Clutter Mitigation for Target Tracking

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Clutter Mitigation for Target Tracking

Edmund Førland Brekke

Faculty of Information Technology, Mathematics and Electrical Engineering Department of Engineering Cybernetics

Department of Engineering Cybernetics

PhD thesis 2010:131

Department of Engineering Cybernetics

Norwegian University of Science and Technology Faculty of Information Technology, Mathematics and Electrical Engineering Department of Engineering Cybernetics

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Clutter Mitigation for Target Tracking

c 2010 Edmund Førland Brekke

http://www.ntnu.no Norwegian University of Science and Technology Faculty of Information Technology, Mathematics and Electrical Engineering Department of Engineering Cybernetics NTNU Philosophiae Doctor 2010:131 ISBN 978-82-471-2229-7 (printed) ISBN 978-82-471-2230-3 (electronic) ISSN 1503-8181

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To Finoop

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Summary

Traditionally, the literature on target tracking assumes that the targets of interest are embedded in homogenous Rayleigh distributed background noise. It is most often assumed that purely kinematic point measurements are extracted from the sensor images, so that the tracking problem can be phrased in terms of data association. The tracker has to decide which point measurements are likely to have been caused by the target, and update the track correspondingly. All other measurements are discarded as clutter. This thesis concerns clutter which does not conform to this framework. The following three challenges are addressed: dim targets, heavy-tailed clutter and wakes. For very dim targets it is impossible to extract point measurements, and tracking can only be done by working directly on the raw sensor images. Several methods for such track-before-detect can be found in the literature. In recent years methods based on sequential Monte Carlo have gained strong popularity. This thesis demonstrates that such methods cannot be expected to perform well unless the amplitudes of sensor cells are treated in a robust way. More precisely, it is demonstrated that a satisfactory performance only can be achieved when the uncertainty of the background noise estimate is taken into account. This is done by means of marginalization over a flat prior distribution for the unknown background power. Similar issues arise in the treatment of heavy-tailed clutter. A heavy-tailed background is a background which generates more frequent occurrences of target-like outliers than what one would expect under the Rayleigh assumption. In other words, more false alarms must be accepted if we shall have any hope of detecting the target, and this causes an inevitable degradation of performance. This thesis presents the first systematic study of this performance loss from a target tracking point of view. It is demonstrated that the performance loss may be substantially mitigated by usage amplitude information, depending on how this information is treated. v

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Summary

Another troublesome source of clutter is wakes that appear behind the target. This problem arises in sonar tracking of human divers, in the tracking of ships using surveillance radars, and also in radar tracking of ballistic missiles. This thesis suggests a new solution to this problem. While previous research has used an approach described as probabilistic editing, the new solution solves the wake problem in a Bayesian framework by means of marginalization. These three advances are all tested and compared to traditional approaches using Monte-Carlo simulations. A theoretical analysis of the gains from amplitude information in heavy-tailed clutter is also carried out by means of the modified Riccati equation. The recurrent theme of this thesis is that clutter in general increases uncertainty, and that this uncertainty should be marginalized out by tracking method. The results of this thesis therefore contribute to the viewpoint that tracking methods should be developed by careful probabilistic modeling. The gains from appropriate modeling of the clutter are considerable, even with very simple models and in the presence of large estimation uncertainties. Track-loss rates are typically reduced by half compared to existing methods, and often by more.

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Preface

This thesis is submitted in fulfillment of the requirements for the degree Philosophiae Doctor (PhD) at the Norwegian University of Science and Technology (NTNU). The work has mostly been carried out at University Graduate Center (UNIK), Kjeller, and partly at the National University of Singapore (NUS), Singapore, and at McMaster University, Hamilton, Canada. I have been part of the Sea, Air and Land Surveillance (SEALS) project which is funded by the Norwegian Research Council (NFR), Kongsberg Maritime (KM), Kongsberg Defence & Aerospace (KDA), Norcontrol IT, Park Air Systems and UNIK.

Acknowledgements The work reported in this thesis was carried out under the joint supervision of Professor Oddvar Hallingstad at UNIK and Dr. John Glattetre at KM. First of all I want to thank them both for giving me the opportunity to pursue PhD studies at UNIK. I am thankful for all their support and guidance, and in particular for asking the important and critical questions which always forced me to think twice about which issues were the important ones. The research was also partially carried out under supervision of Professor Shuzhi Sam Ge at NUS and Professor Thiagalingam Kirubarajan at McMaster University. While Prof. Ge taught me how to focus in my research, Prof. Kirubarajan taught me how to be creative, and I am grateful for having had the opportunity to work with them both. I want to express my gratitude for both academic and social activities at UNIK to my former lab colleagues at UNIK: Are Willumsen, Kjetil Ånonsen, Kjell Magne Fauske, Morten Stakkeland, Morten Topland, Øyvind Hegrenæs, Anders Rødningsby and Roar Tungland. I am also thankful to the PhD students of Prof. Kirubarajan for receiving vii

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Preface

me well at McMaster University. This especially applies to Ratnasingham Tharmarasa who patiently introduced me to the state of the art of target tracking, and Nandakumaran Nadarajah with whom I had several conversations which to a large degree triggered the research presented in this thesis. I also had great benefit from conversations with Øyvind Overrein at Applied Radar Physics, who taught me the importance of looking carefully at the data. Last, but not the least, I am greatly indebted to my wife Rita. Without her support and never-ending confidence that I would eventually reach the goals I was aiming for, this thesis would not have been finished.

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Nomenclature

Notations The following list of notations is not exhaustive, but contains most symbols that are repeatedly used. a

amplitude (absolute value) of measurement or acceleration

A

Local ACF range parameter or width of wake region

b

scale parameter or un-norm. zeroth association weight

B

Local ACF bearing parameter or length of wake region

Bernoulli(. . .)

Bernoulli pmf

β

association probability or manoeuver strength constants

cM

volume of M -dimensional unit-ball

c1 , c2

lower and upper track-loss thresholds

C

part of validation gate behind target

card(·)

cardinality of a set

d

target power

D(x)

support area for PSF

D

part of validation gate in front of target

δ

zeroth association weight before normalization (Chapter 6)

∆x , . . .

resolution cell sizes

e

binary existence state or the transcendental number 2.71828 . . . ix

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Nomenclature

e(. . .)

root mean square error

E[·]

expectation

erfc(·)

complementary error function

exp(·)

exponential function

Exponential(. . .)

exponential pdf

η

background power or texture

f (·)

kinematic transition mapping

F

kinematic transition matrix

g

gate size measured in standard deviations

G(x)

set of auxiliary cells for clutter estimation

G

validation gate

Gamma(. . .)

gamma pdf

γ

prior association probability or lower incomplete gamma function

Γ(·)

gamma function

h

point spread function

h(·)

mapping relating measurement vector to state

H

measurement matrix

H0

target absent hypothesis

H1

target present hypothesis

I0 (·)

modified Bessel function of the second kind

k

time index

Kν (·)

modified Bessel function of the third kind

Kcdf(. . .)

K-distribution cdf

Kpdf(. . .)

K-distribution pdf

l

history index or likelihood ratio

L

time lag for amplitude estimation

LΓ

correlation length of gamma-distributed process

λ

clutter intensity

Λ

eigenvalue matrix of gate matrix

m

number of measurements at time k

m

realization of the random variable mk

M

dimension of measurement vector z or number of auxiliary cells

M

eigenvector matrix of gate matrix

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xi µ(·)

pmf of clutter measurements

n

number of clutter measurements

n

realization of the random variable n

N

number of resolution cells in sensor image

N (. . .)

Real or multivariate Gaussian pdf

Nc (. . .)

Complex Gaussian pdf with expectation zero

ncdf(. . .)

Real Gaussian cdf

ν

shape parameter

ν

innovation

pb (·)

birth pdf (Chapter 3) or ramp-shaped wake pdf (Chapter 6)

pm (·)

wake + background mixture pdf used in WPDAF-1

p0 (·)

pdf under clutter only hypothesis

p1 (·)

pdf under target + clutter hypothesis

P

number of particles

P P˜

covariance of state estimate

PB

wake strength probability

Pbirth

birth probability

PD

detection of probability

Pdeath

death probability

PFA

false alarm rate

PG

gate probability

PGB

wake pdf normalization constant

Poisson(. . .)

Poisson pmf

φ

bearing measured in resolution cells

ϕ

number of background measurements

ψ

number of wake measurements

ψ

full state vector including kinematics, amplitude etc.

Ψ

gate matrix

q

background description for a measurement

q2

information reduction factor

Q

plant noise covariance matrix

r

range measured in meters or wake intensity ratio

spread of the innovations

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Nomenclature

R

measurement noise covariance matrix

R(·)

rotation matrix

RG (·)

Autocorrelation function of Gaussian process

RΓ (·)

Autocorrelation function of gamma-distributed process

RL (·)

Local autocorrelation function

Rayleigh(. . .)

Rayleigh pdf

Rice(. . .)

Rice pdf

ρ

range measured in resolution cells

ρ

position

s

target signal

S

innovation covariance matrix

σ

generic scalar standard deviation

Σ

blurring parameter

ς

amplitude state (square root of target power d)

t

detection threshold

T

time step length

θk (i)

current association hypothesis number i

θl,k

measurement history hypothesis number l

ϑ

bearing measured in radians

τ

number of target measurements

τnm

perpendicular acceleration time constant

τtm

tangential acceleration time constant

u

wake velocity

Uniform(. . .)

Uniform pdf

v

process noise

V

volume of gate

w

complex background noise or particle weight

w

measurement noise

W

Kalman gain

W

wake region

ω

turn rate

x

instantaneous power of background noise

x, y

Cartesian coordinates

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xiii x ˜, y˜

wake-oriented coordinates

x

kinematic target state

ξ

diagonalized measurement vector

y

value of auxiliary cell

z

complex data

z

kinematic measurement vector

Z

measurement set

ζ

full measurement vector including amplitude

(ˆ·)

generic estimate

(·)(i)

with regard to measurement number i

(·)a

assumed

(·)k

at time k

(·)k|j

estimate at time k given measurements up to and including time j

(·)l

lag number

(·)n

perpendicular

(·)ϑ

bearing range

(·)r (·)s ,

(·)s

simulated (a.k.a. true)

(·)t

tangential

(·)v

kinematic (plant noise)

(·)w

wake

(·)0

clutter

(·)1

target

(·)a

with regard to amplitude (truncated)

(·)i , (·)j

cell number

(·)l

history number l

(·)T

transposition

(·)z

with regard to kinematics

(·)−1

inverse of matrix

(·)(i)

sample (particle) number

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Nomenclature

Acronyms ACF

autocorrelation function

AI

amplitude information

BLUE

best linear unbiased estimator

CFAR

constant false alarm rate

CLT

central limit theorem

CRLB

Cramer-Rao lower bound

dB

decibel

EKF

extended Kalman filter

IPDA

integrated Probabilistic Data Association

JPDAF

joint Probabilistic Data Association filter

KF

Kalman filter

MHT

multiple hypothesis tracker

MLE

maximum likelihood estimator

MMSE

minimum mean square error

MNLT

memoryless non-linear transform

MoM

method of moments

MRE

modified Riccati equation

OBA

optimal Bayesian approach

PCRLB

posterior Cramer-Rao lower bound

PDAF

Probabilistic Data Association filter

PDAFAI

Probabilistic Data Association filter with amplitude information

pdf

probability density function

pmf

point mass function

PSF

point spread function

RMSE

root mean square error

ROC

receiving operating characteristic

SMC

sequential Monte Carlo

SNR

signal-to-noise ratio

TBD

track-before-detect

TOC

tracking operating characteristic

WPDAF

wake Probabilistic Data Association

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Contents

Summary

v

Preface

vii

Nomenclature

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1

Introduction 1.1 Motivation and problem description 1.1.1 Dim targets . . . . . . . . . 1.1.2 Heavy-tailed clutter . . . . . 1.1.3 Wake clutter . . . . . . . . 1.2 Scientific contributions of the thesis 1.3 Publications . . . . . . . . . . . . . 1.4 Thesis outline . . . . . . . . . . . .

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A review of single-target tracking 2.1 The tracking system . . . . . . . . . . . . . . . . . . 2.2 Signal processing . . . . . . . . . . . . . . . . . . . 2.3 Sensor modeling and detection theory . . . . . . . . 2.3.1 Background amplitude models . . . . . . . . 2.3.2 Target amplitude models . . . . . . . . . . . 2.3.3 Combined models for target and background 2.3.4 The Neyman-Pearson test . . . . . . . . . . 2.3.5 CFAR detection . . . . . . . . . . . . . . . . 2.4 Measurement modeling . . . . . . . . . . . . . . . . 2.4.1 Cardinality of background measurements . .

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Contents

2.5

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2.4.2 Cardinality of target measurements . . . . . . . . . . . . . 2.4.3 Pdf’s of background measurements . . . . . . . . . . . . . 2.4.4 Pdf’s of target measurements . . . . . . . . . . . . . . . . . Linear and non-linear filtering . . . . . . . . . . . . . . . . . . . . 2.5.1 The Kalman filter . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Numerical approximation of the posterior . . . . . . . . . . Tracking methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Thresholded measurements: Optimal approach . . . . . . . 2.6.2 Thresholded measurements: Probabilistic Data Association . 2.6.3 Unthresholded measurements: Track-before-detect . . . . . 2.6.4 Other tracking methods . . . . . . . . . . . . . . . . . . . . Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Performance analysis using real data . . . . . . . . . . . . . 2.7.2 Performance analysis using Monte-Carlo simulations . . . . 2.7.3 Theoretical performance analysis . . . . . . . . . . . . . .

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Tracking dim targets using integrated clutter estimation 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.2 Bayesian solution and target model . . . . . . . . . . 3.3 Sensor models . . . . . . . . . . . . . . . . . . . . . 3.3.1 Gaussian background . . . . . . . . . . . . . 3.3.2 Rayleigh background . . . . . . . . . . . . . 3.3.3 Unknown Gaussian background . . . . . . . 3.3.4 Unknown Rayleigh background . . . . . . . 3.4 Particle filter tracking algorithm . . . . . . . . . . . 3.5 Test design and simulation results . . . . . . . . . . 3.6 Concluding remarks . . . . . . . . . . . . . . . . . .

4

Target tracking in heavy-tailed clutter using amplitude information 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Conceptual framework . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Measurement model . . . . . . . . . . . . . . . . . . . . 4.2.3 The CLT and the Rayleigh model . . . . . . . . . . . . . 4.2.4 Heavy-tailed clutter and the K-distribution . . . . . . . . 4.2.5 Models for target plus clutter . . . . . . . . . . . . . . . . 4.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Measurement extraction for Rayleigh case . . . . . . . . . 4.3.2 Measurement extraction for the K-distribution case . . . . 4.3.3 Probabilistic Data Association . . . . . . . . . . . . . . . 4.3.4 PDAF with Amplitude Information . . . . . . . . . . . . 4.3.5 The target power . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Amplitude likelihoods for Rayleigh case . . . . . . . . . .

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4.4

4.5

4.3.7 Amplitude likelihood for K-distribution case Test design and simulation results . . . . . . . . . . 4.4.1 Background simulation . . . . . . . . . . . . 4.4.2 Simulation of target signal . . . . . . . . . . 4.4.3 Simulation of target kinematics . . . . . . . 4.4.4 Filter model . . . . . . . . . . . . . . . . . . 4.4.5 Scenario . . . . . . . . . . . . . . . . . . . . 4.4.6 Performance measures . . . . . . . . . . . . 4.4.7 Simulation results and their interpretation . . Concluding remarks . . . . . . . . . . . . . . . . . .

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The modified Riccati equation with amplitude information 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Conceptual framework . . . . . . . . . . . . . . . . . . 5.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . 5.2.2 Measurement model . . . . . . . . . . . . . . . 5.2.3 The joint measurement pdf . . . . . . . . . . . . 5.3 Probabilistic Data Association . . . . . . . . . . . . . . 5.4 The MRE with AI . . . . . . . . . . . . . . . . . . . . . 5.5 Derivation of the MRE with AI . . . . . . . . . . . . . . 5.6 Numerical evaluation of the MRE with AI . . . . . . . . 5.6.1 Sampling of amplitude components . . . . . . . 5.6.2 Sampling of radial components . . . . . . . . . 5.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 System setup . . . . . . . . . . . . . . . . . . . 5.7.2 Performance analysis using simulations . . . . . 5.7.3 Performance analysis using the MRE with AI . . 5.8 Concluding remarks . . . . . . . . . . . . . . . . . . . .

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6

Target tracking in the presence of wakes 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Kinematics and measurement model . . . . . . . . . . 6.2.2 Probabilistic Data Association . . . . . . . . . . . . . 6.2.3 Mitigation of a wake by means of probabilistic editing 6.3 Mitigation of a wake by means of marginalization . . . . . . . 6.3.1 Geometry of the wake model . . . . . . . . . . . . . . 6.3.2 Pre-validated measurements . . . . . . . . . . . . . . 6.3.3 Post-validated measurements . . . . . . . . . . . . . . 6.3.4 WPDAF-2 . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Comparison with exact approach . . . . . . . . . . . . 6.4 Test design and simulation results . . . . . . . . . . . . . . . 6.4.1 Filter model . . . . . . . . . . . . . . . . . . . . . . .

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Contents

6.4.2 Simulation of target kinematics 6.4.3 Measurement generation . . . . 6.4.4 Performance measures . . . . . 6.4.5 Interpretation of the results . . . Concluding remarks . . . . . . . . . . .

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Discussion 149 7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

A Derivation of the OBA 153 A.1 Prior association probabilities . . . . . . . . . . . . . . . . . . . . . . . 153 A.2 Posterior association probabilities . . . . . . . . . . . . . . . . . . . . 154 B Evaluation of heavy-tailed integrals B.1 Evaluation of the K-Swerling I likelihood integral . . . . . . . . . . . . B.2 The detection probability integral . . . . . . . . . . . . . . . . . . . . . B.3 The conservative Rayleigh likelihood integral . . . . . . . . . . . . . .

157 157 161 161

C Wake probability calculations 163 C.1 The role of likelihood ratios in the PDAF . . . . . . . . . . . . . . . . 163 C.2 The normalization constant PGB in WPDAF-1 . . . . . . . . . . . . . . 166 C.3 Single point clutter likelihood in WPDAF-2 . . . . . . . . . . . . . . . 168 Bibliography

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1 Introduction

Target tracking is the estimation of the state of a moving object based on remote measurements obtained by one or more sensors. A tracking method must solve two tasks. First, it must determine which measurements should be used in this estimation process, and which should be discarded as irrelevant clutter. This is known as data association. Second, it must use these measurements to estimate where the targets of interest are believed to be located. This is known as filtering or state estimation. These measurements are obtained by one or several sensors, which measure energy emitted or reflected by the target. The most popular sensors are perhaps the radar and the sonar. Both can be used in a stand-alone active mode, or in a multistatic mode where several emitters and receivers are combined. The sonar is also often used in a passive mode, where a single sensor is only listening to its surroundings. Other passive sensors include cameras and infrared sensors. Target tracking is an essential requirement for surveillance and control systems to interpret the environment. It is not only a field in itself, but also a core component in higher level information fusion systems. Closely related to target tracking is navigation, which concerns the tracking of the platform on which the sensors are located. Advances in autonomous navigation are more and more built upon previous advances in target tracking, since increased autonomy requires improved data association. The driving force behind advances in target tracking has largely been military applications. The tracking of dogfighting or low flying aircrafts is very challenging, both with regard to filtering and to data association. Although the tracking literature often takes for granted that the targets to be tracked are aircrafts, most of the methods discussed are readily applicable to other targets as well. Targets commonly encountered include ballistic missiles, submarines, various ground based vehicles and unmanned autonomous vehicles. In civilian applications target tracking is used to prevent collisions and guide 1

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2

1

Introduction

Figure 1.1: Illustration of targets motivating the research presented in this thesis. Courtesy Torbjørn Pedersen Brekke. traffic smoothly. This is important both in harbor surveillance and air traffic control. The purpose of this introductory chapter is to briefly outline the contributions of this thesis, as well as explaining the background and motivation for this research. The problems addressed by this research are described in Section 1.1, while the contributions of the thesis are listed in Section 1.2. The organization of the thesis is summarized in Section 1.4.

1.1

Motivation and problem description

The research reported in this thesis was motivated by the desire for improved harbor surveillance. In particular this includes the tracking of human divers using active sonar and the tracking of small boats using radar. Such targets, illustrated in Figure 1.1, differ from the targets most frequently discussed in the tracking literature (e.g. military and civil aircraft) in several ways. On the one hand, these targets do typically move relatively slowly compared to the scan rate. Therefore it is often only of secondary importance to provide a refined model for the target’s kinematics. On the other hand, their target strengths can be rather low, and the backscattered target return will be accompanied by abundant clutter from different sources. Therefore the key success factor for tracking of such targets is reliable data association. Data association is a well studied problem, to which new solutions are being proposed continuously. The aim of this thesis is not to add another candidate to this wealth of tracking methods. Instead its aim is to improve the discrimination against clutter within the framework of already established tracking methods. An underlying idea of this thesis is that one should not rely on signal processing alone to mitigate clutter, but instead let the tracking method assist in this important task. Three sources of clutter are considered in this thesis: Arbitrary background noise, heavytailed backscatter and wake clutter.

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1.1

Motivation and problem description

1.1.1

3

Dim targets

Any sensor will to some degree be affected by undesired disturbances in addition to the desired target signal. This background noise does not pose a significant challenge if the target signal is strong. A detection procedure can easily extract sufficiently strong target measurements while suppressing the background noise by means of thresholding. However, conventional detectors run into difficulties if the target is weak. Successful tracking of dim targets can only be achieved by means of so-called track-before-detect (TBD) [49, 92], which avoids the thresholding operation altogether. TBD is a relatively new methodology, whose strengths and weaknesses are not yet fully understood. This thesis aims to improve the robustness of TBD methods, and also illuminate challenges hitherto ignored.

1.1.2

Heavy-tailed clutter

Arbitrary background noise is typically Gaussian or Rayleigh distributed. It has, however, been observed in radar data that backscatter from the sea surface often has a more heavy-tailed character [108]. That is, the backscatter has a higher frequency of targetlike outliers than one would have expected under the Gaussian assumption. Recent research in sonar signal processing and underwater acoustics has also shown that similar effects are exhibited by reverberation in sonar data [2]. It is clear that heavy-tailed clutter deteriorates detector performance, since the increased frequency of outliers forces the detector to be more sceptical, thereby decreasing its detection probability. Despite a vast body of research on this, nothing has so far been said about the impact of heavytailed clutter on tracking methods. This thesis presents the first treatment of heavy-tailed clutter from a tracking perspective.

1.1.3

Wake clutter

Clutter can also be generated by the target itself. Such clutter includes electronic countermeasures [106], multipath and wakes. The former one is deliberately used by the target to evade caption, while the latter two are of a more spontaneous nature. It is well known that wakes behind ships [83] or ballistic missiles [5] can make the tracking of these targets difficult. Similar phenomena can also occur behind human scuba divers [87]. Divers using ordinary scuba equipment release a stream of air bubbles, of which some remain in the water behind the diver. Since air bubbles have a very strong ability to reflect sound waves, this bubble cloud is likely to be more visible to the sonar than even the diver himself. Wakes do in general cause an abundance of clutter measurements right behind the target, thereby dragging the state estimate of the tracking method backwards. Previous research has attempted to combat this by instructing the tracking method to exercise increased scepticism against measurements behind the predicted target location [87]. This thesis aims to develop a more systematic and rigorously founded treatment of wake clutter.

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4

1

1.2

Introduction

Scientific contributions of the thesis

The main scientific contributions of this thesis are briefly summarized below: • Evidence is presented that recursive Bayesian TBD methods have huge difficulties coping with unknown background noise. This problem is solved by assigning the unknown background strength a non-informative prior, over which it is marginalized. • This approach is furthermore transferred to the Probabilistic Data Association filter with amplitude information (PDAFAI). While this is not necessary for the PDAFAI to work, it is shown in this thesis that such an approach yields improved performance. • The first systematic treatment of heavy-tailed clutter from a tracking point of view is presented. Efficient techniques for mitigating such clutter using amplitude information (AI) are developed. Monte-Carlo simulations demonstrate significant reductions in track-loss rates achieved by these techniques. • The modified Riccati equation (MRE) is extended to predict the performance of the PDAFAI. In other words, this thesis presents the first treatment of the MRE that involves AI. This analysis can also be used in order to determine optimal nominal false alarm rates in heavy-tailed clutter. • A new method for target tracking in the presence of wakes is presented. This method is a modification of the Probabilistic Data Association filter (PDAF) in the same way as the method proposed in [87], but it is derived in a different way. Monte-Carlo simulations demonstrate that the new method achieves significantly lower track-loss rates than the method of [87]. In addition to these major contributions, the thesis also contains several minor contributions. These include a presentation of the optimal Bayesian approach to single target tracking with amplitude, an extensive discussion of performance evaluation, a testbed for the evaluation of tracking methods in heavy-tailed clutter, an estimator of the mean power of a target, a new criterion for determining when a track is to be considered lost and recipes for the simulation from a truncated K-distribution.

1.3

Publications

This thesis is based on research presented in the following publications: B E. B REKKE , T. K IRUBARAJAN AND R. T HARMARASA, “Tracking Dim Targets Using Integrated Clutter Estimation”, in Proceedings of Signal and Data Processing of Small Targets, SPIE, San Diego, CA, USA, 2007.

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1.4

Thesis outline

5

B E. B REKKE , O. H ALLINGSTAD AND J. G LATTETRE, “Target Tracking in HeavyTailed Clutter Using Amplitude Information”, in Proceedings of the 12th International Conference on Information Fusion, Seattle, WA, USA, 2009. B E. B REKKE , O. H ALLINGSTAD AND J. G LATTETRE, “Performance of PDAFbased Tracking Methods in Heavy-Tailed Clutter”, in Proceedings of the 12th International Conference on Information Fusion, Seattle, WA, USA, 2009. B E. B REKKE , O. H ALLINGSTAD AND J. G LATTETRE, “Tracking Small Targets in Heavy-Tailed Clutter Using Amplitude Information”, in IEEE Journal of Oceanic Engineering, vol. 35, no. 2, pp. 314-329, May 2010. B E. B REKKE , O. H ALLINGSTAD AND J. G LATTETRE, “Improved Target Tracking in the Presence of Wakes”, submitted to IEEE Transactions on Aerospace and Electronic Systems. B E. B REKKE , O. H ALLINGSTAD AND J. G LATTETRE, “The Modified Riccati Equation for Amplitude-Aided Target Tracking in Heavy-Tailed Clutter”, submitted to IEEE Transactions on Aerospace and Electronic Systems. B E. B REKKE , O. H ALLINGSTAD AND J. G LATTETRE, “Target Tracking in State Dependent Wake Clutter”, in Proceedings of IEEE Oceans Conference 2010, Sydney, NSW, Australia. Nominated for student poster competition. In addition to these publications, the author has also contributed to the following publications during the work with this thesis: B Y. PAN , S. S. G E , A. A. M AMUM AND E. B REKKE, “"Sound Source Recognition for Human Robot Interaction”, in Proceedings of the 17th IEEE International Symposium on Robot and Human Interactive Communication, Munich, Germany, 2008. B M. S TAKKELAND , Ø. OVERREIN , O. H ALLINGSTAD AND E. B REKKE, “Tracking of targets with state dependent measurement errors using recursive BLUE filters”, in Proceedings of the 12th International Conference on Information Fusion, Seattle, WA, USA, 2009. B E. B REKKE , O. H ALLINGSTAD AND J. G LATTETRE, “The Signal-to-Noise Ratio of Human Divers”, in Proceedings of IEEE Oceans Conference 2010, Sydney, NSW, Australia. Nominated for student poster competition.

1.4

Thesis outline

This introductory chapter has given an informal introduction to this thesis, and to target tracking in general. In Chapter 2 a more systematic treatment of single-target tracking

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6

1

Introduction

is given. The pre-processing preceding target tracking is outlined before the bulk of the chapter is devoted to the derivation of some common tracking methods. The aim of the chapter is to prepare the reader for Chapters 3 - 6, in which novel material is presented. The topic of Chapter 3 is recursive Bayesian TBD, and the impact of unknown background noise on such methods. In this chapter the idea of a conservative amplitude likelihood obtained through marginalization over an uninformative prior is introduced. The chapter is based on the conference publication [21], with some notational alterations and additional concluding remarks. The topic of Chapter 4 is target tracking in heavy-tailed clutter. The chapter presents the first systematic treatment of heavy-tailed clutter from a tracking perspective. Trackers using AI under assumptions of both the K-distribution and the Rayleigh distribution are developed and tested in a realistic testbed. The chapter is based on the journal article [27]. The bulk of this chapter has also been previously published in the two peerreviewed conference publications [23] and [22]. The treatment of heavy-tailed clutter and AI is continued in Chapter 5, where the modified Riccati equation is extended to predict the performance of the PDAFAI. The material of this chapter is presented in the article [25], which currently is under review. A cursorial summary has been given in [22]. In Chapter 6 the topic of wake clutter is treated. A new tracker tailored to deal with this kind of clutter is developed and tested using Monte-Carlo simulations. The chapter is based on the article [24], which also is under review. The conclusions from this research are summarized in Chapter 7, where several paths for future research also are suggested. The thesis comes with 3 appendices. Details regarding the derivation of conventional tracking methods such as the PDAF are provided in Appendix A. In Appendix B it is explained how integrals involved in the treatment of heavy-tailed clutter can be evaluated efficiently. Appendix C contains non-trivial calculations underlying the treatment of wake clutter.

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2 A review of single-target tracking

The aim of this chapter is to present the single-target tracking problem in a systematic manner, before any novelties are introduced in Chapters 3 - 6. The chapter contains 7 sections which progressively outline the tasks carried out by a tracking system, from reception of raw sensor input to data interpretation. Section 2.1 presents a cursorial outlook on the entire tracking system. Basic signal processing that precedes detection and target tracking is treated in Section 2.2. Detection theory, in the traditional sense of measurement extraction, is discussed in Section 2.3. How extracted measurements are modeled from the tracking method’s point of view is carefully explained in Section 2.4. Target tracking is most naturally understood as a generalized filtering problem, and linear as well as non-linear filtering is therefore dealt with in Section 2.5. Tracking methods are treated in Section 2.6. Particular attention is given to the PDAFAI and a TBD method advocated by [92]. In Figures 2.1 and 2.2 it is illustrated how these methods interact with the preceding signal processing. The performance evaluation of filtering and tracking methods is finally discussed extensively in Section 2.7. The tasks carried out by a tracking system can roughly be divided into signal processing and information processing. The first of these categories concerns the conversion of raw physical input into measurements that can be interpreted by a human operator or a target tracking method. The second category concerns this interpretation, and constitutes the setting in which tracking methods are developed. Also, higher level information fusion should be viewed as information processing. Traditionally, more research efforts have been devoted to signal processing than to information processing. However, as the theory and methods of signal processing have reached a mature stage, it has become more and more evident that the tools of signal processing only can be fully utilized if accompanied by corresponding tools of information processing. 7

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8

2

A review of single-target tracking

Signal processing

Tracking

Reception and beamforming

Measurement update

Background estimation Set Threshold Measurement extraction

Kinematic measurements

Sensor input

Merging

Prediction

Association Amplitude data

Figure 2.1: Workflow of the PDAF and PDAFAI methods together with preceding signal processing. The stapled connection represents the flow of amplitude information, which only is used in the PDAFAI.

Signal processing

Tracking


Measurement update

Background estimation

Prediction

Sensor input

Figure 2.2: Workflow of a Bayesian TBD method. TBD is conceptually simple, but nevertheless a challenging problem since it is more difficult to implement a sensible measurement update procedure for raw sensor data than for extracted measurements as in conventional tracking methods.

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2.1

2.1

The tracking system

9

The tracking system

Target tracking is the estimation of the state of a moving object based on remote measurements obtained by one or more sensors. Such sensors may be of a passive nature, that is they only “listen” to the environment, hoping to spot the signature of a target. Cameras, infrared sensors and passive sonars are typical examples of passive sensors. On the other hand there are active sensors, which emits some kind of radiation whose backscatter is received and analyzed. Examples of active sensors are active sonars and radars. Active sensors differ from passive sensors in that they are able to measure the two-way travel time, and thereby provide measurements of the position of a target. Such kinematic measurements can sometimes be supplemented by the backscattered amplitude, which often is referred to as a feature measurement. This is an important topic in this thesis. A tracking system consists of the sensors as well as software interpreting the input from these sensors. This software must necessarily perform several tasks. First of all, the sensor inputs must be translated into measurements which can be interpreted by the tracking method. The tracking system should be able to initiate new tracks on targets recently discovered and kill dead tracks on targets no longer visible. Furthermore the tracking system should be able to decide which of these targets are of interest, for example for the purpose of threat assessment. All these tasks must somehow be solved for a tracking system to be practically applicable. A tracking method is on the other hand an algorithm whose major task is to maintain existing tracks. Nevertheless, the trend in recent years has been to include more and more of both lower and higher level tasks into the tracking method, which therefore has become the core component of the tracking system.

2.2

Signal processing

The information in the scene or surveillance area in a tracking system is embedded in a form of energy depending on the surrounding transmission media. The form of energy is typically electromagnetic in a radar system or acoustic in an underwater surveillance system. In active sensors an electrical signal is transformed into an appropriate type of energy, like acoustic for an active sonar or electromagnetic for a radar. This energy is transmitted into the scene to impose reflections from possible targets, and the reflected energy is transformed back into an electrical signal in the sensor. Then the electrical signal is processed to extract the desired information. Processing of the received signal using modern digital software divides the scene into smaller parts called resolution cells or pixels. Each resolution cell covers a certain area or volume in the surveillance region. For a 2-dimensional sensor each resolution cell typically represents a specific range (distance from the sensor) and bearing (direction) interval. A detector can now “look” for a target in each of these resolution cells (cf. Figure 2.7).

2.1

Resolution Cells and Beamforming classtest: 2010-06-08 17:36 — 10(28)

10

Instead of dealing with the total amount of information in the whole scene in one operation, the scene is divided into smaller parts, called resolution cells, covering a certain area in the surveillance region. An example of resolution cells in a two-dimensional sensor system is shown in Figure 2.4(a) where each cell represents a specific range (distance 2 A A review of can single-target tracking from the sensor) and bearing (direction) interval. detector now “look” for a target in each of these resolution cells. Sensor

s(t)

Target

r(t) range 2.1: transmission Signal transmission the and sensor and atarget pointmodel. target model. Figure Figure 2.3: Signal betweenbetween the sensor a point The The transmitted and reflected signals are denoted s(t) and r(t), respectively. transmitted and reflected signals are denoted s(t) and r(t), respectively. Courtesy Anders Rødningsby. The time between transmission of the signal s(t) and reception of r(t) in Figure 2.1 is given by the transmission distance to the target (range) speed of ϑ of theinsignal. The time between of the signal s(t) and and the reception r(t) Figure This 2.3 delay us distance thereforetoinformation about the range,ϑ assuming the speed ϑ is known, is givengives by the the target (range) andtarget the speed of the signal. This delay but it gives no directional information. The principles for how to obtain information gives us therefore information about the target range, assuming the speed ϑ is known, aboutnowhat direction the signal is coming from, called beamforming, is illustrated in but it gives directional information. Figure 2.2. Here, an array Nr receivers are information located withcan a given distanceThe d between There are two common ways inofwhich directional be acquired. them. Assuming the incoming signal has a plane wavefront with angle ϕ to first way, used in many radars, is to rotate the sensor antenna physically so that thethe row receivers, as shown Figure 2.2, wavefront willthen reachgradually the different receivers at desiredofsurveillance region is in scanned. Thethis sensor array can be filled different times depending on the signal direction ϕ. In beamforming, these receivers with signal values as the beam passes by the bearings θj of the resolution cells. Several delay their inputs to for observe signal atsoathat given “look” direction. In Figure pulses are typically emitted each abearing, each resolution cell contains an2.2, let Receiver 1 have no delay (∆t = 0). The wavefront arrives at Receiver 2 ∆t time units 1 2 integrated output over these pulses, thereby increasing the SNR. d sin(ϕ) arrives at in Receiver 1, where . Inasgeneral, Receiver n receives the 2 = is known ϑ Thebefore seconditway, used sonars and some∆t radars, beamforming. The idea wavefront ∆t time units before Receiver 1, where of beamforming is tonfocus an array of omnidirectional receivers (e.g. hydrophones) to look for incoming waves in directions θj corresponding to the resolution cells. A delayand-sum beamformer [55 p. 112] delays the outputs from different receivers so that the (n − 1)d sin(ϕ) ∆tn = (2.1) signals from the steering direction θj are added constructively while signals from other ϑ directions are canceled. In practice, beamforming is carried out using phase-shifting instead Therefore, of delays. For further discussion beamforming, the reader is referred to [55]. by delaying the signalofwith ∆tn at Receiver n, each receiver samples the Notice wave that infrom active sonar only a single pulse, a.k.a. ping, is used to generate the entire this given direction at the same phase. Since the beamformed signal is the scan. sum of all these samples from the individual delayed receivers, the resulting signal is Similar principles cannoise be used to obtain elevation measurements as well, thereby and enhanced and the is suppressed. For further information about beamforming makingarray the measurement space 3-dimensional. It is also sometimes possible to supplesignal processing, see [38]. ment range and direction measurements with Doppler measurements, from which the velocity of the target can be estimated. Neither elevation nor Doppler measurements are considered in this thesis. Both the reception process and the beamforming process induce correlations in the sensor image. The reflection from a point target may not only affect the resolution cell in which it is located, but also its surrounding cells as quantified by the sensor’s point spread function (PSF). In Figure 2.4 it can be seen how real sonar data are correlated along the bearing direction according to the PSF. Both the target itself and the background noise inherit these correlations. This is not surprising, since the background

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2.3

11

Sensor modeling and detection theory 1

Normalized amplitude

Data a(φ) PSF h(φ) 0.8 0.6 0.4 0.2 0 -30

-20

-10 20 0 10 Bearing φ relative to target measured in cells

30

Figure 2.4: Sonar data containing a human diver plotted together with theoretical PSF along bearing direction. The correlation properties are given by the PSF and are the same for both target and background. The bearing PSF can be calculated from the beamforming matrix. noise largely consists of backscatter from rocks and other scatterers which also can be considered targets depending on what one is looking for. Nevertheless, in practice the background noise is often treated as white, and this is done in this thesis as well.

2.3

Sensor modeling and detection theory

There are two kinds of detection going on in a tracking system. The first kind of detection concerns the extraction of measurements to be fed to the tracking method. The second kind of detection concerns the decision about whether a target is present or not. The latter kind can only be done reliably by means of target tracking. In this section we only deal with the first kind of detection, i.e. measurement extraction. The term measurement refers to what is processed as input by the tracking method. In most tracking systems measurements are extracted by means of a detection process, and consequently the tracking method processes only this limited set of point measurements instead of the entire sensor image. An important counter-example to this conventional setup is given by TBD methods, which indeed process the entire sensor image as their input measurement. Since such methods exploit more data than conventional tracking methods, they are in principle more robust and can handle lower values of the signal-tonoise ratio (SNR). Ideally the extraction process should be able to detect the presence of a target with a good confidence. However, there is no such thing as a perfect detector since high detection probabilities inevitably require1high false alarm rates. For moderate values

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12

2


of the SNR (10dB to 20dB) one may need to allow false alarm rates in the range of 10−4 to 10−2 per resolution cell in order to be able to detect the target with a reasonable probability (cf. Figure 2.8). All such false alarms which do not originate from the target of interest are considered to be clutter. This includes detections due to thermal noise in the sensor, ambient noise as well as backscatter from the surroundings. A set of received measurements do therefore typically consist of several clutter detections in addition to a few detections from the targets of interest. In order to distinguish target originating measurements from clutter one must devise probabilistic models for the amplitude under both the hypotheses H0 : The data is caused by background noise only H1 : The data is caused by a target as well as background noise.

(2.1)

The term data in (2.1) can be understood as the data in a single resolution cell at a given time, in a sequence of resolution cells, in the current sensor image or in the entire set of data at all times up to and including the current time. When conventional tracking methods are discussed (Chapters 4 - 6), measurements are extracted by carrying out a test between these hypotheses for all cells j = 1, . . . , N in the sensor array. A detection in cell number j is declared whenever H1 is chosen instead of H0 . When we discuss TBD methods (Chapter 3), a likelihood ratio between these two hypotheses is evaluated for all cells, and then used in the TBD method. A more precise formulation of the detection problem (2.1) can be expressed as follows: H0 : z j ∼ p(z j |H0 ) , p0 (z j ) (2.2) H1 : z j ∼ p(z j |H1 ) , p1 (z j ). Here p0 (z j ) is the probability density function (pdf) of the data in cell number j under the hypothesis H0 and p1 (z j ) is the pdf under H1 . Some commonly encountered models for p0 (z j ) are introduced in Section 2.3.1, while models for p1 (z j ) are treated in Section 2.3.2.

