation . . . . . . . . . . . . . . . . . . 144

2.2 Ba kground . . . . . . . . . . . . . . . . . . . . . . . 145

3 EM for multi-target tra king . . . . . . . . . . . . . . . . . . 148

3.1 Introdu tion to expe tation maximization . . . . . . 148

3.2 Derivation for data asso iation estimation using EM . 149

3.3 Dis ussion . . . . . . . . . . . . . . . . . . . . . . . . 153

4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.1 Des ription . . . . . . . . . . . . . . . . . . . . . . . 154

4.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . 154

5 Tra k hypothesis swit hing . . . . . . . . . . . . . . . . . . . 156

5.1 Comparing tra k hypotheses . . . . . . . . . . . . . . 158

5.2 Dete ting targets in lose proximity . . . . . . . . . . 159

6 Simulation and results . . . . . . . . . . . . . . . . . . . . . 160

6.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . 160

6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7 Con lusion and future work . . . . . . . . . . . . . . . . . . 168

A Derivation of the M�step of EM algorithm . . . . . . . . . . 169

B Smoothing with the RTS smoother . . . . . . . . . . . . . . 170

C Modi�ed au tion algorithm . . . . . . . . . . . . . . . . . . . 171

Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

xii

Contents

Paper V Generalized optimal sub�pattern assignment metri 179

1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

2 Ba kground . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

2.1 Metri . . . . . . . . . . . . . . . . . . . . . . . . . . 182

2.2 OSPA metri . . . . . . . . . . . . . . . . . . . . . . 182

2.3 Drawba k with the normalization in OSPA . . . . . . 184

3 Generalized OSPA metri . . . . . . . . . . . . . . . . . . . 184

3.1 Dis ussion . . . . . . . . . . . . . . . . . . . . . . . . 185

4 Choi e of α in MTT . . . . . . . . . . . . . . . . . . . . . . 186

4.1 Motivating the hoi e of α = 2 . . . . . . . . . . . . . 186

4.2 Reformulating GOSPA for α = 2 . . . . . . . . . . . 188

5 Performan e evaluation of MTT algorithms . . . . . . . . . . 189

6 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

7 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

A Proof of the triangle inequality of GOSPA . . . . . . . . . . 195

B Proof of the triangle inequality of OSPA . . . . . . . . . . . 198

C Proof of the average GOSPA metri . . . . . . . . . . . . . . 199

Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Paper VI A metri on the spa e of �nite sets of traje tories

for evaluation of multi�target tra king algorithms 207

1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 207


2.1 Fundamental properties . . . . . . . . . . . . . . . . 210

2.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . 211

2.3 Bento's natural and omputable metri s . . . . . . . 212

3 Assignment metri . . . . . . . . . . . . . . . . . . . . . . . 214

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . 215

3.2 Multi-dimensional assignment metri . . . . . . . . . 215

3.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . 217

3.4 Computation . . . . . . . . . . . . . . . . . . . . . . 218

4 LP relaxation metri . . . . . . . . . . . . . . . . . . . . . . 219

4.1 Binary linear programming formulation . . . . . . . . 219

4.2 LP relaxation metri . . . . . . . . . . . . . . . . . . 220

4.3 Implementation of the LP relaxation metri using ADMM221

5 Extension to random sets of traje tories . . . . . . . . . . . 222

6 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . 223

6.1 Comparison between ADMM and `linprog' implemen-

tations . . . . . . . . . . . . . . . . . . . . . . . . . . 223

6.2 Example in MTT using sets of traje tories . . . . . . 225

7 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

xiii

Contents

A Proof for Proposition VI.9 . . . . . . . . . . . . . . . . . . . 229

A.1 Proof for omputability using LP . . . . . . . . . . . 229

A.2 Proof for metri properties . . . . . . . . . . . . . . . 229

B Formulation of the LP metri in onsensus form . . . . . . . 235

Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

xiv

Part I

Introdu tory hapters

Chapter 1

Introdu tion

In many appli ations, the obje tive is to systemati ally and sequentially es-

timate quantities of interest from a dynami system using indire t and ina -

urate sensor observations. For instan e, in radar tra king, the aim is often

to determine the position and velo ity of a moving or stationary air raft or

ship. In ommuni ation systems, the on ern is to determine the messages

transmitted through a noisy hannel. In driver assistan e systems, the ob-

je tive is to monitor several features about the driver, the vehi le and the

surroundings. There are also several other appli ations su h as fore asting

weather or �nan ial trends, predi ting house pri es, handwriting re ogni-

tion, speaker identi� ation, and positioning in navigation systems.

The sequential estimation problem an be ategorized into three di�er-

ent problem formulations: predi tion, �ltering and smoothing. The predi -

tion problem is to fore ast the values of the parameters of interest, given

information up to an earlier time, whereas the �ltering problem is about

estimating the parameter at the urrent time, given information up to and

in luding that time. The smoothing problem is to estimate the past state of

the parameter using all the observations made. An example from [1℄ an be

used to explain these di�erent problem formulations, in layman terms. As-

sume that we have re eived a garbled telegram and that the task is to read

it word-by-word and make sense of what the telegram means. The �ltering

formulation would be to read ea h word and understand the meaning so far.

The predi tion formulation would be to guess the oming words, based on

what have been read thus far. In the smoothing formulation, the reader is

allowed to look ahead one or more words. Clearly, as the idiom quoted in

the book goes �it is easy to be wise after the event�, the smoothing formula-

tion will give the best result on average, given that a delay an be tolerated.

In many of the above-mentioned appli ations, the aim is not only to

1

Chapter 1. Introdu tion

estimate the parameters of interest termed `states', but also to des ribe the

un ertainties in the states. The un ertainties are used to des ribe the re-

liability or trustworthiness of the produ ed estimates. Mathemati ally, an

estimate and its asso iated un ertainty is quanti�ed using either a probabil-

ity density fun tion (for ontinuous states) or a probability mass fun tion

(for dis rete states). In sequential estimation, the probability fun tion of the

state, whi h has the information about its estimate and the orresponding

un ertainty, are propagated a ross time to estimate the subsequent states.

For ontinuous states, one of the most ommonly used density fun tions is

the Gaussian density fun tion, whi h is often referred to as the `bell-shaped'

urve. The famous Kalman �lter [2℄ is developed as a solution to the �l-

tering problem when the un ertainties are modelled using Gaussian density

fun tions. There also exist (analyti al) solutions to the smoothing problem

with Gaussian densities.

Even though the Gaussian density models and the Kalman �lter solu-

tions work well for a wide range of appli ations, this may not be enough

for omplex systems. There are many appli ations where the un ertainty

in the evolution of the state or the observation noise annot be a urately

modelled using Gaussian densities. For instan e, in the data asso iation

problem, observations are often re eived from obje ts that are not of inter-

est and the information regarding whi h measurement belongs to the target

of interest is not available. In su h ases, the un ertainty about the states

are lustered in several small regions where ea h region orresponds to ea h

measurement. This happens in ship surveillan e, when false measurements

are re eived from re�e tions of the sea, and in air tra� surveillan e, where

extraneous observations from louds and birds are re eived. In these kinds

of s enarios, instead of a single Gaussian density, the system or the obser-

vations are often modelled using what is alled a Gaussian-mixture density.

In essen e, the un ertainty of the state an be des ribed using a Gaus-

sian mixture where we have a Gaussian omponent for ea h luster/region

around whi h the un ertainty/data is entered, along with a weight that

aptures the intensity. The advantage of using a Gaussian mixture (GM)

is that it is made up of several Gaussian omponents, whi h allows one to

extend the Kalman �lter solutions to these problems as well. However, in

most problems, the number of possibilities and thus the number of Gaussian

omponents in the mixture grows with time, whi h adds to the omplexity

of the algorithms.

In the data asso iation problem, the interpretation of a GM density of

the state is that we have Gaussian un ertainty about the state for every

2

possibility of mat hing the obje ts (also referred to as targets) of interest to

the individual observations from the sensors. To give a sense of the number

of possibilities, assume that there are k targets and n measurements. Then,

the number is

n!(n−k)!

where n! = n× (n− 1)× . . .× 1, assuming that every

target produ es a measurement at ea h time instant. Thus, at ea h time

instant, we have a GM with a large number of omponents. When this

density is propagated to the next time to perform the Bayesian inferen e

we dis ussed in the beginning of the hapter, the omplexity of the problem

explodes. Even for single target, i.e. for k = 1, the omputation of the

optimal solution be omes intra table. Thus, approximations are inevitable.

In this thesis, we provide e� ient and e�e tive solutions for both single and

multi-target s enarios.

Another aspe t onsidered in this thesis is the performan e evaluation

of traje tory estimation algorithms. The main obje tive of this part is to be

able to quantify the similarity between the ground truth and the estimates

returned by an algorithm. We might observe that there is a good mat h

between some of the states in the ground truth and the estimates. We will

want to quantify the similarity to judge how di�erent algorithms perform.

Besides this kind of error, it is possible that there are ertain states in the

ground truth that do not have any good mat h in the estimates, or vi e

versa. We would like to take into a ount these kinds of errors as well when

de�ning the similarity between the ground truth and the estimates. In this

thesis, we have fo ussed on mathemati ally quantifying these kind of simi-

larities for traje tory estimation.

