Particle Filters for Random Set Models

183 Pages·2013·5.005 MB·English

by  Branko Ristic (auth.)

Branko Ristic
Particle Filters for
Random Set Models
Particle Filters for Random Set Models
Branko Ristic
Particle Filters for
Random Set Models
123
Branko Ristic
DSTO
PortMelbourne
Australia
ISBN 978-1-4614-6315-3 ISBN 978-1-4614-6316-0 (eBook)
DOI 10.1007/978-1-4614-6316-0
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013933594
(cid:2)SpringerScience+BusinessMediaNewYork2013
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Preface
This book is about state estimation of stochastic dynamic systems (objects or
phenomena) from noisy measurements, also known as stochastic filtering. The
commonlyacceptedframeworkforstochasticfilteringistheBayesianprobabilistic
framework, founded around the concept of a random variable. The exact con-
ceptual solution in this framework, referred to as the sequential Bayesian esti-
mator, computes recursively the posterior density of the state of the dynamic
system. This time-varying posterior density provides the complete probabilistic
characterization of the dynamic system of interest over time.
In the general non-linear/non-Gaussian context, the sequential Bayesian esti-
mator cannot be solved analytically in closed from and is therefore implemented
byapproximations.ParticlefiltersareaclassofMonteCarlostatisticalsimulation-
based methods which can provide very accurate approximations of the sequential
Bayesian estimator. In the last decade, particle filters have become exceptionally
popular,mainlyduetotheeaseoftheirimplementationandeverincreasingspeed
of computers. Applications of particle filters span many disciplines of human
endeavor: radar/sonar, communications, navigation, bio-informatics, finance, and
ecology to name a few. Not surprisingly, several good tutorials and books have
been published in the last decade on the topic of particle filters.
While the formulation of the standard sequential Bayesian estimator is general
in the sense that it provides the exact solution in the non-linear/non-Gaussian
context, it is nevertheless limited in the following sense:
• itassumesasingledynamicsystem(object,phenomenon)whichispermanently
active (or present)
• it assumes perfect detection (i.e., no false or miss detections)
• it assumes precise measurements and measurement models.
TherecenttheoreticaladvancesinsequentialBayesianestimationcarriedoutin
the framework of random set theory provide new opportunities which are not
widely known. These recent developments, (due to Mahler, Goodman, Vo, and
others)haveovercomeallthelimitationslistedabove,andthushavedramatically
widened the scope of applications of particle filters: from single to multiple sys-
tems; from permanently active systems (i.e., present objects) to on/off systems
v
vi Preface
(i.e., appearing/disappearing objects); from perfect to imperfect detection; from
precise to imprecise measurements and measurement models.
This book attempts to review some of the theoretical aspects of sequential
Bayesian estimation in the random set theoretic framework, describe the particle
filter implementation of the most popular algorithms and illustrate the enormous
practical importance of this framework for inference in applications. The book is
divided into seven chapters. After the introductory chapter, Chap. 2 reviews the
theoretical foundations necessary for understanding the remaining five chapters.
This includes non-standard measurement models, multi-object filtering, sensor
control and standard particle filtering. Chapter 3 is dedicated to applications of
non-standard measurements, such as imprecise observations, imprecise measure-
ment models and implication rules. Chapter 4 presents particle filter implemen-
tation ofmulti-object filters such asthe PHD/CPHDfilter andthe Bernoullifilter.
Sensor control for random set-based particle filters is covered in Chap. 5. Multi-
target tracking, including algorithms, the axiomatic performance metric and the
application to pedestrian tracking, is presented in Chap. 6. Chapter 7 is dedicated
to more advanced topics of extended object tracking and calibration of multi-
sensor tracking systems.
The book is the outcome of 3 years of active collaborative research funded by
the DSTO fellowship programme. The help and guidance from the experts in the
field, primarily D. E. Clark (Heriot Watt University), B.-N. Vo, B.-T. Vo (Curtin
University),andR.P.S.Mahler(LockheedMartin),wasofcrucialimportanceon
this project and hence greatly appreciated. Other collaborators whose contribu-
tions are included in the book are: A. Bishop (NICTA), A. Gning and S. Julier
(University College London), Y. Petetin (Telecom Sup Paris), L. Mihaylova
(Lancaster University), T. Wood (The University of Oxford), M. Uney (Heriot
Watt University), A. Farina (Selex SI), Á. F. García-Fernández (Chalmers
University of Technology), S. Arulampalam, N. Gordon, J. Sherrah and
A. Skvortsov (DSTO). The author is grateful to all of them.
Melbourne, October 2012 Branko Ristic
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 A Brief Review of Particle Filters. . . . . . . . . . . . . . . . . . . . . . 5
2.2 Online Sensor Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Non-standard Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Imprecise Measurements . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Imprecise Measurement Function . . . . . . . . . . . . . . . . . 11
2.3.3 Uncertain Implication Rules. . . . . . . . . . . . . . . . . . . . . 12
2.3.4 Particle Filter Implementation. . . . . . . . . . . . . . . . . . . . 13
2.3.5 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Multiple Objects and Imperfect Detection. . . . . . . . . . . . . . . . . 15
2.4.1 Random Finite Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.2 Multi-Object Stochastic Filtering . . . . . . . . . . . . . . . . . 19