2.3.1

Background amplitude models

Very much of the research on sonar and radar background models is focused on correlation properties in the sensor image. Such correlations are induced both by the beamforming and reception processes, and by underlying correlations in the sea surface, seafloor etc. This kind of modeling is out of the scope for this thesis. Instead we are, in accordance with (2.2), exclusively concerned with the amplitude pdf of the background in an single cell. In other words, we are solely concerned with first-order statistics. This restriction relies on an implicit assumption of whiteness. Sometimes this can be justified by prewhitening [15]. In other situations the whiteness assumption can be justified from lack of knowledge about the autocorrelation function (ACF). It is safer to

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2.3


13

assume a white background than a colored background whenever the correlations cannot be modeled with sufficient reliability. Matched filtering [60] is often used to remove white noise, or more generally noise with a known covariance matrix. The concept of matched filtering is, briefly explained, to convolve the received data with a conjugated time-reversed copy of the expected target return. The matched filter is the optimal linear filter for maximizing the SNR in the presence of additive noise. However, matched filtering will have difficulties removing non-Gaussian (e.g. heavy-tailed) noise, and it requires both the covariance matrix of the noise as well as the expected target return to be known. Furthermore, matched filtering can only filter out background noise with different signal properties than the target return. Since arbitrary backscatter and reverberation often is a far more dominant noise component than thermal noise in the sensor, most of the noise will quite often be indistinguishable from the target signal as far as signal properties are concerned. Therefore, although matched filtering often can be used to increase the SNR, this kind of techniques cannot mitigate the background noise entirely. The simplest possible background model is the Gaussian distribution. Research on first-order background statistics in radar and sonar images has primarily been concerned with devising more flexible alternatives to the simple Gaussian model. Many of the models encountered are nevertheless based on the Gaussian model, either through simple transformations such as taking the absolute value, or as compound distributions in which the Gaussian is a component. In this thesis the following models are employed: the real and the complex Gaussian, the Rayleigh distribution, the exponential distribution and the K-distribution. All of these models are Gaussian-based in one of these two ways. The prevalence of the Gaussian model is largely due to mathematical convenience. Gaussianity does typically yield closed form expressions, while other distributions often require approximations or numerical treatments. Assuming Gaussianity is in some ways equivalent to restricting attention to the first two moments of the background, since the Gaussian distribution is the least informative distribution when only these are known [85 p. 669]. Therefore, one should always assume Gaussianity as a first approximation unless more information is available. A more physical argument for Gaussianity is provided by the central limit theorem (CLT), which states that the sum of many independent random variables tends to behave like a Gaussian random variable [85 p. 278]. Therefore, by assuming that there are enough independent scatterers in any given resolution cell contributing to the received signal, one is justified in modeling the background noise as Gaussian. Both a real and a complex version of the Gaussian distribution is commonly used to model background data. The real version is 2 1 −w p0 (w | η) = N (w ; 0, η) = √ e 2η 2πη

(2.3)

where η is the variance. The expectation is assumed to be zero. This can normally be justified by removing steady-state components through some kind of pre-processing.

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14

2


While (2.3) often is used as a model for optical images [15], it is less appropriate for radar or sonar data which typically have two channels: the in-phase and the quadrature channels [60 ch. 13]. The complex version of the Gaussian distribution must then be used. The complex univariate Gaussian is really a bi-variate Gaussian given by Re(w) 0 η 0 p0 (w | η) = Nc (w ; η) = N ; , (2.4) Im(w) 0 0 η where the multivariate Gaussian with expectation µ and covariance matrix Σ is given by N (x ; µ, Σ) =

1 1 T −1 p e− 2 (x−µ) Σ (x−µ) . (2π) |Σ| n 2

(2.5)

For detection purposes it is often more convenient to work with the absolute value of the data, since this allows one-sided tests to be carried out. A complex Gaussian random variable becomes Rayleigh or exponentially distributed after this transformation, depending on whether the amplitude itself or its square is used. Thus, if w ∼ Nc (w ; η) and a = |w|, s = |w|2 then 2 a −a p0 (a | η) =Rayleigh(a ; η) = exp η 2η 1 −s p0 (s | η) =Exponential(s ; η) = exp . η η

(2.6) (2.7)

The models (2.3 - 2.7) are solely governed by the background power η and provide no flexibility regarding higher order moments. Real data is often observed to have higher kurtosis and skewness [1] than assumed by the Gaussian or Rayleigh distributions. The development of plausible as well as tractable 2 or 3-parameter background distributions has therefore been a major focus for the radar and sonar signal processing communities. Some of the models frequently encountered in the literature are the K, Weibull and lognormal distributions. Other models worth mentioning are the multivariate normal inverse Gaussian, Rayleigh mixtures and the α-stable distribution. In this thesis, attention is restricted to the K-distribution as a representative for heavy-tailed background models. The K-distribution was proposed as a model for radar clutter in [53], and has been extensively treated in [108]. A K-distributed amplitude a has the pdf 4aν 2a p0 (a | ν, b) = Kpdf (a ; ν, b) = √ ν+1 Kν−1 √ (2.8) b b Γ(ν) where ν is termed the shape parameter and b is termed the scale parameter. For ν → ∞ the K-distribution turns into the Rayleigh distribution, while ν < 1 indicates a very

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2.3


15

heavy-tailed background. By Γ(·) we denote the Gamma function while Kν (·) refers to the modified Bessel function of the second kind [3 p. 374]. There are four reasons why this thesis advocates the K-distribution as a model for heavy-tailed clutter: First, experimental research has shown that the K-distribution provides a good fit to real data [1, 2, 79, 108 p. 111]. This is a necessary, but not in itself sufficient, reason for the K-distribution to be preferred. It has also been shown that its rivaling candidates can provide a good fit to reality [1]. Second, reasoning similar to the CLT gives the K-distribution physical plausibility. To the author’s knowledge, similar results have not been established for any of the alternative models. More precisely, the amplitude is K-distributed if either of the following assumptions holds: A1: The number of scatterers per resolution cell is random with a negative binomial distribution while the scatterer amplitudes are constant and non-random [54], A2: The number of scatterers per resolution cell is constant and non-random while the scatterer amplitudes are random with an exponential distribution [2]. Third, the K-distribution has finite moments. This may appear trivial, but there exist models (e.g. the α-stable distribution) whose variance is infinite. Finite moments are tremendously helpful from a practical perspective, since they are necessary for the use of moment based parameter estimators, which can be much easier to implement than other estimators such as the maximum likelihood estimator (MLE). Also, it is rather audacious to claim that the background noise has infinite variance. Such claims should be accompanied by solid evidence of both a theoretical and a practical nature. Fourth, the K-distribution can be developed inside a compound framework where the complex Gaussian or Rayleigh distribution is modulated by a Gamma-distributed local power [107, 108 pp. 108-113]. This compound formulation allows us to view a K-distributed process as the absolute value of an underlying complex process. Physical interpretations of the compound process have been discussed in several references including [33] and [81]. The underlying complex process is essential for evaluation of detection probabilities and target likelihoods, since the background and the target signal is most correctly modeled as additive in the complex domain, and not after the absolute value has been taken. The alternative candidates lack such complex versions, which makes the evaluation of target-plus-clutter statistics difficult, if well-defined at all.

2.3.2

Target amplitude models

Small targets in radar and sonar data are often classified into what is known as Swerling models. These are special cases of the general Gamma-distributed family [108 p. 245] p(|s|2 ; d, κ) =

(|s|2 )κ−1 |s|2 exp − (2d)κ Γ(κ) 2d

(2.9)

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16

2

4 2 -5

0 5

0

0 φj − φ(xk )

-5

5

ρj − ρ(xk )

Target + clutter

4 2 -5

0 5

0

0 φj − φ(xk )

-5

5

ρj − ρ(xk )

Amplitude a

Target only

Amplitude a

Amplitude a

Clutter only


4 2 -5

0 5

0

0 φj − φ(xk )

-5

5

ρj − ρ(xk )

Figure 2.5: Addition of background and target signal. where s is the complex target backscatter and |s|2 is proportional to the cross-section of the target. Although a target may fluctuate between the scans, it can often be treated as constant for all the pulses during a single scan. The sensor output does then contain the random variable s multiplied by the number of pulses in a single scan. A target may also exhibit more rapid fluctuations, so that the cross-section instead should be modeled as independent between the pulses. The sensor output will then contain a sum of independent random variables distributed according to (2.9). For κ = 1 and only scan-to-scan fluctuations we obtain the popular Swerling I model. The case with κ = 1 and pulse-to-pulse fluctuations is referred to as the Swerling II model. Under these models s has a complex Gaussian distribution with power d. The amplitude |s| is Rayleigh distributed while (2.9) turns into an exponential distribution. The Swerling I and II models are appropriate when the target scatterers obey the CLT. The gaussianity of these models makes them mathematically convenient. At the same time they represent some kind of worst-case scenario where the target signal is very noise-like. For κ = 2 the Swerling III and IV models are obtained for the cases of scan-to-scan and pulse-to-pulse fluctuations, respectively. These models approximate the backscatter from an object with one large scatterer and several smaller scatterers. As κ increases, the pdf in (2.9) becomes narrower and the target fluctuations become increasingly negligible. For κ → ∞ the model (2.9) leads to a constant amplitude signal, often referred to as the Swerling 0 model. When only a single pulse is emitted per scan, as for an active sonar, there is no difference between the Swerling I and II models or between the Swerling III and IV models. In this thesis this is always assumed to be the case, and no distinction is therefore made between the Swerling I and II models. In reality even a point target is likely to affect several cells in the sensor array due to blurring effects caused by the reception and beamforming processes (cf. Section 2.2). In some situations (e.g. TBD and modeling of sensor images) the sensor PSF should therefore be included in the target amplitude model. This is further elaborated in Section 4.4.2. 1

1

1

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2.3

17


2.3.3

Combined models for target and background

The amplitude statistics of the target on its own is of limited utility. In order to specify the detection problem we need the amplitude statistics of target plus noise, that is of z = s + w. For a Swerling 0 target in real or complex Gaussian noise we have √ p1 (z | d, η) = N (z ; 2d, η)

(2.10)

(2.11)

where the subscript 1 refers to the H1 hypothesis in (2.2). The corresponding amplitude a = |z| of a Swerling 0 target in Rayleigh (or equivalently complex Gaussian) noise obeys a Rice distribution: √ ! √ a −(a2 + 2d) a 2d p1 (a | d, η) = Rice(a ; 2d, η) = exp . (2.12) I0 η 2η η Here I0 (·) is the modified Bessel function of the first kind with order zero [3 p. 374]. The amplitude statistics for a Swerling I target in Gaussian or Rayleigh noise are less complicated. Since the sum of two Gaussian random variables also is a Gaussian random variable, we obtain p1 (z | d, η) = Nc (z ; d + η)

(2.13)

p1 (a | d, η) = Rayleigh(a ; d + η).

(2.14)

and Another situation considered in this thesis is a Swerling I target embedded in Kdistributed noise. The compound formulation allows the amplitude pdf of a Swerling I target in K-distributed noise to be evaluated as p1 (a | d, ν, b) =

Z∞

Rayleigh(a ; d + η)Gamma(η; ν, b/2)dη

0

a = ν b Γ(ν)

Z∞ 0

a2 η ν−1 η exp − − dη. η+d b 2(η + d)

(2.15)

This is more carefully explained in Section 4.2.5. An important contribution of this thesis is an efficient scheme for numerical evaluation of (2.15), whose details are given in Appendix B.1. Models such as (2.15) are based on crude mathematical assumptions, and only secondarily on physical reasoning. Both the K-distribution and the Swerling I model can be justified physically, but it does not immediately follow that the target plus noise statistics obey (2.15). A more refined and physics-based treatment would include interference

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18

2


between the target signal and the clutter. However, it is questionable whether attempts at modeling such effects will lead to tractable models that can be used as part of a tracking method. It is important to keep in mind that the involved pdf’s can only be estimated with large uncertainties in most practical situations. One should therefore not replace the simple models of this section by more refined models unless the improvement due to a more refined model is visible despite these uncertainties.

2.3.4

The Neyman-Pearson test

A detection problem such as (2.2) is characterized by its detection probability PD (also know as power function) and its false alarm rate PFA (also known as size or type II error). PD is the probability of choosing H1 when H1 is correct, while PFA is the probability of choosing H1 when H0 is correct. The possible outcomes are summarized in Figure 2.6. For a fixed threshold test a ≷ t with threshold t, the detection probability and false alarm rate are given by the following integrals: PD =

Z∞

p1 (a)da

t

PFA =

Z∞

(2.16)

p0 (a)da.

t

Clearly it is desirable to maximize PD while at the same time minimizing PFA . The Neyman-Pearson theorem [60 p. 61] states that the test that maximizes PD for a given PFA is obtained if the likelihood ratio is used as a test statistic: l(a) =

Decision

p1 (a) ≷ t. p0 (a)

H0

(2.17)

H1

Truth False alarm

H0 H1

Missed detection

Detection

Figure 2.6: Possible outcomes of the hypothesis test (2.2).

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2.3

19


Cell under test Auxiliary cells

58

Latitude y [m]

56 54 52 50 48 50

55

60 Longitude x [m]

65

70

Figure 2.7: Illustration of cells used in CFAR detection.

2.3.5

CFAR detection

In practice the test (2.17) is not readily applicable, since both p0 (a) and p1 (a) tend to be unknown. Even when the functional form of these distributions can be specified, the underlying parameters will most often be highly uncertain. The background strength as represented by the first or second moment of p0 (a) is for instance non-stationary in most applications, and must therefore be estimated. In order to ensure that the detection system is not overflowed with false alarms, the conventional approach is to focus on keeping the false alarm rate PFA under control. This is the rationale behind constant false alarm rate (CFAR) detectors which determine the detection threshold according to a criterion whose primary objective is to keep PFA constant. Deciding a good detection threshold is most critical when the target is in danger of being too weak to be reliably detected. One must then either consciously accept a higher false alarm rate with the problems that may cause, or acknowledge that no reliable detection decision can be made. Lowering the threshold uncritically will certainly make sure that something is detected, but one has no guarantee that this is the desired target. The CFAR principle should be viewed as a corollary of the established scientific principle that the burden of evidence rests on the one who claims the existence of some phenomena, and not on the one who claims that there is nothing new under the sun. A CFAR detector used for measurement extraction obtains its threshold as a function of a set of M auxiliary cells, in which the background noise obeys the same distribution as in the cell under test (CUT). This is illustrated in Figure 2.7. For Rayleigh (as well as Gaussian and Exponential) background noise, the MLE ηˆ of the parameter η exists in 1 statistic for setting the threshold. closed form and can be used as a sufficient

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20

2


Since the estimator ηˆ is random, it is either larger or smaller than the true parameter η. If it is larger, the detector risks missing the target when it should have been detected. Similarly, the detector will also be more likely to suppress a false alarm in such a situation. On the other hand, a smaller ηˆ increases the risk of accepting a false alarm. Since the rationale behind CFAR detection is maintenance of a constant false alarm rate and not a constant detection probability, this latter situation is considered more adverse than the first one. Therefore a CFAR detector must in general apply a higher threshold than the optimal Neyman-Pearson detector. The resulting loss in detection probability is called CFAR loss. The correct CFAR threshold t can for the case of Rayleigh background noise be calculated as the estimator ηˆ times a factor α which accounts for the uncertainty of ηˆ. If the amplitude a is Rayleigh distributed with parameter η, then the power x = a2 is exponentially distributed with parameter 2η. The MLE of η is given by ηˆ =

M M 1 X 2 1 X aj = xj . 2M 2M j=1

(2.18)

j=1

Since the sum of independent exponentially distributed variables is Gamma distributed, this estimator follows a Gamma distribution, M −1 exp − ηˆM η ˆ η η ηˆ ∼ Gamma ηˆ ; M, . (2.19) = M η M Γ(M ) M

In the power domain we want to determine a threshold t(ˆ η ) = αˆ η so that the test x ≷ t(ˆ η ) has the desired false alarm rate PFA . The false alarm rate is related to α and the number of auxiliary cells M as follows: PFA =

Z∞ Z∞

η Exponential(x ; 2η)Gamma ηˆ ; M, dxdˆ η M

0 t(ˆ η) Z∞ Z∞

1 = α

Exponential(x ; 2η)Gamma

0 t(ˆ η)

=

=

M αη

M

1 Γ(M )

0

(M/α)M [1/2 + M/α]M

Solving for α yields

Z∞

M −1

t

t η ; M, α M

dxdt

1 1 exp −t + dt 2η αη

.

−1/M α = 2M PFA −1 .

(2.20)

(2.21)

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2.3

21

Sensor modeling and detection theory 1

15dB 0.9

0.7

15dB

9d B

0.8

dB

12

Swerling 0 Swerling 1

B 12d

0.5

9dB

6d

0.4

B

PD

0.6

B

B

0d

0.1 0 10−6

B

B

3d

0.2

3d

B

6d

10−5

10−4

10−3 PFA

0d

0.3

10−2

10−1

100

Figure 2.8: ROC curves for Swerling 0 and Swerling 1 targets embedded in Rayleigh background noise when a cell-averaging CFAR with M = 16 i.i.d. auxiliary cells is used as detector. Therefore the test a≷t with threshold t=

r

−1/M 2M PFA − 1 · ηˆ

(2.22) (2.23)

has the CFAR-property in Rayleigh distributed background noise. It is commonly referred to as the cell averaging CFAR (CA-CFAR). This detector is according to [43] optimal, or more precisely uniformly most powerful, for detecting a Swerling I target in Rayleigh distributed noise. The detection probability can for this detection problem be found as η −M −1/M PD = 1 + PFA −1 . (2.24) d+η

The mean target power d is related to the SNR, which is measured on a logarithmic scale. For this particular detection problem it is given by SNR d SNR = 10 log101 ⇔ d = η10 10 . η

(2.25)

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22

2


Without imposing further constraints there exists no well-defined optimal (PFA , PD )pair. For any detector, PD will be a monotonously increasing function of PFA which takes all values in [0, 1] for some value of PFA . It is always zero for PFA = 0 and unity for PFA = 1. This function is known as the receiving operating characteristic (ROC). Although the ROC cannot by itself tell us what the false alarm rate should be, it is obvious that the tracking performance must depend on the choice of PFA . Figure 2.8 shows the ROC curves for a Swerling I target in Rayleigh background noise when a CA-CFAR detector with M = 16 auxiliary cells are used. The ROC curves for a Swerling 0 target (non fluctuating target) are also shown for comparison. If we assume that a detection probability above 0.5 is necessary for reliable target tracking, we can see that it is going to be very difficult to track targets whose SNR are below 6dB unless very high false alarm rates (> 10−2 ) are accepted. However, such an abundance of false alarms increases the risk of the track being pulled away from the true target. This kind of trade-offs are further discussed in Chapter 5. This section has only treated CFAR detection in Rayleigh distributed background noise. Although numerous papers have been published on CFAR detection with other background models [e.g. 4, 64, 108 ch. 9], this thesis makes no attempt at specifying optimal detectors when the K-distribution is used instead of the Rayleigh distribution. This is partly because we have found that straightforward threshold setting based on moment-based parameter estimates in most cases is able to maintain an acceptable false alarm rate (cf. Table 4.9). Also, the treatment of K-distributed clutter in Chapter 4 relies on knowledge of the entire distribution so that its parameters necessarily must be estimated. It seems most appropriate to ensure that the detection threshold harmonizes with these parameter estimates.

2.4

Measurement modeling

Measurements extracted by means of the detector (2.22) or any other thresholding process constitute a random set Zk = {ζk (1), . . . , ζk (mk )} which is used as input to the tracking method. A conventional (i.e. non-TBD) tracking method has only access to information conveyed by this set. It is therefore important to model all properties of this set adequately. There are three questions that should be answered by the model: Q1: What are the sources from which elements in Zk originate? Q2: What is the distribution of the number of elements in Zk ? Q3: What are the pdf’s of elements in Zk ? Answers to Q2 are presented in Sections 2.4.1 and 2.4.2, while answers to Q3 are presented in Sections 2.4.3 and 2.4.4.

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2.4

23


The standard answer to Q1 is that measurements originate from the target or from the background1 . We can therefore write Zk = Zk1 ∪ Zk0

(2.26)

where Zk1 contains measurements from the target and Zk0 contains measurements from the background. Clearly, the stochastic properties of Zk1 and Zk0 must be of very different natures. If several targets are present it is necessary to replace the set Zk1 with several sets, one for each target. Such considerations will not be pursued further here, since this thesis only concerns single-target tracking.

2.4.1

Cardinality of background measurements

There are three candidate distributions that are commonly considered for the clutter measurement set cardinality card(Zk0 ). For a sensor image consisting of N cells, of which any cell j is independently extracted with probability PFA , the number of detections will obey a binomial point mass function (pmf) with parameters PFA and N [6 p. 102]: N φ 0 PFA (1 − PFA )N −φ . (2.27) µ(φ) = P {card(Zk ) = φ} = φ In practice the binomial model (2.27) is never used. Since N in general is large, a Poisson approximation can be justified [85 p. 113]. The Poisson model is more amenable for mathematical convenience as it provides the simplest possible realistic model for the clutter cardinality. According to the Poisson model, clutter measurements are drawn from a spatial Poisson process with intensity λ: µ(φ) = Poisson(φ ; λV ) = e−λV

(λV )φ φ!

(2.28)

where V is the volume of the surveillance region measured in the units of the measurements. In reality it is seldom possible to assure that the true false alarm rate is equal to the design false alarm rate everywhere, and so the clutter intensity λ will vary both spatially and temporally, yielding a non-stationary Poisson process. One is often primarily interested in the clutter intensity near the target of interest, and attempts at assuming a single intensity for the entire sensor image will at best lead to confusion. This is an obvious reason why the binomial model (2.27) is of limited utility, but it also raises the difficult challenge of clutter estimation. The parameter λ is not readily available, and must in practice be estimated. Since limited information is available for this task, huge uncertainties must be accepted. Two very different approaches to clutter estimation can be found in [69] and [78]. 1

In Chapter 6 we introduce sources (e.g. wakes) that cannot very well be categorized as target or background, but rather should be considered a third kind of source.

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24

2


Due to the difficulties of clutter estimation, a diffuse (a.k.a. flat, non-informative or non-parametric) clutter cardinality model is often used. The diffuse clutter model assumes that all values of the clutter cardinality are equally likely: µ(0) . . . = µ(φ − 1) = µ(φ) = µ(φ + 1) = . . . = .

(2.29)

This is not a proper probability distribution, since it does not sum to unity. However, one can always replace the flat cardinality distribution with a distribution which is constant for all cardinalities smaller than some constant, and zero for all larger cardinalities, and thus obtain a proper probability distribution.

2.4.2

Cardinality of target measurements

The target measurement set cardinality is typically assumed to be Bernoulli distributed. That is, the set Zk1 contains at most one element with probability PD , where PD typically is given by (2.24) or as a tuning constant:  if τ = 1  PD 1 − PD if τ = 0 (2.30) P {card(Zk1 ) = τ } = Bernoulli(τ ; PD ) =  0 otherwise.

This assumption, that at most one detection can originate from the target, is primarily based on mathematical convenience and not on physical modeling of reality. It is obviously often the case that a single target radiates into several resolution cells, thereby rendering (2.30) invalid unless clustering [26, 82, 87, 97] is performed to extract a single centroid. On the other hand, the assumption (2.30) is pivotal in the derivation of almost all tracking methods. Especially for tracking methods that attempt to estimate the number of targets, (2.30) plays a key role, since the track management problem would be ill-posed if both card(Zk1 ) and the number of targets were allowed to take arbitrary values. This being said, limited violations of (2.30) can most often be accepted, and this assumption is therefore maintained in this thesis.

2.4.3

Pdf’s of background measurements

Individual measurement vectors ζ can contain two kinds of information: kinematic information and feature information. The only features considered in this thesis are amplitudes, and we can therefore partition measurement vector number i at time k as ζk (i) = [zk (i)T , ak (i)]T

(2.31)

where zk (i) is the kinematic component and ak (i) is the amplitude component. Assuming independence between these components, we can write the likelihood of an arbitrary background measurement as p0 (ζ|x) = pz0 (z|x)pa0 (a|x) = pz0 (z)pa0 (a).

(2.32)

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2.4

25


Several candidates for the background amplitude pdf were discussed in Section 2.3.1. When any of these are used to model the amplitude of a detected measurement, we need to condition on the event that the measurement was detected. The fact that a measurement is detected indicates that its amplitude must lie above the detection threshold t. Consequently, only the upper tail with probability PFA models the amplitude of a detected background measurement. For measurements obtained from a fixed threshold test we can compensate for this by dividing by the false alarm rate PFA , so that 0 if a < t a p0 (a) = (2.33) 1 p (a) if a ≥ t 0 PFA where p0 (a) is the background amplitude pdf in the cell from which ζ is extracted. In reality, the measurements are extracted from a test such as (2.23), whose threshold t is random (cf. Figure 2.9). The sharp truncation of (2.33) should then be replaced by a soft truncation given by marginalization over t as illustrated in Figure 2.10. For simplicity we ignore this issue and simply divide by PFA (and correspondingly PD for p1 (a)) in such cases as well. Notice that the notation pa0 (a) refers to a truncated pdf, while the notation p0 (a) refers to the untruncated pdf. The kinematic or spatial pdf of clutter measurements is conventionally assumed to be uniform over the entire surveillance region, pz0 (z|x) = pz0 (z) =

1 . V

(2.34)

Here V is the volume of the surveillance region, or (more often) the volume of the validation gate [6 p. 94] to which attention is restricted. In Chapter 6, the assumption (2.34) is relaxed and state dependent clutter is discussed.

2.4.4

Pdf’s of target measurements

In the same way as for background measurements we partition measurements from the target into a kinematic part and an amplitude part p1 (ζ|x) = pz1 (z|x)pa1 (a|x) = pz1 (z|x)pa1 (a).

(2.35)

The amplitude part pa1 (a) is evaluated as explained in Section 2.3.2. Again, we need to condition on the event that the measurement was actually detected, i.e. that it was in the upper PD -tail of p1 (a). Therefore, 0 if a < t a p1 (a) = (2.36) 1 if a ≥ t PD p1 (a) where p1 (a) is the target plus clutter amplitude pdf in the cell from which ζ is extracted. Since pa1 (a) concerns the target, one could expect it to depend on the state. This is, however, only necessary if the SNR is included in the state vector. Even though this

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26

2


1

p0 (a) p1 (a) p(t)

Value of pdf

0.8 0.6 0.4 0.2 0

0

1

2

3 4 6 5 7 8 Amplitude a or corresponding threshold t

9

10

Figure 2.9: Untruncated amplitude pdf’s p0 (a) and p1 (a) together with the pdf p(t) of the CA-CFAR threshold t for a Swerling I target in Rayleigh noise.

1

p(t) Sharp trunc. p1 (a) Cont. sharp trunc. p1 (a) Soft trunc. p1 (a)

Value of pdf

0.8 0.6 0.4 0.2 0

0

1

2

3 4 6 5 7 8 Amplitude a or corresponding threshold t

9

10

Figure 2.10: Different ways of truncating p1 (a) in order to compensate for the fact that the measurement is detected. A rigorous treatment of CFAR uncertainty would lead to the dot-dashed curve labeled “soft truncation”. Since this curve is given by complicated integral, the simpler sharp truncation (solid curve) is used in practice.

1

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2.4

27


is sometimes desirable (cf. Chapter 3), care should be exercised regarding the degree to which target amplitude fluctuations are modeled by the measurement model or by the transition prior. There is no need to include the SNR in the state vector unless there are strong arguments why variation in the amplitude should be modeled by the transition prior. In reality, especially for fluctuating targets, it is not possible to get even a rather crude estimate of the SNR unless several samples are used [31]. It can therefore be argued that it is overkill to attempt to model changes in the SNR together with the kinematics. Instead, the SNR parameter d should be treated as constant. This is discussed in further detail in Section 4.3.5. The kinematic or spatial pdf pz1 (z|x) is commonly referred to as the measurement model in the filtering literature, although the treatment in this section reveals that it is only a part of the actual measurement model for the tracking problem. Measurements modeled by pz1 (z|x) include position measurements and Doppler measurements. Only 2-dimensional position measurements are considered in this thesis. These are typically obtained in polar coordinates. Assuming additive measurement noise wk , such a measurement model can be written zk = ρk + wk = h(xk ) + wk . The point ρk is decomposed in as (ρxk , ρyk ) Cartesian coordinates, and as (ρrk , ρϑk ) in polar coordinates. Spelling out the details of the Cartesian to polar mapping then yields zkp

= h(xk ) + wk =

ρrk ρϑk

" q # (ρxk )2 + (ρyk )2 + wk = + wk . atan ρyk /ρxk

(2.37)

The measurement noise wk depends on several quantities such as the sensor resolution, the SNR and the beamwidth. For extended targets it is clearly necessary to use some kind of clustering, typically a variation of single linkage [56 p. 681], to ensure validity of the target cardinality assumption (2.30). Approximations to the error statistics of the resulting centroids have been discussed in [82] and [97] for such targets. As can be seen from Figures 2.4 or 2.5, similar issues are also relevant for point targets when the beamwidth exceeds the size of a single resolution cell. It is the author’s opinion that evaluation the measurement noise for small targets still is a somewhat unresolved topic. Since a target detection contains both signal from the background and from the target, it is not obvious which detections are to be considered from the target and which are to be considered from the background. Therefore it is not entirely clear how the measurement noise should be defined in the first place. Furthermore, single linkage can potentially extract very long clusters, which will blow up the second moment of the error pdf, possibly to the level of undefined variance. It is common to ignore the cumbersome issues regarding clustering, and proceed without clustering assuming that (2.30) holds anyway. This was done in [89], and it is also done in this thesis. Further research is required to investigate when this is safe. Assuming that (2.30) actually holds without clustering, it follows that wk has a uniform distribution inside the chosen resolution cell. This uniform distribution is inconvenient to work with, so it is approximated by a Gaussian. By matching the first and

classtest: 2010-06-08 17:36 — 28(46)

28 second moments we obtain r 2 ρk σr 0 p p p(zk |xk ) = N zk ; , =N 0 σϑ2 ρϑk

2


zkp ;

ρrk ρϑk

" ,

∆r2 12

0

0 ∆ϑ2 12

#!

.

The non-linearity of (2.37) threatens to violate the assumptions underlying linear estimation methods. However, for high resolution this challenge is illusory. Linearization can safely be used for (2.37) as long as the resolution is high enough. In order to retain full linearity an alternative is to convert the polar measurements z p = [r, ϑ]T into Cartesian measurements z = [˜ x, y˜]T which are related linearly to xk . According to the recipe of [6 pp. 38 - 55] this is done using the formulas # " 2 2 x ˜ r cos ϑ r cos ϑ(e−σϑ − e−σϑ /2 ) = − . 2 2 y˜ r sin ϑ r sin ϑ(e−σϑ − e−σϑ /2 ) The covariances are converted as well 2 c R11 = r2 e−2σϑ cos2 ϑ(cosh 2σϑ2 − cosh σϑ2 ) + sin2 ϑ(sinh 2σϑ2 − sinh σϑ2 ) 2 +σr2 e−2σϑ cos2 ϑ(cosh 2σϑ2 − cosh σϑ2 ) + sin2 ϑ(sinh 2σϑ2 − sinh σϑ2 ) 2 c R22 = r2 e−2σϑ sin2 ϑ(cosh 2σϑ2 − cosh σϑ2 ) + sin2 ϑ(sinh 2σϑ2 − sinh σϑ2 ) 2 +σr2 e−2σϑ sin2 ϑ(cosh 2σϑ2 − cosh σϑ2 ) + sin2 ϑ(sinh 2σϑ2 − sinh σϑ2 ) h i 2 2 c R12 = sin ϑ cos ϑe−4σϑ σr2 + (r2 + σr2 )(1 − eσϑ )

assuming that the original and converted measurement covariance matrices are c 2 c R11 R12 σr 0 c p . and R = R = c c 0 σϑ2 R12 R22

This provides us with a Gaussian-linear model for target originating measurements. Notice that the measurement noise accordingly varies between the measurements. In practice one may find it more convenient to convert the covariances using the predicted target position, and assigning this covariance to all the measurements in the vicinity (e.g. validation gate) of the target. Such an approach causes dependency between the state and measurement noise, violating the basic assumptions for the Bayesian filtering equations (cf. Section 2.5). This has been discussed in [98], but will not be pursued further here. Further discussion of measurement conversion can also be found in [36], [71] and [99]. It is illustrated in Figure 2.11 that the polar measurement model is essentially linear for reasonably high sensor resolution. There exist several other measurement models which induce more serious non-linearities. Removing the range measurements does for instance lead to bearing-only tracking which hardly can be addressed adequately with Gaussian-linear techniques [92 pp. 103-151]. Such sensor models will not be treated in this thesis.

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2.5

29

Linear and non-linear filtering

Clutter measurements Target measurement Prediction Track covariance Validation gate

58

Latitude y [m]

56 54 52 50 48 50

55

60 Longitude x [m]

65

70

Figure 2.11: Illustration of the sensor resolution for the scenario simulated in Chapter 4 (based on the experiments reported in [87]). The very low curvature shows that the measurements are practically linearly related to the target. The small ellipses represent the covariance of the measurement noise in Cartesian coordinates.

2.5


Target tracking is commonly viewed as a filtering problem, where the state of a stochastic dynamic system is to be estimated from a series of noisy measurements. This state-space approach focuses attention on the state vector of the system. All information describing the system is contained in the state vector. For example, the state vector may contain the position and velocity of the target. The filtering problem is phrased in terms of two models, which in the Bayesian approach are assumed to be available in a probabilistic form. First, a model is required to describe the evolution of the state with time. This model is termed process model, plant model or transition prior in various literature. In this thesis the term plant model is used when we want to emphasize that we are talking about the evolution of the kinematic state. Otherwise the term transition prior is used in order to emphasize its a priori nature. Although a continuous-time model often provides the most accurate modeling of reality, discrete-time approximations are most frequently discussed in the tracking and filtering literature. Second, a model is required to relate the noisy measurements to the state. This model is called the measurement model. Another important distinction is the one between simulation and filter models. No model can be made refined enough to describe the real world with absolute accuracy, and the model assumed by the filter or tracking method is often grossly simplified. Nevertheless, it is often desirable to test a filter under more realistic circumstances, and test data may then be generated according to a1 more refined model, which we refer to as the

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30

2


simulation model. This is discussed in more depth in Section 2.7. The generic filtering problem can be phrased as in several ways. In [92], which should be considered a standard reference on non-linear filtering, it is phrased as xk = fk (xk−1 , vk ) , zk = hk (xk , wk )

(2.38)

where vk and wk are noise sequences. It is commonly assumed that x0 and all vk and wk are mutually independent. The filtering problem can then be written in terms of a Markovian pdf p(xk |xk−1 ) representing the transition prior and a measurement pdf p(zk |xk ) in which the measurement only depends on the current state xk . Violations of the independence assumptions can often be mended by including some extra states in the state vector xk . Solutions to the filtering problem (2.38) can possess different kinds of optimality. It is for instance common to talk about optimality in the mean square sense, which entails that the estimator has the smallest mean square error possible. A stronger kind of optimality is Bayes optimality, which is possessed by the posterior distribution p(xk |Z k ). This solution is optimal in the sense that it embodies all available statistical information, and consequently underpins all other kinds of optimality. It is given on a functional form according to the Bayes equations: Z p(xk |Z k−1 ) = p(xk |xk−1 )p(xk−1 |Z k−1 )dxk−1 (2.39) p(xk |Z k ) ∝ p(zk |xk )p(xk |Z k−1 ).

(2.40)

We call (2.39) the prediction while (2.40) is called the measurement update. If either the kinematic prior or the measurement model is nonlinear, the posterior pdf will be non-Gaussian. Also, if the initial state or any of the noise sequences vk or wk are nonGaussian, the posterior pdf will be non-Gaussian as well.

2.5.1

The Kalman filter

The optimal Bayesian solution to (2.38) is given by the Kalman filter (KF) under the following assumptions: K1: The state evolves according to a linear plant model xk = F xk−1 + vk where the plant noise sequence vk is white and Gaussian with mean zero and known covariance Q. K2: Measurements are obtained according to a linear measurement model zk = Hxk + wk where the measurement noise sequence wk is white and Gaussian with mean zero and known covariance R. K3: The initial state is Gaussian with known mean x0 and covariance P0 . K4: The initial state and the noise sequences are independent for all k.

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2.5

31


Under these assumptions the posterior pdf is also Gaussian, and therefore given uniquely ˆ k|k and covariance Pk|k as calculated by the KF: by its expectation x (2.41)

ˆ k|k−1 = F x ˆ k−1|k−1 x Pk|k−1 = F Pk−1|k−1 F + Q

(2.42)

ˆ k|k = x ˆ k|k−1 + Wk νk x

(2.43)

Pk|k = Pk|k−1 − Wk Sk WkT

(2.44)

T

where ˆ k|k−1 = zk −zˆk|k−1 . Wk = Pk|k−1 H T Sk−1 , Sk = HPk|k−1 H T +R , νk = zk −H x Equations (2.41) and (2.42) constitute the time update or prediction, while (2.43) and (2.44) constitute the measurement update of the KF. The matrix Wk is known as the Kalman gain and the vector νk is called the innovation. ˆ k|k is the minUnder the given assumptions it can be shown that the expectation x imum mean square error (MMSE) estimator. The KF is the standard estimator used in tracking systems, and extensive treatments of it can be found in several books including [7] or [44]. The KF is typically derived using least-squares arguments. An alternative derivation, which lucidly emphasizes the Bayesian optimality of the KF, can be found in [72 pp. 33-40]. Violations of Gaussianity and linearity may or may not pose a serious threat to the performance of the KF. Even without Gaussianity, the KF is still the best linear MMSE estimator, that is the best estimator within the class of linear estimators [7 p. 207]. Nonlinearities such as those due to a sensor providing measurements in polar or spheroidal coordinates can easily be treated without resorting to non-linear estimators, cf. Section 2.4.4. Some recent research has attempted to quantify the degree of non-linearity in various tracking problems, and concluded that it often is insignificant [74]. The simplest nonlinear state estimation method is the extended Kalman filter (EKF), which simply linearizes the plant and measurement models such that the KF update equations can be used [7 pp. 381-387]. That is, we obtain F and H as Jacobians of the nonlinear plant and measurement mappings f (·) and h(·).

2.5.2

Numerical approximation of the posterior

There are several tracking problems for which the KF or the EKF cannot be expected to perform well enough. When strong nonlinearities are present it may be necessary to calculate the full Bayesian solution of (2.39) and (2.40). This solution can in general only be approximated by evaluating the posterior pdf p(xk |Z k ) for a finite number of samples. The most straightforward method is to evaluate the posterior pdf over a grid in the state space. This is known as a point mass filter (PMF). This can be very expensive and is practically impossible in higher dimensions due to the curse of dimensionality.

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32

2 (1)

(2)


(N )

Assuming that the samples xk , xk , . . . , xk are regularly spaced over the support of p(xk |Zk ), the PMF equations are simply a discretisation of (2.39) and (2.40): (j)

p(xk |Z k−1 ) = (j)

N X i=1

(j)

(i)

(i)

p(xk |xk−1 )p(xk−1 |Z k−1 ) (j)

(j)

p(xk |Z k ) ∝p(zk |xk )p(xk |Z k ).

(2.45) (2.46)

A computationally simpler approach is found in the unscented Kalman filter, also known as the sigma-point Kalman filter [57]. It generates strategic samples based on the first and second order moments of the assumed probability distributions and propagates these samples through the nonlinear mappings to obtain state estimates. Currently, the most popular approach to non-linear filtering is to use so-called particle filters or sequential Monte-Carlo (SMC) methods. The fundamental idea is to evaluate the integral of (2.39) via importance sampling: p(xk |Z

k−1

)= ≈

Z

p(xk |xk−1 )

N X i=1

p(xk−1 |Z k−1 ) q(xk )dxk−1 q(xk )

(i)

(i)

(2.47)

p(xk |xk−1 )wk−1 . (i)

Here q(xk ) is a proposal density from which the samples xk are drawn. In [92] it is shown that if q(xk ) = q(xk |xk−1 , zk ) then the weights can be updated according to (i)

(i)

(i)

wk = wk−1

(i)

(i)

p(zk |xk )p(xk |xk−1 ) (i)

(i)

q(xk |xk−1 , zk )

.

(2.48)

The posterior pdf is then approximated as p(xk |Z k ) ≈

N X i=1

(i)

(i)

wk δ(xk − xk )

(2.49)

where δ( · ) is the Dirac delta function. In order to prevent degeneracy the samples must be resampled. This is done by drawing each sample with a probability proportional to its weight. Although a particle filter cannot be expected to beat the curse of dimensionality for an arbitrary proposal density [34], the required number of samples can be kept reasonably low if q(xk ) approximates the posterior pdf p(xk |Z k ) well. However, it is often difficult to come up with a better candidate for q(xk ) than the predicted pdf p(xk |Z k−1 ). Particle filtering is very well treated in [92]. In this thesis particle filtering is used to implement TBD ( cf. Section 2.6.3 and Chapter 3).