The resear h presented in this thesis has been sponsored by Vinnova

(The Swedish Governmental Agen y for Innovation Systems), under the

resear h program �Nationella �ygtekniska forskningsprogrammet" (NFFP

6), and by Ele troni Defen e Systems, Saab AB.

Outline of the thesis

The thesis is divided into two parts, where Part I presents the theoreti-

al ba kground of the Gaussian mixture smoothing problem and Part II

ontains a set of appended papers. Part I ontains six hapters, among

whi h Chapter 2 presents the mathemati al formulations of the �ltering

and smoothing problems and di�erent models used. Chapter 3 dis usses

in detail the Gaussian mixture smoothing problem in the ontext of single

traje tory estimation, and the di� ulties involved. In Chapter 4, we dis uss

the data asso iation problem in multi target tra king. Chapter 5 presents

3

Chapter 1. Introdu tion

a summary of the metri problem. In Chapter 6, we provide a summary of

the ontributions in the appended papers and also dis uss possible future

resear h dire tions.

In Part II, the ontributions of the thesis are presented in Papers I

through VI. In Paper I and Paper II, we onsidered the data asso iation

problem in estimating a single target traje tory in the presen e of lut-

ter. Typi ally, this is arried out using pruning approximations on GMs.

In the papers, we present an in-depth study of how to design forward�

ba kward smoothing and two-�lter smoothing for Gaussian mixtures, based

on both merging and pruning approximations. In Paper III and Paper IV,

we onsider the data asso iation problem when estimating multiple target

traje tories in the presen e of lutter. We present two solutions based on

expe tation maximization (EM) that are iterative. The major onsequen e

of applying EM is that the problem of GM smoothing redu es to Gaussian

smoothing in ea h iteration. In Paper V and Paper VI, we address the prob-

lem of de�ning metri s for sets of target states and for sets of traje tories,

respe tively.

4

Chapter 2

Models, obje tives and

on eptual solution

In traje tory estimation, the goal is to sequentially estimate an unknown

variable, given noisy observations of the variable. A ording to the Bayesian

estimation prin iple, whi h is ommonly used for these problems, the idea

is that based on our prior knowledge of the pro ess, we predi t the vari-

able with some un ertainty. Then, at a point when new information is

available, the predi tion is updated to get the `posterior'. For traje tory

estimation, the term 'posterior' takes di�erent meanings depending on the

set of measurement that we ondition on. When the most re ent state of

the traje tory is newer than the omplete olle tion of measurements over

time, we have a predi tion problem; for equally new information we have

a �ltering problem; and, for a state that is older than the newest pie e of

information, we have a smoothing problem.

In this hapter, we brie�y dis uss the mathemati al representation of

the variables and how the traje tory estimation problem an be posed. We

also present a brief summary of the models used in traje tory estimation.

2.1 State spa e representation

In a state-spa e representation, the unknown variable to be estimated is

termed the `state'. The state variable at time k is here denoted as xk ∈Rn

and the observed data as zk ∈ Rm. The time variability of the state

is des ribed by a motion model while the relation between the state and

the measurements are given by a sensor model. The impli it Markovian

assumption of the state spa e is that the state xk at time k, given all the

states until time k − 1, depends only on the state xk−1 at time k − 1. The

5

Chapter 2. Models, obje tives and on eptual solution

motion model an then be written as

xk = gk(xk−1, vk), (2.1)

where vk is the pro ess noise. Using this motion model one an des ribe the

transition model fk(xk|xk−1). It is assumed that we have some knowledge

of the state at time 0, de�ned by the prior density p0(x0).

The measurement zk is given by the sensor model

zk = hk(xk, wk), (2.2)

where wk is the measurement noise random variable. The sensor model is

used to obtain the likelihood fun tion pk(zk|xk). In the remainder of the

introdu tory hapters, the subs ript k in the notation of the fun tions gk(·),hk(·), and pk(·|·) will be dropped without loss of generality and for ease of

writing, and be represented as g(·), h(·), and p(·|·).

2.2 Problem statement and on eptual solu-

tion

The posterior density of the state xk is used to determine our estimates of

the state and also des ribes our un ertainties in the state. The obje tive

is to re ursively ompute the posterior density of the state ve tor xk using

the Bayesian prin iple [1℄. In the predi tion problem, the goal is to obtain

the density p(xk|z1:l) for l < k, given the measurements obtained from time

1 to time l, denoted z1:l. In the �ltering problem, the goal is to obtain the

posterior density p(xk|z1:k) of the state xk. In the smoothing problem, we

are interested in omputing the posterior density p(xk|z1:K), where K > k.Below, we present the on eptual solutions to these problems.

2.2.1 Predi tion and �ltering

The predi tion and �ltering densities an be obtained re ursively in two

steps, namely, predi tion and update, using the prior p(x0), the pro ess

model f(xk|xk−1) and the likelihood p(zk|xk). The one-step predi tion

(where l = k − 1 in p(xk|z1:l)) gives the predi tion density at time k by

evaluating the integral

p(xk|z1:k−1) =

∫p(xk−1|z1:k−1)f(xk|xk−1)dxk−1, (2.3)

6

2.2. Problem statement and on eptual solution

where p(xk−1|z1:k−1) denotes the �ltered density at time k−1, and p(xk|z1:k−1)

the predi tion density at time k. An update of the predi tion density at

time k gives the �ltered density at time k as

p(xk|z1:k) ∝ p(xk|z1:k−1)p(zk|xk). (2.4)

The onstant of proportionality in the above equation is

1p(zk|z1:k−1)

, where

p(zk|z1:k−1) =

∫p(xk|z1:k−1)p(zk|xk)dxk. (2.5)

It should be mentioned here that the equations in (2.3), (2.4) and (2.5) pro-

vide the theoreti al solutions but, in pra ti e, these equations are in general

not tra table. For instan e, the integrals annot be omputed a urately,

or the representation of the di�erent densities in these equations an be

intra table and so on.

2.2.2 Smoothing

Similar to the �ltering problem, sequential estimation of the smoothed pos-

terior an be obtained using the Bayesian prin iple. Though the approa hes

dis ussed here have been designed towards �xed-interval smoothing, they

are in their ontextual form, appli able to the �xed-lag and �xed point

smoothing as well [1℄. One an also refer to [3℄ for a umulated state den-

sity formulation of the smoothing problem.

The �rst approa h is forward�ba kward smoothing (FBS) [4℄. As the

name suggests, the �rst step is forward �ltering from time 1 to K, to ob-

tain the �ltered density p(xk|z1:k) at ea h k. This is followed by ba kward

smoothing from time K to time 1. The ba kward smoothing step at time k

uses the smoothed density at time k+1 together with the �ltering densities

at time k as

p(xk|z1:K) = p(xk|z1:k)∫p(xk+1|z1:K)p(xk+1|z1:k)

f(xk+1|xk)dxk+1. (2.6)

The integral in the above equation is proportional to p(zk+1:K |xk), termed

as the ba kward likelihood in this thesis. Therefore, it is possible to in-

terpret the division in the ba kward smoother as omputing the ba kward

likelihood impli itly.

The se ond approa h to smoothing is the two-�lter smoothing (TFS)

method [5℄. To obtain the smoothed density at time k by this method,

forward �ltering is performed from time 1 to k to get the �ltered density

7

Chapter 2. Models, obje tives and on eptual solution

p(xk|z1:k) and ba kward �ltering is run from time K to time k to get the

ba kward likelihood p(zk+1:K |xk). The produ t of the two �lter outputs

gives the smoothed density,

p(xk|z1:K) ∝ p(xk|z1:k)p(zk+1:K |xk). (2.7)

The ba kward �ltering, similar to the forward �lter, is performed re ursively

using two steps: update and retrodi tion. The update step omputes the

likelihood

p(zk+1:K|xk+1) = p(zk+1|xk+1)p(zk+2:K|xk+1) (2.8)

and the retrodi tion step omputes the ba kward likelihood as

p(zk+1:K |xk) =∫p(zk+1:K |xk+1)f(xk+1|xk)dxk+1. (2.9)

Comparing the individual terms in (2.6) and (2.7) and using (2.9), one

an observe that the di�eren e between the two smoothing methods arises

due to the di�eren e in the ways the term p(zk+1:K |xk+1) is omputed. FBS

omputes it from the division of the predi tion and smoothing densities

p(zk+1:K |xk+1) ∝p(xk+1|z1:K)p(xk+1|z1:k)

. (2.10)

whereas the TFS method uses a �ltering approa h as in (2.8).

2.3 Models

In this se tion, we dis uss the di�erent models relevan e to traje tory esti-

mation. We present them in di�erent ategories based on their properties.

All these models determine whether or not the integrals in the on eptual

solutions presented in the last se tion are tra table or not. For instan e,

as we dis uss in the next hapter, when we have a single target with lin-

ear pro ess and measurement models, along with Gaussian noise terms and

Gaussian prior densities, the predi tion, �ltered and smoothed densities are

all Gaussian densities. In this ase, the Kalman �lter and the Rau h-Tung-

Striebel (RTS) smoother provide a re ursive solution to obtain the mean

and the ovarian e of these densities in losed form. Below we summarize

some of the possible models used in the traje tory estimation literature.