2.4.3 OSPA Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Specialized Multi-Object Filters . . . . . . . . . . . . . . . . . . . . . . . 22
2.5.1 Bernoulli Filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5.2 PHD and CPHD Filter. . . . . . . . . . . . . . . . . . . . . . . . . 24
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Applications Involving Non-standard Measurements. . . . . . . . . . . 29
3.1 Estimation Using Imprecise Measurement Models. . . . . . . . . . . 29
3.1.1 Localization Using the Received Signal Strength . . . . . . 29
3.1.2 Prediction of an Epidemic Using Syndromic Data. . . . . . 34
3.1.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Fusion of Spatially Referring Natural Language Statements . . . . 39
3.2.1 Language, Space, and Modeling. . . . . . . . . . . . . . . . . . 40
3.2.2 An Illustrative Example. . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Classification Using Imprecise Likelihoods. . . . . . . . . . . . . . . . 44
3.3.1 Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.2 Classification Results. . . . . . . . . . . . . . . . . . . . . . . . . . 47
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
vii
viii Contents
4 Multi-Object Particle Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1 Bernoulli Particle Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.1 Standard Bernoulli Particle Filters. . . . . . . . . . . . . . . . . 53
4.1.2 Bernoulli Box-Particle Filter. . . . . . . . . . . . . . . . . . . . . 56
4.2 PHD/CPDH Particle Filters with Adaptive Birth Intensity . . . . . 62
4.2.1 Extension of the PHD Filter. . . . . . . . . . . . . . . . . . . . . 63
4.2.2 Extension of the CPHD Filter. . . . . . . . . . . . . . . . . . . . 66
4.2.3 Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.4 A Numerical Study. . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.5 State Estimation from PHD/CPHD Particle Filters . . . . . 76
4.3 Particle Filter Approximation of the Exact
Multi-Object Filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Sensor Control for Random Set Based Particle Filters . . . . . . . . . 85
5.1 Bernoulli Particle Filter with Sensor Control. . . . . . . . . . . . . . . 85
5.1.1 The Reward Function . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1.2 Bearings Only Tracking in Clutter
with Observer Control. . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.3 Target Tracking via Multi-Static Doppler Shifts. . . . . . . 96
5.2 Sensor Control for PHD/CPHD Particle Filters. . . . . . . . . . . . . 104
5.2.1 The Reward Function . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.2 A Numerical Study. . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Sensor Control for the Multi-Target State Particle Filter . . . . . . 112
5.3.1 Particle Approximation of the Reward Function. . . . . . . 112
5.3.2 A Numerical Study. . . . . . . . . . . . . . . . . . . . . . . . . . . 113
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Multi-Target Tracking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1 OSPA-T: A Performance Metric for Multi-Target Tracking . . . . 122
6.1.1 The Problem and Its Conceptual Solution . . . . . . . . . . . 123
6.1.2 The Base Distance and Labeling of Estimated Tracks. . . 125
6.1.3 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2 Trackers Based on Random Set Filters. . . . . . . . . . . . . . . . . . . 132
6.2.1 Multi-Target Trackers Based on the Bernoulli PF. . . . . . 132
6.2.2 Multi-Target Trackers Based on the PHD
Particle Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.3 Error Performance Comparison Using
the OSPA-T Error. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.3 Application: Pedestrian Tracking. . . . . . . . . . . . . . . . . . . . . . . 137
6.3.1 Video Dataset and Detections. . . . . . . . . . . . . . . . . . . . 138
6.3.2 Description of Algorithms . . . . . . . . . . . . . . . . . . . . . . 139
6.3.3 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Contents ix
7 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.1 Bernoulli Filter for Extended Target Tracking. . . . . . . . . . . . . . 145
7.1.1 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.1.2 Equations of the Bernoulli Filter
for an Extended Target . . . . . . . . . . . . . . . . . . . . . . . . 148
7.1.3 Numerical Implementation. . . . . . . . . . . . . . . . . . . . . . 150
7.1.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.1.5 Application to a Surveillance Video . . . . . . . . . . . . . . . 155
7.2 Calibration of Tracking Systems . . . . . . . . . . . . . . . . . . . . . . . 157
7.2.1 Background and Problem Formulation. . . . . . . . . . . . . . 160
7.2.2 The Proposed Calibration Algorithm. . . . . . . . . . . . . . . 161
7.2.3 Importance Sampling with Progressive Correction . . . . . 163
7.2.4 Application to Sensor Bias Estimation. . . . . . . . . . . . . . 165
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Acronyms
2DA Two-dimensional assignment
BPF Bernoulli particle filter
BPF-X Bernoulli particle filter for extended target
CDF Cumulative distribution function
CPHD Cardinalized probability hypothesis density
CP Constraints propagation
CRLB Cramér-Rao lower bound
FISST Finite set statistics
GLF Generalized likelihood function
GM Gaussian mixture
HOG Histogram of oriented gradients
KDE Kernel density estimation
KL Kullback-Leibler
LM Linear multitarget
MCMC Markov chain Monte Carlo
MoE Measure of effectiveness
NL Natural language
OID Optimal importance density
OSPA Optimal sub-pattern assignment
OSPA-T Optimal sub-pattern assignment metric for tracks
PDF Probability density function
PF Particle filter
PHD Probability hypothesis density
POMDP Partially observed Markov decision process
RFS Random finite set
RSS Received signal strength
SIR Susceptible-infectious-recovered
SIS Sequential importance sampling
SMC Sequential Monte Carlo
TBM Transferrable belief model
TPM Transitional probability matrix
xi