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2.6

Tracking methods

2.6

33

Tracking methods

The tracking problem is more complicated than the filtering problem due to non-unity detection probability and the presence of clutter. A tracking algorithm must not only process a single measurement zk at each time k. Instead it must process a set Zk = {zk (1), . . . , zk (mk )} whose cardinality is zero or a natural number. Target tracking is therefore essentially a problem of data association, and the term tracking method is used as equivalent to the term data association algorithm. This does not mean that state estimation is unimportant, since reliable state estimation is a prerequisite for successful data association. However, this thesis is primarily concerned with scenarios where data association is the main challenge due to low SNR and abundant clutter. The main purpose of this section is to put the tracking methods used in Chapters 3 - 6 on a firm ground. These are the TBD method of [92, 95] and various versions of the PDAF [6]. These methods represent widely different conceptual viewpoints, and are therefore treated separately. The PDAF belongs in the conventional framework of singletarget tracking, while TBD skips the entire measurement modeling of Section 2.4. First of all, the optimal Bayesian solution to conventional single-target tracking is presented in Section 2.6.1. The PDAFAI (of which the PDAF is a special case) is then derived in Section 2.6.2 by applying certain suboptimal assumptions to the optimal solution. In Section 2.6.3 TBD is treated. Finally, a very cursorial survey on other tracking methods is given in Section 2.6.4.

2.6.1

Thresholded measurements: Optimal approach

In this section the optimal Bayesian approach (OBA) to target tracking is developed under the standard assumptions G1-G4 listed below. This treatment is similar to the one in [89], but differs in two important ways. First, the treatment presented here does not invoke any Gaussian-linear assumptions before very late. Second, amplitudes are included here. G1: There is one and only one target present whose track is already initialized. The target moves according to a Markovian plant model p(xk |xk−1 ). G2: There is a single measurement from the target with probability PD at time k as in (2.30). This measurement ζk has, when it exists, a spatial component zk with pdf p(zk |xk ) and an amplitude component ak with pdf pa1 (ak ) as in (2.36). G3: There are φk i.i.d. measurements from the background with probability µ(φk ). Each of these measurements ζk (j) has a spatial component zk (j) with pdf equal to 1/Vk as in (2.34), and an amplitude component ak (j) with pdf pa0 (ak (j)) as in (2.33). G4: Both the measurement processes are independent across time, and also independent of each other. Furthermore, the spatial component of a measurement is independent of its amplitude.

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34

2


Assumptions G1-G4 are invoked explicitly or implicitly in most of the literature on single-target tracking. Since this chapter is meant as an introduction to the established framework for target tracking, these assumptions are adhered to in the derivation of the OBA and the PDAFAI. Nevertheless, these assumptions cannot always be accepted despite their generality. In Chapter 6 we discuss what happens when the measurement set cannot be decomposed into target and background generated subsets. The OBA is implemented by consideration of all possible association hypotheses. The possible association hypotheses at time k constitute the posterior pdf at time k together with their corresponding probabilities. The association hypothesis with the highest probability can be considered a maximum a posteriori (MAP) estimate. In practice this means that every time a new set of measurements is received, each existing data association hypotheses is split into one children hypothesis for each of these measurements. Consequently the number of association hypotheses, or histories, grows exponentially. Therefore the OBA will quickly become unfeasible even for the strongest computers, and can not be implemented for anything but toy problems unless suboptimal simplifications such as hypothesis merging are used. Let Z k = (Z0 , Z1 , . . . , Zk ) be the list of measurement sets up to and including time k. From Z k , let Z k,l be the lth sequence of measurements out of all possible combinations of measurement sequences up to time k. The sequence Z k,l , also called measurement history number l, is composed of the current measurement ζk (i) at time k and the history Z k−1,s , which is the prior part of the history Z k,l up to time k: Z k,l = Z k−1,s , ζk (i) .

(2.50)

The total number of such histories at time k is Lk =

k Y

(1 + mj )

(2.51)

j=1

where mj is the number of measurements at time j. The event that the lth history at time k is the correct sequence of measurements is denoted as θk,l . Its probability, conditioned on all the measurements up to time k, is denoted as β k,l = P {θk,l |Z k }. (2.52) The partitioning of a prior association hypothesis θk−1,s into its children θk,l is generated by the following mutually exclusive and exhaustive events (cf. assumption G2): θk (0) θk (1) .. .

No measurement originates from the target Measurement 1 originates from the target

θk (mk ) Measurement mk originates from the target.

(2.53)

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2.6

35

Tracking methods

The OBA can be decomposed into two tasks. First, the history-conditioned posterior pdf p(xk |Z k , θk,l ) must be evaluated for all histories Z k,l at time k. Second, their respective probabilities β k,l must be evaluated. By the total probability theorem we have k

p(xk |Z ) =

Lk X l=1

p(xk |Z k , θk,l )P {θk,l |Z k }

(2.54)

where the pdf p(xk |Z k , θk,l ) is to be obtained from filtering and P {θk,l |Z k } is an association probability. Both these quantities must be evaluated by means of the joint measurement pdf, which can be written as  1 Qmk a for i = 0   V mk j=1 p0 (ak (j))  p(Zk |xk , mk , θk (i)) = (2.55) pz (z (i)|x )pa (a (i))    1 1k Qmkk 1a k · V mk −1 j6=i p0 (ak (j)) for i = 1, . . . , mk

when conditioned on the state xk as well as the current association hypothesis θk (i). This result follows immediately from Assumptions G2-G4. The independence invoked by assumption G4 allows the spatial and amplitude related terms of target and background measurements to be simply multiplied together. Furthermore, according to assumptions G2 and G3, only the kinematic pdf of target measurements pz1 (zk (i)|xk ) does depend on the state. The filtering part of the OBA follows from Bayes’ rule according to p(xk |Z k , θk,l ) = p(xk |Zk , mk , Z k−1 , θk (i), θk−1,s ) 1 = p(Zk |xk , mk , θk (i), θk−1,s , Z k−1 )p(xk |mk , θk (i), θk−1,s , Z k−1 ) c 1 = p(Zk |xk , mk , θk (i))p(xk |θk−1,s , Z k−1 ). (2.56) c

Most of the terms in the joint measurement pdf p(Zk |xk , mk , θk (i)) can by means of (2.55) be included in the normalization constant since they do not depend on xk . Consequently, the posterior pdf of xk conditioned on measurement history number l is updated according to p(xk |θk−1,s , Z k−1 ) for i = 0 k k,l p(xk |Z , θ ) ∝ (2.57) pz1 (zk (i)|xk )p(xk |θk−1,s , Z k−1 ) for i = 1, . . . , mk . Notice that (2.57) is a filtering equation similar to (2.40), and p(xk |θk−1,s , Z k−1 ) is a predicted density which can be evaluated by means of (2.39). Also notice that since the amplitudes do not depend on the state, the filtering of (2.57) does only depend on the kinematic measurement component zk (i). The association part of the OBA concerns the posterior association weights, which are given by the joint measurement pdf and the prior association weights. The latter ones

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36

2


are prior in the sense that they are based on no further information about the current measurement set Zk than its cardinality mk . As shown in Appendix A.1, the prior association probability of measurement number i is  µ(mk ) (1−PD ) µ(m −1)   k  i=0  µ(mk ) PD +(1−PD ) µ(m −1) k P {θk (i)|mk } = (2.58) 1 P  mk D   i = 1, . . . , m . k µ(mk )  PD +(1−PD ) µ(m

k −1)

In some cases the prior association probabilities P {θk (i)|mk } are important in themselves, and (2.58) is then useful (cf. Chapter 5). However, most often the denominators in (2.58) are of no concern, since they are the same for all i = 0, . . . , mk . The posterior association probability of measurement history number l is under assumptions G1-G4 given by β k,l ∝ mk (1 − PD )µ(mk )β k−1,s β

k,l

lka (i)lkzs (i)PD µ(mk

∝

− 1)β

k−1,s

for i = 0

(2.59)

for i = 1, . . . , mk

(2.60)

where the following short-hand notations have been used: lka (i) =pa1 (ak (i)) / pa0 (ak (i)) Z zs lk (i) =Vk pz1 (zk (i) | xk )p(xk | θk−1,s , mk , Z k−1 )dxk .

(2.61) (2.62)

The proof of (2.59) and (2.60) can be found in Appendix A.2. As discussed in Section 2.4.1, there are two common candidates for the clutter cardinality pmf µ(mk ). For the Poisson clutter cardinality pmf of (2.28) the posterior association probabilities become β k,l ∝ (1 − PD )λV β k−1,s

β k,l ∝ lka (i)lkzs (i)PD β k−1,s

for i = 0

(2.63)

for i = 1, . . . , mk .

(2.64)

For the diffuse clutter cardinality pmf of (2.29) they become β k,l ∝ mk (1 − PD )β k−1,s

β

k,l

∝

lka (i)lkzs (i)PD β k−1,s

for i = 0

(2.65)

for i = 1, . . . , mk .

(2.66)

By comparing (2.63) and (2.65) one can see that using the diffuse clutter pmf produces a tracker which is equivalent to a tracker that uses mk /V as an estimator for the Poisson clutter intensity λ: ˆ = mk . λ (2.67) V The term non-parametric is in this thesis used for trackers which assume the diffuse clutter cardinality pmf. All tracking methods which assume any kind of informative clutter

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2.6

37

Tracking methods

cardinality pmf are labeled parametric. Although this terminology is in agreement with the standard reference [6 p. 103], it is not universally adhered to. According to some, a non-parametric tracker is a tracker which does not know the clutter intensity a priori, and therefore must estimate it. It is the author’s opinion that such a terminology is unfortunate, since it may obscure the difference between parametric and non-parametric as defined in [6]. The important difference is not whether we know the clutter intensity or not. In reality we never know it, and it must therefore always be estimated or guessed. Rather, the important difference is whether we attempt to obtain and use such an estimate or not. It is in this context important to understand that the non-parametric association formulae (2.65) and (2.66) do not employ such an estimate. Quite often one is concerned with Gaussian-linear scenarios where the Kalman filter assumptions K1 - K4 from Section 2.5.1 also apply. This allows the treatment of kinematic information to be written explicitly both in the OBA and the PDAFAI. By applying the Kalman filter to measurement history number l in (2.57), one obtains the Gaussian-linear filtering equations k

k,l

p(xk |Z , θ ) =

(

s ˆ sk|k−1 , Pk|k−1 N (xk ; x ) l l ˆ k|k , Pk|k ) N (xk ; x

for i = 0 for i = 1, . . . , mk .

(2.68)

where the updated quantities are ˆ lk|k =x ˆ sk|k−1 + Wk νkl x l s Pk|k =Pk|k−1 − Wk Sks Wk s Wk =Pk|k−1 H T (Sks )−1

s H T + Rk Sks =HPk|k−1

ˆ sk|k−1 νkl =zk (i) − H x

(2.69)

and the predicted quantities are ˆ sk|k−1 =F x ˆ sk−1|k−1 x s s Pˆk|k−1 =F Pˆk−1|k−1 F T + Q.

(2.70)

The posterior probability that history number l is the correct one is in the Gaussianlinear case given by mk (1 − PD )µ(mk )β k−1,s Vk ˆ sk|k−1 , Sks )PD µ(mk − 1)β k−1,s ∝ lka (i)N (zk (i) ; H x

β k,l ∝ β k,l

i=0

(2.71)

i = 1, . . . , mk

(2.72)

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38

2


where the innovation covariance matrix Sks is given by (2.69). The result (2.72) follows from Z 1 zs lk (i) = pz1 (zk (i)|xk )p(xk |θk−1,s , Z k−1 )dxk p0 (zk (i)) Z s ˆ sk|k−1 , Pk|k−1 =Vk N (zk (i) ; Hxk , R)N (xk ; x )dxk s ˆ sk|k−1 , HPk|k−1 H T + R) =Vk N (zk (i) ; H x

(2.73)

where the third equality is a well-known identity for Gaussian distributions whose proof can be found in [72 pp. 699-703]. It follows that the output of the OBA under the Gaussian-linear assumptions K1-K4 is a Gaussian mixture. Its components are given by (2.68) while its weights are given by (2.71) and (2.72). In Figure 2.12 this mixture is illustrated for the OBA both with and without AI. At k = 1 it can be seen how both trackers place their main lobes around the true measurement. The usage of AI allows the tracker to increase the weight of this lobe while relying less on the predicted lobe. At k = 2 there are no target-originating measurements, and this causes some confusion. The ordinary OBA attempts to hedge on clutter measurements. The AI-OBA does on the other hand exercise a high degree of scepticism towards these measurements due to low amplitudes, and places most of its probability weight on the prediction of the previous main lobe. When another true measurement arrives at k = 3, the AI-OBA is therefore able to confidently discard all other histories but the true one, while the ordinary OBA still believes that the target has moved much further to the right than it actually has. This simple example conveys the message that AI in general increases tracking performance, but some caution should be exercised. Although the ordinary OBA tends to go for the wrong measurements in Figure 2.12, it also hedges on the true measurement with a low, but not negligible, association probability at time k = 3. The high confidence offered by AI can on the other hand make it difficult, even for a brute force enumerative OBA, to revoke previously wrong decisions. The Gaussian mixture of the OBA, whose number of components increases exponentially with time, is difficult to interpret. The conditional mean accompanied by its covariance can be used as more easily interpretable output: ˆ k|k =E[x|Z k ] = x

mk X

ˆ lk|k β k,l x

(2.74)

l=1

Pk|k =

mk X l=1

β

k,l

l Pk|k

+

mk X l=1

T ˆ lk|k − x ˆ lk|k − x ˆ k|k x ˆ k|k . β k,l x

(2.75)

An obvious simplification that should be done in order to make the OBA computationally feasible is validation gating. There is no need to check measurements that are

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39

Tracking methods

Prior pdf at k = 0

2.6

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

2

2.2

2.4

2.6

2.8

3.2 3 Position x

3.4

3.6

3.8

1

Posterior pdf at k = 3



-1

Figure 2.12: Example of OBA with and without AI. The Gaussian mixtures are 1 plotted using red and stapled curves, respectively. Red dots are clutter measurements while the true target measurement is depicted by a black dot when it exists. The true target state is depicted by blue dots.

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40

2


very far away from where we expect the target to be. Therefore, attention is most often restricted to measurements that fall inside a limited validation gate G. It is possible to ˆ sk|k−1 . A simpler apdefine one validation gate for each history-conditioned prediction x proach advocated in [6] is to use a gate based on (2.74) and (2.75) which is common for all histories, T −1 ˆ k|k−1 Sk|k−1 ˆ k|k−1 < g 2 G : zk − H x zk − H x (2.76)

where Sk|k−1 = HPk|k−1 H T + R = H(F Pk|k F T + Q)H T + R and g is a gate threshold which measures the size of the gate in unitless standard deviations. Typical values for g are somewhere between 3 and 10. The probability that the true target originating measurement is inside G is denoted PG . Calculations of PG as a function of g are found in [6 pp. 96], where it can be seen that for g ≥ 4 this probability is as good as unity. In practice, gating is not just a suboptimal modification, but often a necessity. As pointed out in [66], gating is necessary for certain tracking methods (e.g. the strongest measurement filter) to work at all. Also, in parametric tracking methods, the clutter intensity λ is most often assumed constant over the entire surveillance region. Such an assumption is only realistic if the surveillance region is kept reasonably small, for example by defining it as the gate. Therefore, whenever we talk about the volume Vk of the surveillance region, this quantity should be interpreted as the volume of the gate. Gating affects the machinery of the OBA in two ways. First, the target cardinality pmf in (2.30) is changed from Bernoulli(τ ; PD ) to P {card(Zk1 ) = τ } = Bernoulli(τ ; PD PG ).

(2.77)

This means that PD must be replaced by PD PG in (2.58). Second, the kinematic measurement likelihood pz1 (z|x) becomes truncated by the gate, which we account for by altering (2.62) to Z Vk lkzs (i) = pz1 (z(i)|xk )p(xk |θk−1,s , Z k−1 )dxk . (2.78) PG G

Assume that only measurements within a gate with gate size g and corresponding gate probability PG are considered. The posterior probability of measurement history number l being the correct one is then under the general assumptions G1-G4 together with the Gaussian-linear assumptions K1-K4 as well as the Poisson clutter cardinality pmf of (2.28) given by p 1 − PD PG k−1,s 2π|Sk | β PD 1 l T s −1 l a ∝ lk (i)exp − (νk ) (Sk ) νk β k−1,s 2

β k,l ∝ λ β k,l

for i = 0

(2.79)

for i = 1, . . . , mk .

(2.80)

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2.6

41

Tracking methods

The corresponding probabilities under the diffuse clutter cardinality pmf of (2.29) are given by the same expressions with λ replaced by mk /Vk . The amplitude likelihood ratio lka (i) is for the conventional case of a Swerling I target in Rayleigh background given by Rayleigh(ak (i) ; d + η)/PD PFA η a2 d a lk (i) = = · . (2.81) · exp Rayleigh(ak (i) ; η)/PFA PD η + d 2η(η + d) In Chapter 4 more refined alternatives to (2.81) are considered. These expressions are developed to deal with estimation uncertainties and heavy-tailed clutter. In addition to gating, several other simplifications may also be done to increase the practical utility of the OBA. These include history pruning and merging. Furthermore, a limited time horizon is necessary for the OBA to work. One approach is to merge all measurement histories at the beginning of a sliding window that covers the current and the previous N sampling times by means of (2.74) and (2.75). The well-known PDAF and PDAFAI methods, where no measurement histories are carried along and the past is approximated by a single Gaussian distribution, correspond to N = 0 and are described next. Less drastic, although similar, strategies for mixture reduction have been proposed in [96].

2.6.2

Thresholded measurements: Probabilistic Data Association

In this section the PDAF and the PDAFAI are presented. The PDAF was first developed in [8], while the PDAFAI can be traced back to [63]. Both have been extensively treated in [6]. The principles underlying these methods are the same, but the PDAFAI has more information available to carry out the association task, since it uses the amplitudes in addition to kinematic components of the measurements. Since the PDAF thus can be considered a special case of the PDAFAI, the latter method is derived here. The PDAFAI is based upon the general assumptions G1-G4 of Section 2.6.1 as well as the Gaussian-linear assumptions K1-K4 of Section 2.5.1. PDAF-based methods can perfectly well be used in many scenarios where these assumptions are more less violated, but care must then be exercised. It is often possible to modify the PDAF to deal with tracking problems that appear to be out of its domain. An example of this is how the interacting multiple model PDAF (IMMPDAF) was developed to deal with manoeuvering targets [6, pp. 232-277]. Another example is how the PDAF has been modified to deal with wake clutter in [87] and in Chapter 6 of this thesis. The PDAFAI is a suboptimal algorithm which at each time step collapses the posterior pdf of the target state into a single Gaussian which then is propagated to the next time step. Thus the past information about the target at time step k is summarized by ˆ k|k−1 , Pk|k−1 ). p(xk |Z k−1 ) ≈ N (xk ; x

(2.82)

ˆ k|k−1 and its associated covariance Pk|k−1 are treated In other words, the prediction x as sufficient statistics for the prior pdf at time k. The posterior pdf is then, assuming a

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linear-Gaussian measurement model, given as a Gaussian mixture which contains one component centered at the prediction and mk components centered at state estimates conditioned on the latest measurements zk (i). This mixture is forced back to a single Gaussian using moment-matching. Finally the cycle is completed by propagating this Gaussian to the next time step k + 1. The mixture can be written as k

p(xk |Z ) =

mk X

ˆ k|k (i), Pk|k (i)). βk (i)N (x ; x

(2.83)

i=0

The component corresponding to the prediction is 2 ˆ k|k (0), Pk|k (0)) p(xk |θk (0), Z k ) = N (x ; x

(2.84)

where ˆ k|k (0) =x ˆ k|k−1 x (2.85)

Pk|k (0) =Pk|k−1 . The measurement-conditioned components are ˆ k|k (i), Pk|k (i)) p(xk |θk (i), Z k ) = N (x ; x

(2.86)

where ˆ k|k (i) =x ˆ k|k−1 + Wk νk (i) x (2.87)

Pk|k (i) =Pk|k−1 − Wk Sk Wk and νk (i) =zk (i) − zˆk

Wk =Pk|k−1 H T Sk−1 (2.88)

Sk =HPk|k−1 H T + R.

The combined state estimate and its covariance are the first two moments of the mixture in (2.83). The combined state estimate is ˆ k|k =E[xk |Z k ] = x

mk X

βk (i)E[x|θk (i), Z k ] =

i=0 mk X

ˆ k|k−1 + =βk (0)x

i=1

ˆ k|k−1 + Wk νk =x 2

mk X

ˆ k|k (i) βk (i)x

i=0

ˆ k|k−1 + Wk νk (i) βk (i) x

(2.89)

According to [66], the covariance Pk|k (0) should be slightly larger than in (2.85), since additional information is carried by the fact that the measurement is not in the gate.

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where the last line is obtained by defining the combined innovation νk =

mk X

(2.90)

νk (i).

i=1

The covariance of the combined state estimate is [7 p. 56] h i T ˆ k|k xk − x ˆ k|k |Z k Pk|k =E xk − x =

mk X

βk (i)E

i=0 mk X

+

i=0

h

ˆ k|k (i) xk − x

ˆ k|k (i) − x ˆ k|k βk (i) x

=βk (0)Pk|k−1 +

mk X i=1

T i ˆ k|k (i) xk − x ˆ k|k (i) − x ˆ k|k x

T

βk (i) Pk|k−1 − Wk Sk Wk + P˜k

=Pk|k−1 − (1 − βk (0))Wk Sk Wk + P˜k

where the so-called spread of the innovations term is [6] "m # k X P˜k = Wk βk (i)νk (i)νk (i)T − νk ν T W T . k

k

(2.91)

(2.92)

i=1

The association probabilities βk (i) are determined as in (2.79) and (2.80). Since all past histories Z s have been merged, the term β k−1,s is simply unity. It follows that the association probabilities can be written on the form βk (0) = βk (i) = where we have defined

b+

b a j=1 lk (j)ek (j)

la (i)e (i) Pkmk ak b + j=1 lk (j)ek (j)

1 ek (i) , exp − νk (i)T Sk−1 νk (i) 2 and b,

(2.93)

Pmk

2π g

M 2

λVk

1 − PD PG cM PD

(2.94)

(2.95)

(2.96)

for the parametric PDAFAI. The gate volume is [6 p. 96] Vk = cM g M |Sk |1/2

(2.97)

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where M is the dimension of zk , and cM denotes the volume of an M -dimensional unit-radius hypersphere: π M/2 cM = . (2.98) Γ(M/2 + 1) The non-parametric version of the PDAFAI assumes µ(mk ) = µ(mk − 1) so that the clutter cardinality pmf’s cancel out. As shown in Section 2.6.1 this implies that λVk should be replaced by mk in (2.96). Recall from the discussion in Section 2.6.1 that it is not this substitution in itself that makes the parametric PDAFAI non-parametric. Rather, the non-parametric tracker happens to be equivalent to a parametric tracker using mk as an estimator of λVk . The non-parametric PDAFAI is used in Chapter 4 while the parametric PDAFAI is used in Chapter 5. The non-parametric PDAF without amplitudes is used in Chapter 6. To summarize, the PDAF and the PDAFAI can be implemented by means of the state update formula (2.89) and the covariance update formula (2.91) together with the association formulae (2.93) and (2.94). The amplitude likelihood ratio lka (i) is thoroughly discussed in Chapter 4, and will therefore not be treated any further in this introductory chapter. Setting it equal to unity reduces the PDAFAI to the PDAF. Finally, we remark that the conventional notation of (2.93) and (2.94) is discarded in Chapter 6 and replaced by a notation more similar to (2.65) and (2.66) in order to show how the PDAF can be modified to deal with state dependent clutter. The association weights of this alternative PDAF are derived in Appendix C.

2.6.3

Unthresholded measurements: Track-before-detect

Any measurement extraction procedure runs into difficulties when the SNR is very low. As pointed out in Section 2.3.5, the false alarm rate must then be increased in order to ensure target detection, thereby flooding the tracking system with false alarms. This abundance of false alarms is likely to confuse the tracker and ultimately lead to loss of track. In such a situation the extracted measurement modeling of Section 2.4 becomes inadequate and should be discarded entirely. It makes more sense to let the tracking method process raw sensor images directly. This is known as track-before-detect (TBD). In the Bayesian formulation of the TBD problem one does not have a set Zk of measurements. Instead, all the cells of the sensor image are included in a single measurement vector zk . The single-target TBD problem is consequently reduced to a non-linear filtering problem which can be solved using the techniques described in Section 2.5. The following assumptions are used to formulate the problem of recursive Bayesian TBD in [92]: D1: There is maximally one target present whose state ψk in addition to the kinematic components xk , x˙ k , . . . contains the target amplitude state ςk (or possibly the target power dk = ςk2 ) and a binary existence variable ek , so that ψk = [xk , ek ] = [xk , x˙ k , . . . , ςk , ek ] .

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D2: The existence variable ek ∈ {0, 1} is propagated according to a Markov model: P {ek = 1|ek−1 = 0} = Pbirth

P {ek = 1|ek−1 = 1} = 1 − Pdeath P {ek = 0|ek−1 = 1} = Pdeath

P {ek = 0|ek−1 = 0} = 1 − Pbirth . Pbirth is the probability of a target coming into existence and Pdeath is the probability of a target ceasing to exist. D3: The real-valued part of the state xk is propagated according to a Markov model: (a) When ek−1 = 0 and ek = 1, it is given by a birth pdf pb (xk ) which governs where a newborn target can appear in the state space. (b) When ek−1 = 1 and ek = 1, it is given by a conventional transition prior on the form p(xk |xk−1 ) which also can be written on the form xk = f (xk−1 , vk ) as in (2.38). D4: When ek = 0, the measurement vector zk consists of i.i.d. noise wki distributed according to p0 (wki ) where the index i represents resolution cell number i: zk = wk1 , wk2 , . . . , wkN .

D5: When ek = 1, the measurement vector zk does in addition contain a target signal sik = hik (xk ) so that zk = wk1 + s1k , wk2 + s2k , . . . , wkN + sN k .

Although the PSF hik (·) in principle allows all sensor cells to be affected by the target, only a limited set D(xk ) ⊂ {1, . . . , N } is affected in practice. This means that the measurement model can be summarized by the likelihood ratio Q i Y p(z|x, η) i∈D(x) p(z |x, η) l(z|x, η) = = l(z i |x, η). (2.99) = Q i |η) p (z p0 (z|η) 0 i∈D(x) i∈D(x)

The problem defined by assumptions D1-D5 is summarized and solved by the Bayes equations Z p(ψk |Z k−1 ) = p(ψk |ψk−1 )p(ψk−1 |Z k−1 )dψk−1 (2.100) p(ψk |Z k ) ∝ p(zk |ψk )p(ψk |Z k−1 ).

(2.101)

These equations can be approximated both using deterministic filtering [75] or using the particle filtering techniques discussed in Section 2.5.2 [92].

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Although the TBD problem thus has been reduced to a filtering problem, one should not be misled into believing that this has removed the data association. Data association is always necessary in target tracking. However, TBD performs the data association implicitly, in contrast to the explicit approach taken by the PDAFAI. A major strength of the filtering approach to TBD is that it, in contrast to many previous TBD methods such as [29] or [101], is able to track manoeuvering targets. At least for modest degrees of maneuverability, the method discussed in [92] is able to track targets which are impossible to spot with the human eye. A potential weakness of the filtering approach to TBD is that information is allowed to flow between far away parts of the state space in no time. A consequence of this is that an emerging track in one region of the state space suddenly may be discarded when the tracker finds something more interesting in another region.3 This problem appears to be well combated by the conservative treatment of amplitude information proposed in Chapter 3 of this thesis.

2.6.4

Other tracking methods

The clutter problem addressed in Sections 2.6.1 - 2.6.3 is the most basic aspect of data association. In practice one is likely to meet several other aspects, which each has the potential of complicating things a lot more. Most commonly encountered is the problem of multi-target tracking, that is the association of measurements to several targets which are present at the same time. The initiation and termination of tracks, which can be referred to as track management, is also most naturally viewed as a data association problem. Multiple sensors add another layer to the complexity. A single target may generate several measurements and thus further complicate the problem. A vast number of tracking methods have been developed to address such issues [6, 12, 86]. Although it is beyond the scope of this thesis to provide a survey on these, some brief remarks are appropriate in order to put the methods hereto discussed into their proper context. The single-target methods discussed in Sections 2.6.1 - 2.6.3 do all have their multitarget equivalents. The multi-target version of the OBA is known as multiple hypothesis tracking (MHT) [11, 91]. Although any practical MHT implementation must be grossly simplified by suboptimal heuristics and approximations, the MHT is optimal in principle. Unfortunately, many papers discussing the MHT do not provide the reader with any information about the nature of these suboptimalities in their particular implementation. Therefore it happens occasionally that another method appears to beat the MHT. Unless reasons for the MHT’s suboptimality are addressed, great care should be shown when interpreting such results. The multi-target version of the PDAF is known as the joint Probabilistic Data Association filter (JPDAF) and was originally proposed in [40]. It is also thoroughly treated in [6]. The JPDAF calculates the posterior probabilities of all measurement-to-track as3

A suitable name for such a problem could be “spooky action at a distance” Recent research [102] has put attention to similar problems in tracking methods based on finite set statistics (cf. Section 2.6.4).

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47

sociations at a given time step, and then updates the target states as weighed averages in the same way as in the PDAF. The computational cost of the JPDAF can be formidable if many targets are considered at once. However, a wise implementation of the JPDAF will use the PDAF as long as targets are well separated, and only use the JPDAF when two or more targets are so close that their validation regions intersect. The PDAF can be used in such cases as well, although with loss of optimality and thereby increased risk of track loss, switching or coalescence [14]. The TBD method outlined in Section 2.6.3 can be viewed as a special case of the socalled joint multi-target posterior density (JMPD) filter [62]. The JMPD filter attempts to estimate the number of targets together with their states using an “association-free” measurement model which evaluates the likelihood of the measurement (being a set in conventional mode or a vector in TBD-mode) conditional to the multi-target state. A troublesome aspect of the JMPD filter is that it is ill-posed in the absence of further restrictions. A realistic sensor model for TBD must allow any target to radiate into several resolution cells due to sensor blurring etc. But if several targets can be present at any location, it becomes impossible to determine whether a peak in the sensor image is caused by one target or by several closely spaced targets. TBD-implementations of the JMPD filter must, in order to cope with this challenge, require that the targets are well separated. Then the problem is essentially treated as a single-target problem, and the multi-target formalism of the JMPD filter appears to be superfluous. A related problem which recently has attracted some attention is the mixed-labeling problem [20], which concerns the difficulty of maintaining separate track identities for closely spaced targets when particle filtering is used to carry out the data association. There are several alternative approaches to multi-target TBD. Both [9] and [28] have proposed application of dynamic programming to multi-target TBD. A more recent approach to multi-target TBD is the histogram probabilistic multiple hypothesis tracker (PMHT) which has shown promising performance in [35]. It is reasonable to expect that a particle filtering-based TBD method using one particle cloud per target along the lines of [51] will be proposed soon. It is clearly difficult to track a manoeuvering target when the SNR is low, but what if the target moves in a straight line or along a parameterized deterministic path? If this is a reasonable assumption, one may consider discarding the Bayesian formalism and use a maximum-likelihood (ML) approach instead. The ML approach to target tracking has been developed for both TBD [101] and for extracted measurements [13]. In both cases the joint likelihood function of the data over several scans is optimized with respect to the unknown target kinematics parameters. Thus the target tracking problem is converted into a relatively low-dimensional optimization problem. The resulting methods are among the most robust tracking methods that exist. For weak targets the problem of tracking necessarily becomes intertwined with the problem of detection, where detection in this context should be understood as the confirmation or termination of a track. In most practical tracking systems this is done by means of heuristic rules, such as the M/N -logic [6 p. 104], which says that a prelimi-

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nary track should be confirmed if validated measurements have been received in at least M of the last N scans. A more rigorous approach to track confirmation is given by the sequential probability ratio test (SPRT), a.k.a. Wald test, which has been employed in batch-processing tracking methods [16,104,109]. How this approach can be used for recursive tracking methods such as the PDAF is not immediately obvious. Instead, the most popular approach in recent years has been to model track existence as a discrete state to be estimated along with target kinematics. This is done in the integrated Probabilistic Data Association (IPDA) of [77] and the TBD-method outlined in Section 2.6.3. A rigorous generalization of the concept of track existence to multi-target tracking leads to the random set approach, also known as finite set statistics (FISST), which is developed in [47] and [72]. Practitioners of FISST argue that not only should the measurements be modeled by random sets as in Section 2.4, but one should also model the multi-target state as a random set whose elements and cardinality both are governed by a transition prior. This allows multi-target tracking to be phrased using a generalized version of the Bayes equations (2.39) and (2.40). In contrast to the brute-force approach taken by the MHT or the JMPD filter, FISST practitioners have developed suboptimal approximations to the optimal Bayesian multi-target solution. Great attention has been received by the probability hypothesis density (PHD) filter [73], and the cardinalized probability hypothesis density (CPHD) filter [72]. The core idea of these methods is to estimate a non-stationary target intensity function known as the PHD, which corresponds to the clutter intensity λ in (2.28). This avoids the need for explicit target-tomeasurement associations and thus the combinatorial explosion of the MHT. There are several aspects of the PHD and CPHD filters that are problematic. The lack of explicit target-to-measurement associations means that a rigorous treatment of track identity hardly can be offered by these methods, since they never attempt to establish any tracks. This task must therefore be left to clustering methods which operate outside of the FISST framework [32, 70]. Another serious issue regarding the PHD filter is its so-called mis-detection problem. It was shown in [37] that even a single mis-detection can cause the PHD filter to declare that no target is present. The CPHD filter was developed to mend this problem, but recent research has revealed that this improved tracking method also suffers from similar problems [102]. However, other applications of FISST, for instance to the problem of state-dependent clutter [106], demonstrate the fruitfulness of this approach. Also, some attempts at applying FISST to the multi-target TBD problem have recently been carried out [105], although so far this research has only managed to treat well separated targets. It is the author’s belief that FISST will gradually become the conventional framework of target tracking, but several further advances are necessary before this can happen.

2.7

Performance evaluation

There are three ways in which the performance of a tracking method can be investigated. The ultimate test is implementation on real data. It is often more convenient to test it

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49

using Monte-Carlo simulations. Sometimes it is possible to predict the performance of the method theoretically. The kind of performance evaluation needed depends on what kind of research is being done. On the one hand, the performance of a practical tracking system must necessarily be verified by real world experiments. On the other hand, purely theoretical developments are typically too simple to be tested in this way, but can often be analyzed in a theoretical manner. A middle ground consists of primarily methodological developments, including the bulk of the target tracking literature. Before a new tracking method is ready for the real world, it should be tested extensively in simulations. Since the contributions of this thesis mainly are of a methodological nature, the performance evaluation will mostly be done using Monte-Carlo simulations.

2.7.1

Performance analysis using real data

A tracking method’s applicability should be questioned until it has demonstrated satisfactory performance in a real world field study. This does not necessarily mean that it has to be implemented in a commercial or military tracking system monitoring a harbor or an airport etc. It will most often be sufficient to test it on real data recorded during a field experiment. This involves several challenges, of which four are discussed here. First, one does in general not know the ground truth in real world experiments. This is especially so for underwater targets, which cannot be equipped with responders reporting to the global positioning system (GPS). Even if the target path is known, one does still not know the ground truth underlying other noise disturbances such as clutter. Second, real data allows no systematic investigation of model parameters. In this thesis, the strengths heavy-tailed clutter and wake clutter are represented by their respective parameters ν and λw . Tests on real data can hardly investigate the dependence on these parameters systematically, since one will just have to accept the parameter values exhibited by the data. To complicate things further, these parameters can be rather challenging to estimate. Third, it is in general prohibitively expensive to carry out enough experiments to achieve statistical significance. One is often concerned about performance measures such as the number of lost tracks. Since track-loss hopefully should not occur too frequently, thousands of experiments may be needed to obtain statistically relevant information. In spite of this, experimental papers often report only a single successful run. Fourth, to which degree such a single experiment demonstrates satisfactory performance of a tracking method can only be assessed if one has some idea about the difficulty of the experiment. This is complicated by the fact that only limited knowledge about the ground truth is available. To compensate for this, the researcher should as much as possible provide a statistical analysis of SNR, maneuverability, fluctuation pattern and so on. If the test run is too easy, it is hardly worth boasting that the proposed method works well, and the critical reader should not feel assured that the method will work satisfactorily under more challenging circumstances. If on the other hand the test run is too challenging, there is reason to suspect that good performance is due to luck. In order

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to avoid such suspicions, the performance of newly developed methods should always when possible be compared to the performance of established methods. Notwithstanding these challenges, the importance of real world experiments can hardly be overestimated. While simulations can test whether a method solves the anticipated challenges, real world experiments test whether the method solves the actual challenges, which not necessarily coincide with the anticipated challenges. Real data puts the robustness of a tracking method under a much sterner test than the most sophisticated simulation can do. Another reason for using real data is that it prevents tuning of the test data. It is more difficult for a researcher working with real data to choose a particularly favorable scenario, than it is for a researcher working with simulations.

2.7.2

Performance analysis using Monte-Carlo simulations

Monte-Carlo simulations are always necessary when real data are not available. This is often the case in the tracking community since experimental data tend to be treated as highly classified. Even when real data are available, real-world implementations should be accompanied by Monte-Carlo simulations due to the reasons listed in Section 2.7.1. This is especially important when a systematic study of model parameters is desired. Therefore, Monte-Carlo simulations are used by most papers in the tracking literature to investigate performance. There are two approaches to the Monte-Carlo investigation of tracker performance. Most common is what we may term the clean approach, in which the simulated data are generated from the filter model which is assumed to be valid by the tracking method. On the other hand, one can use a realistic approach in which the simulated data are generated from a more refined simulation model (cf. Section 2.5). An objection against this distinction between filter and simulation models is that the existence of a more refined model than the filter model indicates that the filter model should be replaced by this model. This is not necessarily a good idea. No model can be made complex enough to depict reality perfectly. Attempting to do employ such a refined filter model will cause difficulties both with regard to computational costs and with regard to robustness. While it is acceptable to spend days and weeks simulating a scenario, a tracking method is useless unless its filter model can be processed faster than the scan rate. With regard to robustness, one should be careful not to parameterize the filter model in a way that makes it overly sensitive to arbitrary tuning parameters. Since this kind of sensitivity is not always predictable, it is in general advisable to keep the filter model as simple as possible.4 Such issues are much less important for the simulation model as long as it provides sufficiently realistic and diverse outputs. Both the clean and the realistic approaches are susceptible to tuning. The researcher must necessarily choose values of the model parameters which will affect the apparent performance of the method to be tested. These values should primarily be chosen in order to make the simulated scenario as realistic as possible. Secondary, they should be 4

But not simpler, as Albert Einstein, Yaakov Bar-Shalom and Ronald Mahler are fond of reminding us.

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chosen in order to emphasize how the method to be tested differs from rivaling methods. For the most important model parameters a wide range of values should be used instead of a single value. This range should include parameter values for which the tracking method encounters difficulties, so that the reader is given some impression of both its strengths and its weaknesses. The researcher must also choose the performance measures used. These may include both discrete measures such as the number of lost tracks and continuous measures such as the root mean square error (RMSE). More advanced measures such as the Wasserstein distance [84] or the optimal sub-pattern assignment (OSPA) metric [31] have been employed for multi-target tracking. The performance measures chosen should be fair to all the methods being tested. Furthermore, the measures should be relevant. If the aim is to maintain a very accurate track on the target, then an RMSE-based performance measure certainly makes sense. However, in a more challenging scenario it may be overambitious to aim for more than to be able to maintain a somewhat accurate track at all. Then the rate of track-loss is a more instructive performance measure.

2.7.3

Theoretical performance analysis

For a Gaussian-linear filtering problem as given by K1-K4 in Section 2.5.1, the Kalman filter is optimal in the sense that it is the minimum mean square error (MMSE) estimator. In other words, the root mean square error (RMSE) of the Kalman filter is lower than for any other estimator, provided that the system has been modeled correctly. Furthermore, the Kalman filter itself calculates the RMSE by means of (2.42) and (2.44). This covariance evolution is also known as the Riccati equation. The actual RMSE of the Kalman filter can be predicted in the same way even when the system has been wrongly modeled, provided that both models are Gaussian-linear [44]. The lowest possible error covariance that can be attained for a given parameter estimation problem is known as the Cramer-Rao lower bound (CRLB). For a Bayesian filtering problem the corresponding lower bound is referred to as the posterior CramerRao lower bound (PCRLB). Since the PCRLB coincides with the Kalman filter RMSE for Gaussian-linear filtering problems, the PCRLB of any such problem is given by the Riccati equation. For non-linear filtering problems one can in general not expect the state estimator to reach the PCRLB. Neither can one expect the PCRLB to be available through closed form equations such as (2.42) and (2.44). Instead, numerical techniques such as the method of [100] must be used to evaluate the PCRLB for non-linear filtering problems. In [92] this technique has been used to analyze several filtering problems including single-target TBD. Data association is, due to the measurement origin uncertainty, a non-linear problem even if the underlying kinematics and measurement processes are linear. The conventional Riccati equation can therefore not be employed to determine the expected performance of a tracking method. It is, however, possible to modify the Riccati equation in such a way that the impact of measurement uncertainties are taken into account.