The pro ess model g(xk−1, vk) and/or the measurement model h(xk, wk)

an be non-linear fun tions of the state and the noise variables. A ommonly

used non-linear measurement model is when we have range and bearing

measurements and the state that we are interested in is Cartesian position.

8

2.3. Models

These kind of non-linear fun tions are often handled using extended Kalman

�lter (EKF) [1,6℄, uns ented Kalman �ler (UKF) [7,8℄, quadrature Kalman

�lter [9℄, ubature Kalman �lter (CKF) [10℄ or the parti le �lter [11�16℄.

Often in appli ations, we re eive measurements, not only from the tar-

get of interest, but also from sour es that are not of interest to us. These

measurements are termed lutter and often yield un ertainties in the data

asso iations. The problem arises when it is not immediate whi h measure-

ments are from the single target and whi h are from lutter. The data

asso iation problem also arises when we have multiple targets in the region

of interest. Again, we re eive a set of measurements whi h may not have

the target identity. The problem is only aggravated when we also have the

lutter data in addition.

9

Chapter 3

Single traje tory estimation

In this hapter, we brie�y dis uss the various hallenges in �ltering and

smoothing problems that arise in the single�traje tory estimation problem.

We dis uss in detail the Gaussian mixture �ltering and smoothing problems,

whi h are fo us areas of the thesis.

3.1 Linear and Gaussian �ltering and smooth-

ing

Assume that the prior, p(x0), is a Gaussian density, and that the motion

and sensor models are linear fun tions of the state ve tor xk, with additive

Gaussian noise, i.e.,

xk = Fkxk−1 + vk (3.1)

and

zk = Hkxk + wk, (3.2)

where Fk ∈ Rn×n, Hk, ∈ Rm×n

, vk ∼ N (0, Qk) and wk ∼ N (0, Rk). Then,it an be shown that the posterior densities are Gaussian and have losed

form expressions [2℄. Again, for onvenien e of writing, the subs ript kwill be dropped from the matrix notations. In this se tion, we dis uss the

algorithms to obtain the mean and ovarian e of the �ltered and smoothed

densities.

3.1.1 Kalman �ltering

Let the predi tion density, p(xk|Z1:k−1), and �ltered density, p(xk|Z1:k), be

denoted as N(xk;µk|k−1, Pk|k−1

)and N

(xk;µk|k, Pk|k

), respe tively. The

notationN (x;µ, P ) denotes a Gaussian density in the variable x with mean

µ and ovarian e P . The goal of predi tion and �ltering is then to �nd the

11

Chapter 3. Single traje tory estimation

�rst two moments of the orresponding Gaussian densities. The ubiquitous

Kalman �lter equations [2℄ provide losed�form expressions for the �rst two

moments of the predi tion and �ltered densities in (2.3) and (2.4). The

predi tion equations are given by

µk|k−1 = Fµk−1|k−1 (3.3)

Pk|k−1 = FPk−1|k−1FT +Q (3.4)

and the update equations by

Sk = HPk|k−1HT +R (3.5)

Kk = Pk|k−1HTS−1

k (3.6)

z̃k = zk −Hµk|k−1 (3.7)

µk|k = µk|k−1 +Kkz̃k (3.8)

Pk|k = (In −KkH)Pk|k−1. (3.9)

where In is an n-by-n identity matrix. The so� alled innovation z̃k, and

Sk, the innovation ovarian e, des ribe the expe ted measurement distribu-

tions. Kk is the Kalman gain, whi h an be viewed as the weight for new

information in the innovation ompared to the predi tion.

3.1.2 Smoothing

Below we provide the two versions�forward�ba kward smoothing (FBS)

and two��lter smoothing(TFS)�of the smoothing algorithm dis ussed in

Se tion 2.2.2 to obtain the mean and ovarian e of the smoothed density

under linear and Gaussian assumptions.

For the FBS method, the Rau h-Tung-Striebel (RTS) [4℄ smoother gives

the losed-form expressions for the mean and ovarian e of the smoothed

density. Using notations similar to the ones for the predi tion and �ltered

densities, the smoothed density at time k is denoted as N(xk;µk|K, Pk|K

).

The RTS equations are

µk|K = µk|k + Ck

(µk+1|K − µk+1|k

)(3.10)

Pk|K = Pk|k + Ck

(Pk+1|K − Pk+1|k

)CT

k , (3.11)

where

Ck = Pk|kFTP−1

k+1|k (3.12)

is similar to the Kalman gain in the Kalman �lter equations (3.3) to (3.9).

12

3.2. Non-linear models

For the TFS method, the work in [17℄ provides the losed-form solu-

tion for the moments of the smoothed density. Let the likelihoods be de-

noted as p(Zk+1:K|xk+1) = N (Uk+1xk+1;ψk+1, Gk+1) and p(Zk+1:K|xk) =

N (Jkxk; ηk, Bk). Let the starting onditions at time K be JK = [ ],ηK = [ ] and BK = [ ]. The update step in (2.8) of the ba kward �lter is

then given by

Uk+1 =

[Jk+1

H

](3.13)

ψk+1 =

[ηk+1

zk+1

](3.14)

Gk+1 =

[Bk+1 0

0 R

](3.15)

while the retrodi tion step in (2.9) is given by

Jk = Uk+1F (3.16)

ηk = ψk+1 (3.17)

Bk = Uk+1QUTk+1 +Gk+1. (3.18)

Using the outputs of the forward �lter and the ba kward �lter at time k,

the smoothed density in (2.7) is given by

µk|K = µk|k +Wk

(ηk − Jkµk|k

)(3.19)

Pk|K = (In −WkJk)Pk|k, (3.20)

with gain

Wk = Pk|kJTk

(JkPk|kJ

Tk +Bk

)−1. (3.21)

Note that the above three equations have similarities to the Kalman update

equations in (3.8) and (3.9). Here, the �ltering parameters, µk|k and Pk|k,

are updated with the innovation from the future measurements from time

k + 1 to K, whereas in the Kalman �lter the predi tion parameters are

updated with the innovation from the measurement at time k.

3.2 Non-linear models

When the motion model g(·) and/or the measurement model h(·) are non-linear or when the noise is not additive Gaussian, the posterior density for

the predi tion, �ltering and smoothing is, in general, not a Gaussian den-

sity. One example is when we obtains range and bearing measurements

13


from a radar and want to tra k the position and the velo ity of the target.

The optimal solution then be omes intra table as the integrals annot be

omputed in losed form. Approximations and sub�optimal approa hes are

therefore inevitable. There are several sub-optimal approa hes to estimate

the �ltered density in this ase, some of whi h are dis ussed in this se tion

su h as the Gaussian and parti le �lters. We dis uss Gaussian mixtures,

whi h is appli able when the posterior has multimodal shape, in more detail

in subsequent se tions of this hapter.

The smoothing problem has additional hallenges ompared to the �l-

tering problem. First, the equation in the FBS method involves division of

densities, whi h is di� ult to ompute for arbitrary densities. Se ond, the

a ura y of the approximations in the forward �ltering highly a�e ts the

ba kward smoothing and the smoothed density. The TFS method, on the

other hand, does not involve density divisions and the two �lters an ideally

be run independently of ea h other. However, the likelihood p(zk+1:K |xk)is not, in general, a normalizable density fun tion, whi h limits the possi-

bilities to apply onventional approximation te hniques for densities during

the ba kward �ltering. Due to these additional ompli ations, applying the

te hniques used for non-linear �ltering to non-linear smoothing does not

always produ e fruitful results. In this se tion, we also dis uss the hal-

lenges in extending te hniques su h as sequential Monte Carlo sampling,

linearization and sigma-point methods to the smoothing problem.

3.2.1 Gaussian �lters

One approa h to handle non-linear models is to approximate the posterior

density as a Gaussian density. The methods that use this approa h are

alled Gaussian �lters/smoothers, named appropriately. There are several

methods to make a Gaussian approximation of the �ltered density and to

ompute its �rst two moments. One method is based on linearizations of

the fun tions, g(·) and h(·), after whi h the Kalman �lter equations in (3.3)

to (3.9) an be used to obtain the mean and ovarian e of the Gaussian

approximation of the �ltered density. The famous extended Kalman �l-

ter [1,6,18℄ and the many variants of it are based on this approa h. Though

these algorithms work for a good number of models, their performan e de-

teriorates when the fun tions are highly non-linear.

Another type of methods used to obtain a Gaussian approximation of

the �ltered density is based on sigma-points, su h as the uns ented Kalman

�lter [7, 8℄, the quadrature Kalman �lter [9℄ and the ubature Kalman �l-

14

3.2. Non-linear models

ter [10℄. In these methods, a handful of points, termed sigma-points, are

hosen deterministi ally based on the �rst two moments of the prior den-

sity. The sigma points are then propagated through the non-linear models

to obtain the means and ovarian es used to ompute the moments of the

Gaussian approximation of the �ltered density. The sigma-point methods

also impli itly perform a linearisation using statisti al linear regression [19℄.

The analogue of Gaussian �ltering methods, su h as the extended Kalman

�lter and the uns ented Kalman �lter, exists for TFS of non-linear mod-

els. The extended Kalman smoother [20℄, similar to its �ltering ounter-

part, has poor performan e when the non-linearity is severe. The uns ented

Kalman smoother [21℄, [20, Chap. 7℄ needs the inverse of the dynami model

fun tions, whi h may not be feasible in all s enarios. The uns ented RTS

smoother [22℄ is the FBS version of a Gaussian smoother and is shown to

have similar performan e as the uns ented Kalman smoother, but without

the need of inverting the model fun tions.

3.2.2 Parti le �lters

Parti le �lters or sequential Monte Carlo �lters [11�16℄ are based on repre-

senting the density p(x) with a set of randomly drawn samples x(m), termed

`parti les', along with their orresponding weights. The parti les de�ne the

positions of Dira delta fun tions su h that the weighted sum of the Dira

delta fun tions of the parti les provides a good approximation of the true

density:

p(x) ≈ 1

N

N∑

m=1

δ(x− x(m)

). (3.22)

These methods use the on ept of importan e sampling, where the parti-

les are generated from a proposal density, whi h is simpler to generate

the samples from, instead of the true density. The parti les are propagated

through the pro ess model and the weights are updated using the likelihood

to obtain the posterior density. The hoi e of proposal density is ru ial to

parti le �lters, and the proposal density should have the same support as

the true density and should be as similar to the true density as possible.

The advantages of parti le �lters are that the performan e of su h �lters

is una�e ted by the severity of the non-linearity in g(·) and h(·), that theyare are asymptoti ally optimal also when the fun tions are non-linear, and

that they are often easy to implement. However, they an be omputa-

tionally demanding as the dimension of the state ve tor in reases. Another

problem with parti le �lters is that they degenerate, whi h means that

15


the weights of most parti les be ome zero. This an be over ome by re-

sampling [16℄ frequently, where multiple opies of the `good' parti les with

signi� ant weights are retained and the `poor' parti les are removed.

Similar to the �ltering method, sequential Monte Carlo smoothing is

based on approximating the smoothed posterior density using a set of par-

ti les. In parti le Markov hain Monte Carlo (MCMC) methods [23℄, par-

ti le �lters are use to approximate the joint posterior distribution, whi h

is then used to generate proposals for MCMC. In ase of FBS based on

these methods, a vanilla version works well when k ≈ K, where K is the

bat h duration. However, when k ≪ K [24℄, be ause of su essive resam-

pling steps, the marginal density be omes approximated by a single parti le

whi h leads to deteriorated performan e. This is the degenera y problem

that is inherent in parti le �lters [11℄. One simple approa h is to use the

forgetting properties of the Markov model, i.e., to approximate the �xed-

interval smoothed density p(xk|z1:K) using the �xed-lag smoothed density

p(xk|z1:k+δ) [25, 26℄. Unfortunately, automati sele tion of δ is di� ult.

In ase of TFS, it is not straightforward to approximate the output of

the ba kward �lter using parti les, as it it not a normalizable density. The

arti� ial prior method [5℄ uses the auxiliary probability density p̃(xk|zk+1:K)

instead of the likelihood p(zk+1:K |xk). The auxiliary density is obtained

using what is alled arti� ial prior densities. The hoi e of the arti� ial

prior plays a major role in the performan e of the TFS algorithm for parti le

methods.

3.3 Gaussian mixture �ltering and smoothing

There are many appli ations in whi h we re eive several measurements on

the state variable, where the reliability of the measurements an vary. The

likelihood in these appli ations are onveniently modelled as mixtures. In

the lassi data asso iation problem, where we do not have information

about whi h measurement orresponds to whi h target upon re eiving a set

of measurements, the posterior density is a Gaussian mixture, even if we

assume the motion and measurements models are linear and Gaussian. We

dis uss more on this problem in the next hapter.

Gaussian mixtures are weighted sum of Gaussian densities, whi h usually

make a good approximation for the multi-modal densities. In this se tion,

we explain that when the likelihood and/or the state transition density are

Gaussian mixtures, the true posterior densities after �ltering and smoothing

16

3.3. Gaussian mixture filtering and smoothing

are also Gaussian mixtures. The number of terms in the GM usually grows

exponentially with time, and we therefore need to onstrain the number of

terms. In these situations, redu tion algorithms an be used to approximate

the posteriors. In this se tion, we provide a brief overview of the most

ommonly used mixture redu tion methods and dis uss the hallenges in

applying these to smoothing problems.

3.3.1 Optimal solution

It was presented in the last hapter that the forward-ba kward smooth-

ing (FBS) method is based on forward �ltering and ba kward smoothing

while the two-�lter smoothing (TFS) method involves forward �ltering and

ba kward �ltering. These steps involve the predi tion, update and retrod-

i tion steps stated in equations (2.3), (2.4), (2.8) and (2.9). One an noti e

that all these equations involve produ ts of fun tions, whi h in this ase

are Gaussian mixtures. When the state transition densities and the likeli-

hoods are both GMs, one an use the fa t that a produ t of GMs yields a

GM and show that the posterior densities are all Gaussian mixtures. The

number of omponents in the resulting GM is the produ t of the number of

omponents in the individual mixtures, whi h explains why the number of

omponents grows exponentially with time.

Forward �ltering

One an show that, starting with a GM prior, the predi tion and the up-

date steps of forward �ltering result in a Gaussian mixture posterior density.

Evaluating these steps with GMs is equivalent to using Kalman �lters, one

for every triplet of Gaussian omponents in the prior p(xk−1|z1:k−1), the tran-

sition density f(xk|xk−1) and the likelihood p(zk|xk), yielding a Gaussian

term in the posterior p(xk|z1:k). The term in the onstant of proportion-

ality p(zk|z1:k−1) in (2.5), is not al ulated expli itly in the update step of

the Kalman �lter, whi h involves produ t of Gaussian densities. However,

in the ase with GMs, this onstant of proportionality is used in the up-

dated weight al ulation. The updated weight for the resulting Gaussian

omponent is given by the produ t of the individual weights of the ompo-

nents in the predi tion density and the likelihood along with the onstant

of proportionality.

Ba kward smoothing of FBS

The ba kward smoothing of FBS involves a division of the smoothed and

the predi tion GM densities as in equation (2.6). Starting from time K,

17


using the prin iple of mathemati al indu tion, it an be shown that the

division results in a GM and therefore the smoothed posterior p(xk|z1:K)is also a GM whi h has the same number of omponents for k = 1, . . . , K.

The weights of the omponents in the the smoothed density at time k are

the same as the weights of the omponents in the smoothed density at time

k + 1. Instead of performing the division, an equivalent way of obtaining

the smoothed posterior is as follows [3, Se . V A℄: starting from k = K−1,

the RTS re ursions are used for every triplet of asso iated omponents in

the smoothed density at time k + 1, the predi tion at time k and in the

�ltered density at time k, to ompute the smoothed density at time k.

Ba kward �lter of TFS

In the ba kward �lter of TFS, we need to ompute the ba kward likelihood

as in equations (2.8) and (2.9). The ideas in forward �ltering annot be

applied dire tly to the ba kward �lter be ause often the likelihoods an be

of the form

w0 +∑

i

wiN (Hixk;µi, Pi) (3.23)

where di�erent Hi an apture di�erent features of the state xk. Stri tly

speaking, these are not Gaussian mixture densities; they are neither Gaus-

sian nor densities in xk. We refer to them as redu ed dimension Gaussian

mixtures in this thesis. To ompute the produ t of likelihoods, one an use

the following general produ t rule:

wiN (Hix;µi, Pi)× wjN (Hjx;µj, Pj) = wijN (Hijx;µij, Pij) (3.24)

where

wij = wiwj (3.25)

Hij =

[Hi

Hj

](3.26)

µij =

[µi

µj

](3.27)

Pij =

[Pi 0

0 Pj

]. (3.28)

Using this in equations (2.8) and (2.9), one an show that the output of

the ba kward �lter has a stru ture similar to the inputs as in (3.23). The

smoothed density is given by the produ t of the outputs of the two �lters,

that an be omputed similarly to the update step in the forward �lter of

GMs, in luding the weight update using the proportionality onstant as

dis ussed in Se tion (3.3.1).

18

3.4. Gaussian mixture redu tion

3.4 Gaussian mixture redu tion

The number of omponents in the resulting GM, after update, predi tion

and retrodi tion iterations, grows exponentially with time. Therefore, ap-

proximations are ne essary to redu e the number of omponents. There are

several Gaussian mixture redu tion (GMR) algorithms that are well stud-

ied in the literature, whi h an be used for �ltering and smoothing. The

GMR algorithms are based on pruning insigni� ant omponents from the

GM and/or merging similar omponents.

3.4.1 Pruning

The number of omponents in the posterior GM an be prevented from

growing exponentially by pruning some of the omponents after ea h iter-

ation. There are several pruning strategies that an be adopted. Three

methods that are ommonly used are threshold-based pruning [27℄, M-best

pruning [28�30℄ and N-s an pruning. In threshold-based pruning, only the

omponents that have a weight greater than a prede�ned threshold are

retained and used for predi tion in the next iteration. The number of om-

ponents in the resulting GM an vary based on the threshold. The idea

behind the M-best pruning algorithm is that only the nodes with the Mhighest weights (or asso iation probabilities) are retained.