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There are two approaches to this. On the one hand, the PCRLB of a data association problem can be calculated. This has been done in [50, 110] for the single-target case and in [52] for the multi-target case. On the other hand, the expected performance of the PDAF can be evaluated by means of the MRE [39]. In both cases the performance degradation due to clutter and non-uniform detection probability can be quantified by a scalar known as the information reduction factor (IRF). Further discussion of theoretical performance prediction is left for Chapter 5 where the MRE is extended to predict the performance of the PDAFAI. .

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3 Tracking dim targets using integrated clutter estimation In this chapter, based on the conference paper [21], we address the problem of detecting and tracking a single dim target in unknown background noise. Several methodologies have been developed for this problem, including TBD methods which work directly on unthresholded sensor data. The utilization of unthresholded data is essential when SNR is low, since the target amplitude may never be strong enough to exceed any reasonable threshold. Several problems arise when working with unthresholded data. Blurring and non-Gaussian noise can easily lead to very complicated likelihood expressions. The background noise also needs to be estimated. This chapter proposes a robust treatment of the background estimate in accordance with its random nature. The background noise is estimated by averaging over nearby sensor cells not affected by the target. The uncertainty of this estimate is incorporated in the likelihood evaluation by marginalizing over the true but unknown background strength. The resulting TBD method is implemented using sequential Monte Carlo evaluation of the optimal Bayes equations, also known as particle filtering. Monte-Carlo simulations show that accounting for estimation uncertainty in this way has a crucial impact on whether the TBD method succeeds or fails.

3.1

Introduction

In a conventional tracking system the sensor (e.g. a radar, a sonar, an infrared sensor etc.) provides an array of observations to be further analyzed through signal processing and data analysis. After preprocessing, such as matched filtering, a detector is employed to extract point measurements from the array. This is done by setting a threshold according to a CFAR criterion [42]. Most tracking methods, including the PDAF [6] and the MHT [91], need such thresholded measurements as input. 53

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3

Tracking dim targets using integrated clutter estimation

Another option is to exploit all the information in the data by letting the tracking method work directly on the unthresholded data. This is referred to as TBD. The rationale behind TBD is that for low SNR it is impossible to set a reasonable detection threshold. Due to the weak nature of the target signal, the threshold must be set very low, leading to a very high false alarm rate. By using the TBD approach we avoid this trade-off and make our detection decision based on tracking results obtained from several data frames instead. The most obvious way of doing this is perhaps by using the Hough transform [29], which essentially integrates the data for several time frames along all possible paths. Another classical approach is the dynamic programming algorithm (DPA) used by Barniv [9]. In DPA the state space is discretized so that we get a hidden Markov model. This allows one to treat the problem in rigorous probabilistic terms. A third approach proposed by Tonissen and Bar-Shalom [101] is to treat TBD as a parameter estimation problem instead of a state estimation problem. The MLE of the target parameter is found using a numerical search procedure. Related to this approach is the ML-PDA [13] which works on thresholded measurements. The approaches mentioned above are very computationally demanding, and they require a lot of memory. This is because they compare several scans of data using batch processing. A recursive TBD algorithm was therefore proposed by Salmond and Birch [95], and further developed by Ristic et al [92]. It uses particle filtering to propagate the single-target posterior probability density function (pdf). Several papers based on this approach have been published during the last decade. Rutten et al. discuss TBD in Rayleigh background noise [94] and TBD when the target amplitude is fluctuating [93]. Multiple targets and extended targets have been addressed by Boers and Driessen [17, 18]. Another approach to multitarget TBD is the JMPD filter of Kreucher et al [62]. With exception of the Hough transform, all the TBD methods mentioned compare the likelihood of the current data scan under a target present hypothesis with the likelihood of the data scan under a noise only hypothesis. This requires an estimate of the background noise. We treat the terms background noise and clutter as equivalent in this chapter. To avoid susceptibility to spatial or temporal non-stationarity this estimate should be based on test cells close to the sensor cells affected by the target. Unless the number of test cells used is rather large, the estimate will be accompanied by a high uncertainty. Our main contribution is to treat this estimate as a random variable. Although the classical detection literature is full of such considerations [60], we have not found any treatment of this topic in the TBD context. In the Gaussian and Rayleigh cases to be discussed the MLE of the background noise and its corresponding pdf can both be written in closed form. By marginalizing over the true, but unknown, noise parameter we obtain new likelihoods which take the uncertainty of the background noise estimate into account. We show that when little information is available to estimate the background noise, this not only improves tracking performance, but is also crucial in order to get the correct detection performance. Also, in the Gaussian case, we show mathematically

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3.2

Bayesian solution and target model

55

that the traditional likelihood of previous TBD literature is regained in the limit case when infinitely many cells are used to obtain a perfect estimate of the background noise parameter. The remainder of this chapter is organized as follows: In section 3.2 the Bayesian TBD framework is presented. In section 3.3 we present sensor models for both known and unknown background noise. In section 3.4 a particle filter implementation is outlined, while section 3.5 describes a test scenario and simulation results.

3.2

Bayesian solution and target model

We phrase the single target TBD problem in terms of the optimal Bayesian equations, Z p(ψk |Z k−1 ) = p(ψk |ψk−1 )p(ψk−1 |Z k−1 )dψk−1 (3.1) p(ψk |Z k ) ∝ p(zk |ψk )p(ψk |Z k−1 ).

(3.2)

The state vector at time step k is denoted ψk and Z k = [z1 , z2 , . . . , zk ] is the cumulative set of measurements at time step k. In our treatment of TBD the full state vector ψk consists of the following variables, T ψk = [xk , x˙ k , yk , y˙ k , ςk , ek ]T = [xT k , ek ]

(3.3)

where xk contains all real valued components of the state including the target amplitude state ςk . Changing dimensionality of the state vector is only a matter of notation. The existence variable ek is defined such that 1 if target is present (3.4) ek = 0 if target is not present. The state space of ψk is the Cartesian product of a real valued and a discrete valued space and therefore a hybrid space. It should at this point be mentioned that the above formalism can easily be extended to multi-target scenarios, leading to the JMPD filter described in [62]. However, the estimation problem then becomes ill-posed unless the issue of merged measurements is addressed. To avoid these problems we restrict attention to a single target scenario. The transition prior of the discrete valued component ek is given by Markov transition probabilities, P (ek = 1|ek−1 = 0) = Pbirth P (ek = 1|ek−1 = 1) = 1 − Pdeath P (ek = 0|ek−1 = 1) = Pdeath

P (ek = 0|ek−1 = 0) = 1 − Pbirth

(3.5)

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3


where Pbirth is termed the birth probability and Pdeath is termed the death probability. The quantity PE = P (ek = 1|Z k ) is called the probability of existence and is our main indicator of whether a target is present or not. The transition prior of the continuous component xk must be specified for each of these four cases. However, when ek = 0 the exact value of xk is of no interest since a non-existent target clearly has no kinematic attributes. Therefore it suffices to specify the transition prior of xk in the first two cases only. Assuming white plant noise, we write p(xk |xk−1 , ek−1 = 0, ek = 1) = p(xk |ek−1 = 0, ek = 1) , pb (xk )

p(xk |xk−1 , ek−1 = 1, ek = 1) = p(xk |xk−1 ).

(3.6) (3.7)

Here pb (·) is a birth pdf governing where a newborn target can appear in the state space. The kinematic plant equation (3.7) can alternatively be written as xk = f (xk−1 , vk ). For the case of Gaussian plant noise vk ∼ N (vk ; 0, Q), the transition prior p(ψk |ψk−1 ) is given by f (·, ·), Q, Pbirth , Pdeath and pb (·).

3.3

Sensor models

We consider a sensor providing two-dimensional images of the surveillance region. Changing dimensionality of the sensor array is only a matter of notation. At each discrete time k we can view the current image as one big measurement vector n n

zk = [zk11 , zk12 , . . . , zk x y ]T

(3.8)

where zkij is the value of pixel or cell number (i, j). In order to update the estimate of ψk from the measurement zk we need to relate the value in each cell zkij to the reduced state vector xk when ek = 1. Each cell may contain contributions both from the target signal given by xk and from the background noise. In this work the target signal is given deterministically conditioned on xk . Another option is to treat the target signal as stochastic conditioned on xk . This must be done if a Swerling I target fluctuation model is to be used as in [93]. Here we will instead account for target amplitude fluctuations by using the plant noise. We write the sensor model as   g hij (xk ) + wij if target is present k k zkij = (3.9) ij  g w if target is not present. k ij The mapping hij k (xk ) = sk represents both the received signal and the blurring properties of the sensor. In the TBD literature it is common to use the following PSF given by a blurring parameter Σ and the cell sizes ∆x and ∆y : ∆ x ∆ y ςk (i∆x − xk )2 + (j∆y − yk )2 hij (x ) = exp − . (3.10) k k 2πΣ2 2Σ2

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Sensor models

The mapping g(·) is typically the identity mapping or the absolute value operation | · | depending on whether we assume Gaussian or Rayleigh background noise. In any case we assume wkij Gaussian and spatially as well as temporally white: wkij wkij

∼ N (wkij ; 0, ηijk ) for Gaussian background

∼ Nc (wkij ; 0, ηijk ) for Rayleigh background.

(3.11) (3.12)

The notation Nc (· ; 0, η) represents a circularly symmetric complex Gaussian with variance η. In contrast to previous work on TBD, we will in sections 3.3.3 and 3.3.4 assume that there may be some variation in the noise parameter η. The notation ηijk is used when we want to stress this aspect. The spatial variation of the noise parameter is assumed to have approximately unity correlation over an area of cells covering the PSF area and the test cells used for clutter estimation. No assumptions are made regarding temporal correlations or correlations between noise parameters in cells far apart. For the rest of this section we suppress the indices of ηijk , treating it as constant within the PSF area. The time index k is dropped for all other quantities as well. We also assume η unknown so that it must be estimated. This estimate is a random variable with a certain distribution that should be taken into account when the likelihood p(zk |xk ) is evaluated. We start by looking at the standard framework used when η is known and constant. Since the target signal is close to zero outside of the PSF area we can factorize the likelihood as Y Y p(z|x, η) = p(z ij |x, η) p0 (z ij |η). (3.13) ij∈D(x)

ij ∈D(x) /

D(x) is the set of sensor cells affected by the PSF of a target with state x, and p0 (z ij |η) is the likelihood of z ij under the hypothesis of no target present. In order to restrict attention to the cells actually affected by the target we define the likelihood ratio Q ij p(z|x, η) ij∈D(x) p(z |x, η) l(z|x, η) = = Q = ij p0 (z|η) ij∈D(x) p0 (z |η)

Y

ij∈D(x)

l(z ij |x, η).

(3.14)

Since l(z|x, η) ∝ p(z|x, η) we use the likelihood ratio instead of the likelihood when (3.2) is implemented.

3.3.1

Gaussian background

When wij ∼ N (wij ; 0, η) for a known η we immediately get the likelihood used in [92]:

sij (sij − 2z ij ) l(z |x, η) = exp − 2η ij

.

(3.15)

Instead we obtain the background estimate from a set of test cells G(x), which for instance can be chosen to lie around D(x). This is similar to what is done in conventional CFAR detection [1]. See Figure 1 for an illustration. If the estimate of x and y is wrong, G(x) may contain contributions from the target signal. When this classtest: 58(76) happens the noise estimate will be too high. However, in a2010-06-08 particle 17:36 filter—implementation this can be accepted, as a large number of particles still will get the correct estimate. Even if G(x) is chosen correctly the noise estimate will fluctuate between both lower and higher values than the true noise parameter. It is particularly important to account for this uncertainty when the estimate is low, since the straightforward usage of (8) or (9) leads to spurious false track declarations in such cases. 58 3 Tracking dim targets using integrated clutter estimation

G(xk )

D(xk )

xk

Figure 3.1: Neighborhoods for clutter estimation and for support of the PSF.

Figure 1. Neighborhoods for clutter estimation and for support of the PSF.

3.3.2 Rayleigh background 2 is a random variable with a corresponding PDF that should be accounted for in the likelihood The estimate σ In some the data z ij can only attain positive values. Then the Gaussian like2 |σ 2 ) applications evaluation. Its PDF p(σ does however depend on the true noise parameter σ 2 . To utilize the information lihood (3.15) is clearly not appropriate. However, such a situation can often be modeled carried in this PDF requires us to treat σ 2 as random as well and assign it a prior distribution. By marginalizing ij circularly by assuming w symmetric complex Gaussian and letting g(·) = | · |. The 2 we obtain the joint distribution of z and σ 2 given σ data will then be Rayleigh distributed in the noise only case and Rice distributed in the of a target signal [85, 94]: presence 2 |x, σ 2 )p(z, σ 2 |x) p(z, σ 2 2 2 2 2 p(z|x, σ ) = p(z, σ |x, σ )dσ = dσ 2 ij 2 ij p(z, z σ |x) −(z ) exp p0 (z ij |η) = Rayleigh z ij ; η = 2 2 2 2 |σ )p(z|x)p(σ ) η 2η p(z|σ , x)p(σ 2 2 2 22 |σ 2 )p(σ 2ij)dσ dσ = ∝ p(z|x, )p(ijσ (10) ij ij )2 σ ij . z −(z − (s ) z s 2 ij ij ij p(z|x)p( σ ) p(z |x, η) = Rice z ; s , η = exp I0 η 2η η ij ij ij 2 2 as independent of −(sG(x) ) used zto sobtain the noise estimate σ To obtain this result we have ijtreated the test cells ⇒ l(z |x, η) = exp I0 . (3.16) 2 2 the cells z used in the likelihood calculation. This2ηallows us toη factorize p(z, σ |x, σ ) as done. Furthermore, 2 are treated as independent of x since the noise strength is considered locally constant and the target σ 2 and σ The function I0 (·)find is a the modified Besselratio function signal does not affect G(x). We then likelihood as of the second kind [3 p. 374]. 2 ) p(z|x, σ Unknown Gaussian 2background , l(z|x, σ )= 2 ) p (z| σ N In this section we develop the strategy for coping with unknown noise, and Section 3.3.4

3.3.3

which concerns unknown Rayleigh noise is therefore also based on what follows. When η is unknown we have no choice but to use p(z|x, ηˆ) and p0 (z|ˆ η ) instead, where ηˆ is an estimator of η. Ideally the target amplitude and the background noise should be jointly estimated. This is complicated by the fact that only the tracking output tells us whether the PSF area D(x) actually contains the target signal. Furthermore such a direct estimation of the target amplitude is difficult to carry out when the number of cells in D(x) is small.

(11)

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59

Sensor models

Instead we obtain the background estimate from a set of test cells G(x), which for instance can be chosen to lie around D(x). This is similar to what is done in conventional CFAR detection [42]. See Figure 3.1 for an illustration. If the estimate of the target position (x, y) is considerably wrong, the test cell set G(x) may contain contributions from the target signal. When this happens the noise estimate will be too high. However, in a particle filter implementation this can be accepted, as a large number of particles still will get the correct estimate. Even if G(x) is chosen correctly the noise estimate will fluctuate between both lower and higher values than the true noise parameter. It is particularly important to account for this uncertainty when the estimate is low, since the straightforward usage of (3.15) or (3.16) leads to spurious false track declarations in such cases. The estimate ηˆ is a random variable with a corresponding pdf that should be accounted for in the likelihood evaluation. Its pdf p(ˆ η |η) does however depend on the true noise parameter η. To utilize the information carried in this pdf requires us to treat η as random as well and assign it a prior distribution. By marginalizing the joint distribution of z and η given ηˆ we obtain Z Z p(z, ηˆ|x, η)p(z, η|x) p(z|x, ηˆ) = p(z, η|x, ηˆ)dη = dη p(z, ηˆ|x) Z p(z|η, x)p(ˆ η |η)p(z|x)p(η) = dη p(z|x)p(ˆ η) Z ∝ p(z|x, η)p(ˆ η |η)p(η)dη. (3.17) To obtain this result we have treated the test cells G(x) used to obtain the noise estimate ηˆ as independent of the cells z used in the likelihood calculation. This allows us to factorize p(z, ηˆ|x, η) as done in (4.38). Furthermore, η and ηˆ are treated as independent of x since the noise strength is considered locally constant and the target signal does not affect G(x). The likelihood ratio is then found as l(z|x, ηˆ) =

p1 (z|x, ηˆ) p0 (z|ˆ η)

(3.18)

where the denominator is given by evaluating (4.38) with no target signal present. The factorization of (3.13) no longer holds because all the cells in z are affected by the common random variable η. We should, however, not assume that η is constant over the entire image. Our estimate is only supposed to be valid for the PSF cells in D(x), and therefore we still maintain the assumption p(z|x, ηˆ) = p1 (z α |x, ηˆ)p0 (z β |ˆ η)

(3.19)

where z α and z β denote the parts of z that are inside and outside of D(x), respectively. This makes it possible to define the unknown background likelihood ratio of (4.41) by evaluating (4.38) using only the cells z α in D(x). The practical implication of this

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PDF of ηˆ for real univariate Gaussian background

Normalized PDF of MLE

M=16 M=50 M=1000

0

0.5

1 1.5 MLE of η in units of η

2.5

2

Figure 3.2: Probability density functions of the MLE ηˆ in cases of Gaussian background for various numbers M of test cells.

PDF of ηˆ for Rayleigh background

Normalized PDF of MLE

M=16 M=50 M=1000

0

0.5

1 1.5 MLE of η in units of η

2

2.5

Figure 3.3: Probability density functions of the MLE ηˆ in cases of Rayleigh background for various numbers M of test cells.

1

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61

Sensor models

simplification is to make our likelihood ratio more conservative, since information that otherwise could be used is discarded as unreliable. In the Gaussian case we can use the MLE of η which is given by 1 X ηˆGauss = (z ij )2 (3.20) M ij∈G(x)

where M is the number of test cells in G(x). It has the pdf M M (ˆ 2 exp −ˆ η ) η 2η M 2η p(ˆ η |η) = Gamma ηˆ ; = M , . 2 M 2η 2 M Γ M 2

(3.21)

In Figures 3.2 and 3.3 the pdf of ηˆ is plotted for various M . It can be seen that when the number M of test cells is low, the MLE becomes very uncertain. Recall that when noise is spatially and temporally non-stationary one cannot use a large number of test cells in G(x), so it is important that the algorithm works for low M . Since any prior assumptions regarding the value of η can mislead the tracking algorithm, we want to use p(ˆ η |η) without being bothered by p(η). The prior p(η) should therefore be made as uninformative as possible. We choose a flat prior p(η) = Uniform(η ; [0, ξ])

(3.22)

where ξ is a very large number. The likelihood in (4.38) becomes   Zξ Y M 2η 1 ij ij   p(z|x, ηˆ) ∝ N (z ; s , η) Gamma ηˆ ; , dη 2 M ξ ij∈D(x) 0     Zξ X 1 ηˆM  1  √ −m−M (z ij − sij )2 + dη. ∝ η exp −  2 2 η 0

ij∈D(x)

(3.23)

Here m is the number of cells over which the product of Gaussians is taken, that is the number of elements in D(x). Since we are only interested in p(z|x, ηˆ) up to proportionality we have discarded the factor 1/ξ. This makes it possible to evaluate the integral in the maximally uninformative limit ξ → ∞. Using the substitution 1 u = √ ⇒ dη = −2σ 3 du η and taking the limit ξ → ∞, transforms (3.23) into a more tractable integral,     Z∞ X 1 ηˆM  2  p(z|x, ηˆ) ∝ um+M −3 exp −  (z ij − sij )2 + u du 2 2 0

ij∈D(x)

(3.24)

(3.25)

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whose solution is known to be [3 p. 302] Γ 2

P 1 2

ij∈D(x)

m+M −2 2

(z ij

−

sij )2

+

ηˆM 2

m+M −2 .

(3.26)

2

The likelihood in target absent case is given by the same expression with sij = 0 ∀ (i, j). The likelihood ratio is then found as P m+M −2 2 ηˆM 1 ij 2 ij∈D(x) (z ) + 2 2 p1 (z|x, ηˆ) l(z|x, ηˆ) ∝ = m+M −2 p0 (z|ˆ η) 2 1P ij − sij )2 + ηˆM (z ij∈D(x) 2 2 ! m+M −2 P ij 2 2 ˆM ij∈D(x) (z ) + η = P . (3.27) ij ij 2 ˆM ij∈D(x) (z − s ) + η

When infinite information is available to estimate the background noise, we regain the familiar likelihood ratio (3.15):   ! m+M −2 P ij )2 + η 2 X (z ˆ M 1 ij∈D(x) P −−−−→ exp  (sij )2 − 2z ij sij  . ij ij 2 M →∞ ˆM 2ˆ η ij∈D(x) (z − s ) + η ij∈D(x)

3.3.4

Unknown Rayleigh background

Extension to the unknown Rayleigh case is straightforward in principle, following along the same lines as in section 3.3.3. In the Rayleigh case the background noise MLE is X 1 ηˆRayleigh = (z ij )2 (3.28) 2M ij∈G(x)

with corresponding pdf

M M (ˆ η ) exp −ˆ η η η p(ˆ η |η) = Gamma ηˆ ; M, = . M η M Γ(M )

(3.29)

M

The equivalent of (3.23) then becomes   Zξ Y η 1  p(z|x, ηˆ) ∝ Rice z ij ; sij , η  Gamma ηˆ ; M, dη M ξ ij∈D(x) 0     Zξ X 1 1 ∝ η −m−M exp − ηˆM + (z ij )2 + (sij )2   2 η ij∈D(x) 0 ij ij Y z s I0 dη. (3.30) η ij∈D(x)

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63

Sensor models

5

0 Assumed SNR

Log Likelihood Ratio

10

-5

-5

0

Known η Unknown η, M = 16 Unknown η, M = 32 Unknown η, M = 128

5 10 True SNR [dB]

15

20

Figure 3.4: Convergence of LLR curve: The log likelihood ratio ln(l(z|x, ηˆ)) is plotted for various M in Rayleigh case. When M is large the likelihood ratio using clutter estimation converges to the conventional likelihood ratio treating η as constant. Here we have assumed ηˆ = η. The curves are generated using a target signal given by (3.10) and a common realization of the random noise.

5

0 Assumed SNR


10

-5

-5

0

Correct η treated as known correct η treated as unknown Wrong η treated as known Wrong η treated as unknown

5 10 True SNR [dB]

15

20

Figure 3.5: Sensitivity to model mismatch: When the estimate happens to be wrong, the proposed treatment of the amplitude likelihood exhibits better robustness than the conventional treatment. These curves were generated with ηˆ = 0.8η, which, as revealed by Figures 3.2 and 3.3, is not an uncommon situation. 1

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The product of Bessel functions makes it impossible to obtain any closed form solutions to the optimal expression in (3.30). We must therefore resort to a numerical solution. Although the integrand is rather heavy-tailed, taking the limit ξ → ∞ and employing the substitution (3.24) transforms (3.30) into the more well-behaved integral,

p(z|x, ηˆ) ∝

Z∞ 0

 

u2m+2M −3 exp − ηˆM +

Y

I0

ij∈D(x)

z ij sij u2 du.

1 2

X





(z ij )2 + (sij )2  u2 

ij∈D(x)

(3.31)

The target absent likelihood p0 (z|ˆ η ) is found by evaluating (3.31) with sij = 0 ∀ (i, j). The likelihood ratio l(z|x, ηˆ) is then found as the ratio between these quantities. The integral in (3.31) is difficult to attack using importance sampling since it is not straightforward to determine the support of the integrand. We have found a grid based numerical evaluation with > 50 samples on the interval u ∈ [0, √2ηˆ ] more reliable.

Our main objective is to track in uncertain environments where the noise estimate cannot be very accurate. Therefore the general shape of the likelihood ratio as a function of z and a is just as important as the exact value. This is illustrated in Figure 3.4. The conventional likelihoods of sections 3.3.1 and 3.3.2 increase monotonously as the actual amplitude affecting the data z gets higher than the assumed amplitude a. This means that outliers due to the noise can easily be given too high likelihood, especially if our noise estimate is in the lower tail of its Gamma distribution. The likelihoods of sections 3.3.3 and 3.3.4 are more conservative and refuse to conclude anything in the presence of too extreme outliers, since the presence of an outlier indicates a poor noise estimate.

3.4

Particle filter tracking algorithm

The direct implementation of (3.1) and (3.2) is a highly nonlinear problem. Furthermore low SNR makes direct inference difficult. The target can only be tracked by some kind of search approach, justifying particle filtering for this problem. The particle filter (p) (p) represents the posterior pdf by a set of samples {ψk|k , wk|k }Pp=1 such that p(ψk |Z k ) ≈

X p

(p)

(p)

wk δ(ψk − ψk )

(3.32)

where δ(·) is Dirac’s delta function in the hybrid state space of ψk . Particle filtering approximates the integral of (3.1) by importance sampling. To briefly show how this

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3.5

65

Test design and simulation results

works we combine (3.1) and (3.2) to obtain Z k p(ψk |Z ) ∝ p(zk |ψk ) p(ψk |ψk−1 )p(ψk−1 |Z k−1 )dψk−1 Z p(zk |ψk )p(ψk |ψk−1 ) = p(ψk−1 |Z k−1 )q(ψk |ψk−1 , zk )dψk−1 q(ψk |ψk−1 , zk ) ≈

(p) X p(zk |ψk(p) )p(ψk(p) |ψk−1 ) p

(p) (p) q(ψk |ψk−1 , zk )

(p)

(p)

(3.33)

wk−1 δ(ψk − ψk ).

The importance density q(ψk |ψk−1 , zk ) is used to sample a new particle at time k for each particle at time k − 1. It is well known that one can increase performance and decrease the necessary number of particles by tailoring the importance density in a strategic way. However, since this is not our main focus we will confine ourselves to the standard sequential importance sampling and resampling (SIR) filter which simply uses the kinematic prior, (p)

(p)

(p)

q(ψk |ψk−1 , zk ) = p(ψk |ψk−1 ) ⇒ wk ∝ p(zk |ψk )wk−1 .

(3.34)

Thus, the particle weights are updated by multiplying the likelihood, which as explained in section 3.3 is replaced by the likelihood ratio. Expressions for the various likelihoods that can be used were given in sections 3.3.1 - 3.3.4. We remark that our likelihoods must not necessarily be used inside a particle filter, although this seems to be the most straightforward implementation. To avoid degeneracy the particle cloud is resampled when the effective sample size P (p) 2 Pˆ = 1/ w goes below a certain threshold. In our simulations we have used p

k

systematic resampling [92]. Another options is to use deterministic resampling as described in [45], which preserves a higher degree of diversity in the particle cloud. We believe that this can prevent fluctuations from deteriorating the performance.

3.5


We use a standard linear kinematic model xk = f (xk−1 , vk ) = F xk−1 + vk where vk ∼ N (vk ; 0, Q), and with matrices given as  2  σv 3 σv2 2   T T 0 0 0 1 T 0 0 0  σ3v2 2 2 2    0 1 0 0 0  σv T 0 0 0     2T  σv2 3 σv2 2    . (3.35) F = 0 T T 0  0 0 1 T 0 , Q =  0  3 2   2  0 0 0 1 0  σv 2 2  0 0 σv T 0  2 T 0 0 0 0 1 0 0 0 0 σa2 T

Let σv2 be 0.01 and σa2 be 0.1. We specify the birth and death probabilities to be Pbirth = Pdeath = 0.05. The birth pdf pb (·) is assumed uniform over the surveillance region.

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For convenience we let T = 1. The particle cloud is initialized with all particles in state ep0 = 0. The unknown background is simulated as follows. Denote the true noise parameter in cell number ij at time k by ηijk . We construct it as ηijk =

1 Mx My

X

mn∈Hij

2 ζmnk where ζmnk ∼ N (ζmnk ; 0, 1).

(3.36)

Here Hij is a sliding window centered around cell (i, j) with dimensions Mx = My = 11 in our simulations. The resulting background noise has correlations on a large scale but is approximately white on a small scale. Inside the windows D(xk ) and G(xk ) we treat it as stationary and Gaussian or Rayleigh with a “parameter” ηijk whose expectation is unity. The background is generated independently for each time. This somewhat artificial construction is chosen for illustratory purposes only. The resulting background can in the Rayleigh case be said to follow a mild K-distribution since ηijk will be Gamma-distributed (c.f. Sections 4.2.4 and 4.4.1). However, as argued in Section 4.4.7, such a mild degree of heavy-tailedness does not justify abandonment of the Gaussian or Rayleigh assumption. A target is simulated to appear in the middle of a surveillance region of 20 × 20 cells at time step k = 15 with initial position [8, 8]T and initial velocity [0.18, 0.20]T . It disappears at k = 50. A frame of data at time k = 49 for this scenario is shown in Figure 3.6. We performed 60 Monte-Carlo runs for the likelihoods of sections 3.3.2 and 3.3.4. In both cases the background noise was estimated using the cells G(xpk ) surrounding the PSF cells D(xpk ) as illustrated in Figure 3.1. For the Rayleigh case with initial amplitude a0 we define the SNR as ! E[max{(sij )2 }] 1 a0 ∆x∆y 2 SNR = 10 log10 = 10 log10 E[(wij )2 ] 2E[η] 2πΣ2 a0 ∆x∆y . (3.37) = 20 log10 2πΣ2 Results for the conventional Rayleigh likelihood can be seen in Figure 3.7. Although the method manages to track at the rather high SNR of 8dB, its performance is severely deteriorated for lower SNR. More serious is that its detection performance is not reliable. It concludes that a target is more likely to be present than not, irrespectively of whether there actually is a target present. These graphs should be compared with the graphs in Figure 3.8 obtained by using the likelihood of section 3.3.4. One can see how the clutter estimation allows one to track the targets with a higher accuracy, and also how the existence probability PE gives a much more reliable indication on whether a target is present. When the SNR goes below 4dB we can no longer expect to detect the target. The performance is thus slightly lower than the performance reported by [92] or [94]. This price is paid both for working with Rayleigh noise (instead of the less confusing Gaussian noise) and for working with unknown noise.

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3.5

67


20

Position y

15

10

5

0

0

5

10 Position x

15

20

(a) Data at k = 10.

(b) Particles at k = 10.

20

Position y

15

10

5

0

0

5

10 Position x

15

20

(c) Data at k = 30.

(d) Particles at k = 30.

20

Position y

15

10

5

0

0

5

10 Position x

15

(e) Data at k = 60.

20

(f) Particles at k = 60.

Figure 3.6: Data and tracking results for a target of SNR 5dB. The red curve represents the mean position estimated from the particle cloud while the other curve (white or black) represents the true track. Although impossible to tell by visual inspection of the data, the target is only present in the middle frame at k = 30. However, we see in Figure 3.6b that possible candidates are found in the data at k = 10 as well (where the particle cloud thickens). If the uncertainty of ηˆ was not accounted for, we would risk a track being declared at one of these spots. Also notice in Figure 3.6f that the particle cloud is kept for some time after the target disappears, in case it should appear again.

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68

3


7 1 0.9

4 dB 6dB 8dB

6

0.8 5 RMS error [pixels]

0.7

PE

0.6 0.5 0.4 0.3

4

3

2

0.2

4 dB 6dB 8dB Truth

0.1 0 0

10

20

30 Time step k

1

40

50

0

60

0

10

(a) Probability of existence

20

30 Time step k

40

50

60

(b) Position RMSE

Figure 3.7: Probability of existence and position RMSE for conventional Rayleigh likelihood. From the beginning the tracker is unable to decide whether a target is present or not. As the true target appears, the tracker gradually builds up its confidence that something is present, but the graphs in 3.7b reveal that it only has a very rough idea where this something is located.

7 1 0.9

4 dB 6dB 8dB

6

0.8 5 RMS error [pixels]

0.7

PE

0.6 0.5 0.4 0.3

4

3

2

0.2

4 dB 6dB 8dB Truth

0.1 0 0

10

20

30 Time step k

1

40

(a) Probability of existence

50

60

0

0

10

20

30 Time step k

40

50

(b) Position RMSE

Figure 3.8: Probability of existence and position RMSE for unknown Rayleigh likelihood. The conservative treatment of amplitude information allows the tracker to correctly discard arbitrary background fluctuations, so that no target is declared present before the actual target appears. Whenever this method succeeds in detecting the target, it also localizes the target within sub-pixel accuracy.

60

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3.6

3.6

Concluding remarks

69

Concluding remarks

A recursive TBD method has been developed that integrates the uncertainty of the background estimate as part of its likelihood evaluation. Simulation results show how this algorithm works when clutter is not known. The conventional algorithm treating the clutter estimate as a deterministic constant fails. The proposed approach is thus a step towards recursive TBD methods that work on real data where the clutter power cannot be treated as a priori known. Unknown background noise is only one of several problems related to likelihood ratio based TBD. Satisfactory behavior of such methods on real data will also require topics such as Swerling fluctuation models, colored background noise and multiple targets to be addressed. Especially the last topic is conceptually challenging since, as argued in Section 2.6.4, the multi-target TBD problem is ill-posed without further restrictions. It is possible that the stabilizing effects of such restrictions would be sufficient to deal with the unknown background problem. It can be argued that a multi-target TBD method should not allow what happens in one part of the state space to immediately affect what happens in a completely different part of the state space. If this principle is adhered to, it would prevent spontaneous estimation errors in one region from claiming probability mass occupied in another region. It is nevertheless interesting to see that the marginalization over estimation uncertainty is sufficient to solve this problem. Another interesting problem is the application of TBD methods to targets in heavytailed backgrounds. However, before such a challenge can be addressed, the behavior of conventional tracking methods such as the PDAF should be investigated in heavy-tailed clutter. This is the topic of the next chapter.

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70

3


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4 Target tracking in heavy-tailed clutter using amplitude information The radar and sonar signal processing communities have during the last decades devoted increasing efforts to the treatment of clutter which is less benign than the Gaussian and Rayleigh models used in the previous chapter. Analysis of experimental data have revealed that real environments often are characterized by heavy-tailed backgrounds. Target tracking in heavy-tailed environments is challenging even for moderate signalto-noise ratios due to the increased frequency of target-like outliers. A tracking method operating in such an environment should exploit as much of the data as possible in order to ensure robustness. Still, conventional tracking methods rely on kinematic measurements such as range, bearing and Doppler only. The performance of the tracking method can be improved by using the backscattered signal strengths together with the kinematic measurements. This is done in the PDAFAI. This chapter, which is based on the journal article [27], proposes new conservative amplitude likelihoods for the PDAFAI with improved robustness compared to existing methods. The first likelihood works by incorporating the uncertainty of the background estimate under the conventional Rayleigh assumption. The second likelihood explicitly treats the background as heavy-tailed using the K-distribution. Extensive and realistic Monte-Carlo simulations show that both the conservative likelihoods give significant reductions in track-loss. Furthermore, this chapter provides a quantitative evaluation of the difficulties encountered by tracking methods in heavytailed clutter. This is the first such analysis that can be found in the open literature.

4.1

Introduction

In harbor surveillance, sensors such as radars and sonars are used to detect and track various targets. The sensor provides an array of observations to be further analyzed through 71

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72

4

Target tracking in heavy-tailed clutter using amplitude information

signal processing and data analysis. After preprocessing, such as matched filtering, a detector is employed to extract point measurements from the array. This is done by setting a threshold according to a CFAR criterion [42]. Then tracks are established and maintained by feeding these extracted point measurements to a target tracking algorithm. The targets we have in mind are small boats in radar data or human divers in sonar data. Both can be expected to have a value of the SNR somewhere between 0dB and 20dB [26]. This means that if the target is to be detected with a reasonably high probability one must allow some false alarms to occur. In order to establish and maintain a track on such a target one must determine which detections are actually from the target, and which detections are to be considered as false alarms or clutter. A popular solution to this problem of data association is found in the PDAF and its multi-target version the JPDAF [6]. One way to improve the performance of the PDAF is the utilization of amplitude information (AI). In [63] the PDAFAI (PDAF with amplitude information) was therefore presented. While the PDAF only uses kinematic measurements zk (i), the PDAFAI also uses the corresponding amplitudes ak (i) in order to decide which measurement is most likely to originate from the target. Of course the utilization of AI can be incorporated in other tracking methods as well. Although AIbased tracking methods have been around for some time, the usage of AI does not appear to be commonplace in practical systems. Most work on AI assumes that the background is Gaussian or Rayleigh distributed [63,93] with a known power. Such assumptions may or may not be adequate in the real world. Experimental evidence [1] has indicated that more heavy-tailed background models should be considered. In particular, the K-distribution [108] is popular in the radar and sonar communities. The radar and sonar communities have published many papers on detectors and their performance in heavy-tailed clutter. However, these papers make no mention of target tracking. This is problematic, since a reliable decision regarding the presence of a target only can be carried out after target tracking. A conventional detector does not really attempt to determine the presence of a target. It only extracts measurements to be used by a tracking method. This is especially so for a detector in heavy-tailed noise, which may need to operate at quite high false alarm rates (≈ 10−2 ). One way to overcome this challenge is to integrate the data before the detection decision is made, as done in TBD. For very low SNR, when the target return cannot be expected to exceed any reasonable threshold, this is the only feasible approach to target tracking. Otherwise TBD is not worthwhile the very large computational expenses. In the previous chapter it was observed that effects similar to the well known phenomena of CFAR loss [42] could deteriorate the performance of TBD methods. Since TBD only relies on amplitude information, it is more sensitive to the interpretation of the background than conventional tracking methods. To prevent the method from being misled by misinterpretation of the background it was necessary to make its amplitude likelihood more conservative. This was done by marginalizing over the true, but unknown, noise parameter.

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4.2

73

Conceptual framework

The contributions of this chapter are twofold. First, the developments of the previous chapter are adapted to the PDAFAI. The conservative PDAFAI thus obtained is more robust than the conventional PDAFAI in near Rayleigh clutter. Second, the first systematic investigation of target tracking in heavy-tailed clutter is presented in this chapter. Extensive Monte-Carlo simulations show that a substantial performance gain can be achieved by accounting for heavy-tailedness. We tailor the PDAFAI to cope with K-distributed clutter by specifying the appropriate amplitude likelihoods and explaining how they can be evaluated efficiently. The remainder of the chapter is organized as follows: Section 4.2 presents the framework in which our tracking methods are developed. In Section 4.3 we describe the tracking methods to be used, with particular attention to the amplitude likelihoods. We also describe the detection process preceding target tracking, and how the various quantities involved are estimated. Readers familiar with this material may proceed directly to the main results in Sections 4.3.6 and 4.3.7. In Section 4.4 the tracking methods are tested on simulated data with varying degrees of heavy-tailedness. A conclusion and some topics for further research are given in Section 4.5. Details regarding the numerical evaluation of the amplitude likelihoods are left for Appendix B.

4.2


The aim of single-target tracking is to evaluate the posterior probability density function (pdf) p(xk |Z k ) from the set of received measurements Z k = {Z1 , . . . , Zk } where each k Zk contains mk measurement vectors: Zk = {ζk (i)}m i=1 .

4.2.1

Kinematics

In accordance with the assumptions underlying the PDAF we assume a linear kinematic transition prior, xk = F xk−1 + vk , vk ∼ N (0, Q) (4.1)

where xk is the kinematic state and vk is the plant noise. The state does typically contain the position and velocity of the target. The mean target power dk should in principle also have been included in the target state. On the other hand it is very difficult to estimate a time-varying target power accurately for fluctuating targets [93]. It makes more sense to talk about the target power as something being constant over several (say ≥ 20) data frames. Therefore we treat the target power state as a constant parameter and write dk = dk−1 = d.

4.2.2

Measurement model

We parameterize the measurement vector into a kinematic part zk (i), an amplitude part ak (i) and a background part qk (i) so that ζk (i) = [zkT (i), ak (i), qkT (i)]T .

(4.2)

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74

4


The background part qk (i) contains a description of the background in the vicinity of zk (i), and is further elaborated in Sections 4.3.1 and 4.3.2. In order to make the estimation problem well-posed we assume that at most one measurement ζk (i) can originate from the target. All the other measurements are thus considered as false alarms. The false alarms are uniformly distributed over the surveillance region. The true measurement is linearly related to the target according to (4.3)

zk = Hxk + wk , wk ∼ N (0, R)

where wk is the measurement noise. The amplitudes ak (i) are under the hypotheses of noise or a target embedded in noise assumed to be distributed according to likelihoods p0 (ak (i) | qk (i)) or p1 (ak (i) | dk , qk (i)) respectively. These are thoroughly discussed in Sections 4.3.6 and 4.3.7.

4.2.3

The CLT and the Rayleigh model

The background model most often encountered in the tracking literature is the Rayleigh distribution, which is valid under the CLT. Assume that the complex data in resolution cell j at time k is a sum of njk contributions from independent scatterers, j

zkj ≈

nk X

Ajlk e−iθlk .