To explain the N-s an pruning [27℄, whi h is designed for redu tion dur-

ing �ltering, let us say we are interested in performing pruning at time k.We pi k the omponent that has the maximum weight. Starting from this

omponent, we tra e ba kwards N steps to �nd its parent omponent, at

time k−N . Only the o�spring at time k, of this parent node at time k−N ,

are retained. To be mentioned here is that the multiple hypothesis tra king

(MHT) �ltering [27℄ is often based on N-s an pruning.

3.4.2 Merging

One an also use merging of similar omponents to redu e the number

of omponents in a GM . There are several merging algorithms su h as

Salmond's [31�33℄, Runnalls' [34℄, Williams' [35℄ algorithms and many more

[36�39℄. These algorithms work based on the following three steps:

1. Find the most suitable pair of omponents to merge a ording to a

`merging ost' riterion.

2. Merge the similar pair and repla e the pair with the merged Gaussian

omponent.

19


3. Che k if a stopping riterion is met. Otherwise, set the redu ed mix-

ture to the new mixture and go to step 1.

The merging ost in step 1 looks for similarity of the omponents and it an

be di�erent a ross algorithms. A few of the ommonly used merging osts

are the Kullba k-Leibler divergen e [40℄ and the integral-squared error [41�

43℄. The merging of the omponents in step 2 is usually based on moment

mat hing [40℄. That is, the moments of the GM before and after merging are

the same. The stopping riterion an also vary a ross algorithms, e.g., it an

be based on if the omponents in the redu ed mixture is at a manageable

number. In ertain algorithms, it is he ked based on that the omponents

in the redu ed GM are not similar.

3.4.3 Choi e of GMR

Two main riteria in hoosing the appropriate GMR algorithm are the om-

putational omplexity involved and the a ura y. Most of the pruning al-

gorithms are usually simpler to implement, ompared to merging. There is

information about the un ertainty of the estimate in the ovarian e matri-

es of the pruned omponents. So, as a result of pruning, we might have

underestimated un ertainties. In ontrast, for merging, the un ertainty is

preserved be ause of moment-mat hing. However, the merging algorithms

are more omputationally intensive than pruning. As a trade-o� between

omplexity and a ura y of the un ertainty, it may be more feasible to use a

ombination of pruning and merging. Pruning ensures that the omponents

with negligible weights are removed, without being aggressive. Merging re-

du es the number of omponents further, but keeping the moments of the

retained density the same as before.

3.4.4 GMR for FBS and TFS

Applying GMR, both pruning and merging for the forward �ltering is straight-

forward. When the forward �ltering is based on pruning, it is trivial to

perform the ba kward smoothing of the FBS similar to the optimal solu-

tion, using the �ltered densities. Starting from the last time instant, RTS is

performed ba kwards on the individual retained omponents. This method

su�ers from degenera y similar to parti le smoothing. This is be ause for

k ≪ K, the number of omponents in the forward �lter that orresponds

to the omponents in the smoothed posterior will be one. A solution to

the degenera y is to perform FBS based on merging, something that has

not been explored mu h in the literature. The main hallenge is that for

the ba kward smoothing, the asso iations a ross omponents are no longer

20

3.4. Gaussian mixture redu tion

simple, to use RTS dire tly and ompute the weights of the smoothed den-

sity. In Paper I [44℄ of this thesis, the problem of FBS based on merging is

investigated.

The literature on TFS for Gaussian mixture densities is also sparse. The

two �lters of the TFS an be run independently of ea h other. This allows

the GMR algorithms to be used on both the �lters. Then the di� ulty is

in using the Gaussian mixture redu tion te hniques in the ba kward �lter,

sin e its output is not a density fun tion. So, the GMR algorithms dis ussed

here annot be applied dire tly. In Paper II [45℄ of this thesis, we propose a

method alled smoothed posterior pruning, through whi h pruning an be

employed in the ba kward �lter.

21

Chapter 4

Multi�traje tory estimation

In the previous hapter, we presented a brief dis ussion of the data as-

so iation problem. Even with linear Gaussian assumptions, the number of

Gaussian terms in the Gaussian mixture form of the posterior density grows

exponentially. In this hapter, we dis uss the problem further in the multi�

target setting. Additional hallenges are posed by having a set of data from

multiple targets, where the target identities are not available.

4.1 Data asso iation

In the presen e of multiple point targets, ea h of whi h follows a linear-

Gaussian pro ess and measurement model, at ea h time we observe a set

of measurements. If the identity of a target is not available in the mea-

surements, then there is un ertainty about whi h measurement belongs to

whi h target. In ase of a single traje tory, ea h hypothesis is an event of

assigning a target to a measurement from the set. In this ase, the number

of possibilities at ea h time is the same as the number of measurements,

whi h when multiplied a ross time be omes exponential.

With multiple targets, at ea h time, the problem is even worse. Let

us say we have n targets and m measurements at a parti ular time. Now,

ea h hypothesis, often referred to as a global hypothesis, is the event of

asso iating the n targets to the m measurements (assuming n ≤ m) su h

that a target is assigned to at most one measurement and a measurement is

assigned to at most one target. The number of possibilities is ombinatorial

given by

m!(m−n)!

; for instan e, if we assume n = 2 and m = 3, the number of

ombinatorial possibilities is 6; if we double them up, n = 4 and m = 6, thenumber of possibilities is 360. If we also onsider the possibility that a tar-

get does not need to generate a measurement, in other words, a target an

23

Chapter 4. Multi�traje tory estimation

be missed, the number of possibilities is even higher, and again ombinato-

rial. A ross time, the omplexity of the problem gets multiplied. Therefore,

the optimal way of solving the problem onsidering all the possibilities is

intra table. Below we dis uss brie�y the traditional approa h taken.

Let K denote the bat h duration, k the time index, NK the number of

targets in the entire bat h duration and Mk the number of measurements

obtained at k. Assume the state variable is X = (Xk,i : k = 1, . . . , K, i =

1 . . . , NK) and the measurement is Z = (Zk,j : k = 1, . . . , K, j = 1 . . . ,Mk)where i stands for the target index and j for the measurement index.

If we know the posterior density p(X|Z), we an estimate the states,

whi h are tra table and straightforward if there is no un ertainty in the

measurement origin. However, in the multi�target tra king problems, the

measurement set omprises the measurements from the targets that are

dete ted as well as the lutter measurements, and the origin of the mea-

surements in the set are not known. To handle this un ertainty in the

measurement origin, one traditional way is to introdu e a data asso ia-

tion variable φ = {φk,i, ∀k, i}, where φk,i = j denotes the assignment of

the target i at time k to the measurement Zk,j. With these variables,

the density of interest be omes p(X, φ|Z), using whi h the estimates an

be omputed. Though the introdu tion of this variable makes it easier

to represent the measurement un ertainty, the estimation problem is still

intra table due to the sheer number of possibilities of φ. For instan e, on-

sider X̂MAP

= argmaxX p(X|Z) = argmaxX maxφ p(X, φ|Z). The optimal

point is sear hed over all the possibilities of φ, whi h is exponential in the

number of measurements. Thus, sub-optimal approa hes are inevitable. In

the remainder of this se tion, we give a brief overview of some of the ex-

isting sub-optimal algorithms to estimate X . Adhering to the onventional

terminology, we refer to an instan e of φ as the data asso iation hypothesis.

4.2 Tra king algorithms

The existing algorithms for tra king an be broadly ategorised into two:

the ones that jointly estimate X and φ, and the others that estimate Xwhile marginalizing φ. Global nearest neighbour (GNN) [27℄ and multiple

hypothesis tra king (MHT) [27,46,47℄ belong to the �rst ategory, whereas

joint probabilisti data asso iation (JPDA) [27, 48�55℄, probabilisti MHT

(PMHT) [56�59℄ and their variants belong to the se ond one. There are

also sampling�based algorithms like Markov hain Monte Carlo data asso-

iation (MCMCDA) [60℄. In this algorithm, an estimate of a hypothesis is

24

4.2. Tra king algorithms

obtained by making several random hanges to the existing hypothesis. The

omputational and memory requirements of this algorithm are very high,

in general. In this se tion, we �rst fo us on the �rst ategory of algorithms

that estimate φ, while estimating X , followed by the se ond ategory that

estimate X .

4.2.1 Global nearest neighbour

In ase of the global nearest neighbour algorithm, at ea h time instant k,the best data asso iation hypothesis is hosen to be the one with the largest

Pr{φk|Z1:k}, where φk stands for the data asso iation at time k and Z1:k

for the set of measurements from time 1 to k, respe tively. This hypothesis

is propagated to the next time and asso iated with the new set of mea-

surements to form a new set of hypotheses. The best hypothesis is hosen

to be the one with the largest Pr{φ1:k+1|Z1:k+1} and the whole pro edure

is repeated for subsequent time instants. This algorithm is very simple to

implement using 2-D assignment algorithms [61℄ su h as the au tion algo-

rithm [62℄ or Jonker-Volgenant-Castanon (JVC) [63℄, but sin e it makes

hard de isions every time instant, it underestimates the ovarian e and an

often lead to tra k loss.

4.2.2 Multiple hypothesis tra king

Similar to GNN, MHT generally makes hard de isions when estimating φ.

However, unlike GNN, the MHT algorithms make hard de isions based on

multiple s ans of data. The ommonly used N-s an pruning based tra k�

oriented MHT algorithm hooses the best set of hypotheses at time k based

on the last N s ans of data and, propagates them, and repeats the pro e-

dure again. In essen e, the algorithm estimates the best hypothesis at time

k − N based on the information until the urrent time instant k. The N-

s an pruning algorithm is typi ally implemented using the N-dimensional

assignment algorithms as in [64�66℄. Another popular version of MHT is

to retain the M-best hypotheses at ea h time and propagate only these M

hypotheses to the next time instant. This is typi ally implemented using

Murty's algorithm [28�30,67℄.

As an be noti ed, MHT maintains multiple data asso iation hypotheses

every time instant and, hen e the name `multiple hypothesis tra king'. The

larger the N (orM) is, the more a urate the estimates are. However, larger

N leads to higher omplexity; and smaller N leads to the `short' history

problem of MHT. That is, the di�erent hypotheses that are maintained

ea h time instant k di�er only in the most re ent N (k − (N + 1), . . . , k)

25


asso iations and are identi al from time 1 to k−N . Therefore, there is only

one data asso iation sequen e maintained from time 1 to k − N and any

new information from future measurements annot be used to update the

data asso iation in those time instants.

4.2.3 Joint probabilisti data asso iation

Let us now shift our fo us to the other lass of algorithms whi h estimate

X by integrating out φ. The JPDA algorithm integrates out φ at ea h time

instant and performs moment mat hing to approximate the distribution

over X to a Gaussian density. That is, the possibly multi�modal density

p(X|Z) =∑φ

p(X, φ|Z) is approximated to a uni-modal density p̃(X|Z)where the in lusive KL divergen e KL(p(X|Z)||p̃(X|Z)) [68℄ is minimized.

Again, this algorithm is omputationally simpler than MHT, but when the

approximated posterior density is propagated a ross time, the performan e

degrades.

4.2.4 Probabilisti multiple hypothesis tra king

Another popular tra king algorithm is the probabilisti multiple hypothesis

tra king (PMHT), whi h aims to ompute the maximum a posteriori (MAP)

estimates of the entire sequen e of target states. The idea is to obtain these

estimates using the expe tation maximization (EM) algorithm, where the

solution is iterative and involves several lo al optimisations:

X̂(n+1) = argmaxX

∑

φ

Pr{φ|Z, X̂(n)} ln p(X, φ, Z). (4.1)

PMHT allows a target to be asso iated to multiple measurements, whi h

enables losed�form expressions to ompute the marginal asso iation prob-

abilities of the di�erent traje tories. However, this approximation is more

suitable for extended target models than the point target model assump-

tions presented in the beginning of this se tion. Paper III [69℄ of this thesis

also uses EM similar to PMHT to estimate the states; however, we retain

the point�target onstraints that a target is assigned to at most one mea-

surement and a measurement is assigned to at most one target.

4.3 EM for data asso iation

Expe tation maximization (EM), �rst dis ussed in [70, 71℄, is an iterative

te hnique, widely used to obtain approximate maximum�likelihood estima-

tion (MLE), or MAP estimates of parameters from observed data. In the

26

4.3. EM for data asso iation

EM solution, the model is assumed to have hidden variables that relate the

measurements with the states. This makes the posterior density analysis

simpler, whi h is otherwise intra table. In traje tory estimation, we are

interested in obtaining estimates of the state ve tor X . In this se tion, we

propose two versions of EM to obtain the state estimates using the joint

density p(X, φ, Z). To start with, we present a brief introdu tion to the EMalgorithm for MAP estimation.

To derive EM in its general form, we adhere to a notation that is ommon

in the EM literature. A ording to the notation, θ is the parameter to

be estimated, Z the observed data and γ the hidden variable. The MAP

estimation of θ is given by,

θ̂ = argmaxθp(θ|Z) = argmax

θln

∫p(θ, Z, γ)dγ. (4.2)

Note that the logarithm that has been introdu ed in the maximization is a

monotoni ally in reasing fun tion and does not a�e t the MAP estimation.

In many appli ations (in luding tra king, as will be shown), the integral

in the MAP estimation a ording to (4.2) is not always tra table. To get a

tra table approximation, qγ(γ) over the hidden variable γ is introdu ed in

the obje tive fun tion in (4.2):

ln p(θ, Z) = ln

∫qγ(γ)

p(θ, Z, γ)

qγ(γ)dγ (4.3)

≥∫qγ(γ) ln

p(θ, Z, γ)

qγ(γ)dγ (4.4)

, F(qγ(γ), θ). (4.5)

Jensen's inequality is used to go from (4.3) to (4.4). As an be observed,

the term F(qγ(γ), θ) on the right�hand side of (4.4) is a lower bound on the

logarithm of the joint density ln p(θ, Z) [71℄ and is a fun tional of qγ and θ.

In EM, this lower bound is in reased with iterations su h that the di�eren e

between the bound and the logarithm of the density is de reased [70℄. This

is a hieved by performing the following set of operations in ea h iteration

(n + 1):

q(n+1)γ (γ) = argmax

qγ(γ)F(qγ(γ), θ(n)) (4.6)

θ(n+1) = argmaxθF(q(n+1)

γ (γ), θ). (4.7)

The �rst step is alled the the E-step, where the best q(n+1)γ (γ) is omputed

given θ(n). The se ond step, alled the M-step, omputes the best θn given

27


q(n+1)γ (γ).

In Paper III [69℄ and Paper IV [72℄, we have used two approa hes in EM

to obtain the estimates. In Paper III, we use the EM problem formulation

to estimate the state X dire tly from the joint density p(X, φ, Z). In other

words, we set θ = X and γ = φ. In Paper IV, we reverse the roles of X and

φ. That is, we use EM to estimate the data asso iation φ, from whi h we

an obtain the state estimates X immediately.

28

Chapter 5

Metri s

Metri s are important in multi-target tra king (MTT) for performan e eval-

uation and algorithm design. In essen e, metri s are ne essary to quantify

the loseness between a ground truth and an estimate thereof. When design-

ing metri s for MTT, there are spe i� hallenges that must be addressed,

su h as lo alisation error, error due to missed and false targets and penalty

for tra k swit hes.

In this hapter, we dis uss the need for metri s in MTT, followed by a

dis ussion on the basi metri properties. We also present a summary on

the ommonly used metri s, and brie�y dis uss the hallenges in designing

a metri for traje tory estimation.

5.1 Need for a metri

Metri s are needed in traje tory estimation for two main reasons: designing

algorithms and performan e evaluation. In algorithm design, one wants an

estimate that is lose to the true state in some sense. It is reasonable that a

metri is used to de�ne this loseness. When evaluating the performan e of

an algorithm, one needs a similarity measure to quantify the error between

the obtained estimates and the ground truth. On e again, it seems natural

that a metri is used to quantify this error.

In multi-target tra king (MTT), the estimation is often formulated as a

Bayesian �ltering problem where the ground truth is a random quantity and

the estimates depend deterministi ally on the observed data. For perfor-

man e evaluation, in many ases, we average over several realizations of the

data, so estimates are random as well. In both the s enarios we dis ussed

above�designing algorithms and performan e evaluation�the obje tives

29

Chapter 5. Metri s

involve an averaging of the metri over di�erent instan es of the random

quantities involved, i.e., the ground truth and the estimate. In other words,

the similarity between the ground truth and the estimate is omputed in

an average sense. It is again important that this similarity whi h involves

averaging is also a metri .

5.2 Metri properties

The de�nition of a metri varies slightly based on if the variables involved

are random or not. In this se tion, we summarize the properties of a metri

on general spa es and on probability spa es. We also dis uss the signi� an e

of these properties brie�y.

De�nition 5.1. A metri dA(·, ·) on a set A is a fun tion that satis�es the

following properties for any x, y, z ∈ A [73, Se . 2.15℄:

1. Non-negativity: dA(x, y) ≥ 0.

2. De�niteness: dA(x, y) = 0 ⇔ x = y.

3. Symmetry: dA(x, y) = dA(y, x).

4. Triangle inequality: dA(x, y) ≤ dA(x, z) + dA(z, y).

For metri s in a probability spa e A, the de�niteness between random

variables is in the almost�sure sense [74, Se . 2.2℄, as des ribed in the fol-

lowing de�nition.

De�nition 5.2. A metri dA(·, ·) on a set A is a fun tion that satis�es the

following properties for random variables x, y, z ∈ A:

1. Non-negativity: dA(x, y) ≥ 0.

2. De�niteness: dA(x, y) = 0 ⇔ Pr(x = y) = 1.

3. Symmetry: dA(x, y) = dA(y, x).

4. Triangle inequality: dA(x, y) ≤ dA(x, z) + dA(z, y).

In the above de�nition, Pr(x = y) = 1 implies that the event that x and ytake the same value has probability 1.

We now pro eed to des ribe and dis uss the signi� an e of these prop-

erties, with more emphasis on the triangle inequality property. The non-

negativity property ensures that the distan e annot be negative. The de�-

niteness property ensures that a distan e between a point to itself is 0. For

30

5.3. Common metri s

random variables, this property is in essen e ensured for all the points that

have non-zero probability. The symmetry property on�rms that the dis-

tan e from point x to y should be the same as the distan e from point y to x.