(4.4)

l=1

where i is the imaginary unit. It is obviously reasonable to assume the phases θlk uniformly distributed on [0, 2π). If njk is non-random and large enough, the Central Limit Theorem applies. Then zkj has a complex Gaussian distribution, implying that the envelope or amplitude ajk is Rayleigh distributed, ajk

=

|zkj |

∼

Rayleigh(ajk

aj ; η) = k exp η

−(ajk )2 2η

!

(4.5)

where η is the power of the background.

4.2.4

Heavy-tailed clutter and the K -distribution

In reality the number of significant scatterers per resolution cell may not be high enough for the CLT to hold. One often observes a more heavy-tailed background than expected according to the CLT. Several distributions have been suggested as a replacement of the Rayleigh distribution in such cases. We mention here the Weibull distribution [64], the log-normal distribution [46], stable distributions [30], Rayleigh mixtures [65] and the normal inverse Gaussian distribution [80]. A comparison of various heavy-tailed distributions can be found in [1].

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4.2

75


The possibly most popular model for non-Rayleigh clutter is the K-distribution, which was proposed as a model for radar clutter in [53]. If the amplitude a is Kdistributed (we drop the cell index j and the time index k in the remainder of Section 4.2 for notational simplicity) it has the pdf 4aν 2a Kpdf (a ; ν, b) = √ ν+1 (4.6) Kν−1 √ b b Γ(ν) and the cumulative distribution function (cdf) Kcdf (a ; ν, b) = 1 − √

2aν ν

b Γ(ν)

Kν

2a √ b

.

(4.7)

By Γ(·) we denote the Gamma function while Kν (·) refers to the modified Bessel function of the second kind [3 p. 374]. For ν → ∞ the K-distribution turns into the Rayleigh distribution. The K-distribution is popular for two reasons: Physical plausibility and mathematical convenience. The first reason has to do with the fact that reasoning similar to the CLT leads to the K-distribution [2,54]. The second reason is that the K-distribution has certain mathematical benefits that other heavy-tailed distributions lack. In contrast to so called stable distributions it has finite moments, √ nΓ 1 + n Γ ν + n n 2 2 mn = E [a ] = b . (4.8) Γ(ν) It is just as convenient that the K-distribution can be viewed as a Rayleigh distribution modulated by a Gamma distribution. In mathematical terms, η ν−1 exp − 2η b p(η ; ν, b) = Gamma(η ; ν, b/2) = ν (b/2) Γ(ν) p(a|η) = Rayleigh(a ; η) ⇒ p(a ; ν, b) = Kpdf (a ; ν, b) .

(4.9)

This has motivated researchers to introduce the compound K-model [107], in which radar sea clutter is modeled by treating η as a highly correlated “texture” component, while a|η is an uncorrelated “speckle” component. We let η refer to both the deterministic Rayleigh parameter in (4.5) and to the texture random variable in (4.9). The context will make it clear how η is supposed to be interpreted.

4.2.5

Models for target plus clutter

The compound interpretation of the K-distribution allows us to evaluate the pdf of a Swerling I target in K-distributed clutter as follows. Conditioned on the texture η, the background “noise” w has a complex zero-mean Gaussian pdf, Re(w) 0 η 0 p(w|η) = Nc (η) = N ; , . (4.10) Im(w) 0 0 η

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76

4


Heavy-tailed case (ν = 0.5)

Rayleigh case 100 pdf (logarithmic scale)

pdf (logarithmic scale)

100

10−5

10−5

clutter only clutter + target .

clutter only clutter + target . 10−10

0

5 amplitude a

10

10−10

0

5 amplitude a

10

Figure 4.1: Pdf’s of target plus clutter for a 15dB target in both Rayleigh and K-distributed background noise compared with the corresponding pdf’s of clutter only. The complex backscatter signal s from a Swerling I target with power d also is Gaussian, p(s|d) = Nc (d).

(4.11)

Therefore the sum z = s + w of signal and noise conditioned on η is Gaussian as well, and the corresponding amplitude a = |z| is Rayleigh, p(z|d, η) = Nc (d + η) ⇒ p(a|d, η) = Rayleigh(d + η).

(4.12)

Under the compound K-model the texture η is random, so the pdf of the amplitude must be found by marginalizing over η,

p(a|d, ν, b) =

Z∞ 0

b Gamma η; ν, Rayleigh(a; d + η)dη 2

a = ν b Γ(ν)

Z∞ 0

η ν−1 η a2 exp − − dη. η+d b 2(η + d)

(4.13)

Figure 4.1 shows that although the pdf’s of target and clutter are more similar for heavytailed background noise than for Rayleigh distributed background noise, there is still a noticeable difference. 1

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4.3

77

Methodology

4.3

Methodology

Most tracking systems employ a modularized architecture in which a detector extracts measurements to be fed to the data association and state estimation methods. This does not necessarily have to be the case, as exemplified by TBD methods which do not carry out any measurement extraction at all. Although our work is relevant for such methods as well [21], we are here content with the modularized architecture as represented by the PDAF and the PDAFAI methods.

4.3.1

Measurement extraction for Rayleigh case

For any cell i a detection is declared if its amplitude exceeds a certain threshold tiRk . The detection threshold tiRk is determined from a set of M auxiliary cells GiR according to the closed form formulas 1 X j 2 ηbki = (4.14) (ak ) 2M i j∈GR r tiRk =

−1/M

2M PFA

− 1 · ηbki

(4.15)

where ηˆki is the MLE of the background strength. The detector given by (4.14) and (4.15) gives an actual false alarm equal to the design false alarm rate PFA as long as the background is Rayleigh. It is therefore referred to as a CFAR detector [42]. The set of auxiliary cells is chosen to lie in the vicinity of the cell under test, although with guardbands to prevent them from being affected by signal leakage. Each extracted measurement vector contains the location of the detected cell, as well as the value of the cell and the clutter estimate from its auxiliary cells. As discussed in Section 4.2.2, range and bearing can be converted to cartesian coordinates using simple formulas [6 pp. 38-41]. The full measurement vector of (4.2) is under the assumption of a Rayleigh background parameterized as ζ(i) = [z T (i), a(i), ηˆ(i)]T .

(4.16)

Here z(i) = [˜ x(i), y˜(i)]T is the centroid of resolution cell number i in the cartesian coordinate system, which may or may not be related to the target through a model such as (4.3).

4.3.2

Measurement extraction for the K -distribution case

If the background is significantly heavy-tailed, the detector given by (4.15) becomes increasingly inadequate. Instead of attempting to estimate the local background power η, we will estimate the K-distribution parameters ν and b from a somewhat larger set of N auxiliary cells, and then set the threshold according to the design false alarm rate for a K-distribution with these parameters.

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78

4


Range ρ

Cell under test

Guard band

Auxiliary cells GiR

Bearing φ

Figure 4.2: The auxiliary cells used to estimate the background under Rayleigh assumption. Abraham [2] recommends an MoM estimator using the first and the second moments of the data given by mi1 =

1 X j 1 X j 2 ak and mi2 = (ak ) . N N i i j∈GK

(4.17)

j∈GK

This estimator attempts to solve the equation (again we drop time and cell indices) m2 4νΓ2 (ν) . = m21 Γ ν + 21

(4.18)

Since no closed form solution can be found the equation must be solved numerically. As explained in [2] a second-order approximation can be found as πm2 −1 1 νˆ = log . 4 4m21

(4.19)

While [2] recommends (4.19) as a starting point for the numerical search, we simply use (4.19) as our estimator of ν. This is justified by the fact that more advanced estimators require numerical searches whose runtime easily will exceed the entire tracking machinery of Sections 4.3.3 - 4.3.7 by far. Figure 4.5 shows that for very heavy-tailed

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4.3

79

Methodology

Range ρ

Cell under test

Guard band Auxiliary cells GiK

Bearing φ

Figure 4.3: The auxiliary cells used to estimate the background under Kdistribution assumption. clutter (ν ≈ 0.1) the estimator (4.19) has a significant bias, and the numerical search is necessary in order to obtain an unbiased estimator. But for moderately heavy-tailed clutter (ν ≥ 0.5) the numerical search appears to be overkill. Having estimated ν, we estimate b as ˆb = m2 /ˆ ν. (4.20) We can now for any resolution cell i determine a threshold tiK according to the criterium 2aν 2tK P (a > tK ) ≈ 1 − √ ν Kν √ = PFA . (4.21) b b Γ(ν)

This is a nonlinear equation in tK which must be solved numerically. Instead of working directly with the cumulative distribution function as in (4.21), we recommend using the square of its logarithm for numerical stability. A test using the threshold tK is carried out in order to decide whether cell i should be stored among the detected measurements. When a K-distributed background is assumed we must store two background estimates for each measurement vector. In addition it is convenient to store the detection threshold tK (i): ζ(i) =[z T (i), a(i), νˆ(i), ˆb(i), tK (i)]T =[z T (i), a(i), q(i)]T .

(4.22)

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80

4


Samples per histogram bin

Performance of MoM-estimators for ν = 0.1 Several iterations One iteration

1500

1000

500

0

0.05

0.1

0.2 0.15 0.25 Estimated shape parameter νˆ

0.3

0.35

0.4

Figure 4.4: Histograms illustrating the distribution of νˆ when the true shape parameter is ν = 0.1.

Samples per histogram bin

Performance of MoM-estimators for ν = 1 Several iterations One iteration

800 600 400 200

0

0.5

1.5 1 2 Estimated shape parameter νˆ

2.5

3

Figure 4.5: Histograms illustrating the distribution of νˆ when the true shape parameter is ν = 1.

1

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4.3

81

Methodology

4.3.3

Probabilistic Data Association

The PDAF [6 pp. 129-161] is perhaps today’s most popular tracking algorithm due to its pragmatic compromise between efficiency and robustness. It is a suboptimal algorithm which at each time step collapses the target state posterior pdf into a single Gaussian which then is propagated to the next time step. Thus the past information about the target at time step k can be summarized by ˆ k|k−1 , Pk|k−1 ). p(xk |Z k−1 ) ≈ N (xk ; x

(4.23)

The lumping (cf. Figure 4.6) implied by (6.3) is done by expressing the state estimate ˆ k|k−1 and state estimates conditioned at time k as a weighted average of the prediction x on the latest measurements zk (i). This leads to the following Kalman Filter-like equaˆ k|k and its associated tions for prediction and measurement update of the state estimate x covariance Pk|k : ˆ k|k−1 = F x ˆ k−1|k−1 x Pk|k−1 = F Pk−1|k−1 F T + Q ˆ k|k = x ˆ k|k−1 + Wk x

mk X

βk (i)νk (i)

i=1

where

Pk|k = Pk|k−1 − (1 − βk (0))Wk Sk WkT + P˜k

(4.24)

Wk = Pk|k−1 H T Sk−1 Sk = HPk|k−1 H T + Rk ˆ k|k−1 = zk (i) − zˆk|k−1 νk (i) = zk (i) − H x "m # k X P˜k = Wk βk (i)νk (i)νk (i)T − νk ν T W T k

k

i=1

νk =

mk X

βk (i)νk (i).

(4.25)

i=1

The index i ranges over all mk extracted and validated measurements inside a validation gate around zˆk|k−1 . The validation gate is defined by the criterium T zk (i) − zˆk|k−1 Sk−1 zk (i) − zˆk|k−1 < g 2 (4.26) where the gate threshold g typically has a value in the interval between 3 and 10. Derivations of the association weights βk (i) can be found in [6 pp. 134-157]. In the conventional case without AI they are written  e(i)  P mk i = 1, . . . , mk b+ j=1 e(j) βk (i) = (4.27)  Pmbk i=0 b+ e(j) j=1

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4


Signal processing

Tracking

Sensor input

zk (i) Measurement update

[a1k , . . . , aN k ] Background estimation qk (i)

ˆ k|k−1 , Pk|k−1 x


ˆ k|k (i) x

Pk|k (i)

Merging ˆ k|k x

Pk|k

Set Threshold

Prediction

Measurement extraction

Association

βk (i)

82

ak (i), qk (i)

Figure 4.6: Schematic description of the PDAF and PDAFAI methods. The dashed connection represents the flow of amplitude information from detector to tracker, and is only present in the PDAFAI.

where

T e(i) = exp − zk (i) − zˆk|k−1 Sk−1 zk (i) − zˆk|k−1 /2 .

(4.28)

√ b = (2π/g)mk (1 − PD PG )/( πPD ).

(4.29)

The value of b depends on the model for the false alarm point process. Here we use the non-parametric PDAF which assumes no knowledge about the PMF of false alarms. For 2-dimensional zk (i) it can then be shown that

The constant PD is the probability of detection while PG is the probability that a true detected measurement is inside the gate. PD is a tuning parameter assumed known (cf. Section 4.3.5), while PG can be calculated from g as explained in [6 p. 96].

4.3.4

PDAF with Amplitude Information

In [63] it was suggested to improve the performance of the PDAF algorithm by using the amplitude of the measurements as part of the tracking algorithm. The amplitude

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4.3

83

Methodology

information only affects the association weights and not the state estimation itself, as illustrated in Figure 4.6. The modified association weights are given by T e(i) = exp − zk (i) − zˆk|k−1 Sk−1 zk (i) − zˆk|k−1 /2 · [p1 (ak (i)|dˆk , qˆk (i)) / PD ] / [p0 (ak (i)|qˆk (i)) / PFA ] T = exp − zk (i) − zˆk|k−1 Sk−1 zk (i) − zˆk|k−1 /2 · l(ak (i)|dˆk , qˆk (i))

(4.30)

where the amplitude likelihood ratio l(ak (i)|dˆk , qˆk (i)) depends on the observed amplitude ak (i), the target power estimate dˆk and a description of the background which is contained in qˆk (i). The likelihoods p0 (. . .) and p1 (. . .) are the pdf’s of the amplitude under the two hypotheses of only clutter and of clutter plus a target with power dˆk . Their exact expressions depend on how the background is modeled. Notice that p0 (. . .) and p1 (. . .) are divided by PFA and PD respectively in order to compensate for the fact that ak (i) has exceeded the detection threshold.

4.3.5

The target power

The mean target power parameter d is in general not readily available. One possible option is to set the power parameter according to the lowest SNR one expects to be able to track [16]. This is not advisable in heavy-tailed clutter, since it can make the clutter appear more target-like than necessary. Another option is to determine the power parameter from off-line tools such as the sonar equation [103 p. 29]. The problem with this approach is that the expected SNR thus obtained will depend on several factors which may or may not provide an accurate model of reality. A third approach adopted here is to estimate the power parameter over the last L consecutive frames using momentbased estimators. We discuss how to do this by means of an example. Assume we have L amplitude samples {al }L l=1 in which a target with power parameter d is supposed to be present. Let each al have M auxiliary cells {ylj }M j=1 as discussed in Section 4.3.1. Furthermore we assume that al ∼ Rayleigh(ηl + d)

ylj ∼ Rayleigh(ηl ) for j = 1, . . . , M.

(4.31)

The power parameter d and the background strength parameters ηk = [η1 , . . . , ηk ] should ideally be jointly estimated. An MLE can be found by maximizing the joint j M L likelihood of the target cells {al }L l=1 and the set of auxiliary cells {{yl }j=1 }l=1 . Even in this very simple case the MLE requires an inconvenient numerical search, and the MLE is not particularly superior to other estimators. A much simpler Best Linear

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Unbiased Estimator (BLUE) [59 p. 133] performs equally well. The BLUE is given by ηbl =

M 1 X j 2 (yl ) 2M j=1

L L M 1 XX j 2 1 X 2 al − (yl ) . db = 2L 2LM

(4.32)

l=1 j=1

l=1

Both the MLE and the BLUE suffer from the possibility that db may turn out negative. The BLUE is obviously unbiased (given that the Rayleigh assumption holds) and its mean square error is not significantly worse than the mean square error of the MLE. Based on this we suggest the following moment-based estimator dˆk of the target power d at time step k, Ck B k − Ck b dk = max , (4.33) Dk Dk where

1 Bk = 2L 1 Ck = L

L X

l=1 L X l=1

mk−l+1

1 mk−l+1 1

mk−l+1

X

a2k−l+1 (i)βk−l+1 (i)

i=1 mk−l+1

X

ηˆk−l+1 (i)βk−l+1 (i)

i=1

!

!

L 1X Dk = (1 − βk−l+1 (0)). L l=1

This estimator is also used to estimate the target power when the background is Kdistributed. Then ηˆk−l+1 (i) is found as half the second moment, or equivalently ηˆk−l+1 (i) =

νˆk−l+1 (i)ˆbk−l+1 (i) . 2

(4.34)

The time lag L is a tuning constant whose optimal value hardly can be determined rigorously. On the one hand the motivation for using a Swerling I model instead of a Swerling 0 model is that the target return may change abruptly from scan to scan without any change in the “expected” power of the target. Also, too few samples will make this estimator unacceptably inaccurate. On the other hand, more permanent changes in the target amplitude may occur quite abruptly. As a compromise we have found L = 20 to be a reasonable value. When the track is younger than 20 time steps (i.e. k < 20) we necessarily use a shorter lag. The amplitude estimate plays a fundamental role in the likelihood ratio introduced in Section 4.3.4. It is also useful in order to determine the expected probability of detection PD . For the case of a Swerling I target in Rayleigh clutter we find the detection

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Methodology

probability as

−1/M PD = 1 + PFA −1

η d+η

−M

.

(4.35)

When the clutter is K-distributed we find the detection probability by reasoning analogous to (4.13) as (2/b)ν PD = Γ(ν)

Z∞ 0

t2K 2 η ν−1 exp − η − dη b 2(η + d)

(4.36)

where tK is the detection threshold as obtained from (4.21). How to evaluate (4.36) efficiently is explained in Appendix B.2.

4.3.6

Amplitude likelihoods for Rayleigh case

Under the Rayleigh-Swerling I assumption used in [63] the likelihoods of Section 4.3.4 become 2 a −a p0 (a|η) = exp η 2η a −a2 p1 (a|d, η) = exp (η + d) 2(η + d) η a2 d PFA · · exp . ⇒ l(a|d, η) = PD η + d 2η(η + d)

(4.37)

In reality one must replace the parameters η and d by their corresponding estimators ηˆ ˆ Both dˆ and ηˆ (c.f. Figure 4.7) suffer inevitably from a high uncertainty. While and d. the uncertainty of dˆ has been discussed in [31], we are here more concerned about the uncertainty of ηˆ. The background power estimator ηˆ may attain different values for different measurements and therefore change the relative association weights. In this section we treat the uncertainty of ηˆ in a way analogous to what is done in the CFAR detector (4.15). In the context of TBD this was shown to have a crucial impact on performance in Chapter 3. The estimator ηˆ given by (4.14) is a random variable with a corresponding pdfthat should be accounted for in the likelihood evaluation. Its pdfp(ˆ η |η) depends on the true noise parameter η. To utilize the information carried in this pdfrequires us to treat η as random as well and assign it a prior distribution p(η). By marginalizing the joint distribution of a and η given ηˆ we obtain p(a|d, ηˆ) ∝

Z

p(a|d, η)p(ˆ η |η)p(η)dη.

(4.38)

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The background estimate ηˆ given by (4.14) is under the assumption of independent Rayleigh samples known to follow a Gamma distribution, M −1 exp − ηˆM η ˆ η η p(ˆ η |η) = Gamma M, . (4.39) = η M M Γ(M ) M

Since we do not have any more prior information than the value of ηˆ itself we use a flat and thus non-informative prior in the Rayleigh case: (4.40)

p(η) = Uniform(η; [0, ξ]).

Here ξ is a very large number. The flat prior allows us to use p(ˆ η |η) without being bothered by p(η). The likelihood ratio becomes PFA p(a|d, ηˆ) · PD p(a|0, ηˆ) Rξ η PFA 0 Ra (a ; η + d) Gamma ηˆ ; M, M dη = . Rξ PD Ra (a ; η) Gamma ηˆ ; M, η dη

l(a|d, ηˆ) =

0

For ξ → ∞ we obtain PFA l(a|d,ˆ η) = ·h PD

Γ(M ) ηˆM +

a2 2

iM

(4.41)

M

·

Z∞ 0

1 ηˆM a2 exp − − dη. (η)M (η + d) η 2(η + d) (4.42)

Appendix B.3 explains how this expression can be evaluated in an efficient way. The likelihood ratio given by (4.42) is more conservative than the conventional likelihood ratio of (4.37). While (4.37) increases fast and unboundedly with a, the conservative likelihood ratio of (4.42) saturates and even falls back to unity for very large a. In other words it refuses to make any decision if a is poorly described by the models of target and clutter. An alternative approach could be to develop a formula for l(a|d, ηˆ) in the framework of robust statistics [58]. A simple technique of saturation thresholds have been recommended by [75] in the TBD setting. Unfortunately no rigorous recipe for setting the saturation threshold is known. On the other hand, (4.42) does a similar job automatically.

4.3.7

Amplitude likelihood for K -distribution case

In Section 4.3.6 we treated η as random, knowing that this was only done for mathematical convenience. When the background contains significant heavy-tailedness the uncertainty of η becomes more severe than the one induced by the CFAR background

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Uncertainty of background estimate 14 M = 16 M = 50 M = 1000

PDF p(ˆ η |η) of the MLE

12 10 8 6 4 2 0

0

0.2

0.4

0.6

1 1.2 0.8 1.4 MLE ηˆ of η in units of η

1.6

1.8

2

Figure 4.7: Probability density functions of the MLE ηˆ for various numbers M of auxiliary cells.

PFA = 0.005, SNR = 15dB 40


30 20 10 Conventional Rayleigh ν=8 Conservative Rayleigh ν=1 ν = 0.1

0 -10 0

2

4

6

8

12 10 Amplitude a

14

16

Figure 4.8: Various likelihoods for SNR = 15.

1

18

20

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estimate (4.14). It then makes sense to treat η as fully random and replace the flat prior of (4.40) by the Gamma prior of (4.9). This informative prior is in practice given by our local K-distribution parameter estimates νˆ and ˆb as obtained from (4.19) and (4.20). Since the expressions involved already are intractable enough we do not make any attempts at accounting for the estimation error as was done in (4.38) or (4.42). Rather we simply evaluate p1 (a|d, ν, b) and p0 (a|ν, b) under the compound K-model (4.9) as done in (4.13). This yields 4aν 2a p0 (a) = √ ν+1 Kν−1 √ b b Γ(ν) ∞ Z ν−1 a η η a2 dη p1 (a) = ν exp − − b Γ(ν) η+d b 2(η + d) 0

√ Z∞ ν−1 PFA (a b)1−ν a2 η η ⇒ l(a) = · exp − − dη. 2a PD 4K η+d b 2(η + d) ν−1 √b 0

(4.43)

Again, we encounter an integral with no closed form solution. We solve it numerically using the same techniques as used for the integral in (4.36). This is elaborated in Appendix B.1. Figure 4.8 shows the log likelihood ratios ln l(a| . . .) corresponding to (4.37), (4.42) and (4.43) for various shape parameters ν. Assuming that our estimate of the target power d corresponds to a target with SNR = 15dB, the curves show the log likelihood ratios as functions of the actual amplitude value. Heavy-tailed noise is seen to lead to a more conservative likelihood. For really spiky clutter (ν ≈ 0.1) the log likelihood ratio curve corresponding to (4.43) hardly exceeds zero, which tells us that tracking a 15dB target in such noise is pretty much futile. For ν = 8 the behavior of (4.43) is not very different from (4.37), which means that the Rayleigh approximation may be considered adequate. The behavior of (4.42) is on the other hand, as explained in Section 4.3.6, always qualitatively different from (4.37).

4.4


In this section several PDAF-based tracking methods are tested in more or less heavytailed environments using Monte-Carlo simulations. The simulator generates raw sensor data according to the recipe of [108 pp. 145-165], so that the entire machinery of detection and tracking methods described in Section 4.3 can be tested. The simulated scenarios are supposed to be tough; pushing all the methods discussed to their limits.

4.4.1

Background simulation

In order to provide the simulated data with correlations similar to those encountered in the real world we use the compound K-model (4.9). The “texture” η is supposed to have

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89


Value of Gamma random variable η

The MNLT ξ → η

100

10−2 ν = 0.1 ν=1 ν=8 10−4

-3 0 2 3 -2 -1 1 Value of unit variance Gaussian random variable ξ

-4

4

Figure 4.9: The transform ξ → η for various shape parameters ν. Notice that the Gamma variable η is plotted in logarithm domain. a “global” ACF RΓ (z) which accounts for inhomogeneities in the medium or scattering surface such as wave patterns. In order to also account for correlations induced by signal processing we let a be correlated according to a “local” ACF RL (z). For notational brevity we let z both refer to 3-tuples [ρ, φ, k]T and to the “lags” between them. The coordinates ρ and φ refer to range and bearing measured in resolution cells. The correlated Gamma process η(z) is simulated by generating a correlated Gaussian process ξ(z) and then mapping it into a Gamma process using a so-called Memoryless Non-Linear Transform (MNLT). In order to do this one must determine what ACF ξ should follow in order to provide η with the desired ACF. The MNLT is given by solving the following equation with respect to η: Zξ

−∞

N (u; 0, 1)du =

Zη

Gamma(v ; ν, b/2)dv.

(4.44)

0

This amounts to solving 1 1− γ Γ(ν)

2η ,ν b

− erfc

ξ √ 2

/2 = 0

(4.45)

where γ(·, ·) is the lower incomplete Gamma function and erfc(·) is the complementary error function as defined in [48]. The MNLT is in practice carried out using linear interpolation between tabulated values of ξ and ν. In Figure 4.9 the MNLT is plotted for various values of ν. The 1

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treatment of correlations is done using an expansion in Hermite polynomials Hn (ξ). Denoting the ACF of ξ by RG (z), it is shown in [108 p. 155] that  ∞ Z ∞ n X 1 RG (z)  RΓ (z) = exp(−ξ 2 )Hn (ξ) π 2n n! n=0 −∞ erfc(ξ) b −1 1− ; ν, dξ Gammacdf 2 2 ∞ X = αn RG (z)n . (4.46) n=0

The integrals in (4.46) depend on ν, and must be evaluated off-line for a suitable grid of values of ν before the MNLT can be constructed. When this is done we have the Gamma correlation function RΓ (z) expressed as a polynomial in the Gaussian correlation function RG (z). Inverting this polynomial for all lags z yields a mapping from the desired Gamma ACF to the Gaussian ACF used in simulation. The simulation of correlated Gaussian noise is done using standard techniques, and will not be elaborated. To summarize, we generate correlated K-distributed noise with a given shape parameter ν and ACF RΓ using the following recipe: 1. Store the mappings ξ → η from (4.45) in an interpolation table. 2. Store the coefficients from (4.46) in another table. 3. Convert the desired Gamma ACF RΓ (z) into the corresponding Gaussian ACF RG (z) using the output from step 2. 4. Draw Gaussian noise ξ(z) with ACF RG (z). 5. Convert ξ(z) into a Gamma process η(z) with shape parameter ν using the interpolation table from step 1. 6. Draw complex unit-variance Gaussian noise w∗ (z) with ACF RL (z) according to (4.49). 7. p Modulate the complex Gaussian noise with the Gamma process: w(z) = w∗ (z) · η(z).

The amplitude a(z) = |w(z)| is K-distributed with the desired correlation properties. A simple Markov model is used as the global ACF, |φ| |k| |ρ| RΓ (z) = exp − exp − exp − . (4.47) LΓ LΓ 4

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91


ν = 0.5, LΓ = 1 5

Amplitude a

4 3 2 1 0

0

10

20

30

50 60 40 Range cell number ρ

70

80

90

100

Figure 4.10: Example K-distributed data generated with the recipe of [108]. A target is embedded in the background at ρ = 9. The shape parameter is ν = 0.5 and the global correlength is LΓ = 1. ν = 0.5, LΓ = 1 5

Amplitude a

4 3 2 1 0

0

10

20

30

50 60 40 Range cell number ρ

70

80

90

100

Figure 4.11: Another time series of K-distributed data generated with the recipe of [108]. The global correlation length has now been altered to LΓ = 8. As a consequence, the effective degree of heavy-tailedness is reduced and the target can be detected with a higher confidence. Intuition tells us that such a scenario should be less challenging for a tracking method than the one depicted in Figure 4.10, but a moment-based estimator of the K-distribution will not be able to convey this information. 1

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The local ACF RL (z) is related to the sensor’s PSF, which can be deduced from the beamforming and the reception processes. To keep things simple we approximate the range PSF as a sinc function and the bearing PSF as a Gaussian, φ2 h(ρ, φ) = hρ (ρ)hφ (φ) = sinc exp − 2 . A 2B ρ

(4.48)

This implies that the local ACF in the above recipe is φ2 RL (ρ, φ) = sinc exp − 2 . A 4B ρ

(4.49)

Clutter or reverberation is supposed to outweigh thermal noise by far, so that there is no delta function component of RL . The coordinates ρ and φ represent the sensor’s coordinate system normalized to the sensor’s resolution. Examples of data generated using this recipe can be seen in Figures 4.10 and 4.11. Notice how the correlation length has a significant impact on the effective degree of heavy-tailedness. To mimic jamming and other sudden disturbances we have multiplied the background noise w(z) by two in time steps 5, 9, 15, 32, 37, 38 and 45. Thus the methods to be tested are not allowed to rely on temporal stationarity or correlation of the background.

4.4.2

Simulation of target signal

When a target is present, it is mapped into the data by the PSF h(ρ, φ). For any cell j in the vicinity of zk = h(xk ) its complex value zkj can be related to the state ψk by zkj =hj (xk )sk + wkj =h(ρj − ρ(xk ), φj − φ(xk ))sk + wkj

(4.50)

where sk ∼ Nc (0, d) under the Swerling I model and wkj is the complex noise generated as explained in Section 4.4.1. The notations ρj and φj refer to the centroids of cell j, while ρ(xk ) and φ(xk ) refer to the location of the point target in discrete sensor coordinates. The targets we simulate are only point targets, meaning that the target itself covers less than one resolution cell or thereabout. Nevertheless, due to blurring as modeled by (4.50), the point target will in general be visible in more than one sensor cell. This means that the mathematically convenient assumption of maximally one detection per target, which underlies the PDAF and the PDAFAI, does not hold. Although this does not appear to be an important violation for the scenario simulated here, it may lead to difficulties and require clustering of detections for other scenarios with coarser sensor resolutions.

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4.4.3

Simulation of target kinematics

Although we assume that the target moves according to a linear model on the form (6.1), we generate trajectories from a more complicated curvilinear model developed in [10] and discussed in [68]. The curvilinear model is arguably more realistic; modeling colored plant noise, maneuvers and sudden accelerations. It is especially in such situations that amplitude information may be useful, enabling the measurement update to override a more or less inadequate plant model. In the curvilinear model the state is parameterized as xk = [xk , yk , x˙ k , y˙ k , atk , ank ] where atk is acceleration tangential to the target trajectory and ank is acceleration perpendicular to the trajectory. The kinematic plant model can then be written as   Fcv Gt (xk ) Gn (xk )  xk + vk . βt 0 (4.51) xk+1 =  0 0 0 βn The matrices Gt (xk ) and Gn (xk ) are explicitly given by  − ω12 cos ϕk+1 + ω12 cos φk − ω1k T sin φk k k  1 1  ωk sin ϕk+1 − ωk sin φk Gt (xk ) =   − ω12 sin ϕk+1 + ω12 sin φk + ω1k T cos φk k

and



− ω12 sin ϕk+1 − k

k

− ω1k cos ϕk+1 + 1 ωk2

1 ωk

cos φk

sin φk −

1 ωk T

cos φk

     

  − ω1k cos ϕk+1 − ω1k cos φk   Gn (xk ) =    − ω12 cos ϕk+1 + ω12 cos φk + ω1k T sin φk  k k 1 1 ωk sin ϕk+1 − ωk sin φk q where ϕk+1 = φk + ωk T , ωk = ank /| x˙ 2k + y˙ k2 | and φk = tan−1 (y˙ k /x˙ k ). The constants βt and βn are related to the maneuver time constants τt and τn by βt = exp(−T /τt ) and βn = exp(−T /τn ).

(4.52)

In our simulations we draw the accelerations according to first order Markov models, so that vk ∼ N (0, Q) where   0 0 0 . 0 Q =  0 σt2 (1 − βt2 ) (4.53) 2 2 0 0 σn (1 − βn )

The noise matrix in (6.43) is more sparse than the matrix in (5.82) since the mapping of noise (i.e. acceleration) onto xk , x˙ k etc. is taken care of by (6.41) in the simulation model. The four tuning constants σt2 , τtm , σn2 and τnm can be chosen to accommodate for a wide range of scenarios corresponding to various types of targets.

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Table 4.1: Simulation model parameters Parameter

Value

Specification

T

1.0 s

Sampling period

N

140 × 80

Number of resolution cells

48.4 m

Innermost sensor range

rmax

158.6 m

Outermost sensor range

∆ϑ

55.5 ◦

Sensor bearing coverage

σt2

(0.1 m/s3 )2

Tan. acceleration power

τtm

2.0 s

Tan. acceleration time constant

σn2

(0.05 m/s3 )2

Perp. acceleration power

τnm

1.0 s

Perp. acceleration time constant

A

1.0 cells

Local range ACF parameter

B

1.3 cells

Local bearing ACF parameter

ν

{0.1, 0.5, 1, 2, 4, 8}

Shape parameters tested

LΓ

{1, 8} cells

Correlation lengths tested

SNR

{12, 15, 18} dB

Values of SNR tested

rmin

Table 4.2: Filter model parameters Parameter

Value

Specification

g

6

Gate size

PFA

0.005

Design false alarm probability

σr2 σϑ2 σv2

(0.23 m)2

Measurement noise (range)

(3.5 · 10−3 rad)2

Measurement noise (bearing)

(0.0125 m/s2 )2

Plant noise in filter model

L

20

Amplitude estimation lag

c1

5

Lower track-loss threshold

c2

15

Upper track-loss threshold

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4.4.4

Filter model

The linear models (6.1) and (4.2.2) are implemented using the following matrices 

1 T  0 1 F =  0 0 0 0  2 σv 3 T  σ32 2  vT 2 Q =   0 0 1 0 H= 0 0

 0 0 0 0   1 T  0 1 σv2 2 2 T σv2 T

0 0 0 0 1 0

(4.54)

0 0 σv2 3 3 T σv2 2 2 T

0 0 σv2 2 2 T σv2 T

.

     

(4.55)

(4.56)

The measurement noise matrix R is given by a conversion from polar to cartesian coordinates as described in [6 pp. 36-41]. In polar coordinates the covariance is Rpolar =

"

∆2r 12

0

0 ∆2ϑ 12

#

=

σr2 0 0 σϑ2

(4.57)

where ∆r and ∆ϑ are the sizes of a resolution cell in range and bearing respectively. The uniform distribution over a resolution cell is approximated by a Gaussian distribution with the same variance. Thus the number 12 appears.

4.4.5

Scenario

The target appears with initial kinematic state x0 = [64m, 82m, 0m/s, −0.75m/s] at time k = 0. It then moves according to the model given by (6.41) and (6.43) for the next 50 seconds. The sensor observes the region given by 48.36m < r < 158.57m and 0◦ < ϑ < 56◦ with a resolution of 140 range cells and 80 bearing cells. This is supposed to mimic a cutout of the images provided by the sonar used in [87]. Parameters used in the simulation of the scenario are summarized in Table 6.1. In order to distinguish the filter model from the simulation model we summarize parameters related to the filter model (i.e. governing the detection and tracking methods) in a separate Table 6.2. In order to test the performance of our tracking methods this scenario is run 5000 times for various SNR and shape parameters ν. The SNR is measured in decibel (dB), and can for a Swerling I target in K-distributed noise be related to the target power d as follows, E[max{|sj |2 }] 2d SNR = 10 log10 = 10 log10 . (4.58) j 2 E[|w | ] νb

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4.4.6

4


Performance measures

The most important measure of the performance of a tracking algorithm is whether or not it manages to maintain a track on the target. Only as long as it stays on the target does it make sense to discuss other measures such as root mean square error (RMSE). However, track-loss is a rather subjective phenomenon which cannot be defined rigorously. In this chapter we treat track-loss as a two-stage process. A track is considered q tentatively lost at time k if the position error (xk|k − xsk )2 + (yk|k − yks )2 exceeds a threshold c1 . If the error later goes below c1 the lost label is removed. On the other hand, if the error never manages to go below c1 again, we consider it lost at time k. If the error exceeds the higher threshold c2 > c1 we immediately consider it lost at time k, irrespectively on whether the error later goes below c1 . In our analysis we have used the values c1 = 5m and c2 = 15m. For a different scenario different values may be more suitable.

4.4.7

Simulation results and their interpretation

Several PDAF-based trackers are tested in more or less heavy-tailed environments simulated as explained in Sections 4.4.1 - 4.4.6. The trackers to be tested are: T1: The standard PDAF with Rayleigh detector, T2: The standard PDAF with K-based detector, T3: The PDAFAI with Rayleigh detector and standard Rayleigh likelihood, T4: The PDAFAI with Rayleigh detector and conservative Rayleigh likelihood, T5: The PDAFAI with K-based detector and K-based likelihood. Here Rayleigh detector means that detections are provided by (4.15) while K-based detector means that detections are provided by (4.21). For standard Rayleigh, conservative Rayleigh and K-based likelihoods the amplitude information is processed using (4.37), (4.42) and (4.43) respectively. Our aim is to investigate whether there is anything to gain from a more advanced treatment than that offered by the conventional Rayleigh model. Therefore the Rayleighbased trackers do not have access to any information conveyed by the K-distribution framework. The Rayleigh-based trackers assume that their design PFA is correct, although the actual PFA will be significantly higher in spiky clutter. Also notice that the trackers know the exact target state at time k = 0. In reality tracks must also be initialized and terminated. This is easy for high enough SNR, but will most likely pose additional challenges in scenarios such as those simulated here. Tables 4.3 - 4.8 provide a summary of our results. These tables should not be interpreted as a definitive truth of our tracking methods’ performances, but as evidence for the following observations.

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Table 4.3: Number of lost tracks for SNR 12dB and LΓ = 1 ν ν ν ν ν ν

= 0.1 = 0.5 =1 =2 =4 =8

T1 4018 3672 2812 1819 1155 737

T2 4642 3232 2000 1152 650 493

T3 4568 4140 3175 1908 935 454

T4 4980 4665 3434 1625 579 224

T5 4883 3221 1501 622 268 133


= 0.1 = 0.5 =1 =2 =4 =8

T1 2285 1793 1190 797 556 465

T2 2585 1567 1082 671 431 311

T3 2914 1905 1023 538 294 209

T4 3844 1775 637 228 151 116

T5 2357 994 533 254 140 78

First, it can be seen that heavy-tailed clutter (as represented by ν) makes the tracking scenario more challenging. The ν = 0.1 scenario ranges from 5 times as difficult as the ν = 8 scenario (T1 in Table 4.3) to 58 times as difficult as the ν = 8 scenario (T5 in Table 4.7). A non-maneuvering 12dB target can easily be tracked in Rayleigh noise. A maneuvering target in a heavy-tailed background must be much stronger if we shall have any hope of tracking it. Second, the difficulties encountered depend on the correlation length LΓ . The conventional trackers T1 and T2 loose track roughly twice as often for LΓ = 1 as for LΓ = 8. The impact is more severe for the Rayleigh-based trackers T3 and T4 which loose track 5-8 times more often in uncorrelated clutter than in correlated clutter. Third, performance is improved by accounting for heavy-tailedness in the detection and tracking processes. In most cases we see that T2 performs better than T1. The improvement is most noticeable for medium heavy-tailed clutter with 1 ≤ ν ≤ 4. T5 yields further improvements. Fourth, the utilization of amplitude information improves tracking performance. In the near-Rayleigh case both T3 and T4 give fewer lost tracks than T1. In all cases when these outperform T1 it can also be seen that T4 outperforms T3. For ν ≈ 2 and SNR = 15 we see that T3 gives twice as many lost tracks as T4. Overall, T5 has the best performance. Compared to T1 we see that T5 can reduce the number of lost tracks by up to 90% (ν = 1 in Table 4.7). Also notice that the improvement of T5 compared to T2 is stronger than the improvement of T2 compared to T1.