The triangle inequality property, despite its abstra tness, has a major

pra ti al importan e in algorithm assessment [75, Se . 6.2.1℄. Suppose there

are two estimates y and z for a ground truth x. Let us assume that a ording

to dA, the estimate z is lose to the ground truth x and is also lose to the

other estimate y. Then, a ording to intuition, the se ond estimate y should

also be lose to the ground truth x. This property is ensured by the triangle

inequality property. The triangle inequality also has pra ti al impli ations

to ensure the quality of approximate optimal estimators. Consider x to be

the ground truth, and z to be the optimal estimate, a ording to a ertain

riterion. Let us assume that it is di� ult to ompute the optimal estimate

z su h that we resort to an approximation y of the optimal z. This happens

often in pra ti e. If the triangle inequality does not hold, it would mean

that even if we have a good estimate y, lose to the optimal z, whi h in

turn is lose to the ground truth x, it is possible that the distan e from yto the ground truth x is high. This property is learly not desirable.

5.3 Common metri s

In this se tion, we dis uss some of the ommonly used metri s in the litera-

ture. One way of ategorizing the metri s is based on the kind of variables

involved. Below, we summarize the metri s used for ve tors, �nite sets of

ve tors and �nite sets of traje tories. Before we present the metri s, we �rst

dis uss in ea h se tion in whi h s enarios these kinds of metri s are useful.

5.3.1 On ve tor spa es

Metri on ve tor spa es is a well studied problem with the most ommon

one being the Eu lidean metri . The root mean square error (RMSE) is

based on the Eu lidean metri and is ommonly used when the involved

quantities are random. Below we present a slightly general version of the

RMSE.

De�nition 5.3. Given two ve tors x, y in RN, then the p-norm for any

1 ≤ p ≤ ∞ is a metri . It is de�ned as follows:

dp(x, y) ,p

√√√√N∑

i=1

|xi − yi|p. (5.1)

31

Chapter 5. Metri s

De�nition 5.4. If the ve tors x and y are random ve tors with joint dis-

tribution f(x, y), then the following de�nition is also a metri :

d̄p(x, y) ,p

√E[dp(x, y)p], (5.2)

where the expe tation is de�ned with respe t to the joint distribution f(x, y).When we set p = 2 in the above de�nition, we get the ommonly used RMSE

metri .

The Eu lidean metri is also ommonly used in the traje tory estimation

problem for simple s enarios. For instan e, in the single traje tory estima-

tion problem, where there is no un ertainty in the birth time and the death

time of the traje tory, then there is a one-to-one orresponden e between

the estimated state and the ground truth at ea h time instant. In this ase,

one an just use the metri for ve tors at ea h time instant.

5.3.2 On the spa e of �nite sets of ve tors

Let us now onsider s enarios with multiple targets, where we are inter-

ested in how good the lo alisation is at ea h time instant. In this ase, at

ea h time instant, both the ground truth and the estimates are sets of state

ve tors, where there is un ertainty about whi h ve tor in the estimated set

orresponds to whi h ve tor in the ground truth. Now, the quantity of in-

terest is a metri between sets of ve tors.

The study of the metri s in this spa e is relatively new. In the MTT lit-

erature, there are several metri s that have been proposed for this purpose,

su h as the Wasserstein metri [75,76℄, the Hausdor� metri [76℄, the OSPA

metri [77℄ and many more [78�83℄. Among these, the optimal sub-pattern

assignment (OSPA) metri is the most ommonly used one. The metri s

in [82℄ and [83℄ propose a base distan e in the metri that also takes into

the a ount the quality information about the estimated state. Below we

present the OSPA metri .

De�nition 5.5. Let d(·, ·) denote a metri on RNsu h that d(x, y) is the

distan e between x, y ∈ RNand let d(c)(x, y) = min(d(x, y), c) be the ut-o�

metri asso iated with d(x, y) [75, Se . 6.2.1℄. We also refer to d(c)(x, y)

as base distan e. Let Πn be the set of all permutations in {1, . . . , n} for anyn ∈ N. Any element π ∈ Πn is a sequen e (π(1), . . . , π(n)).

Let X = {x1, . . . , x|X|} and Y = {y1, . . . , y|Y |} be �nite subsets of a

bounded observation window W ⊂ RN, where |A| denotes the ardinality of

32

5.3. Common metri s

a set A. For 1 ≤ p <∞ and |X| ≤ |Y |, OSPA [77℄ is de�ned as

d(c)p (X, Y ) ,

1

|Y |

min

π∈Π|Y |

|X|∑

i=1

d(c)(xi, yπ(i))p + cp(|Y | − |X|)

1p

. (5.3)

For |X| > |Y |, d(c)p (X, Y ) = d(c)p (Y,X). The ∞-OSPA is de�ned as

d(c)∞ (X, Y ) ,

minπ∈Π|Y |

max1≤i≤|X|

d(c)(xi, yπ(i)) |X| = |Y |

c otherwise

. (5.4)

In Paper V [84℄, we dis uss the short omings of the above formulation.

We propose a new metri whi h addresses lo alisation error as well as missed

and false targets that are of interest in MTT. We also extend the metri to

ompute the equivalents of the RMSE metri for ve tors.

5.3.3 On the spa e of �nite sets of traje tories

In many tra king algorithms, su h as in multiple hypothesis tra king (MHT)

[27,46,47℄ and joint probabilisti data asso iation (JPDA) [49,50℄, the out-

put of the algorithm is not just the set of states at ea h time. Instead, the

output is a set of time sequen es of states, i.e., traje tories of states. Note

that the theory for sets of traje tories has been well established in [85℄. To

de�ne a metri between sets of traje tories, it is ommon to use the metri

dis ussed in Se tion 5.3.2 or a simpler modi� ation of it. But this strategy

produ es strange and ounter�intuitive behavior. Below, we dis uss some

of those approa hes and their short omings.

One approa h is to use OSPA on the entire sets of traje tories [86,87℄. In

this approa h, one uses the OSPA de�nition where the base metri between

two tra ks is de�ned. To dis uss the problems with this approa h, let us

onsider a new set of examples in Figures 5.1 and 5.2. The ground truth is

the traje tory shown in blue o's in both the �gures and the estimates are

the ones shown in red x's. A ording to the metri we just dis ussed, OSPA

pi ks the one with the minimum of the base distan e between the tra ks

in the ground truth and the tra ks in the estimates. Assuming ǫ is almost

0 and ρ is large, the OSPA distan e indi ates that both the estimates in

Figure 5.1 and 5.2 have the same distan e to the ground truth. This is

learly ounter�intuitive. The estimate in Figure 5.1 is learly a better one

ompared to the one in Figure 5.2. This undesirable behavior is due to

the property that OSPA assigns ea h tra k in the ground truth to exa tly

one tra k in the estimate, assuming the estimate has more tra ks than the

33

Chapter 5. Metri s

ground truth.

1 2 3 4 5 6

∆

∆+ ǫ

time

state

Figure 5.1: If ǫ is small, the estimate indi ated by red ×'s is still a good estimate

ompared to the one in Figure 5.2 for the ground truth in blue o's. The only problem

with this estimate is that it has split a single traje tory into two.

1 2 3 4 5 6

∆

∆+ ǫ

∆+ ρ

time

state

Figure 5.2: If ǫ is small and ρ is large, the estimate tra ks in red olor with×'s should havea larger distan e to the ground truth in blue o's ompared to the estimate in Figure 5.1.

It is ommon to dire tly use the OSPA metri on the set of states of

the traje tories at ea h time instant. A short oming of this approa h an

be illustrated with a simple example. Let us assume that in the ground

truth we have a traje tory of states, i.e., a time sequen e of states. Let us

onsider two di�erent version of the estimates:

34

5.3. Common metri s

• Estimate 1: we obtain a time sequen e of states exa tly identi al to

the one in the ground truth,

• Estimate 2: we obtain two time sequen es of states, one whi h ex-

a tly mat hes the �rst half of the tra k in the ground truth and the

se ond sequen e whi h exa tly mat hes the se ond half of the tra k

in the ground truth. (This orresponds to ǫ = 0 in the example in

Figure 5.1).

Using the strategy we just dis ussed, we get the exa t same value, 0, forthe `metri ' for both these estimates. This property learly violates the

de�niteness property we dis ussed in the beginning of the hapter.

A summary of the learnings from the above two approa hes is that you

an assign a tra k in the ground truth to di�erent tra ks at di�erent times,

but there should be an additional ost for being assigned to di�erent tra ks.

This we refer to as the swit hing ost. We have used this approa h in

Paper VI of the thesis. In the paper [88℄, we ompare our metri to some

of the other metri s [89, 90℄ in the literature that uses the same approa h.

35

Chapter 6

Contributions and future work

The main obje tives of the thesis are to design algorithms for addressing the

data asso iation problem in traje tory estimation and to design a metri to

evaluate the algorithms. The ontributions of ea h paper omprising the

thesis are dis ussed brie�y in this hapter. Furthermore, possible ideas for

future resear h that arose during the writing of this thesis are presented.

6.1 Contributions

In the following se tion, the ontributions of the six papers in the thesis,

and the relations between them, are presented.