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98

4



= 0.1 = 0.5 =1 =2 =4 =8

T1 2399 1564 872 444 248 132

T2 2939 926 452 225 151 100

T3 3383 2123 1047 423 144 48

T4 4867 3018 1046 242 63 21

T5 3404 536 154 62 31 23


= 0.1 = 0.5 =1 =2 =4 =8

T1 1224 603 272 169 114 85

T2 1207 399 240 143 93 63

T3 1666 554 211 80 40 25

T4 2618 476 83 29 17 13

T5 881 173 69 30 22 17


= 0.1 = 0.5 =1 =2 =4 =8

T1 1045 453 215 113 47 30

T2 900 223 121 57 42 33

T3 1787 623 220 79 33 17

T4 4309 1105 184 29 8 8

T5 930 61 19 16 19 16


= 0.1 = 0.5 =1 =2 =4 =8

T1 502 144 51 36 21 16

T2 361 85 53 40 32 19

T3 717 117 28 12 6 10

T4 1360 76 8 6 5 7

T5 224 31 15 13 14 13

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4.4

99


RMSE for position estimate - only tracks on target 1.2

RMSE [m]

1 0.8 0.6 0.4

Standard PDAF (T1) K PDAF (T2) Rayleigh PDAFAI (T4) K PDAFAI (T5)

0.2 0

0

5

10

15

20

25 Time [s]

30

35

40

45

50

Figure 4.12: Position RMSE for ν = 1, LΓ = 1 and SNR = 15. Table 4.9: Actual false alarm rates Rayleigh K

ν = 0.1 0.0671 0.0096

ν = 0.5 0.0442 0.0066

ν=1 0.0299 0.0061

ν=2 0.0190 0.0063

ν=4 0.0121 0.0065

ν=8 0.0080 0.0060

Based on these observations we may conclude that T5 is the preferred tracker in moderately heavy-tailed clutter, while T4 is the preferred tracker in near-Rayleigh clutter. Even when T5 works equally well as T4 we may prefer T4 since T4 uses fewer auxiliary cells and thus is less susceptible to interference from other targets. For very heavy-tailed clutter there is little we can do, and all our trackers fail more or less. In Figure 4.12 and Figure 4.13 the position RMSE is plotted for two different scenarios. The averaging is only done over tracks which stay on target for all the trackers during the entire simulation. Although the trackers have perfect knowledge to begin with, stationary values of The RMSE are quickly reached. It may come as a surprise that the K-distribution trackers T2 and T5 perform worse than the Rayleigh-based trackers T1 and T4 in Figure 4.13. This, together with Tables 4.3 - 4.8, show there is nothing to gain from using the K-distribution when the data are just as well described by the Rayleigh distribution. On the contrary, the more advanced machinery of T2 and T5 should then be avoided since the K-based detector lacks the optimality properties possessed by the conventional CFAR from Section 4.3.1. Table 4.9 justifies the rather primitive shape parameter estimator (4.19). This table was generated by simply counting the number of detections using the two detectors over 50 time steps in 100 Monte-Carlo runs. The 1 actual false alarm rate resulting from the K-

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100

4


RMSE for position estimate - only tracks on target 1.2

RMSE [m]

1 0.8 0.6 0.4

Standard PDAF (T1) K PDAF (T2) Rayleigh PDAFAI (T4) K PDAFAI (T5)

0.2 0

0

5

10

15

20

25 Time [s]

30

35

40

45

50

Figure 4.13: Position RMSE for ν = 8, LΓ = 8 and SNR = 15. based detector in Section 4.3.2 does admittedly have a positive bias, but never more than twice the design false alarm rate. On the other hand, assuming a Rayleigh distribution for low ν leads to false alarm rates more than 10 times too high, which clearly is undesirable. The Rayleigh assumption appears adequate for ν ≥ 8 . The K-distribution based trackers T2 and T5 do not offer significant improvements for ν ≈ 0.1. This is partly due to the rather bad behavior of the estimator (4.19) for very low ν. But as anticipated from Figure 4.8 we cannot expect to track moderately strong targets in such backgrounds anyway.

4.5

Concluding remarks

A systematic treatment of heavy-tailed clutter from a target tracking perspective has been presented in this chapter. Simulation results show that heavy-tailed clutter inevitably leads to deteriorated performance. The performance loss can be substantially mitigated by accounting for heavy-tailedness in the detection and tracking processes, and by usage of amplitude information as explained in Sections 4.3.6 and 4.3.7. The main contributions of this chapter are two new versions of the PDAFAI which treat amplitude information in a more conservative way than the conventional PDAFAI. The first one of these, termed T4, is developed by marginalizing over the estimation uncertainty of the background strength in the conventional Rayleigh framework. The second, termed T5, assumes the background to be K-distributed and thus more heavytailed than under the Rayleigh assumption. T5 yields the best performance in significantly heavy-tailed clutter, but it is beaten1 by T4 when the degree of heavy-tailedness is

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4.5

Concluding remarks

101

insignificant. The recommendation of the trackers labeled T4 and T5 is subject to computational resources. While the conventional PDAF and PDAFAI are hardly more expensive than a simple Kalman filter, significant additional costs are incurred by the novel trackers. This cost increase is of a linear nature, and may thus be considered acceptable if the alternative is more expensive MHT methods. Hopefully this chapter will motivate more researchers to investigate the impact a heavy-tailed background has on target tracking, and how tracking methods can be tailored to cope with such backgrounds. Out of several topics not covered in this chapter we shall here mention two: First, the important task of track management, i.e. the combined task of track initialization, confirmation and termination, has not been dealt with here. This is most commonly done by means of heuristic rules [6 pp. 107-117]. A large bulk of recent research on target tracking has attempted to develop more rigorous approaches to track management [72, 77]. As argued in Section 4.1, it is only through track management that a reliable decision regarding the presence of a target can be made. Therefore the impact of heavy-tailedness can only be fully understood in the context of track management. Second, it is of interest to determine lower bounds on SNR and ν for which tracking is feasible. Clearly this bound cannot be attained by suboptimal PDAF or PDAFAI methods. Such an analysis would instead require more powerful methods such as MHT or TBD, combined with evaluation of the PCRLB. For further suggestions on further research the reader is referred to Section 7.2. The next chapter continues the study of the PDAF and the PDAFAI in heavy-tailed clutter by evaluating the MRE with amplitude information. The MRE is similar to the PCRLB, although it is not a lower bound, but rather a measure for expected performance.

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4


classtest: 2010-06-08 17:36 — 103(121)

5 The modified Riccati equation with amplitude information The performance of tracking methods can most often only be assessed by means of Monte-Carlo simulations. An exception to this rule is the PDAF, whose RMSE can be predicted by means of the modified Riccati equation (MRE). This kind of treatment is in this chapter extended to the PDAFAI. We evaluate the MRE with amplitude information (AI) for the case of a fluctuating target in heavy-tailed background noise. Monte-Carlo simulations are used to verify the MRE. The MRE can be used to determine the nominal false alarm rate according to an RMSE-based optimality criterion. To the best of our knowledge, this chapter contains the first systematic approach to the determination of false alarm rates in heavy-tailed clutter. In particular, it is indicated that the PDAFAI can safely operate in the presence of very abundant clutter, while the PDAF only can cope with limited amounts of clutter.

5.1

Introduction

The MMSE estimator for a linear filtering problem is the Kalman filter, and the covariance of its error is governed by the Riccati equation. For target tracking the Kalman filter alone is no longer an adequate solution, even if the kinematics and the measurement model are linear. The presence of clutter as well as the sporadic absence of target measurements necessitates data association, for example through the PDAF or the PDAFAI. Our intuition tells us that since neither clutter nor missed detections are beneficial, larger estimation errors must be expected when solving a tracking problem than when solving the corresponding filtering problem. The MRE [39] was developed to make this intuitive notion precise. It is a generalization of the Riccati equation [7] to single-target data association problems, or more precisely to the PDAF. The key idea of the MRE is a so-called information reduction factor which quantifies how much information is lost 103

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5

The modified Riccati equation with amplitude information

due to non-unity detection probability PD and false alarms with rate PFA . The MRE is an important tool because it provides a connection between the detection process and the tracking process. For any detection scenario the probability of detection PD is a function of the false alarm rate PFA . This function is known as the receiving operating characteristic (ROC) and traces a curve in the plane spanned by PFA and PD . There are at least two ways in which the MRE can be used to determine an optimal false alarm rate for a given tracking scenario. Either one can minimize the output of the MRE along the ROC curve, or one can use its output in the hybrid averaging technique of [67] in order to minimize the expected track loss probability. The lowest possible error covariance that can be attained for a Bayesian filtering problem is given by the PCRLB [100]. It is a popular tool in the analysis of non-linear filtering problems [92]. It has also been extended to tracking problems with measurement uncertainty in [50, 110] for the single-target case and in [52] for the multi-target case. The PCRLB does not tell us how well the PDAF or any other practical tracking method can be expected to perform, but only how an optimal method would perform. This is kind of redundant, since one has good reason to expect that this optimal performance anyway is reached by a TBD method1 , whose PCRLB was analyzed in [92]. The MRE does on the other hand predict the expected performance of the PDAF, and is therefore from a practical perspective just as important as the PCRLB. Since the PDAF is sub-optimal, the MRE can be viewed as an upper bound of the PCRLB. Recent research on simplified versions of the MRE has discussed this relationship in greater detail [19]. Despite the extensive treatments of the MRE and the PCRLB in the tracking literature, there has been very little discussion regarding the impact of AI in this context. To the best of our knowledge, AI has only been included in such treatments for TBD [92 pp. 251-257] and for the ML-PDA [13 pp. 157-179]. A prerequisite for the usage of AI is adequate modeling of the amplitudes of both clutter and target measurements. The simplest alternative is to model both as Gaussian or Rayleigh distributed with different parameters. However, it is often observed that the background noise is more heavy-tailed or target-like than one would expect under these assumptions [1]. An alternative background model which is more adequate under such circumstances is the K-distribution [108]. See Section 2.3.1 for a comprehensive discussion regarding the benefits of this model. It should be noted that the Rayleigh distribution is a limiting case of the K-distribution. While heavy-tailed clutter has been a major focus area for the radar and sonar signal processing communities, it has barely been treated from a target tracking perspective. The previous chapter presented the first systematic treatment of heavy-tailed clutter from a target tracking perspective by tailoring the PDAFAI to deal with K-distributed clutter. 1

It can be argued that any threshold-based tracking method in principle must be inferior to a corresponding TBD method which uses all the available information. If one really is aiming to reach optimal performance, one will then have to abandon thresholding. However, as demonstrated in Chapter 3, the practical performance of a TBD method may be substantially inferior to its theoretical performance.

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5.2


105

The performance of this tailored PDAFAI together with several other PDAF-based trackers was investigated using rather realistic Monte-Carlo simulations. This chapter continues the investigation of target tracking in heavy-tailed clutter by predicting the performance of the PDAFAI in the same way as the performance of the PDAF was predicted in [39]. The MRE with AI is in this chapter evaluated for a Swerling I target in K-distributed clutter. The chapter is organized as follows: In Section 5.2 the problem to be solved by the PDAFAI is formally described in its proper Bayesian setting. The PDAF and PDAFAI methods are briefly summarized in Section 5.3. The MRE with AI is then presented in Section 5.4. A careful derivation of the MRE with AI is given in Section 5.5. For completeness the entire derivation is given, although it to a large extent recites the original derivation of [41]. The derivation given here differs in the important aspect that amplitudes are included. Details regarding the numerical evaluation of the MRE are outlined in Section 5.6. In Section 5.7 we discuss results from the evaluation of the MRE, and how these results are in agreement with experimental performance evaluation using Monte-Carlo simulations. A conclusion is given in Section 5.8.

5.2


The aim of single-target tracking is to evaluate the posterior probability density function (pdf) p(xk |Z k ) from the set of received measurements Z k = {Z1 , . . . , Zk } where each k Zk contains mk measurement vectors: Zk = {ζk (i)}m i=1 . The PDAFAI employs a key approximation, namely that the posterior pdf can be collapsed into a single Gaussian which then is propagated to the next time step: ˆ k|k−1 , Pk|k−1 ). p(xk |Z k−1 ) ≈ N (xk ; x

(5.1)

ˆ k|k−1 and the matrix The previous data are in other words summarized by the vector x Pk|k−1 , which both are treated as known and non-random quantities during the next estimation cycle.

5.2.1

Kinematics

The kinematic state xk will typically contain position and velocity, and possibly accelerations, heading, maneuver strengths etc. The kinematic transition prior is assumed Gaussian and linear: p(xk |xk−1 ) = N (xk ; F xk−1 , Q). (5.2) In principle, changes in the mean target power dk should also be modeled by the transition prior. However, it is very difficult to estimate a time-varying target power accurately for fluctuating targets [93]. Therefore the target power state is treated as a constant parameter dk = dk−1 = d.

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5

5.2.2


Measurement model

Each measurement vector ζk (i) in Zk can be parameterized into a kinematic part zk (i) and an amplitude part ak (i), (5.3)

ζk (i) = [zkT (i), ak (i)]T . The kinematic part zk (i) has for clutter measurements a uniform distribution pz0 (zk (i)|xk ) =

1 Vk

(5.4)

where Vk is the volume of the surveillance region, i.e. the validation gate. For targetoriginating measurements, the kinematic component is related to the state by the Gaussianlinear model pz1 (zk (i)|xk ) = N (zk (i) ; Hxk , R). (5.5) The amplitude ak (i) has the pdf pa1 (ak (i)|d, q) if ζk (i) originates from the target, otherwise it has the pdf pa0 (ak (i)|q), where q contains parameters describing the background noise. In this chapter the target return is modeled by the Swerling I-model, while the Rayleigh distribution and also the K-distribution are considered as models for the clutter return. The superscript a indicates that these are pdf’s of thresholded measurements, and thus only the upper tails of the un-thresholded pdf’s p1 (ak (i)|d, q) and p0 (ak (i)|q), suitably normalized. The clutter-only amplitude likelihood is for Rayleigh distributed background noise given by 1 1 a −(a)2 a p0 (ak (i) | η) = p0 (ak (i) | η) = exp . (5.6) PFA PFA η 2η

For K-distributed background noise it is pa0 (ak (i) | ν, b)

1 4aν 1 = p0 (ak (i) | ν, b) = Kν−1 √ ν+1 PFA PFA b Γ(ν)

2a √ b

.

(5.7)

For a Swerling I target in Rayleigh distributed background noise, the clutter plus target amplitude likelihood is 1 1 a −(a)2 a p1 (ak (i) | d, η) = exp . (5.8) p1 (ak (i) | d, η) = PD PD d + η 2(d + η) For a Swerling I target in K-distributed background noise it is given by marginalization over the so-called texture η, 1 p1 (a | d, ν, b) PD Z∞ ν−1 1 a η η a2 = exp − − dη. PD bν Γ(ν) η+d b 2(η + d)

pa1 (a | d, ν, b) =

0

(5.9)

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5.2

107


Finally, we assume in this chapter that false alarms are distributed according to a Poisson process with parameter λVk , µ(mk ) = Poisson(mk ; λVk ) = e−λVk

(λVk )mk mk !

(5.10)

and that the true target originating measurement is received with a probability PD PG , where PD is the probability of detection and PG is the gate probability.

5.2.3

The joint measurement pdf

The additional information received between time step k − 1 and time step k is of a random nature as specified by the measurement pdf’s elaborated in Section 5.2.2. These pdf’s can be combined into a joint measurement pdf, which we decompose as follows using the definition of conditional probability: p(Zk |Z k−1 ) = p(Zk , mk |Z k−1 ) = p(Zk |mk , Z k−1 )P {mk }.

(5.11)

The cardinality mk of Zk is distributed according to P {mk } =PD PG µ(mk − 1) + (1 − PD PG )µ(mk ) =e−λVk

(λVk )mk −1 (PD PG mk + (1 − PD PG )λVk ) . mk !

(5.12)

The measurement pdf p(Zk |mk , Z k−1 ) as conditioned on the cardinality mk as well as the previous data Z k−1 is a mixture over the mutually exclusive and exhaustive hypotheses θk (i): θk (0) θk (1) .. .


(5.13)

θk (mk ) Measurement mk originates from the target. For i = 0 (i.e. no target detection) the prior event probability is γ0 =P {θk (0)|mk } =

(1 − PD PG )λVk PD PG mk + (1 − PD PG )λVk

(5.14)

while for all other i it is γ1 =P {θk (1)|mk } = . . . = P {θk (mk )|mk } PD PG = . PD PG mk + (1 − PD PG )λVk

(5.15)

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5


For each θk (i) the corresponding hypothesis-conditioned measurement pdf is p(Zk |θk (0), mk , Z k−1 ) = p(Zk |θk (i), mk , Z

k−1

1 Vkmk

mk Y

pa0 (ak (j))

(5.16)

j=1

) = N (νk (i) ;

0, Sk )pa1 (ak (i))

1 Vkmk −1

mk Y

pa0 (ak (j))

(5.17)

j6=i

where we have introduced the innovation ˆ k|k−1 νk (i) = zk (i) − H x

(5.18)

and its corresponding covariance Sk = HPk|k−1 H T + R.

(5.19)

Combining all this yields p(Zk |mk , Z k−1 ) =

mk X i=0

p(Zk |θk (i), mk , Z k−1 )P {θk (i)|mk }

! mk mk Y X γ1 cM g M = pa0 (ak (j)) b + ek (i)lka (i) . √ M mk Vk PD 2π j=1 i=1

The second line of (5.20) has introduced the short-hand notations 1 T −1 ek (i) = exp − νk (i) S νk (i) 2 and

M

(5.20)

(5.21)

1 − PD PG (5.22) cM PD p where the volume of the validation gate is Vk = cM g M |Sk |, and the constant cM = M π 2 /Γ( M 2 + 1) is the volume of a unit-radius M -dimensional sphere. For target tracking without AI, the amplitude likelihood ratio la (i) is just unity. For a Swerling I target in Rayleigh background noise it is PFA η a2 d PFA p1 (a|d, η) a l (i) = = · · exp . (5.23) PD p0 (a|d, η) PD η + d 2η(η + d) b=

2π g

2

λVk

For a Swerling I target in K-distributed background noise it is √ Z∞ ν−1 PFA p1 (a|d, ν, b) PFA (a b)1−ν η η a2 · l (i) = = · exp − − dη. 2a PD p0 (a|d, ν, b) PD 4K η+d b 2(η + d) √ ν−1 0 b a

(5.24)

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5.3

109


5.3


The PDAF and the PDAFAI express the state estimate at time k as a weighted average of ˆ k|k−1 and state estimates conditioned on the latest measurements zk (i). the prediction x This leads to the following Kalman filter-like equations for prediction and measurement ˆ k|k and its associated covariance Pk|k : update of the state estimate x ˆ k|k−1 = F x ˆ k−1|k−1 x Pk|k−1 = F Pk−1|k−1 F T + Q ˆ k|k = x ˆ k|k−1 + Kk x

mk X

βk (i)νk (i)

i=1

Pk|k = Pk|k−1 − (1 − βk (0))Kk Sk KkT + P˜k

(5.25)

where Kk = Pk|k−1 H T Sk−1 "m # k X P˜k = Kk βk (i)νk (i)νk (i)T − νk ν T K T k

k

i=1

νk =

mk X

(5.26)

βk (i)νk (i).

i=1

The index i ranges over all mk extracted and validated measurements inside a validation gate G, defined by ˆ k|k−1 ) < 1 ˆ k|k−1 )T (g 2 Sk )−1 (zk − H x G : (zk − H x

(5.27)

where the scalar g is called the gate size. The association probabilities or weights βk (i) = P {θk (i)|Z k } in (5.25) are under the Poisson assumption (5.10) given by βk (0) = βk (i) =

b+

b a j=1 lk (j)ek (j)

Pmk

la (i)e (i) Pkmk ak b + j=1 lk (j)ek (j)

(5.28) (5.29)

with ek (i) and b as in (5.21) and (5.22), respectively. For derivations of (5.28) and (5.29) the reader is referred to Sections 2.6.1 - 2.6.2.

5.4

The MRE with AI

The data association problem solved by the PDAFAI is non-linear due to the measureˆ k|k ment origin uncertainty. The covariance Pk|k corresponding to the state estimate x

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5


does therefore depend on the data Zk . Consequently, it is impossible to predict the performance of the PDAFAI in the same way as the conventional Riccati equation predicts the performance of the Kalman filter. A similar recursion can nevertheless be carried out if Pk|k is replaced by its conditional expectation h i ˆ k|k − xk )(x ˆ k|k − xk )T |Z k−1 . Pk = E[Pk|k |Z k−1 ] , E (x

(5.30)

The MRE with AI is then given by Pk|k−1 = F Pk−1 F T + Q Pk = Pk|k−1 − q2 (Sk ; PD , PFA )Kk Sk KkT Sk = HPk|k−1 H T + R.

(5.31)

The information reduction factor q2 is given by q2 =

∞ X

m=1

ηm (Sk )Sk = E

"

(5.32)

η2m (Sk )P {m}

m X i=1

# β(i)2 νk (i)νk (i)T m, Z k−1 .

(5.33)

We find η2m by averaging over the probability density p(Zk |mk Z k−1 ) or equivalently p(νk (1), . . . , νk (mk ), ak (1), . . . , ak (mk ) | mk , Z k−1 ): η2m =

m−1 Zg

Zg Z∞

Z∞



m Y



M mγ1 cM ... . . . pa1 (ak (j))  pa0 (ak (j)) √ M M g PG 2π j=2 t 0 0 t   m Y 1 2 M +1  M −1  × %1 %j exp − %1 2 j=2 la (1)exp − 21 %21 × (5.34) P 1 2 da(m) . . . da(1)d%m . . . d%1 . a b+ m i=1 l (i)exp − 2 %i

Notice that q2 depends on Sk through γ1 and b which,pas can be seen from (5.15) and (5.22), depend on λ and the gate volume V = cM g M |S|. The only way to evaluate (5.34) is by importance sampling. This can be done by averaging the fraction on the last line over distributions proportional to the preceding factors. The main results (5.31), (5.33) and (5.34) are derived in Section 5.5. This derivation is similar to the one found in [41], but differ since amplitudes are included here. The numerical evaluation of (5.34) is discussed in Section 5.6.

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5.5

111

Derivation of the MRE with AI

5.5


By means of the total probability theorem one can write Pk =

∞ X

mk =0

E[Pk|k |mk , Z k−1 ]P {mk }

(5.35)

where the cardinality probabilities P {mk } are given as in (5.12). Taking the conditional expectation with respect to mk and Z k−1 yields E[Pk|k |mk , Z k−1 ] =Pk|k−1 − (1 − E[βk (0)]) Wk Sk WkT   mk X + Wk E  βk (j)νk (j)νk (j)T − νk νkT mk , Z k−1  WkT j=1

=Pk|k−1 − (1 − E[βk (0)]) Wk Sk WkT   mk X + Wk E  βk (j)νk (j)νk (j)T mk , Z k−1  WkT j=1

  mk X − Wk E  βk (j)2 νk (j)νk (j)T mk , Z k−1  WkT j=1

=Pk|k−1 − PD PG Wk Sk WkT

+ Wk U1 (mk )WkT − Wk U2 (mk )WkT

(5.36)

where U1 (m) =

Z

=m

p(Zk |mk , Z Z

...

G

Z Z∞ G

t

k−1

)

mk X

β(j)ν(j)ν(j)T dZk

j=1

...

Z∞

p(ν(1), . . . , ν(m), a(1), . . . , a(m))

t T

× β(1)ν(1)ν(1) da(m) . . . da(1)dν(m) . . . dν(1)

(5.37)

and U2 (m) =

Z

=m

p(Zk |mk , Z k−1 ) Z G

...

Z

G 2

Z∞ t

...

Z∞

mk X

β(j)2 ν(j)ν(j)T dZk

j=1

p(ν(1), . . . , ν(m), a(1), . . . , a(m))

t

× βk (1) ν(1)ν(1)T da(m) . . . da(1)dν(m) . . . dν(1).

(5.38)

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112

5


The second equalities of (5.37) and (5.38) follows from noting that the joint measurement pdf is symmetric in the measurements. Given that there are m measurements, each of these will have the same marginal distribution. The time subscript k is hereafter dropped for notational simplicity. Also, explicit mention of the dependency on mk and Z k−1 is dropped since this is always assumed. Inserting the joint measurement pdf (5.20) together with the expression (5.29) for β(1) yields Z Z Z∞ Z∞ m Y mγ1 cM g M a U1 (m) = ... . . . l (1) pa0 (ak (j)) √ M V m PG 2π j=1 t G G t 1 × exp − ν(1)T S −1 ν(1) ν(1)ν(1)T da(m) . . . da(1)dν(m) . . . dν(1) 2 and Z Z Z∞ Z∞ m Y mγ1 cM g M a ... . . . l (1) U2 (m) = pa0 (ak (j)) √ M m V PG 2π j=1 t G G t 1 × exp − ν(1)T S −1 ν(1) ν(1)ν(1)T 2 la (1)exp − 21 ν(1)T S −1 ν(1) da(m) . . . da(1)dν(m) . . . dν(1). × P 1 a T −1 b+ m i=1 l (i)exp − 2 ν(i) S ν(i)

The validation region G is an ellipsoid given by the matrix Sk and gate size g. It is convenient to perform a change of variables so that the integration can be performed over a spherically symmetric ball instead. This is done by introducing the variables 1 ξj = S − 2 ν(i). It follows from this substitution that dν(i) =

p |S|dξi =

V dξi cM g M

(5.39)

and that ν(1)T S −1 ν(1) =ξ1T ξ1 = kξ1 k2 1

1

ν(1)ν(1)T =S 2 ξ1 ξ1T S 2 .

(5.40)

Furthermore, we denote the origin-centered ball with dimension M and radius g by B M (g). For future reference, notice that the boundary of this ball is the sphere with dimension M − 1 and radius g. We denote it by S M −1 (g) = ∂B M (g), where ∂ is the

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5.5

113


boundary operator. In the new coordinates the integrals U1 (m) and U2 (m) become

U1 (m) =S

1 2

√

Z

mγ1 M

PG 2π (cM

g M )m−1

Z

...

Z∞

...

BM (g) t

BM (g)

Z∞ t

la (1)

m Y

pa0 (ak (j))

j=1

!

1 1 2 × exp − kξ1 k ξ1 ξ1T da(m) . . . da(1)dξm . . . dξ1 S 2 2

(5.41)

and

U2 (m) =S

1 2

mγ1

√

M

g M )m−1

PG 2π (cM 1 × exp − kξ1 k ξ1 ξ1T 2

Z

...

G

Z Z∞ G

...

t

Z∞

la (1)

m Y

pa0 (ak (j))

j=1

t

! la (1)exp − 21 kξ1 k 1 da(m) . . . da(1)dξm . . . dξ1 S 2 . (5.42) × Pm a 1 b + i=1 l (i)exp − 2 kξi k Some observations should now be made about the expressions in (5.41) and (5.42). Since the term ξ1 ξ1T is a matrix, it follows that U1 (m) and U2 (m) are also matrices. Since the remainders of the integrands are spherically symmetric in ξ1 , it follows that U1 (m) and U2 (m) are multiples of the M -dimensional identity matrix IM . For more elaborate justifications of these claims the reader is referred to [41]. It follows that U1 (m) and U2 (m) are given uniquely by any elements along their diagonals, and we can write 1

1

1

1

U1 (m) =S 2 IM η1m (S)S 2 = η1m (S)S (5.43)

U2 (m) =S 2 IM η2m (S)S 2 = η2m (S)S where η1m and η2m are scalar quantities given by

η1m =

√

mγ1 M

g M )m−1

Z

...

Z

Z∞

...

Z∞

la (1)

m Y

PG 2π (cM j=1 t BM (g) BM (g) t 1 × exp − kξ1 k2 (ξ11 )2 da(m) . . . da(1)dξm . . . dξ1 2

pa0 (ak (j)) (5.44)

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114

5


and η2m =

√

mγ1 M

g M )m−1

Z

...

Z

Z∞

...

Z∞

la (1)

m Y

pa0 (ak (j))

PG 2π (cM j=1 t BM (g) t BM (g) 1 × exp − kξ1 k2 (ξ11 )2 2 a l (1)exp − 12 kξ1 k2 da(m) . . . da(1)dξm . . . dξ1 × P 1 a 2 b+ m i=1 l (i)exp − 2 kξi k

(5.45)

where ξ11 denotes the first component of the vector ξ1 = [ξ11 , . . . , ξ1M ]. It can be seen that for neither η1m nor η2m does the integrand depend angularly on ξi for i ≥ 2. This motivates a change to spherical coordinates: ξ 1 =% cos(φ1 ) ξ 2 =% sin(φ1 ) cos(φ2 ) .. . ξ M −1 =% sin(φ1 ) . . . sin(φM −2 ) cos(φM −1 ) ξ M =% sin(φ1 ) . . . sin(φM −2 ) sin(φM −1 ).

(5.46)

The volume element of M -dimensional space is in these coordinates dξ =%M −1 sinM −2 (φ1 ) sinM −3 (φ2 ) . . . sin(φM −2 )d%dφ1 dφ2 . . . dφM −1 =%M −1 dΩM −1 = %M −1 sinM −2 (φ1 )dΩM −2

(5.47)

where the second line defines the solid angle element dΩN corresponding to the last N angular coordinates. Denote the volume of B M (%) by V M (%) and the volume (or generalized surfacearea) of S M −1 (%) by S M −1 (%). It can the be shown that M V M (%) = gS M −1 (%): S M −1 (%) =

dV M (%) M V M (%) 2π M/2 %M −1 = = = M cM %M −1 . d% % ΓM/2

(5.48)

−1 do therefore contribute a factor (S M −1 (1))m−1 = The integrations over φ12 , . . . , φM m M m−1 m−1 (M V (1)) = (M cM ) . Therefore

Zg Zπ Zπ Z2π Zg Zg Z∞ Z∞ m Y mγ1 M m−1 a η1m = . . . . . . . . . l (1) pa0 (ak (j)) √ M M g PG 2π j=1 t 0 0 0 0 0 0 t m Y 1 +1 M −2 −1 2 1 M −2 1 M −3 2 × exp − %21 %M cos (φ ) sin (φ ) sin (φ ) . . . sin(φ ) %M 1 1 1 1 1 j 2 j=2

−1 −2 × da(m) . . . da(1)d%m . . . d%2 dφM dφM . . . dφ11 d%1 1 1

(5.49)

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5.5

115


and Zg Zπ Zπ Z2π Zg Zg Z∞ Z∞ m Y mγ1 M m−1 a η2m = . . . . . . l (1) . . . pa0 (ak (j)) √ M M g PG 2π j=1 t 0 0 0 0 0 0 t m Y 1 +1 M −2 −1 2 1 M −2 1 M −3 2 × exp − %21 %M cos (φ ) sin (φ ) sin (φ ) . . . sin(φ ) %M 1 1 1 1 1 j 2 j=2 1 2 a l (1)exp − 2 %1 −1 −2 da(m) . . . da(1)d%m . . . d%2 dφM dφM . . . dφ11 d%1 . × Pm a 1 1 b + i=1 l (i)exp − 12 %2i (5.50) −1 The integrations over φ21 , . . . , φM contribute another factor (S M −2 (1)) = (M − 1 1)V M −1 (1) = (M − 1)cM −1 , so that the expressions of η1m and η2m become

η1m

Zg Zπ Zg Zg Z∞ Z∞ m Y mγ1 M m−1 a = √ M (M − 1)cM −1 ... . . . l (1) pa0 (ak (j)) M g PG 2π j=1 t 0 0 0 0 t 1 +1 cos2 (φ11 ) sinM −2 (φ11 ) × exp − %21 %M 1 2 m Y −1 × %M da(m) . . . da(1)d%m . . . d%2 dφ11 d%1 (5.51) j j=2

and η2m

Zg Zπ Zg Zg Z∞ Z∞ m Y mγ1 M m−1 a = √ M (M − 1)c . . . . . . l (1) pa0 (ak (j)) M −1 M g PG 2π j=1 t 0 0 0 0 t m Y 1 +1 −1 cos2 (φ11 ) sinM −2 (φ11 ) × exp − %21 %M %M 1 j 2 j=2 1 2 a l (1)exp − 2 %1 da(m) . . . da(1)d%m . . . d%2 dφ11 d%1 . × (5.52) Pm a b + i=1 l (i)exp − 12 %2i

The angular integral over φ11 , which is the same in both (5.55) and (5.56), can now be solved using (3.621.5) and (8.384.1) from [48]: Zπ Γ M2−1 Γ 23 M Γ(M − 1)c2M M −2 1 2 1 1 sin φ1 cos φ1 dφ1 = = . (5.53) 2M π M −1 Γ M 2 +1 0

The second equality in (5.53) is obtained by means of the Gamma function’s doubling formula ((8.335.1) in [48]) together with the definition of cM . Further usage of the doubling formula reveals that (M − 1)cM −1

M Γ(M − 1)c2M = cM . 2M π M −1

(5.54)

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5


Consequently, we arrive at

η1m =

m−1 Zg

Zg Z∞



Z∞

m Y



mγ1 cM M ... . . . pa1 (ak (j))  pa0 (ak (j)) √ M M g PG 2π j=2 t 0 t 0   m Y 1 2 +1  M −1  × %M % exp − % da(m) . . . da(1)d%m . . . d%1 (5.55) 1 j 2 1 j=2

and

η2m =

m−1 Zg

Zg Z∞



Z∞

m Y



M mγ1 cM . . . pa1 (ak (j))  ... pa0 (ak (j)) √ M gM PG 2π j=2 t 0 0 t   m Y 1 2 M +1  M −1  × %1 %j exp − %1 2 j=2 la (1)exp − 21 %21 da(m) . . . da(1)d%m . . . d%1 . × (5.56) Pm a b + i=1 l (i)exp − 12 %2i

For η1m the radial integral can also be expressed in closed form, although only −1 approximately. Each of the terms %M contributes a factor g M /M . The integral of j M +1 %1 exp(%21 /2) is approximated by a Gamma function: Zg 0

M +1

%

exp

%2 2

%2 d% ≈ % exp d% 2 0 √ M M (2π)M/2 = 2 Γ +1 = . 2 cM Z∞

M +1

(5.57)

Let us also notice that for η1m all the radial variables except %1 can be integrated out explicitly. Each of these %i contributes a factor g M /M . Furthermore, the amplitude integrals are all simply unity due to the definition of the thresholded amplitude pdf’s pa0 (·) and pa1 (·). The following approximation follows: η1m ≈

mγ1 . PG

(5.58)

For η2m it is on the other hand not possible to proceed any further due to the fraction on the third line of (5.56). The entire integral of (5.56) must therefore be evaluated numerically. In Section 5.6 it is explained how this can be done. The fact that U1 (m) and U2 (m) are multiples of the identity matrix makes it sensible

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5.6

Numerical evaluation of the MRE with AI

117

to introduce information reduction factors q1 and q2 by q1 (Sk ) = q2 (Sk ) =

∞ X

m=1 ∞ X m=1

η1m (Sk )P {m} η2m (Sk )P {m}

(5.59)

so that E[Pk|k |Z k−1 ] =Pk|k−1 − PD PG Wk Sk WkT

+ q1 (Sk )Wk Sk WkT − q2 (Sk )Wk Sk WkT .

(5.60)

From (5.12) and (5.15) it follows that q1 (S) is approximately PD : ∞ X m PD PG (λV )m−1 q1 (S) ≈ e−λV (PD PG + (1 − PD PG )λV ) PG PD PG + (1 − PD PG )λV m! m=0

=PD

∞ X

m=1

e−λV

(λV )m−1 = PD . (m − 1)!

(5.61)

In practice, the gate size g is most often chosen so high that PG ≈ 1. Then the middle terms of (5.60) cancel, and only q2 makes the MRE differ from the conventional Riccati equation: E[Pk|k |Z k−1 ] =Pk|k−1 − q2 (Sk )Wk Sk WkT .

5.6

(5.62)


The information reduction factor q2 (PD , PFA , λVk ) is a function that depends on both the detection probability PD , the false alarm rate PFA and the clutter intensity as represented by λVk . The volume Vk is given by the innovation covariance Sk , which again is given by the output of the MRE at the previous time step k − 1. The three quantities PD , PFA and λV are intertwined in the scalar quantities η2m , whose integrals cannot be expressed in closed form. The most efficient way of propagating the MRE is by constructing an interpolation table. In the original paper [39] the information reduction factor was evaluated over a discrete grid in the (PD , λVk )-plane, which in the absence of amplitude information is equivalent to the (PD , PFA )-plane. When amplitudes are included this approach becomes too simple, since PFA is no longer equivalent to λVk . Instead a 3-dimensional interpolation table must be constructed over PD , PFA as well as λV . The integral in (5.34) can only be evaluated numerically. This integral is, especially for higher values of mk , so high-dimensional that grid-based techniques will run into difficulties. Instead, importance sampling should be used. We have organized the terms

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5


Non-AI case: ν = 1, SNR = 15dB, λVk = 2, PFA = 10−2 1200

Histogram count

1000 800 600 400 200 0

0

0.2

0.4 0.6 Value of fraction sample

0.8

1

AI case: ν = 1, SNR = 15dB, λVk = 2, PFA = 10−2 4000 3500 Histogram count

3000 2500 2000 1500 1000 500 0

0

0.2

0.4 0.6 Value of fraction sample

0.8

1

Figure 5.1: Distribution of the fraction in (5.65) when the variables are drawn as proposed in Sections 5.6.1 and 5.6.2. AI can be seen to move probability mass towards one, thereby increasing the kurtosis. The clutter Poisson rate λV = 2 indicates a rather tough tracking scenario where AI is necessary to avoid track-loss. For lower values of λV both histograms look more similar to the lower one.

1

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5.6

119


in (5.34) as done in order to demonstrate how the sampling should be carried out. The amplitudes a1 , . . . , am are sampled using the truncated densities on the first line as explained in Section 5.6.1, while the radial variables %1 , . . . , %m are sampled from densities proportional to the quantities on the second line as explained in Section 5.6.2. The ratio on the third line is then averaged over these samples (c.f. Figure 5.1). Thus, assuming (l) (l) N that we have N samples (a1 )N l=1 , . . . , (%m )l=1 , we approximate η2m by an average on the form (l) 1 (l) 2 N (% ) l(a | d, ν, b)exp − X 1 1 2 Cm . η2m ≈ (5.63) Pm (l) (l) N l(a | d, ν, b)exp − 1 (% )2 b+ l=1

i=1

i

2

i

where Cm is a constant. This constant is comprised of the constant in (5.34) together with proportionality constants linking the functions of ρi in (5.34) to their corresponding sampling densities for i = 1, . . . , m . By insterting these proportionality constants as obtained in Section 5.6.2 we find M m−1 mγ1 M m−1 M g mγ1 cM M/2 Cm = = ×2 Γ +1 × . (5.64) √ M M g 2 M PG PG 2π

It follows that η2m

5.6.1

(l) 1 (l) 2 N l(a | d, ν, b)exp − (% ) X 1 1 2 mγ1 . ≈ Pm (l) 1 (l) 2 PG N l(a | d, ν, b)exp − b + (% ) l=1 i=1 i 2 i

(5.65)

Sampling of amplitude components

When amplitudes are ignored there is simply no need to sample the amplitude components, and we may proceed as if the amplitude related terms were all unity. For Rayleighdistributed clutter the amplitudes can be sampled according to the strategy devised in [13 p. 166]. Sampling from the truncated K-distribution pdf’s (5.7) and (5.9) is done using inverse transform sampling, implemented by means of interpolation tables. Denote the survival function corresponding to the truncated density pa (a | d, ν, b) by S a (a | d, ν, b): a

S (a | d, ν, b) =

Z∞ a

pa (u | d, ν, b)du.

(5.66)

Inverse transform sampling treats the value of survival function itself (or equivalently and more commonly the cdf) as a random variable which is drawn according to u = S a (a | d, ν, b) ∼ Uniform(u ; [0, 1]).

(5.67)

Samples of a distributed according to pa (a) can then be obtained by inverting the survival function.

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120

5 a


a

a(l)

T pτ1 (a)

x(l)

p(x)

x

x

Figure 5.2: Sampling technique for amplitude variables. The function f −1 (x) (dashed curve) defines a mapping from the exponential random variable x to the truncated K-Swerling I random variable a. The function is approximated by linear interpolation as depicted by the blue curve. Alternatively, we may instead draw the negative logarithm of the survival function, which in accordance with (5.67) must be exponentially distributed: x = f (a) = − ln S a (a | d, ν, b) ∼ Exponential(x ; 1).

(5.68)

Amplitude samples are then drawn according to a = f −1 (x) ∼ pa (a | d, ν, b).

(5.69)

The sample scheme given by (5.68) is preferred over the sample scheme given by (5.67) due to numerical benefits. Since no closed-form expression is known for S −1 (·), this mapping is most conveniently implemented using linear interpolation. The mapping f −1 (·) is as illustrated in Figure 5.2 easily approximated by piecewise linear segments. An interpolation table for the implementation of a = f −1 (x) consists of control points (x(p) , a(p) )Pp=1 . We first decide where the control point domain values a(p) should be placed, and thereafter calculate the corresponding codomain values x(p) . As argued (1) (P ) in Appendix B, the interpolation grid [ai , . . . , ai ] should have a variable resolution reflecting the fact that the curvature of pa (a | d, ν, b) decreases as a → ∞. This is obtained by requiring the intervals ∆a(p) = a(p) −a(p−1) of this grid to be geometrically

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5.6

121


increasing: (p)

∆ai

= AB p .

(5.70)

While the lowermost point a(0) of the grid should be the threshold T , its uppermost point a(P ) can be any reasonably large value, say 100. The constant A is also a tuning parameter, for which 0.25 has been decided to be an appropriate value. The constant B is then determined by solving a(P ) = T + A

1 − BP . 1−B

(5.71)

The amplitude values of the interpolation grid are then given by   P X T + 0, ∆a(1) , ∆a(1) + ∆a(2) , . . . , ∆a(p)  .

(5.72)

p=1

The corresponding control points x(p) are evaluated using (5.68). The survival functions S1a (a|d, ν, b) and S0a (a|ν, b) corresponding to pa1 (a1 ) and a p0 (a1 ) are truncated versions of the survival functions S1 (a|d, ν, b) and S0 (a|ν, b) corresponding to (5.9) and (5.7), respectively. Mathematically they can be written S1a (a|d, ν, b) = where

S0 (a|ν, b) S1 (a|d, ν, b) and S0a (a|ν, b) = S1 (T |d, ν, b) S0 (T |ν, b)

(5.73)

2 a2 exp − η − dη b 2(η + d)

(5.74)

(5.75)

(2/b)ν S1 (a | d, ν, b) = Γ(ν)

Z∞

η

ν−1

0

and S0 (a ; ν, b) = √

2aν ν

b Γ(ν)

Kν

2a √ b

.

The integral in (5.74) must be evaluated numerically, for example using the scheme described in Appendix B.2. To summarize, we draw samples of a1 using (5.74), while samples of all other aj are drawn using (5.75). The sampling technique is illustrated in Figure 5.2, whose major plot shows the mapping a = f −1 (x) for a 15dB target embedded in K-distributed noise with ν = 1. The leftmost plot shows the pdf of target plus clutter, while the bottom plot shows the exponential distribution used to draw x(l) . It is illustrated how the linear interpolation (blue curve) provides a very good approximation of the exact mapping.

5.6.2

Sampling of radial components

The radial variable %1 is to be drawn from a pdf proportional to the function 2

+1 −%1 /2 f (%1 ) = %M e . 1

(5.76)

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5


This is a Chi-density 2

xM +1 e−x /2 . p(%1 ) = χ(%1 ; M + 2) = M/2 M 2 Γ 2 +1

(5.77)

The sampling density and the actual function to be integrated differ by a proportionality constant 2M/2 Γ M + 1 . In other words, 2 M M/2 f (%1 ) = 2 Γ + 1 p(%1 ). (5.78) 2

The middle factor in (5.64) follows from this. The remaining radial variables %j should be drawn from densities proportional to the −1 monomials %M . This is done by drawing j 1

%j = gu M where u ∼ Uniform([0, 1])

(5.79)

from which it follows that

M M −1 % . (5.80) gM It follows that the sampling density and the function to be integrated differ by a proportionality constant M/g M . Thus the third factor in (5.64) must be included when %j is drawn using (5.79) for j = 2, . . . , m. p(%j ) =

5.7

Results

In this section the performance of the PDAFAI in K-distributed clutter is investigated by means of both simulations and by evaluation of the MRE.

5.7.1

System setup

The linear models (5.2) and (5.5) are in the numerical evaluation of the MRE implemented using the following matrices   1 T 0 0  0 1 0 0   F = (5.81)  0 0 1 T  0 0 0 1  2  σv 3 σv2 2 T T 0 0   σ32 2 2 2  vT  σ T 0 0 v 2  (5.82) Q =  σv2 3 σv2 2  0 T T  0  3 2 σv2 2 2 0 0 T σ vT 2 1 0 0 0 H= . (5.83) 0 0 1 0

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5.7

123

Results

The measurement noise matrix R is in practical applications typically specified in polar coordinates. However, in order to make it meaningful to talk about the steady-state output of the MRE, linearity must be ensured. To keep things simple we use R = σr2 I

(5.84)

where I is the 2 × 2 identity matrix. Values for the constants appearing in (5.81) - (5.84) can be found in Table 5.1.

5.7.2

Performance analysis using simulations

In order to validate the MRE its output should be compared with simulation results. This was done for the conventional MRE in [41], and a similar validation is presented for the AI case in this chapter. In addition to verifying the MRE, this section is also written with the purpose of complementing the simulation results reported in Chapter 4. In that chapter realistic Monte-Carlo simulations were used to illustrate the gains that could be expected from using AI in practical scenarios with heavy-tailed clutter. The realistic approach required that the trackers could not know the model parameters a priori; instead they had to be estimated, causing a corresponding performance loss. For a more lengthy discussion of realistic Monte-Carlo simulation the reader is referred to Section 2.7. The simulations in this chapter are carried out according to what we instead may describe as a clean approach. In this approach, the trackers are given perfect knowledge about all model parameters such as PFA , PD , ν, b and so on. Thus, there is no mismatch between filter and simulation models. The output from the simulations is summarized by two measures: The position error and the rate of track-loss. Caution is required when attempting to estimate the position RMSE from empirical data. Occurrences of track-loss will, if the number of samples is high enough, cause severe outliers in the error pdf which make it impossible to obtain a well-defined value for the RMSE. In [41] this problem was circumvented by using very few (more precisely 10) Monte-Carlo runs. In this paper we take a different approach: Instead of estimating the RMSE empirically we store the entire error pdf as represented by a histogram. Track-loss results can be seen in Tables 5.2 and 5.3. The simulation results were generated using 10000 Monte-Carlo runs for each scenario. The numerical recipe of Appendix B was used to evaluate the amplitude likelihood for the PDAFAI. The results indicate, as one would expect, that the PDAFAI always performs better than the PDAF. Notice that the PDAFAI offers significant improvements irrespective of the false alarm rate. As a general tendency it can be noted that the PDAFAI reaches its best performance for high false alarm rates, while the PDAF is better off with lower false alarm rates. The combination of a weak target (SNR = 10dB) and significantly heavy-tailed clutter (ν = 1) justifies usage of a high false alarm rate for the PDAF as well.

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5.7.3

5


Performance analysis using the MRE with AI

The output of the MRE is most conveniently summarized by the stationary position RMSE as predicted by the MRE: p 11 + P 22 . e(PFA , PD ) = P∞ (5.85) ∞

For a given scenario, this quantity generates a surface over the PFA -PD -plane known as the tracking operating characteristic (TOC). Since the ROC-curve PD (PFA ) is given by the scenario, it makes sense to plot the TOC as a function of PFA along the ROC curve to investigate what would be the optimal false alarm rate for the given scenario. In Figure 5.3 we have plotted the ROC-curves for a 15dB target both significantly heavy-tailed noise (red stapled curve) and in not so heavy-tailed noise (blue solid curve). It can clearly be seen how the more heavy-tailed case is more challenging due to lower detection probability for the same false alarm rate. This is most noticeable for low false alarm rates, while the difference appears negligible for very high false alarm rates. Corresponding TOC-curves have been plotted in Figure 5.4 for both the conventional MRE corresponding to the PDAF, and for the MRE with AI corresponding to the PDAFAI. This figure confirms the observations from Section 5.7.2. It appears to be safest to use low false alarm rates from the PDAF in not so heavy-tailed clutter. In significantly heavy-tailed clutter one should on the other hand consider higher false alarm rates. However, too high false alarm rates (roughly ≥ 10−1 ) cause the MRE to diverge, indicating that the PDAF cannot cope with such large amounts of clutter. The most striking observation to made from Figure 5.4 is that the PDAFAI is able to beat this divergence for much higher false alarm rates (roughly up to PFA = 0.5). On the other hand, for low false alarm rates the MRE predicts only marginal improvements due to AI. The conclusion appears therefore to be that the main purpose of AI should be to make it possible to use so high false alarm rates that target detection can be expected. Validation results are presented in Figures 5.5, 5.6 and 5.7, where error statistics of the PDAF and the PDAFAI have been collected. The histograms were generated using 10000 Monte-Carlo runs as explained in Section 5.7.2. All these three figures reveal a slightly positive (i.e. pessimistic) bias of the MRE as compared to the modes of the error pdfs. On the other hand the error pdfs are also slightly skewed due to upper tails. This is especially visible for the PDAF in the highPFA cases treated in Figures 5.6 and 5.7. We may therefore conclude that the MRE does a reasonably good job in predicting the expected error of both the PDAF and the PDAFAI. One should treat this analysis with some caution. First, it is very optimistic. The error graphs in Section 4.4.7 show a slightly worse performance, even after removal of bad tracks. The obvious explanation for this is that in the real world (or in realistic simulations as in Chapter 4), several parameters are not known a priori. The resulting estimation errors cause an inevitable performance loss which is difficult to account for in a simple equation such as the MRE.

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5.7

125

Results

Table 5.1: Filter model parameters Parameter

Value

Specification

g

6

Gate size

PFA

2 · 10−4 , 2 · 10−2

False alarm rates

1/12m2

Measurement noise

(0.0125 m/s2 )2

Process noise

T

1s

Sampling time

c1

5

Lower track-loss threshold

c2

15

Upper track-loss threshold

SNR

15 dB

Signal-to-noise ratio

N

300s

Duration of scenario

σr2 σv2

Table 5.2: Lost tracks out of 10000 for ν = 1 SNR = 10

SNR = 15

PFA PFA PFA PFA PFA PFA

= 10−4 = 10−2 = 10−1 = 10−4 = 10−2 = 10−1

PDAF 4367 3361 9549 67 482 6727

PDAFAI 3789 834 721 22 12 19

Table 5.3: Lost tracks out of 10000 for ν = 8 SNR = 10

SNR = 15

PFA PFA PFA PFA PFA PFA

= 10−4 = 10−2 = 10−1 = 10−4 = 10−2 = 10−1

PDAF 181 1401 9327 3 275 6611

PDAFAI 74 51 64 0 1 4

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126

5


1 0.8

PD

0.6 0.4 0.2 0

ν=1 ν=8 10−5

10−4

10−3 PFA

10−2

10−1

100

Figure 5.3: ROC curves in K-distributed noise for a target with SNR = 15dB.

11 + P 22 [m] P∞ ∞

0.8

Position RMSE

0.6

0.4

ν ν ν ν

0.2

0

10−5

= 1, = 1, = 8, = 8,

PDAF PDAFAI PDAF PDAFAI

10−3 PFA

10−4

10−2

10−1

100

Figure 5.4: Evaluation of the MRE for a target with SNR = 15dB. The vertical lines indicate the false alarm rates used in Figures 5.5, 5.6 and 5.7.

1

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127

Results

Error pdf (histogram)

1000

PDAF error pdf PDAFAI error pdf PDAF MRE PDAFAI MRE

800 600 400 200 0 0.2

0.3

0.4

0.5 Position error [m]

0.6

0.7

0.8

Figure 5.5: Error pdf for ν = 1 and PFA = 10−4 .

2000 Error pdf (histogram)

5.7


1500

1000

500

0 0.2

0.3

0.4


0.6

0.7


1

0.8

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128

5



Error pdf (histogram)

2000

1500

1000

500

0 0.2

0.3

0.4


0.6

0.7

0.8


Second, the MRE deals only with the expected RMSE. As illustrated in Figures 5.5 and 5.6 this corresponds to the first order moment of the error pdf. Track-loss is on the other hand an outlier phenomena which only can be adequately addressed if the upper tail of the error pdf is studied. Tables 5.2 and 5.3 reveal that even when the MRE predicts the same performance from the PDAFAI as from the PDAF, the PDAFAI is actually vastly superior to the PDAF as measured by such rare but undesirable occurrences. Several criteria must be taken into account when choosing a nominal false alarm rate. Two criteria that we have not discussed here are the computational power of the tracking system and track management. Increasing the number of false alarms clearly increases the computational burden, and it may therefore in practice be impossible to use a higher false alarm rate than, say, 10−4 . Lowering the false alarm rate will in particular reduce the number of false preliminary tracks, which in many tracking systems consume a huge percentage of the computational power. However, rigorous methods for track management (i.e. initiation and termination) will not function properly if the detection probability is too low. An extreme example is the PHD filter, which may run into difficulties due to a single misdetection [37]. The situation is less severe, but still troublesome, for the IPDA. According to [66], the IPDA should terminate a confirmed track after as few as 3 misdetections. For PD ≤ 0.8 this will happen quite frequently. In [26] it is argued that even if it is tuned to be more optimistic, the IPDA will still need a rather high PD to function properly in the real world. Further advances in track management must therefore either use AI to a larger extent, or exhibit better robustness to low1detection probabilities.

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5.8

5.8

Concluding remarks

129

Concluding remarks

In this chapter the MRE has been extended to the case of amplitude information. Using this approach, the performances of the PDAF and the PDAFAI have been predicted in Kdistributed clutter. We have thoroughly described how the MRE is evaluated numerically for this scenario with and without AI. To the best of our knowledge, this chapter has presented the first systematic approach to the determination of nominal false alarm rates in heavy-tailed clutter. In particular, the analysis shows that one may consider setting the false alarm rate quite high when the PDAFAI is used. The MRE also tells us that the improvement of the PDAFAI over the PDAF should be most noticeable when the false alarm rate is high. Comparison with simulation results have validated the performance prediction offered by the MRE, but also called attention to shortcomings of this approach. The MRE is only able to predict the performance of tracks on target. It provides no information regarding the rate of track-loss, unless it is so high that the MRE diverges. Therefore, the MRE is unable to shed light on the improvements of the PDAFAI over the PDAF for low false alarm rates. The significance of this observation is further discussed in Section 7.2.

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130

5


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6 Target tracking in the presence of wakes

The previous chapters have treated clutter which is stationary, at least in a local sense. There are several kinds of clutter which cannot be treated as stationary. The focus of this chapter is on clutter which depends on the target state. More precisely, the chapter treats wakes that appear behind the target. This kind of clutter must often be dealt with in sonar tracking of human divers, in the tracking of boats using surveillance radars, and also in radar tracking of ballistic missiles. Recent research [87] has integrated a solution to this problem in the popular PDAF method. In this chapter a new solution to this problem is proposed in the same framework. While [87] used an approach described as probabilistic editing, the new solution solves the wake problem in a Bayesian framework by means of marginalization. Monte-Carlo simulations show that the new solution achieves significantly increased robustness as compared to both the standard PDAF and the probabilistic editing approach. As the new solution has improved theoretical underpinnings, it can hopefully be useful for further research on tracking in the presence of wake clutter.

6.1

Introduction

In harbor surveillance, sensors such as radars and sonars are used to detect and track various targets. Conventional tracking methods work on point measurements extracted from the radar or sonar images. The tracking method attempts to associate these correctly to targets of interest and to clutter, which by definition is not of interest. Clutter is caused by several sources. There will always be some background noise that is not filtered out by the detection operation, as no detector can achieve unity detection probability for a non-unity false alarm rate [42]. Acoustic reverberation or radar backscatter from the sea surface may cause more troublesome clutter, which often is 131

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132

6

Target tracking in the presence of wakes

characterized by heavy-tailed amplitude distributions (cf. Chapters 4 - 5). Almost all literature on target tracking has so far assumed that the clutter is independent of the target. An important counter-example to this assumption is targets with wakes. Such targets can be scuba divers in sonar data, and ships or missiles in radar data. The wake of a target can be a much more powerful source of clutter than background noise or arbitrary backscatter. The background for our work on wake targets can be traced back to [5], in which it was suggested to treat wake measurements as clutter, thereby enabling the tracking method to filter out such measurements using probabilistic editing. In [87] this approach was integrated into an operational data association algorithm, namely the classical PDAF [6]. Further extensions of this work treated multiple targets [88] and multiple sensors [90]. The intuition behind any solution to the wake problem is that measurements behind the predicted target position are more likely to originate from the wake than measurements in front of the predicted target position. Intuition also tells us that the further behind a measurement is, the more likely it is to be from the wake. In [5] and [87] this intuitive notion was quantified by assigning wake measurements a spatial pdf in the shape of a linear ramp. An objection to that approach is that the intensity of the wake cannot be expected to increase indefinitely as one moves backwards from the target. Rather, physical considerations leads to a decaying wake model, or possibly an approximately flat wake model if the decay is slow. Another objection to the approach of [5] and [87] is their usage of probabilistic editing. The linear ramp does not have any further justification than the ad-hoc reasoning in the previous paragraph, and one may ask whether a more rigorous approach can be developed. In this chapter a different solution to the wake problem is developed. This solution can be summarized by two key ideas. First, the wake pdf is assumed flat. Second, the wake is assumed to begin at the true but unknown target position, and not at the target prediction as implicitly assumed in [87]. A backwards increasing clutter pdf is then obtained by marginalization over the predicted pdf of the target. Tracking methods can be classified into parametric and non-parametric tracking methods. Parametric tracking methods assume knowledge about the clutter cardinality distribution or pmf. Non-parametric tracking methods do not assume any knowledge about it, and assign the clutter cardinality a non-informative or diffuse pmf. Using a flat wake model allows us to specify the wake as a region with a well-defined clutter intensity λw which is higher than the surrounding clutter intensity λ0 . The flat wake model can therefore be integrated into parametric tracking methods. This is important since most tracking methods are parametric. On the other hand, it is also important to validate that the proposed approach is useful for non-parametric tracking methods. Therefore we are here content with the non-parametric PDAF, which also was used in [87]. The work presented here addresses some of the same problems as the references [72 pp. 424-426] or [106], in which a FISST approach to extraneous target measurements

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6.2

133

Background

was developed. This leads to a more general and abstract methodology of which wake clutter can be considered a special case. However, our aim is not to elaborate such a special case, but rather to improve the treatment of wake clutter in the conventional framework of the PDAF. Three versions of the PDAF are investigated: The classical non-parametric PDAF, the modified PDAF of [87] which we refer to as WPDAF-1, and the tracker developed here, which we call WPDAF-2. The chapter is organized as follows: Section 6.2 presents background material, including the PDAF tracking method and how it has been modified in previous research on tracking of wake targets. In Section 6.3 an alternative modification of the PDAF is developed. In Section 6.4 the tracking methods are tested on simulated data. A brief conclusion is found in Section 6.5. Details regarding the derivations of the PDAF and the wake models are left for Appendix C.

6.2

Background

This section presents the conventional PDAF method and a modified version of this method that was used to track wake targets in [87].

6.2.1

Kinematics and measurement model

We assume that there is one and only one target in the surveillance region, whose kinematic state is denoted by the vector xk . The state contains position ρk and possibly other T variables such as velocity, acceleration, heading and so on, so that xk = [ρT k , . . .] . For ease of presentation the kinematic transition prior is assumed linear, xk = F xk−1 + vk , vk ∼ N (0, Q).

(6.1)

Measurement vectors zk of the target are also related linearly to the target state: zk = Hxk + wk = ρk + wk , wk ∼ N (0, R).

(6.2)

Notice that the matrix H is supposed to be constructed in such a way that the measurement vector contains only the position of the target corrupted by the measurement noise wk . We believe that extensions of our work to measurements including Doppler, amplitude and so on are quite straightforward, but for simplicity such cases are not to be considered here. Also notice that measurements in polar coordinates can easily be converted to measurements obeying (6.2) as explained in [6 pp. 38-41]. For uncorrelated vk and wk the filtering problem posed by (6.1) and (6.2) is optimally solved by the Kalman filter [44]. In reality one does not have a single measurement at each time, but a set of validated measurements Zk = {zk (1), . . . , zk (mk )} which may or may not include a true measurement from the target. The rest of the measurements are considered “clutter”, and data association is needed to discriminate against such false measurements.

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134

6.2.2

6



The PDAF is one of today’s most popular tracking algorithms due to its pragmatic compromise between efficiency and robustness. It is a suboptimal algorithm which at each time step collapses the target state posterior pdf into a single Gaussian which then is propagated to the next time step. Thus the past information about the target at time step k is summarized by ˆ k|k−1 , Pk|k−1 ) p(xk |Z k−1 ) ≈ N (xk ; x

(6.3)

where Z k−1 = (Z1 , . . . , Z k−1 ) is the collection of all previous measurements. The lumping implied by (6.3) is done by expressing the state estimate at time k as a weighted ˆ k|k−1 and state estimates conditioned on the latest measureaverage of the prediction x ments zk (i). This leads to the following Kalman Filter-like equations for prediction and ˆ k|k and its associated covariance Pk|k : measurement update of the state estimate x ˆ k|k−1 = F x ˆ k−1|k−1 x Pk|k−1 = F Pk−1|k−1 F T + Q ˆ k|k = x ˆ k|k−1 + Kk x

mk X

βk (i)νk (i)

i=1

Pk|k = Pk|k−1 − (1 − βk (0))Kk Sk KkT + P˜k

(6.4)

where Kk = Pk|k−1 H T Sk−1 Sk = HPk|k−1 H T + Rk ˆ k|k−1 = zk (i) − zˆk|k−1 νk (i) = zk (i) − H x # "m k X T T P˜k = Kk βk (i)νk (i)νk (i) − νk ν K T k

k

i=1

νk =

mk X

βk (i)νk (i).

(6.5)

i=1

The index i ranges over all mk extracted and validated measurements inside a validation gate G, defined by ˆ k|k−1 )T Ψ−1 ˆ k|k−1 ) < 1 G : (zk − H x k (zk − H x

(6.6)

where Ψk = g 2 Sk and the scalar g is called the gate size. The association probabilities βk (i) = P {θk (i)|Z k } in (6.4) are the probabilities of

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6.2

135

Background

the following mutually exclusive and exhaustive events θk (0) θk (1) .. .


θk (mk ) Measurement mk originates from the target. The PDAF comes in a parametric and in a non-parametric version. The parametric version assumes that the number of clutter measurements is Poisson distributed (c.f. Section 6.3.3). However, since it can be very difficult to estimate the parameters of this Poisson process, a non-parametric version is commonly used instead. The non-parametric PDAF treats the clutter intensity as unknown by assigning it a flat pmf: µ(0) = µ(1) = . . . = µ(m − 1) = µ(m) = µ(m + 1) = . . . .

In this chapter the non-parametric PDAF is used. It can be derived under standard assumptions as done in Sections 2.6.1 - 2.6.2. In order to emphasize the treatment of nonstationary clutter, we express the association probabilities in a slightly different form than the conventional expressions of (2.93) and (2.94). As shown in Appendix C.1, the association probabilities can instead be written δ Pmk βk (0) = (6.7) δ + j=1 l(j) βk (i) =

δ+

where

l(i) Pmk

j=1 l(j)

(6.8)

1 − PD PG . (6.9) PD PG The constant PD is the probability of detection while PG is the probability that a true detected measurement is inside the gate. PD is a tuning parameter assumed known, while PG can be calculated from g as explained in [6 p. 96]. The likelihood ratio l(i) is defined as p(zk (i)|zk (i) is from target) l(i) = . (6.10) p(zk (i)|zk (i) is from clutter) For the conventional PDAF we have p1 (i) (6.11) l(i) = p0 (i) where 1 p1 (i) = N (zk (i) ; zˆk|k−1 , Sk ) (6.12) PG and 1 1 p0 (i) = = (6.13) V πg 2 |Sk |1/2 when the dimension of zk (i) is 2. In Sections 6.2.3 and 6.3.3 alternatives to the uniform clutter pdf of (6.13) are suggested. δ = mk

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136

6.2.3

6


Mitigation of a wake by means of probabilistic editing

In [87] it was noticed that the PDAF had trouble tracking a human scuba diver due to a trailing “wake” of air bubbles. The proposed solution was to replace the uniform clutter pdf pi0 = 1/Vk with a mixture pm (i) , (1 − PB )

1 PB pb (zk (i)). + Vk PGB

(6.14)

It consists of the conventional flat clutter pdf and a linear ramp pb (·) which increases backwards behind the predicted target position. PB is a mixture constant related to the wake intensity while PGB is a normalization constant. The ramp is before normalization given by y˜ z /(AB 2 ) if 0 < z y˜ < B and kz x˜ k < A pb (z) = (6.15) 0 otherwise where the wake-oriented coordinates z x˜ and z y˜ are yet to be properly introduced in Section 6.3.1. It is shown in Appendix C.2 that the normalization constant PGB can be written Z

π 2

− 12 T

PGB = pb (z)dz = det(Λ) Λ u (6.16) M R

3AB 2 2 G

where (Λ, M ) is the diagonalization of Ψ as in (6.36). To summarize, the tracking method of [87] results when the likelihood ratio in (6.7) and (6.8) is chosen to be l(i) =

p1 (i) . pm (i)

(6.17)

We refer to this tracking method as WPDAF-1.

6.3

Mitigation of a wake by means of marginalization

The mixture (6.14) is a valid pdf, but it is not the spatial pdf of clutter measurements under any reasonable model of the wake. Previous papers on wake targets [5,88,87,90] have not made any attempts at providing a probabilistic model for the wake. The rationale behind this section is that a more robust tracking method may be obtained if the ramp in (6.15) is replaced by a pdf which actually models the wake.

6.3.1

Geometry of the wake model

As the simplest yet plausible model imaginable we suggest a flat wake model. Wake measurements are uniformly distributed in a rectangular region behind the target with width 2A and length B. The notion of “behind” is given by the heading unit vector uk , which points in the opposite direction of the wake. We assume this vector known. This can to some degree be justified by the fact that for a slowly manoeuvering target several

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6.3

137


W(ρk ) G y˜

zk

C(ρk )

x ˜ ρˆk|k−1 ρk

D(ρk ) y

uk

x

Figure 6.1: Illustration of the validation gate G and the regions W, C and D. Notice that W, C and D depend on the position of the true state ρk , which is unknown. We can decompose the point ρk (as well as any measurement zk ) in fixed coordinates (ρxk , ρyk ) or in wake-oriented coordinates (ρxk˜ , ρyk˜). scans can be used to obtain a good estimate of this vector. This was actually done in [87] as well. It is useful to employ a wake oriented coordinate system (˜ x, y˜) with unit vectors exk˜ = R(π/2)uk , eyk˜ = −uk

(6.18)

in addition to the fixed (x, y)-system. Here R (α) denotes a two-dimensional rotation matrix corresponding to α radians. When a point ρk is decomposed in the (˜ x, y˜)-system we will denote its components by ρxk˜ and ρyk˜: ˆ k|k−1 )T R (π/2) uk ρxk˜ =(z − H x ˆ k|k−1 )T uk . ρyk˜ = − (z − H x

(6.19)

The entire wake region is denoted W, and it splits the validation gate into a back region C and a front region D, C = G ∩ W, D = G \ C,

(6.20)

where “\” is the set difference operator. Notice that since W depends on the state xk , the ˆ k|k−1 , regions C and D depend on xk as well. The regions do of course also depend on x ˆ k|k−1 is given and not random when but this dependency poses less of a challenge since x conditioned on Z k−1 . See Figure 6.1 for an illustration of the geometry.

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138

6.3.2

6


Pre-validated measurements

˜k = Z ˜1 ∪ Z ˜0 ∪ Z ˜ w of measurements from the At time k the tracker is fed a set Z k k k respective sources of target, background and wake. The tildes indicate that these sets have not yet been subject to validation. In contrast to what was done in Section 6.2, this section takes a parametric viewpoint and assigns informative pmf’s to the number of measurements from these sets. In other words, the measurement sets are characterized by their cardinality distributions as well as their distributions of individual member points. The cardinalities are distributed according to ˜ 1 ) = τ } =Bernoulli(τ ; PD ) P {card(Z k 0 ˜ P {card(Z ) = ϕ} =Poisson(ϕ ; V˜ λ0 ) k

where

˜ w ) = ψ} =Poisson(ψ ; 2ABλw ) P {card(Z k

(6.21)

  1 − PD if τ = 0 P if τ = 1 Bernoulli(τ ; PD ) ,  D 0 otherwise

e−λ λm . (6.22) m! The parameters λ0 and λw are the intensities of background and wake, respectively. By V˜ we denote the area of the entire surveillance region. The spatial pdf’s of individual measurements are Poisson(m ; λ) ,

˜ 1 , xk ) =N (zk ; Hxk , R) p(zk |zk ∈ Z k ˜ 0 , xk ) =1/V˜ p(zk |zk ∈ Z k w ˜ , xk ) = 1/(2AB) if z ∈ W p(zk |zk ∈ Z k 0 otherwise

x˜ where z ∈ W if and only if z < A and 0 < z y˜ < B.

6.3.3

(6.23) (6.24) (6.25)

Post-validated measurements

As mentioned in Section 6.2.2 a validation gate is used to prevent the tracking method from wasting resources on measurements very far away from the prediction zˆk|k−1 . This means that the actual set Zk = Zk1 ∪ Zk0 ∪ Zkw encountered by the tracking method is a ˜k from Section 6.3.2. The validated measurement sets are certain subset of the full set Z given by ˜ 1 ∩ G, Z 0 = Z ˜ 0 ∩ G, Z w = Z ˜ w ∩ C. Zk1 = Z (6.26) k k k k k The first two of these sets have obvious cardinality pmf’s

P {card(Zk1 ) = τ | Z k−1 } =Bernoulli(τ ; PD PG )

P {card(Zk0 ) = ϕ | Z k−1 } =Poisson(ϕ ; 2Vk λ0 ).

(6.27) (6.28)

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6.3


139

The statistics of the validated wake measurement set Zkw are less trivial. Difficulties are caused by the fact that the region C is random, since it depends on the state xk . The cardinality of Zkw is no longer Poisson distributed due to the marginalization over xk . Still, a Poisson distribution with parameter λw Vk /2 may serve as an approximation: P {card(Zkw ) = ψ|Z k−1 } ≈ Poisson(ψ ; λw Vk /2).

(6.29)

Since the sum of two Poisson processes also is Poisson [85], the cardinality of general clutter measurements is approximately P {card(Zkw ∪ Zk0 ) = n|Z k−1 } ≈ Poisson(n ; λw Vk /2 + λ0 Vk ).

(6.30)

The spatial pdf’s of a single target or background measurement when conditioned on previous data are Z p(zk |zk ∈ Zk1 , Z k−1 ) = p(zk |zk ∈ Zkt , xk )p(xk |Z k−1 )dxk 1 N (zk ; zˆk|k−1 , Sk ) = p1 (i) PG Z p(zk |zk ∈ Zk0 , Z k−1 ) = p(zk |zk ∈ Zkb , xk )p(xk |Z k−1 )dxk =

(6.31)

=1/Vk = p0 (i).

(6.32)

Our solution to the wake problem is based on the spatial pdf of validated wake measurements. Since these are to be considered clutter, we focus on the spatial pdf of validated clutter measurements zk ∈ Zkw ∪ Zk0 . In Appendix C.3 it is shown that this pdf can be approximated as pw (i) =p(zk (i) | zk (i) ∈ Zkw ∪ Zk0 , Z k−1 ) √ 32 πλ2w α2 σk2 1 2π ≈p λ0 + 2 2λ0 + λw π (2λ0 + λw )3 π 3 |Ψk | √ √ 32 πλ3w α2 σk2 2 πλw y˜ 2 +ncdf(zk (i) ; 0, σk ) + 2 2λ0 + λw π (2λ0 + λw )3 ! !# √ √ zky˜(i)2 16 2λ3w α2 σk zky˜(i) 4 2λ2w ασk + exp − + π(2λ0 + λw )2 π 2 (2λ0 + λw )3 2σk2

(6.33)

where T σk2 = uT k HPk|k−1 H uk

(6.34)

and the scalar αk is found as αk =

k det(Bk )k kBk · [1, 0]T k

(6.35)

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140

6


where Bk is given by an eigenvalue-decomposition −1

Mk Λk MkT = Ψk and Bk = Λk 2 MkT . By ncdf(·) we denote the cumulative Gaussian distribution: 1 z 2 √ ncdf(z ; 0, σ ) = 1 + erf . 2 σ 2

(6.36)

(6.37)

Notice that (6.33) only depends on λ0 and λw through the ratio λ0 /λw . It can therefore be used in a non-parametric context as long as one has some idea what this ratio should be.

6.3.4

WPDAF-2

The tracking method termed WPDAF-2 is obtained when we replace the mixture (6.14) with the pdf approximated in (6.33) of Section 6.3.3: l(i) =

6.3.5

p1 (i) . pw (i)

(6.38)

Comparison with exact approach

WPDAF-2 does not provide the exact association probabilities that would result from rigorous Bayesian calculations (cf. Appendix C.1). The usage of single-measurement likelihood ratios as in (6.7) and (6.8) requires the measurements zk (1), . . . , z(mk ) to be independent when conditioned on the previous measurements Z k−1 . The measurements are in general only independent when conditioned on the state xk , and the exact Bayesian solution can therefore only be evaluated numerically by marginalizing the joint density of the entire measurement set Zk over the predicted density p(xk |Z k−1 ). In Figures 6.2 and 6.3 the association probabilities of the PDAF, WPDAF-1 and WPDAF-2 as well as the exact approach are illustrated for a scenario with two measurements in the validation gate. It can be seen that WPDAF-1 is in danger of increasing the association probability for measurements that are even significantly behind the prediction. WPDAF-2 does on the other hand exhibit a similar behavior as the exact approach in vicinity of the prediction. Also notice that the ratio between β(i) and β(i)|PDAF does not necessarily decrease monotonously as a function of z y˜(i) for the exact approach. We believe this to be due to the coupling between the measurements in the exact approach.

6.4


In this section the trackers known as conventional PDAF, WPDAF-1 and WPDAF-2 (cf. Sections 6.2.2, 6.2.3 and 6.3.4) are tested and compared using Monte-Carlo simulations.

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6.4

141


Association weight β(1)

1 WPDAF-1 WPDAF-2 Exact PDAF

0.8 0.6 0.4 0.2 0

-4

-3

-2

0 -1 1 Along wake position z y˜(1)

2

3

4

Figure 6.2: Comparison of WPDAF-1, WPDAF-2, exact solution and standard PDAF. These curves show the association probability β(1) as a function of y˜ for a scenario where mk = 2, z x˜ (1) = 3, z x˜ (2) = 0, z y˜(2) = 5, Pk|k−1 = I, R = I, g = 6 and PB = λw /λ0 = 0.9.

101

100

Ratio

β(1)/β(0) β(1)/β(0)|PDAF

(log. scale)

WPDAF-1 WPDAF-2 Exact

-4

-3

-2

-1 1 0 Along wake position z y˜(1)

2

3

4

Figure 6.3: Comparison of WPDAF-1, WPDAF-2 and exact solution. These curves show how much the association probability of a single measurement is increased or decreased for the WPDAF’s relative to the conventional PDAF. The parameters are the same as in Figure 6.2. 1

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6


The authors of [87] tested WPDAF-1 on a single set of real sonar data containing a scuba diver. Its multi-target generalizations have performed well in Monte-Carlo simulations [88, 90]. Still no systematic investigation of its ability to discriminate against wake measurements alone has so far been carried out. WPDAF-2 is a novel contribution of this thesis, and its performance has consequently never been investigated before. The methods are here tested on two scenarios which only differ in how artificial wake measurements are generated. In order to test the performance of the tracking methods these scenarios are run 5000 times for 4 different intensities of the wake clutter.

6.4.1

Filter model

The tracking methods are implemented using the linear kinematics model given by (6.1) and (6.2) with standard matrices given by  3  T 0 0 3  T2 0 0   , Q = σv2   2 1 T   0 0 1 0



1 T  0 1 F =  0 0 0 0 and H=

6.4.2

1 0 0 0 0 0 1 0

,R=

T2 2

T 0 0

 0  0  2 T   2 T

0 0 T3 3 T2 2

σr2 0 0 σr2

(6.39)

(6.40)

.

Simulation of target kinematics

The performance analysis of this chapter uses the same simulation model for the target kinematics as in Chapter 4, although with different parameters. In this curvilinear model developed in [10] the state is parameterized as xsk = [xk , yk , x˙ k , y˙ k , atk , ank ], where atk is acceleration tangential to the target trajectory and ank is acceleration perpendicular to the trajectory. The kinematic process model can then be written as xsk+1



F = 0 0

 Gt (xsk ) Gn (xsk )  xsk + vks . βt 0 0 βn

(6.41)

The matrices Gt (xsk ) and Gn (xsk ) are explicitly given by 

− ω12 cos ϕk+1 + k

1 ωk2

cos φk −

1 ωk T

sin φk

 1 1  ωk sin ϕk+1 − ωk sin φk Gt (xk ) =  1 1  − ω2 sin ϕk+1 + ω2 sin φk + ω1k T cos φk k k − ω1k cos ϕk+1 + ω1k cos φk

    

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6.4

143


and



− ω12 sin ϕk+1 −

1 ωk2

sin φk −

1 ωk T

cos φk



k   − ω1k cos ϕk+1 − ω1k cos φk   Gn (xk ) =    − ω12 cos ϕk+1 + ω12 cos φk + ω1k T sin φk  k k 1 1 ωk sin ϕk+1 − ωk sin φk q where ϕk+1 = φk + ωk T , ωk = ank /| x˙ 2k + y˙ k2 | and φk = tan−1 (y˙ k /x˙ k ). The constants βt and βn are related to the maneuver time constants τt and τn by

βt = exp(−T /τt ) and βn = exp(−T /τn ).

(6.42)

We draw the accelerations according to first order Markov models, so that vks ∼ N (0, Qs ) where   0 0 0 . 0 Qs =  0 σt2 (1 − βt2 ) (6.43) 2 2 0 0 σn (1 − βn )

The target moves in a surveillance region covering 500m × 500m for 300 seconds. The tuning constants σt2 , τtm , σn2 and τnm have in Table 6.1 been chosen such that the kinematics shall mimic a human diver swimming relatively straight ahead.

6.4.3

Measurement generation

Measurements are generated according to point processes similar to those described in Section 6.3.2. The simulation of the target and background measurement sets Zk1 and Zk0 are straightforward and need no further elaboration. It is less obvious how a realistic wake measurement set Zkw should be generated. We will therefore provide two different scenarios. In both scenarios the severity of wake clutter is characterized by a wake intensity ratio r which tells us to which degree the wake clutter dominates over ordinary background clutter. ˜ w according to (6.21) and (6.25). That is, wake measureIn Scenario I we draw Z k ˜ w is drawn from ments are drawn from a uniform spatial density and the cardinality of Z k a Poisson distribution with parameter 2rs ABλ0 . ˜ w according to a recipe similar to the one used in [88]. In Scenario II we draw Z k This recipe draws each wake measurement from a density whose marginal densities are Gaussian and exponentially distributed in the cross-wake and along-wake directions respectively, y˜ 1 z p(z) = N (z x˜ ; 0, σs2 ) · exp − . (6.44) λs λs ˜ w is drawn from a Poisson distribution with parameter 2rs σs λs λ0 . The cardinality of Z k The SNR affects the scenario through the detection probability PD . In all our simulations we have used the value PD = 0.55, which corresponds to a 12dB target in Rayleigh noise when the false alarm rate is PFA = 0.005. We assume that the resolution

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6


Table 6.1: Simulation model parameters Param.

Value

Specification

T

1.0 s

Sampling period

∆x × ∆y

1m × 1m

Resolution cell size

σt2

(0.01 m/s3 )2

Tan. acceleration power

τtm

2.0 s

Tan. acc. time constant

σn2

(2.5 · 10−5 m/s3 )2

Perp. acceleration power

1.0 s

Perp. acc. time constant

PFA

0.005

False alarm rate

PD

0.55

Detection probability

A×B

2.5m × 50m

Wake region in Scenario I

2.5m

Wake half-width in Scenario II

λs

10m

Wake length in Scenario II

rs

{1, 5, 10, 50}

True wake intensity ratios

τnm

σs

cells of the sensor are 1m × 1m wide, implying that the background clutter intensity is λ0 = 0.005m−2 . These are values for which PDAF-based trackers are most suitable, and quite realistic in order to model a human diver.

6.4.4

Performance measures

In the same way as was done in Chapter 4, track-loss is used as the primary performance measure in this chapter as well. We treat track-loss as aq two-stage process. A track is considered tentatively lost at time k if the position error (xk|k − xsk )2 + (yk|k − yks )2 exceeds a threshold c1 . If the error later goes below c1 the lost label is removed. On the other hand, if the error never manages to go below c1 again, we consider it lost at time k. If the error exceeds the higher threshold c2 > c1 we immediately consider it lost at time k, irrespectively on whether the error later goes below c1 .

6.4.5

Interpretation of the results

Tables 6.3 and 6.4 provide a summary of our results. We point out the following observations: First, it can be seen that the strength of wake clutter is critical to the performance of the PDAF. The PDAF does cope reasonably well with moderate wake intensities in Scenario I, although a quite significant performance loss is apparent. For very high wake

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6.4

145


Table 6.2: Filter model parameters Param.