Paper I

In this paper, the problem of forward-ba kward smoothing (FBS) of Gaus-

sian mixture (GM) densities based on merging is addressed. The existing

literature provides pra ti al algorithms for the FBS of GMs that are based

on pruning. The drawba k of a pruning strategy is that as a result of ex es-

sive pruning, the forward �ltering an result in degenera y. The ba kward

smoothing on this degenerate forward �lter an lead to highly underesti-

mated data asso iation un ertainties. To over ome this, we propose us-

ing merging of the GM during forward �ltering as well as during ba kward

smoothing. As mentioned before, the forward �lter based on merging is well

studied in the literature. A strategy to perform the ba kward smoothing

on �ltered densities with merged omponents is analysed and explained in

this paper. When ompared to FBS based on an N-s an pruning algorithm,

the two-�lter smoothed densities obtained using the presented approxima-

tions of the ba kward �lter show better tra k-loss, root mean squared error

(RMSE) and normalized estimation error squared (NEES) for lower om-

plexity.

37

Chapter 6. Contributions and future work

Paper II

The obje tive of this paper is to obtain an algorithm for two-�lter smooth-

ing (TFS) of GM densities based on merging approximations. The TFS

involves two �lters, namely the forward �lter and the ba kward �lter, where

the former has been studied extensively in the literature. The latter, i.e.,

the ba kward �lter, has a stru ture similar to a GM, but is not a normaliz-

able density. Therefore, the traditional Gaussian mixture redu tion (GMR)

algorithms annot be applied dire tly in the ba kward �lter. The existing

literature, though providing an analysis of the ba kward �lter, does not

present a strategy for the involved GMR. This paper presents two strate-

gies using whi h the Gaussian mixture redu tion (GMR) an be applied to

the ba kward �lter. The �rst one is an intragroup approximation method

whi h depends on the stru ture of the ba kward �lter, and presents a way

in whi h GMR an be applied within ertain groups of omponents. The

se ond method is a smoothed posterior pruning method, whi h is similar to

the pruning strategy for the (forward) �ltered densities dis ussed in [91℄. In

Paper I, the posterior pruning idea is formulated and proved to be a valid

operation for both the forward and the ba kward �lters. When ompared to

FBS based on an N-s an pruning algorithm, the two-�lter smoothed densi-

ties obtained using the presented approximations of the ba kward �lter are

shown to have better tra k-loss, RMSE and NEES for lower omplexity.

Paper III

This paper address the data asso iation problem in multiple traje tory es-

timation. The obje tive in this paper is to obtain the maximum aposteriori

(MAP) estimate of the state X from the joint density p(X, φ, Z). This prob-

lem is not tra table as the number of possibilities of φ is exponential. In

this paper, we address the problem using expe tation maximisation (EM)

to estimate X , while treating φ as a hidden variable. We show that state

estimates an be obtained by running an iterative algorithm, where in ea h

iteration, a Rau h-Tung Striebel (RTS) smoother is run for ea h target.

The measurements updates in the �lter and smoother is arried out with

the omposite measurements, whi h are weighted sums of the measurements

at ea h time instant. The weights, whi h are the marginal data asso iation

probabilities, are omputed using loopy belief propagation. We show in the

paper, that despite the simpli ity, the algorithm performan e is omparable

to a multiple hypothesis tra king (MHT) algorithm.

38

6.1. Contributions

Paper IV

The data asso iation problem is addressed in this paper by estimating the

data asso iation variable φ from the joint density p(X, φ, Z). On e φ is es-

timated, X is immediate to estimate using an RTS smoother. The strategy

is to use EM to estimate φ. This strategy results in an iterative algo-

rithm, where in ea h iteration, one runs an RTS smoother for ea h target.

The measurements for the RTS smoother are obtained using global nearest

neighbour (GNN) at ea h time. In the paper, we show that the algorithm

outperforms an MHT implementation in terms of mean optimal sub-pattern

assignment(OSPA) performan e.

Paper V

In this paper, we present a metri named generalised OSPA (GOSPA) to

ompute distan e between two sets of ve tors. We show that ompared to

the OSPA metri , our metri addresses the problem by penalising missed

and false targets, whereas OSPA penalises the ardinality mismat h. We

also show that the GOSPA metri an be extended to random �nite sets of

ve tors, whi h is relevant for performan e evaluation and algorithm design.

We show that given a joint distribution over two sets of ve tors, the mean

GOSPA and the root mean squared GOSPA are also metri s.

Paper VI

In this paper, we propose a metri based on multidimensional assignments

in the spa e of sets of traje tories. Besides the lo alisation ost, missed and

false targets [84℄, this metri also addresses the problems of tra k swit hes by

allowing a traje tory to be assigned to multiple traje tories a ross time, but

by penalising it for these multiple assignments. We introdu e the on epts

of half and full swit hes to quantify the penalty. As this multidimensional

assignment metri belongs to the NP hard lass of problems, we also propose

a lower bound for the metri , whi h is omputable in polynomial time using

linear programming (LP). We also show that this lower bound is a metri

in the spa e of sets of traje tories. From simulations, we have observed that

the lower bound omputed using LP often returns the optimal value for the

multidimensional metri . An e� ient way to ompute the LP metri using

alternating dire tion method of multipliers (ADMM) that s ales linearly

with time is also presented. We further adapt this metri to random sets of

traje tories.

39

Chapter 6. Contributions and future work

6.2 Future work

Besides the ideas and algorithms presented in the thesis, we also obtained

a plethora of ideas to investigate in the future. In this se tion, we present

and dis uss the ideas, whi h range from the extensions of GM smoothing to

more omplex s enarios than single-target linear Gaussian pro ess models,

to omputationally heaper GM merging methods and message passing in

generi graphs.

Merging algorithms

The TFS and FBS algorithms presented in the thesis are based on merging.

There are several methods, su h as Runnalls', Salmond's and variants of

these, whi h one an hoose for GM merging implementation. However,

the omputational omplexity of these methods is a serious limitation when

it omes to pra ti al implementations where GM merging is ne essary at

ea h time instant. In both redu tion algorithms, the merging ost must be

omputed for every pair of omponents, whi h involves expensive matrix

multipli ations. Therefore, the omplexity of these algorithms is quadrati ,

if not exponential, in the number of omponents, whi h is still expensive

onsidering predi tion, update and retrodi tion steps. For the results pre-

sented in the thesis, signi� ant amount of e�ort went into devising pra ti al

merging algorithms, whi h resulted in two strategies. One is a ombination

of Runnalls' and Salmond's algorithms, whi h is used in the forward �lter.

The other method is a modi�ed version of Salmond's algorithm. A possible

investigation an be in making fewer ost omputations than omputing the

ost for every pair (i, j). One way of redu ing the number of merging ost

omputations is by obtaining bounds on the ost fun tion. Suppose there

is an upper bound on the least possible ost. And suppose that for some

group of pairs of omponents, we an ompute a lower bound on the osts.

If the group's lower bound is greater than the upper bound on the lowest

ost, the ost omputation for the omponent pairs in the group an be

avoided. The hallenge is thus in obtaining the upper bound on the least

ost, and sele ting the group that an be eliminated. A loser analysis of

the ost fun tion is ne essary to obtain these bounds and a good hoi e of

groups.

Traje tory estimation with random birth and death events

In Paper III and Paper IV, it is assumed that all the tra ks are present

the whole bat h duration. It would be interesting to extend the approa h

to ases where the tra ks births and deaths happen at random times. One

40

6.2. Future work

exhaustive approa h is to onsider all possible birth and death time om-

binations for all the tra ks. Similar to the data asso iation problem, this

is also a ombinatorial problem. A more appealing approa h would be to

model these variables into the joint density and use EM to estimate the

birth and death time variables as well.

Online algorithms

The algorithms proposed in Papers I to IV are all bat h algorithms. Ex-

tending all these algorithms to online algorithms extends the s ope of these

algorithms. There are several possibilities to investigate here. For instan e,

one an use a sliding window approa h, where one an tune the width of

the window and also the overlap a ross the windows based on the appli a-

tion. Another approa h is to extend the idea of smoothed �ltering proposed

in [91℄. That is, to obtain the �ltered density p(xk|z1:k), one an go ba k

and improve all the approximations made at all the previous time instants.

This improvement in approximation an be implemented using an iterative

approa h in the papers.

Metri for sets of traje tories based on distan e-based

swit hing ost

In Paper VI, tra k swit hes are used to penalise when a traje tory in the

ground truth set is assigned to multiple traje tories in the estimate. In

the urrent version, the penalty for the tra k swit h is a �xed parameter.

But there an be s enarios when this penalty must be varied based on the

severity of the swit h. For instan e, onsider a pair of traje tories in the

ground truth that are lose to ea h other for ertain duration and then

move apart. Let us onsider two estimates for the ground truth. First is

an estimate with traje tories su h that the tra k swit h happens when the

traje tories in the ground truth are lose together. The se ond estimate has

traje tories su h that the tra k swit h happens when the two traje tories

in the ground truth are far apart. A ording to intuition, the �rst estimate

is better than the se ond, as the tra k swit h happens in a region where it

might be di� ult to resolve. This di�eren e should be possible to address

by de�ning a penalty for the tra k swit h that depends on the loseness

of traje tories in the ground truth and their orresponding traje tories in

the estimate. The major hallenge here an be in de�ning the penalty in

a onsistent way that still retains the metri properties. This would be an

interesting problem to investigate.

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