Value

Specification

g

6

Gate size

σr2

(0.0833 m)2

Mea. noise

σv2

(0.0036 m/s2 )2

Process noise

L

20

Wake direction estimation lag

c1

3

First stage track-loss threshold

c2

10

Second stage track-loss threshold

ra

{1, 5, 10, 50}

Assumed wake intensity ratios

Table 6.3: Lost tracks for Scenario 1 rs = 1

rs = 5

rs = 10

rs = 50

ra ra ra ra ra ra ra ra ra ra ra ra ra ra ra ra

=1 =5 = 10 = 50 =1 =5 = 10 = 50 =1 =5 = 10 = 50 =1 =5 = 10 = 50

PDAF 5 5 5 5 62 62 62 62 325 325 325 325 4993 4993 4993 4993

WPDAF-1 8 61 152 655 20 56 141 634 50 64 139 606 183 109 132 405

WPDAF-2 1 16 46 190 10 9 31 184 46 23 32 172 336 84 84 109

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6


Table 6.4: Lost tracks for Scenario 2 rs = 1

rs = 5

rs = 10

rs = 50

ra ra ra ra ra ra ra ra ra ra ra ra ra ra ra ra

=1 =5 = 10 = 50 =1 =5 = 10 = 50 =1 =5 = 10 = 50 =1 =5 = 10 = 50

PDAF 43 43 43 43 659 659 659 659 749 749 749 749 543 543 543 543

WPDAF-1 17 51 106 532 105 67 99 374 82 77 99 273 53 75 88 125

WPDAF-2 11 12 24 141 76 37 46 93 48 51 59 87 29 57 66 71

intensities (λw ≥ 50λ0 ) the PDAF looses track inevitably. In Scenario II moderate wake intensities are most troublesome, while stronger wakes are easier to handle, probably because the exponential distribution places a large number of wake measurements close to the target. Second, it can be seen that both WPDAF-1 and WPDAF-2 may reduce the track-loss encountered by the PDAF. Their mitigating effect is as one would expect most prominent for strong wakes. WPDAF-2 does in the vast majority of cases outperform WPDAF-1, and especially so when the true wake intensity is weak. Unfortunately, neither WPDAF1 nor WPDAF-2 can be guaranteed to improve performance. There is a danger that the tracker will push the state estimate too far ahead when it assumes the wake to be stronger than it actually is. This may cause it to loose track. We see this happening more frequently for WPDAF-1 than for WPDAF-2. Third, the WPDAF’s do not necessarily achieve their best performance for the correct wake intensity ratio ra = rs . Although WPDAF-1 was run with rather high values of ra in [88] and [87], the results presented in Tables 6.3 and 6.4 indicate that lower values of ra should be used for WPDAF-1. The novel tracker WPDAF-2 does on the other hand most often attain its optimal performance when ra is equal to or reasonably near rs . It is nevertheless also for WPDAF-2 safer to assume a too low wake intensity than a too high wake intensity.

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6.5

6.5

Concluding remarks

147

Concluding remarks

In this chapter, three PDAF-based methods have been investigated for tracking targets with wakes. These are the conventional PDAF, a recently developed modification based on probabilistic editing (WPDAF-1), and a tracker developed in this chapter by means of marginalization (WPDAF-2). Simulation results have revealed that a significant performance gain can be achieved by accounting for the wake as done in WPDAF-1 and WPDAF-2, and that this gain is most prominent and consistent for WPDAF-2. We briefly mention two alleys for further research on target tracking in the presence of wakes. Since both WPDAF-1 and WPDAF-2 do exhibit some sensitivity to the assumed wake strength, it is of great interest to develop robust estimators for this quantity. WPDAF-1 is purely non-parametric, and it is difficult to see how estimation of the wake strength can be done for it. WPDAF-2 is on the other hand developed in a parametric framework, so that such an estimation does make more sense for this tracker. Exactly what kind of estimators would be preferable is not that obvious. The approaches of [69] or [78] may provide some inspiration for this. This chapter has solely focused on track maintenance, i.e. whether or not the tracker manages to follow the target. Another important topic for future research is track initialization and termination in the presence of wake clutter. While simple heuristic rules were used successfully in [87] and [90], it would be interesting to replace these with a more systematic approach such as the IPDA [77] or FISST [72]. These challenges are further elaborated in Section 7.2.

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148

6


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7 Discussion

This thesis has addressed three challenges to the conventional framework of single-target tracking. These were the challenges of dim targets (Chapter 3), heavy-tailed clutter (Chapters 4-5) and targets with wakes (Chapter 6).

7.1

Conclusion

A detailed presentation of the conventional framework for single-target tracking was given in Chapter 2. The PDAFAI was derived as a special case of the optimal Bayesian approach (OBA). TBD methods that work on raw sensor data were also discussed. In Chapter 3, the impact of unknown background noise on TBD methods was discussed. It was demonstrated that this additional uncertainty could deteriorate the performance of the tracking method to an unacceptable level. As a solution, it was proposed to account for the uncertainty by marginalizing the true but unknown background power over a flat prior distribution. This almost restored the performance achieved for known background noise. In Chapter 4, the same ideas were utilized to improve the performance of the PDAFAI in unknown Rayleigh distributed background noise. Although not as dramatic, rates of track-loss were still reduced by half in scenarios where the Rayleigh assumption was reasonably adequate. Furthermore, Chapter 4 presented the first systematic treatment of heavy-tailed clutter ever done in the context of target tracking. The PDAFAI was tailored to deal with Kdistributed clutter by means of marginalization over a so-called texture variable, which corresponds to the background power in the conventional Rayleigh setting. Simulation results revealed that this modified PDAFAI could reduce track-loss rates by up to 90% when compared to the conventional PDAF. A more theoretical approach to the analysis of the PDAF and the PDAFAI in heavy149

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150

7

Discussion

tailed clutter was taken in Chapter 5. In that chapter, the modified Riccati equation corresponding to the PDAFAI was derived and evaluated for K-distributed background noise. This analysis correctly indicated that the improvement of the PDAFAI over the PDAF is most noticeable for high false alarm rates. It should nevertheless be noted that it failed to reveal the improvement of the PDAFAI for low false alarm rates. In Chapter 6, a new solution to the problem of single-target tracking in the presence of wakes was developed. In contrast to previous works on wake clutter, the new solution was obtained by specifying a Bayesian model for the wake clutter, which led to a closedform approximation for the clutter likelihood. Simulation results showed considerable reductions in track-loss rates as well as increased robustness due to the new solution. All the modified PDAF’s and PDAFAI’s proposed in this thesis can be readily implemented in real world tracking systems. Although the conservative treatments of amplitude information do require substantially more computational power than the conventional PDAFAI, this increase is quite modest compared to the increase incurred by other tracking methods which require exhaustive enumeration, numerical searches etc. The TBD method of Chapter 3 is less ready for the real world. Nevertheless, the discussion of that chapter may still be crucial in order to make Bayesian TBD actually work in the real world. It should be noted that although the conclusions of this thesis are in agreement with common sense, they were only arrived at after the research was carried out. It was anticipated that lack of precise knowledge about the background noise should have some impact on TBD-methods, but it was not expected that the impact would be so severe. After this was revealed, it came as a pleasant discovery that the performance loss could almost entirely be mitigated by the treatment proposed in Chapter 3. Neither was it obvious that amplitude information should be as beneficial in heavy-tailed clutter as observed in Chapters 4 and 5. A reviewer of the conference paper [23] anticipated “little or no improvement” due to the overlap between clutter and target pdf’s for heavy-tailed clutter. The probabilistic edit solution to the wake problem has been around for more than 3 decades. Since it has never been questioned during all this time, it did not seem likely that it could be much improved. Only after a careful study of its counter-intuitive behavior was the alternative solution of Chapter 6 developed. We therefore conclude that a tracking method can be significantly improved by modeling clutter as integral part of the tracking problem. More precisely, such modeling should address the uncertainties involved in a smooth manner. This has been done in this thesis by marginalization over estimation uncertainty (Chapters 3 and 4), over the local clutter power (Chapters 4 and 5), and over the predicted state estimate itself (Chapter 6). To summarize: With the completion of the research reported in this thesis, rigorous and efficient treatments of heavy-tailed clutter and wake clutter have been developed for use in target tracking. Furthermore, the research has contributed towards improved understanding of TBD methods.

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7.2

7.2

Future research

151

Future research

This thesis has exclusively treated the single-target problem. Further research should address how the methods developed here can be extended to multi-target scenarios. This is probably most straightforward for the conservative PDAFAI’s. However, such an extension may require a more careful treatment of the mean target power than the one given in here, possibly along the lines of [31]. The probabilistic editing solution (WPDAF-1) to the wake problem has been extended to multi-target scenarios in [88]. It is not immediately clear how the marginalization solution (WPDAF-2) developed in this thesis can be extended to multiple targets. The development of multi-target TBD is, as argued in Section 2.6.4, far from straightforward. Although several advances have recently been done towards this goal, it is the author’s opinion that this very important problem still is fundamentally unsolved. The developments of this thesis should also be extended towards tracking methods which do track management, i.e. methods which attempt to make inference regarding the presence of a target. A full understanding of the impact of heavy-tailed clutter cannot be achieved without such developments. It would be interesting to compare how several of the methods mentioned in Section 2.6.4 perform in heavy-tailed clutter. Such a comparison should include the IPDA [77], the MHT [91] and FISST methods [72]. The conservative amplitude likelihoods of Chapter 4 may possibly be replaced by better models. On the one hand it is certainly a drawback that they do not allow closed form expressions. It would be very useful if the integrals discussed in Appendix B could be approximated by simpler expressions. One may also be able to replace the K-distribution with other, but similar, models which allow closed form expressions. On the other hand, the treatment of amplitude information presented by this thesis is nevertheless very simple. A more refined treatment may for example treat both spatial and temporal correlations. The truncation of amplitude likelihoods can also be done in a more refined way. Another alley for future research is to investigate the usage of noninformative priors in target tracking. In Chapter 4 the flat prior was used when necessary. It is possible that other choices such as Jeffrey’s prior [31] will yield better performance. The results presented in Chapter 4 show that the PDAF and the PDAFAI face severe difficulties if the clutter is sufficiently heavy-tailed. It would be of interest to determine bounds for the lowest feasible SNR in clutter with various degrees of heavy-tailedness. Such bounds will in general not be attained by these methods, since they are quite suboptimal. The determination of such performance bounds will require evaluation of the PCRLB as well as implementation of stronger tracking methods such as TBD or the MHT. It was argued in Chapter 5 that the MRE alone could not distinguish the performance of the PDAFAI from that of the PDAF for low false alarm rates, since it does not account for the outlier nature of track-loss events. This shortcoming of the MRE will also be shared by other RMSE-based performance measures such as the PCRLB. The recent surge in research on the PCRLB and other continuous performance measures should

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152

7

Discussion

therefore be accompanied with critical investigations of whether conclusions obtained by means of these measures also hold when track-loss is used as the primary criterion. The hybrid averaging technique of [67] may possibly be used in order to include trackloss in the performance prediction. Clutter estimation is an important unsolved challenge with regard to state dependent clutter (e.g. in the form of wakes) in two ways. First, the method proposed in Chapter 6 is not insensitive to the assumed wake-to-background intensity ratio. Although estimators of this quantity and the actual clutter intensities necessarily must suffer from a large uncertainty, it is possible that approaches such as those proposed in [69] or [78] could improve performance. Second, the problem of state dependent clutter has so far not been treated from a track management point of view. This is quite straightforward in principle, but more challenging in practice. Rigorous methods for track management are necessarily sensitive to the cardinalities of target and background measurements, since a high frequency of target measurements compared to background measurements indicates the presence of a target. Consequently, track management methods tend to be parametric in the sense of [6 p. 103]. However, the parameters of the corresponding pmf’s can only be estimated within a very large uncertainty, and a track management method is therefore in danger of being provided highly inadequate models for these cardinalities. When a wake is present, several measurements can be expected to originate from the target indirectly through the wake. The standard approach to track management as represented by the IPDA, MHT or FISST would lead to methods which attempt to draw inference regarding the existence of a target using a model for this cardinality. Such a method will sooner or later terminate the track prematurely due to unfulfilled expectations when fewer than expected wake measurements are observed. Finally, the developments of this thesis should be implemented and tested on real data. When carrying out such an investigation one should exercise great care regarding the issues discussed in Sections 2.7.1 and 2.7.2. It may in the practical world be necessary to combine all the advances of this thesis together. Targets with wakes (e.g. divers) are often so weak that reliable tracking cannot be carried out without amplitude information. It may even be necessary to use TBD in order to track them. As mentioned, TBD should also be developed to deal with heavy-tailed clutter in order to investigate performance bounds in such a challenging environment.

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A Derivation of the OBA

The optimal Bayesian approach (OBA) to single-target tracking was presented in Section 2.6.1. The purpose of this appendix is to elaborate some derivations that were skipped for the sake of brevity. Notice that since the PDAF and PDAFAI are sub-optimal versions of the OBA, the results derived here are also pertinent to these methods.

A.1

Prior association probabilities

The event that measurement number i (possibly the empty measurement represented by i = 0) originates from the target at time k is denoted θk (i). The prior probability of θk (i), conditioned only on the number mk of measurements, is

P {θk (i)|mk } =

        

µ(mk ) k −1) µ(mk ) PD +(1−PD ) µ(m −1) k 1 P mk D µ(mk ) PD +(1−PD ) µ(m −1) k

(1−PD ) µ(m

i=0 (A.1) i = 1, . . . , mk .

The probabilities P {θk (i)|mk } are a priori probabilities in the sense that they are based on no further information about the current measurement set Zk than its cardinality mk . Denote the realization of mk by m. Also denote the total number of clutter measurements at time k by nk . For the special case i = 0, P {θk (0)|mk } is the a priori 153

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154

A

Derivation of the OBA

probability that all of the mk measurements are false, P {θk (0)|mk } =P {θk (0)|mk = m}

=P {θk (0)|mk = m, nk = n − 1}P {nk = m − 1|mk = m} + P {θk (0)|mk = m, nk = n}P {nk = m|mk = m}

=0 · P {nk = m − 1|mk = m} + 1 · P {nk = m|mk = m}

=P {nk = m|mk = m} P {mk = m|nk = m}P {nk = m} = P {mk = m} (1 − PD )µ(m) = . P {mk = m}

(A.2)

For all other cases i = 1, . . . , mk , P {θk (i)|mk } is the a priori probability that measurement number i is the correct one, P {θk (i)|mk } =P {θk (i)|mk = m}

=P {θk (i)|mk = m, nk = n − 1}P {nk = m − 1|mk = m}

+ P {θk (i)|mk = m, nk = n}P {nk = m|mk = m} 1 = · P {nk = m − 1|mk = m} + 0 · P {nk = m|mk = m} m 1 = · P {nk = m − 1|mk = m} m P {mk = m|nk = m − 1}P {nk = m − 1} = mP {mk = m} PD µ(m − 1) = . (A.3) mP {mk = m} The probability of having mk = m measurements at time k is P {mk = m} = PD µ(mk − 1) + (1 − PD )µ(mk ).

(A.4)

Inserting (A.4) into (A.2) and (A.3) yields (A.1)

A.2

Posterior association probabilities

In Section 2.6.1 it was claimed that measurement history θk,l has the posterior probability β k,l ∝ mk (1 − PD )µ(mk )β k−1,s

β k,l ∝ lka (i)lkzs (i)PD µ(mk − 1)β k−1,s

for i = 0

(A.5)

for i = 1, . . . , mk

(A.6)

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A.2

155

Posterior association probabilities

where lka (i) =pa1 (ak (i)) / pa0 (ak (i)) Z 1 pz1 (zk (i)|xk )p(xk |θk−1,s , mk , Z k−1 )dxk . lkzs (i) = Vk In order to show this a few short-hand notations are defined. Let the clutter pdf’s pa0 (ak (j)) and pz0 (zk (j)) be written as pa0k (j) and pz0k (j), respectively. Let the target measurement amplitude pdf pa1 (ak (i)) be written as pa1k (i). The target measurement spatial pdf conditioned on previous data is written pas 1k (i)

=

Z

pz1 (zk (i)|xk )p(xk |θk−1,s , Z k−1 )dxk .

(A.7)

From Bayes’ rule it follows that measurement history number l has the probability β k,l , P {θk,l |Z k }

= P {θk (i), θk−1,s |Zk , mk , Z k−1 } 1 = p(Zk |θk (i), θk−1,s , mk , Z k−1 )P {θk (i)|mk }β k−1,s c

(A.8)

where c is a normalization constant. By inserting the prior association probabilities as given by (2.58) or (A.1), it follows that β k,l ∝ p(Zk |θk (0), θk−1,s , mk , Z k−1 )(1 − PD )mk µ(mk )β k−1,s β

k,l

∝ p(Zk |θk (i), θ

k−1,s

, mk , Z

k−1

)PD µ(mk − 1)β

k−1,s

for i = 0 for i = 1, . . . , mk .

The first term in these expressions is the joint measurement pdf conditioned on previous data Z k−1 . It is found by integrating the state-conditioned measurement pdf of (2.55) over the predicted pdf of the state xk . In the case of i = 0 it becomes p(Zk |θk (0), θk−1,s , mk , Z k−1 ) =

Z

p(Zk |θk (0), mk , xk )p(xk |θk−1,s , Z k−1 )dxk   Z Y mk =  pa0 (ak (j))pz0 (zk (j)) p(xk |θk−1,s , Z k−1 )dxk j=1

=

=

mk Y

j=1 mk Y

j=1

pa0 (ak (j))pz0 (zk (j)) pa0k (j)

mk Y

j=1

pz0k (j).

(A.9)

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156

A

Derivation of the OBA

For all other i = 1, . . . , mk it becomes p(Zk |θk (i), θk−1,s , mk , Z k−1 ) Z = p(Zk |θk (i), mk , xk )p(xk |θk−1,s , Z k−1 )dxk   Z mk Y = pa1 (ak (i))pz1 (zk (i)|xk )  pa0 (ak (j))pz0 (zk (j)) p(xk |θk−1,s , Z k−1 )dxk j6=i

=pa1 (ak (j))

mk Y

pa0 (ak (j))pz0 (zk (j))

j6=i mk Y

=pa1k (i)pas 1k (i)

pa0k (j)

mk Y

Z

pz1 (zk (i)|xk )p(xk |θk−1,s , Z k−1 )dxk

pz0k (j)

j6=i j6=i m m k k Y Y pa0k (j) pz0k (j). =lka (i)lkzs (i) j=1 j=1

(A.10)

The product terms are the same in both (A.9) and (A.10), and can therefore also be included in the normalization constant. The result (A.5) and (A.6) follow from this.

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B Evaluation of heavy-tailed integrals

The tracking methods developed in Chapter 4 rely on evaluations of the integrals in (4.36), (4.42) and (4.43). Although no closed-form solutions exist to any of these integrals, they can be evaluated quite fast with a reasonable accuracy. How this can be done is explained in detail in this appendix.

B.1

Evaluation of the K -Swerling I likelihood integral

The integral in (4.43) does frequently occur in literature concerning the K-distribution since it either describes a Swerling I-target in K-clutter, or thermal noise on top of Kclutter. According to [76] this integral is “readily integrated numerically”. However, a straightforward numerical scheme will require very many samples in order to provide satisfactory accuracy for arbitrary values of a, d, ν and b. The challenge is to determine the effective support of the integrand (i.e. where it differs significantly from zero). The integrand η ν−1 η a2 g(η) = exp − − (B.1) η+d b 2(η + d) has several “bad” properties which must be addressed: B1: It may tend very quickly towards ∞ as η → 0. B2: It may tend very slowly towards 0 as η → ∞. B3: Factorizing it in the obvious way (i.e. a Rayleigh pdf and a Gamma pdf) may lead to factors with disjunct effective supports. B4: Its overall shape (maxima, minima, inflection points etc.) is difficult to characterize in simple formulas. 157

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158

B

Evaluation of heavy-tailed integrals

B1 and B2 means that deterministic sampling using a uniform grid will yield bad accuracy. B3 means that stochastic importance sampling will be difficult to carry out. B4 makes it difficult to determine the parameters of any grid or sampling strategy. We could possibly circumvent these problems by generating a lookup table off-line using brute force numerical integration. However, searching a 3-dimensional lookup table is also quite time consuming. To overcome these challenges the following substitution is useful, 1 2 ⇒ dη = − 3 du. 2 u u Then we express the likelihood ratio as √ (a b)1−ν PFA · l(a|d, ν, b) = 2a PD 2K √ η=

ν−1

·

Z∞ 0

(B.2)

b

2 u2 a2 u1−2ν exp − 2 − du 1 + u2 d u b 1 + u2 d

√ Z∞ PFA (a b)1−ν · g ∗ (u)du. , · 2a PD 2K ν−1 √b 0

(B.3)

In Figure B.1 the integrands g(η) and g ∗ (u) are compared for various values of ν. It is apparent that the behavior of g ∗ (u) is more regular than the behavior of g(η). More precisely, the modified integrand g ∗ (u) has always one and only one peak, given by the root of the cubic equation 0 =(u2peak )3 (1 − 2ν)d2 − 2d2 4 + (u2peak )2 2(1 − 2ν)d − 2d + d2 − 2a2 b 8d 4 + u2peak 1 − 2ν + + . (B.4) b b

For ν ≥ 0.5 uniqueness of the peak follows from the fact that g ∗ (u) can be factorized into one monotonously increasing factor and one monotonously decreasing factor. For ν < 0.5 uniqueness can be seen from re-arranging (B.4) so that two monotonous cubic functions in u2peak are obtained. As u → ∞ the modified integrand g ∗ (u) becomes proportional to u−1−2ν . In order to approximate the integral of such a function a grid with variable resolution should be used. We therefore evaluate g ∗ (u) over two grids. A lower grid with points u u [1, 2, . . . , NLower ]2upeak /NLower is used to catch the behavior of g ∗ (u) around its peak. An upper grid is used to catch the asymptotic behavior of g ∗ (u). This grid can possibly be constructed in many ways. We require it to be on the form 2upeak + A exp(B [1, 2, . . . , NUpper ]).

(B.5)

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B.1

159

Evaluation of the K-Swerling I likelihood integral

Integrands as functions of η 0 ν = 0.1 ν=1 ν=8

Normalized ln(g(η))

-2 -4 -6 -8 -10

0

1

2

3 4 Integration variable η

5

6

Integrands as functions of u

Normalized ln(g ∗ (u))

0 -2 -4 -6 ν = 0.1 ν=1 ν=8

-8 -10

0

1

2

3 4 Integration variable u

5

6

Figure B.1: Example integrands of the K-Swerling I likelihood plotted as functions √ of the texture variable η and u = 1/ η. The amplitude is a = 5 and the expected SNR is 15dB for all three cases.

1

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160

B


Normalized error of likelihood ratio l(a|d, ν, b)

Normalized error

10−1

10−2

10−3

10−4

10−5 −1 10

100 Shape parameter ν

101

Figure B.2: Normalized error of the integration scheme described in Appendix B.1 for ν ∈ [0.1 , 8]. The scheme described in Appendix B.2 has the same error.

In order to cover the effective support of g ∗ (u) we require its upper limit to be upeak (2 + 1 1/ 2ν ) where is a small number. This leads to the following values for the parameters in (B.5):

2upeak exp(−B) NLower ν upeak 2upeak 1 1 B= ln √ − ln . NUpper ν NLower A=

(B.6)

Although this upper grid may waste some resources for u slightly above 2upeak , it guarantees that the asymptotic behavior of g ∗ (u) is understood by the numerical integration procedure. The two grids are then joined together and the trapezoidal rule [61] is used to evaluate (B.3). u For the small number we have found 0.005 to work well. The grid sizes NLower and have been set to 20 and 80 respectively. In Figure B.2 the error of this scheme is plotted for various ν. Keeping all the other uncertainties we are facing in mind, this error is certainly nothing to worry about. 1 u NUpper

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B.2

B.2

161

The detection probability integral

The detection probability integral

Using the substitution (B.2) in (4.36) we express the K-Swerling I detection probability as Z∞ 2 u2 TK 2(2/b)ν 2 1 PD = exp − 2 − du. (B.7) Γ(ν) u2ν+1 u b 2(1 + u2 d) 0

Again the modified integrand has a unique peak, which is given by solving 4d2 2 0 = − (u2peak )3 [2ν + 1] d2 + (u2peak )2 2(2ν + 1)d + − TK b 8d 4 + u2peak − 2ν − 1 + . b b

(B.8)

The modified integrand is also in this case asymptotically proportional to u−1−2ν , and therefore the numerical integration technique of Appendix B.1 is applied with upeak given by (B.8) instead of (B.4).

B.3

The conservative Rayleigh likelihood integral

The substitution (B.2) also makes the integral in (4.42) more well-behaved: PFA l(a|d, ηˆ) = ·h PD

Γ(M ) ηˆM +

a2 2

iM

Z∞ 0

u2M −1 a2 2 exp −u ηˆM + du. 1 + u2 d 2(1 + u2 d)

For this integral it is easier to determine √ the support. We simply spread 100 sample points uniformly on the interval u ∈ [0, 2/ ηˆ].

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162

B


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C Wake probability calculations

Non-trivial calculations underlying the results of Chapter 6 are placed in three appendices. The first appendix explains how the PDAF can be written by means of likelihood ratios. The second appendix provides an alternative calculation of the normalization constant PGB used in WPDAF-1. The third appendix derives the spatial clutter pdf of WPDAF-2, which is the main result of Chapter 6.

C.1

The role of likelihood ratios in the PDAF

The PDAF is derived under the assumption of a uniform spatial clutter pdf [6 pp. 129161]. This assumption conveniently allows us to express the association probabilities (6.7) and (6.8) using likelihood ratios li of the individual measurements conditioned on previous data. When the spatial clutter pdf depends on xk this de-coupling breaks down. Approximations, whose nature is the topic of this appendix, are then necessary to maintain this simplicity. Let mk and nk be the number of total and clutter measurements respectively at time k, and let m be the realization of the random variable mk . Bayes’ rule yields P {θk (0)|Z k } =P {θk (0)|mk , Zk , Z k−1 } 1 = p(Zk |θk (0), mk , Z k−1 )P {θ0 |nk = m, mk = m} c · P {nk = m|mk = m} 163

(C.1)

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164

C

Wake probability calculations

and P {θk (i)|Z k } =P {θk (i)|mk , Zk , Z k−1 } 1 = p(Zk |θk (i), mk , Z k−1 )P {θi |nk = m − 1, mk = m} c · P {nk = m − 1|mk = m}.

(C.2)

The joint measurement pdf’s p(Zk |θk (0), mk , Z k−1 ) and p(Zk |θk (i), mk , Z k−1 ) are conditioned on the past data (i.e. the prediction and its covariance) and not on the state xk . However, since target and clutter measurements depend on the state rather than the prediction, these quantities must be written as integrals over the state, p(Zk |θk (0), . . .) = p(Zk |θk (i), . . .) =

Z Y mk

Z

j=1

pc (zk (j)|xk )p(xk |Z k−1 )dxk

p1 (zk (i)|xk )

mk Y j6=i

pc (zk (j)|xk )p(xk |Z k−1 )dxk .

(C.3)

Here p1 (·|xk ) is the spatial pdf of target measurements and pc (·|xk ) is the spatial pdf of clutter measurements conditioned on the state. In the conventional PDAF only p1 (·|xk ) depends on the state, while pc (·|xk ) = pc (·) is uniform. The integrals in (C.3) are then of a trivial nature. However, in the treatment of wake clutter we allow pc (·|xk ) to depend on the state, which results in non-trivial marginalization integrals. An “exact” Bayesian approach along the lines of [106] would solve these integrals numerically (c.f. Section 6.3.5). A simpler, although somewhat heuristic, approach is to approximate the integrals by assuming that the the measurements zk (1), . . . , zk (mk ) are independent, not only when conditioned on the state xk , but also when conditioned on the previous data Z k−1 . We simply proceed as if the joint measurement pdf’s also in this case can be de-coupled as follows: p(Zk |θk (0), . . .) ≈ =

mk Z Y

j=1 mk Y

pc (zk (j)|xk )p(xk |Z k−1 )dxk (C.4)

pc (j)

j=1

p(Zk |θk (i), . . .) ≈

Z ·

p1 (zk (i)|xk )p(xk |Z k−1 )dxk

mk Z Y

pc (zk (j)|xk )p(xk |Z k−1 )dxk

j6=i mk Y i =p1 pc (j) j6=i

= l(i)

mk Y

j=1

pc (j).

(C.5)

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C.1

165

The role of likelihood ratios in the PDAF

The association probabilities can then be found as mk 1Y P {mk = m|nk = m}P {nk = m} P {θk (0)|Z } = pc (j) c P {mk = m} k

=

=

1 c

j=1 mk Y

pc (j)

j=1 mk Y

m χ

j=1

(1 − PD PG )µ(m) P {mk = m} (C.6)

pc (j)(1 − PD PG )µ(m)

and m

k Y 1 P {mk = m|nk = m − 1}P {nk = m − 1} P {θk (i)|Z } = l(i) pc (j) c mP {mk = m}

k

1 = l(i) c 1 = l(i) χ

j=1 mk Y

j=1 mk Y

j=1

pc (j)

PD PG µ(m − 1) mP {mk = m} (C.7)

pc (j)PD PG µ(m − 1)

where another normalization constant has been introduced as χ=m

mk Y

j=1

pc (j)(1 − PD PG )µ(m) + PD PG µ(m − 1)

mk Y

j=1

pc (j)

mk X

l(j).

(C.8)

j=1

We find the inverse association probabilities to be

and

χ 1 = Qmk k P {θk (0)|Z } m j=1 pc (j)(1 − PD PG )µ(m) PD PG µ(m − 1) X =1 + l(j) m(1 − PD PG ) µ(m) χ 1 Qm = k P {θk (i)|Z } l(i) j=1 pc (j)PD PG µ(m − 1) X 1 m(1 − PD PG ) µ(m) = + l(j) . l(i) PD PG µ(m − 1)

(C.9)

(C.10)

For the case of a non-parametric PDAF the ratio µ(m)/µ(m − 1) is by definition unity. Under that assumption the association probabilities in (6.7) and (6.8) follow immediately.

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166

C.2

C


The normalization constant PGB in WPDAF-1

The normalization constant PGB was calculated in [87]. Since that calculation was rather cursorial, we provide a more detailed calculation here. The wake pdf has (in accordance with assumption A2 of Appendix C.3) a support stretching behind from the x ˜-axis in such a way that its left, right and upper boundaries are outside of the validation ellipse. For WPDAF-1 it is defined by y˜/(ab2 ) if 0 ≤ y˜ ≤ B and − A ≤ x ˜≤A pb (˜ x, y˜) = (C.11) 0 otherwise.

Since wake measurements from outside the gate are discarded, we need to restrict this density to the gate in order for it to be a proper normalized pdf. In other words we need to evaluate Z PGB = pb (˜ x, y˜)d˜ xd˜ y. (C.12) G

This integral will here be solved using diagonalization. Let Λ contain the eigenvalues of Ψ and let the orthogonal matrix M contain the corresponding eigenvectors as in (6.36). Denote the components of these matrices by 2 m11 m12 σ1 0 and M = Λ= . (C.13) 0 σ22 m21 m22 We may transform any measurement vector z = [z x˜ , z y˜]T into a corresponding diagonalized vector ξ defined in such a way that all validated wake-originated ξ lie within the unit circle and above the x ˜-axis. This transform is given by 1

ξ = R(α)Λ− 2 M T z , T z,

(C.14)

where R(α) is the rotation matrix corresponding to an angle α whose tangent is the negative slope of the vector m11 /σ1 − 21 T Λ M ex˜ = . (C.15) m12 /σ2 In other words,

−m21 σ1 . (C.16) m11 σ2 It is well known [85 p. 244] that if z has the pdf pz (z) and ξ = T z, then pξ (ξ) = pz (T −1 ξ)| det T −1 | = pz (Kξ)| det K| where K = T −1 . Since both R(α) and M T are orthogonal matrices we find tan α =

1

| det T | = det R(α) · det Λ− 2 · | det M T | 1/σ1 0 ·1 =1· 0 1/σ2

⇒ | detK| = σ1 σ2 .

(C.17)

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C.2

167

The normalization constant PGB in WPDAF-1

Since pB (z) only depends on the second component of z, we need only care about the lower row of K. The component k21 is proportional to t21 . Therefore k21 must be zero if t21 is zero, which is the case: t21

(C.14)

=

(C.16)

=

m11 sin α m12 cos α + σ1 σ2 m11 sin α m11 σ2 sin α cos α − = 0. · σ1 σ1 cos α σ2

(C.18)

The last component k22 is therefore all that matters, and it is k22

1 = = t22

m21 sin α m22 cos α + σ1 σ2

−1

.

(C.19)

The transformed, but still un-normalized, wake density is p(ξ) =

σ1 σ2 k22 ζ2 , AB 2

(C.20)

and its integral inside the validation gate is (using polar coordinates r and θ so that ζ2 = r sin θ) Z1 Zπ σ1 σ2 2 σ1 σ2 PGB = k22 k22 . (C.21) r2 sin θdrdθ = 2 AB 3 AB 2 0

0

What we have obtained so far is not entirely satisfactory since it contains the angle α, for which we do not yet have a closed form expression. Its cosine is given by 1

cos α =

eTx˜ Λ− 2 M T ex˜ − 21

kΛ

M T ex˜ k

=

m11 /σ1 − 12 T

kΛ

M ex˜ k

.

(C.22)

Its tangent is given by (C.16) so that its sine can be found as sin α = tan α cos α =

−m12 /σ2 1

kΛ− 2 M T ex˜ k

.

(C.23)

Inserting (C.22) and (C.23) into (C.20) yields 1

PGB

2 σ12 σ22 kΛ− 2 M T ex˜ k = 3 AB 2 m221 + m22 · m11

2 σ12 σ22

− 12 T = Λ M e

x ˜ 3 AB 2

π 2

− 21 T

= det(Λ) Λ M R u .

2 3AB 2

(C.24)

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168

C.3

C


Single point clutter likelihood in WPDAF-2

In this appendix we derive the approximation (6.33) of the spatial pdf for elements of Zk0 ∪ Zkw . This treatment will take place in the wake oriented coordinate system (˜ x, y˜) under the following three assumptions A1: The validation gate G is so large that PG ≈ 1. A2: The rectangular wake region W is so large that none of its corners can fall inside G as long as xk ∈ G. A3: The event θk (i) is not true, i.e. zk (i) ∈ Zkw ∪ Zk0 . Clearly these assumptions cannot be guaranteed to hold. Situations where A1 and A2 cannot both hold, will in reality occur quite frequently. Such violations do not primarily have an impact on the qualitative properties of the spatial clutter pdf, rather they mean that the actual number of wake measurements may differ significantly from the expected number of wake measurements. For parametric tracking methods which attempt to draw inference from the number of measurements this could have dangerous consequences, but it should not pose a serious problem in the non-parametric setting. By the total probability theorem we have p(zk |zk ∈ Zkw ∪ Zk0 , Z k−1 ) = =

Z Z

p(zk |xk , Z k−1 )p(xk |Z k−1 )dxk

(C.25)

[p(zk |zk ∈ C)P {zk ∈ C|xk }

+ p(zk |zk ∈ D)P {zk ∈ D|xk }] p(xk |Z k−1 )dxk .

Since the state conditioned clutter likelihoods are uniform inside their (unknown) support regions we will work with expressions for the areas of C and D. Due to assumption A2 the value of ρxk˜ is irrelevant and it suffices to express these areas as functions of ρyk˜. If the areas of C(ρk ) and D(ρk ) are denoted by C(ρyk˜) and D(ρyk˜) respectively, we can write C(ρyk˜) = D(ρyk˜)

=

(

(

C1 (ρyk˜) C2 (ρyk˜)

if ρyk˜ 0

D1 (ρyk˜) D2 (ρyk˜)

if ρyk˜ 0

(C.26)

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C.3

169


where

p 1 y˜ y˜ = |Ψk | π − acos(−αρk ) + sin 2 acos(−αρk ) 2 p 1 y˜ y˜ y˜ D1 (ρk ) = |Ψk | acos(−αρk ) − sin 2 acos(−αρk ) 2 p 1 y˜ y˜ y˜ C2 (ρk ) = |Ψk | acos(αρk ) − sin 2 acos(αρk ) 2 p 1 y˜ y˜ y˜ D2 (ρk ) = |Ψk | π − acos(αρk ) + sin 2 acos(αρk ) 2 C1 (ρyk˜)

(C.27)

with α given by (6.35). The probabilities that a single clutter measurement zk is in C or D can be found as P {zk ∈ C|xk } = P {zk ∈ D|xk } =

(λ0 + λw )C(ρyk˜)

(λ0 + λw )C(ρyk˜) + λ0 D(ρyk˜) λ0 D(ρyk˜) (λ0 + λw )C(ρyk˜) + λ0 D(ρyk˜)

.

Therefore, the spatial clutter likelihood conditioned on xk is  (λ0 +λw )  if zk ∈ C(xk ) (λ0 +λw )C(ρyk˜ )+λ0 D(ρyk˜ ) p(zk |xk ) = λ 0  if zk ∈ D(xk ). y ˜ y ˜

(C.28)

(C.29)

(λ0 +λw )C(ρk )+λ0 D(ρk )

We can then rewrite (C.25) as

p(zk |zk ∈ Zkw ∪ Zk0 , Z k−1 ) Z ˆ k|k−1 , Pk|k−1 ) N (xk ; x χC(xk ) (zk )dxk =(λ0 + λw ) y˜ y˜ ) ) + λ D(ρ (λ + λ )C(ρ 0 0 w k k G∗ Z ˆ k|k−1 , Pk|k−1 ) N (xk ; x + λ0 χD(xk ) (zk )dxk (C.30) (λ0 + λw )C(ρyk˜) + λ0 D(ρyk˜) ∗ G

where G ∗ , G × R2 and χC (·) and χD (·) denote the characteristic functions [38 p. 46] of the regions C and D. Some manipulations of these and integration over all variables in xk but ρyk˜ yields y ˜ y˜ 2 2 Zzk exp −(ρ ) /2σ k λ0 + λw p(zk |zk ∈ Zkw ∪ Zk0 , Z k−1 ) = √ dρyk˜ y˜ y˜ σ 2π (λ0 + λw )C(ρk ) + λ0 D(ρk ) −∞ y˜ 2 2 Z∞ exp −(ρ ) /2σ k λ0 + √ dρyk˜. y˜ y˜ σ 2π (λ0 + λw )C(ρk ) + λ0 D(ρk ) zky˜

(C.31)

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170

C


The challenge posed by the function gl (ρyk˜) =

1 (λ0 + λw )Cl (ρyk˜) + λ0 Dl (ρyk˜)

, l ∈ {1, 2}

(C.32)

can only be overcome by some kind of approximation. In order to obtain a closed form approximation to (C.25), a series expansion approach can be used. Since the Gaussian has the bulk of its probability mass close to zero in wake oriented (˜ x, y˜)-coordinates, y˜ y˜ we expand g1 (ρk ) around ρk = 0 up to second order. This requires the first and second order derivatives of C1 (ρyk˜) and D1 (ρyk˜). The details are messy, so we only summarize the results here: 2 g1 (0) = p π |Ψ|(2λ0 + λw ) 8λw α g10 (0) = p 2 π |Ψ|(2λ0 + λw )2

g100 (0) =

64λ2w α2 p . π 3 |Ψ|(2λ0 + λw )3

(C.33)

We can then approximate gl (ρyk˜) with the function

2 8λw α g(ρyk˜) = p + ρyk˜ p 2 π |Ψ|(2λ0 + λw ) π |Ψ|(2λ0 + λw )2 + (ρyk˜)2

32λ2w α2 p . π 3 |Ψ|(2λ0 + λw )3

(C.34)

R Inserting (C.34) into (C.31) yields 6 terms on the form (ρyk˜)n exp(−(ρyk˜)2 /2σ 2 )dρyk˜, which all can be integrated in closed form [48]. Evaluation of these integrals yields (6.33). The adequacy of this approximation is illustrated in Figure C.1.

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171


Function values

C.3

gl (ρyk˜ ) Approximant Gaussian -4

-3

-1 2 -2 0 1 Along-wake position ρyk˜ in units of σ

3

4

Figure C.1: Adequacy of the approximation (C.34). It can be seen that the 2nd order approximation g(ρyk˜) is as good as identical with gl (ρyk˜) over the effective support of the Gaussian N (ρyk˜ ; 0, σ 2 ). Therefore the integral of g(ρyk˜) times the Gaussian approximates the integral of gl (ρyk˜) times the Gaussian very well.

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172

C


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Bibliography [1] D. Abraham, “Choosing a Non-Rayleigh Reverberation Model,” in Proceedings of Oceans 1999 MTS/IEEE, Riding the Crest into the 21st Century, vol. 1, Seattle, WA, USA, Sep. 1999, pp. 284–288. [2] D. Abraham and A. Lyons, “Novel Physical Interpretations of K-Distributed Reverberation,” IEEE Journal of Oceanic Engineering, vol. 27, no. 4, pp. 800–813, Oct. 2002. [3] M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions. Dover, 1965. [4] B. Armstrong and H. Griffiths, “CFAR detection of fluctuating targets in spatially correlated K-distributed clutter,” IEE Proceedings F Radar and Signal Processing, vol. 138, no. 2, pp. 139–152, Apr. 1991. [5] M. Athans, R. Whiting, and M. Gruber, “A Suboptimal Estimation Algorithm with Probabilistic Editing for False Measurements with Applications to Target Tracking with Wake Phenomena,” IEEE Transactions on Automatic Control, vol. 22, no. 3, pp. 372–384, Jun. 1977. [6] Y. Bar-Shalom and X. R. Li, Multitarget-Multisensor Tracking: Principles and Techniques. Storrs, CT, USA: YBS Publishing, 1995. [7] Y. Bar-Shalom, X. R. Li, and T. Kirubarajan, Estimation with Application to Tracking and Navigation. Wiley, 2001. [8] Y. Bar-Shalom and E. Tse, “Tracking in a Cluttered Environment with Probabilistic Data Association,” Automatica, vol. 11, pp. 451–460, September 1975. 173

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