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Bayesian approaches to sonar performance prediction and breast tumor diagnosis using microwave measurements

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BAYESIAN APPROACHES TO
SONAR PERFORM ANCE PRED ICTIO N AND
BREAST TUM OR DIAGNOSIS USING MICROWAVE
MEASUREMENTS
by
Liewei Sha
Department of Electrical and Computer Engineering
Duke University
^
Date:
Approved:
u J '7 7 ^
Dr. Loren W. Nolte, Supervisor
^
Dr.^effery
r^effery L. K);olik
Knolik
I ,_ L/y .
„
Dr. Qing H. Liu
Dr. Leslie M. Collins
Dr. Donald B. Bliss
Dissertation submitted in partial fulfillment of the
requirements for the degree of Doctor of Philosophy
in the Department of Electrical and Computer Engineering
in the Graduate School of
Duke University
2004
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UMI Number: 3177855
Copyright 2004 by
Sha, Liewei
All rights reserved.
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Copyright (c) 2004 by Liewei Sha
All rights reserved
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ABSTRACT
(Engineering-Electrical and Computer)
BAYESIAN APPROACHES TO
SONAR PERFORM ANCE PREDICTION AND
BREAST TUM OR DIAGNOSIS USING MICROWAVE
MEASUREMENTS
by
Liewei Sha
Department of Electrical and Computer Engineering
Duke University
Date;
Approved:
7
cJ
Dr. Loren W. Nolte, Supervisor
effery'L.^Krolik
|ing H. Liu
Dr. Leslie M. Collins
Aw d
Dr. Donald B. Bliss
An abstract of a dissertation submitted in partial fulfillment of the
requirements for the degree of Doctor of Philosophy
in the Department of Electrical and Computer Engineering
in the Graduate School of
Duke University
2004
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A bstract
The development of effective sonar systems depends on the ability to predict ac­
curately the performance of sonar detection and localization algorithms in realistic
ocean environments. Such environments are typically characterized by a high degree
of uncertainty, thus limiting the usefulness of performance prediction approaches
th a t assume a known environment. Using a statistical model of environmental uncer­
tainty, we derive analytical receiver operating characteristic (ROC) expressions and
probability of correct localization (PCL) expressions for predicting the performance
of optimal and sub-optimal sonar detection and localization algorithms in uncertain
environments. We used d ata collected during the SWellEx-96 experiment and sim­
ulated d ata generated from an NRL benchmark shallow water model to assess the
validity of the performance expressions. The results showed th at 1) Bayesian de­
tection performance primarily depends on the signal-to-noise ratio, the rank of the
signal m atrix th a t captures the effect of environmental uncertainty, and the signal-tointerference coefficient; 2) Bayesian localization performance is primarily determined
by the signal-to-noise ratio, the effective correlation coefficient between the signal
wavefronts th at in part captures the effect of environmental uncertainty, and the
number of hypothesized source positions; 3) the proposed analytical performance ex­
pressions illustrate the importance of and tradeoffs between fundamental parameters
for sonar performance; and 4) it is possible to perform sonar performance prediction
much faster than with commonly used Monte Carlo methods.
An optimal Bayesian signal detection framework is developed for the detection
and localization of breast tumor using microwave measurements. The proposed like­
lihood ratio detection algorithm incorporates the prediction of the random field of
the electromagnetic (EM) measurements using a forward EM propagation model and
iv
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a 2D Markov random field (MRF) model th at characterizes the spatial properties of
both benign and malignant breast tissue permittivity. W ith simulated data, the ROC
and PCL curves for the proposed algorithm were illustrated as a function of local un­
certain tissue perm ittivity characteristics, and tumor contrast, size, and shape, and
were demonstrated improvements over the algorithms th at optimally post-process a
reconstructed image. Simulation results also indicate the convergence of the estima­
tion of the MRF model parameters and the effect of the sensor array configuration
on tumor detection performance.
V
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A cknow ledgem ents
The work with this dissertation has been extensive and hard, but in the first place ex­
citing, instructive, and fun. The dissertation could not have been completed without
the support of many people who are gratefully acknowledged here.
My greatest debt is to Dr. Loren Nolte, who has been a dedicated advisor and
a judicious mentor.
He provided constant guidance to my academic work.
The
questions he posed have undoubtedly shaped my perspective on the right way to
approach a problem. He stood by me at times of difficulty, encouraging me and
showing his confidence in me. His technical and editorial advice was essential to the
completion of this dissertation and has taught me innumerable lessons and insights
on the workings of academic research in general. Appreciate Dr. Nolte for coming
up with interesting research projects and financially supporting my doctoral study
from Sep. 1998 to Aug. 2003.
I would like to express my deepest gratitude to my research committee; Dr.
Jeffrey Krolik, Dr. Qing Liu, Dr. Leslie Collins, and Dr. Donald Bliss. They had
provided many valuable comments during the prelim examination and stimulated
good ideas for performing research projects. Discussions with Dr. Jeffrey Krolik on
the detection model and discussions with Dr. Qing Liu about the forward propagation
model clarified im portant concepts.
I am very grateful to Dr. David Schwartz, Dr. S. L. Tantum, and Yifang Xu for
their valuable time and patience in modifying part of the manuscript. Because of
their help, the structure and the language took on more fluency. Their useful advice
and comments have helped my writing.
I would like to make a special acknowledgement to my PhD colleagues, several
of whom are my closest friends. They have always been there for me, share my joy,
vi
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listen to complain, and ponder my way through the PhD study.
Thanks also to my parents and my brother and his family in China for their love
and emotional support throughout the process. It was a joyful moment when my
parents proudly learned th a t I became Dr. Sha in the family.
Finally, thanks to two very special people in my life, my husband Ding and son
Tim.
Vll
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C ontents
A b s t r a c t ....................................................................................................................
iv
A c k n o w le d g e m e n t s ............................................................................................
vi
List o f T a b le s .........................................................................................................
x iv
List o f F i g u r e s ....................................................................................................
xv
1 I n t r o d u c t io n .....................................................................................................
1
2
3
1.1
Sonar Performance P re d ic tio n ....................................................................
2
1.2
Breast Tumor Diagnosis Using Microwave M easu rem en ts...................
7
1.3
Detection and Localization Performance M e tric s...................................
9
Sonar Perform ance P red iction M e t h o d o l o g y ..................................
10
2.1
Environmental U ncertainty..........................................................................
10
2.2
Signal Model
.................................................................................................
12
2.3
Detection Signal M a t r i x .............................................................................
13
2.4
Localization Signal S e t ................................................................................
14
Sonar D e te ctio n Perform ance Prediction: E nvironm ental U ncer­
ta in ty ....................................................................................................................
16
3.1
In tro d u c tio n ....................................................................................................
16
3.2
Detection P r o b l e m .......................................................................................
17
3.3
Detection A lgorithm s...................................................................................
19
3.3.1
Optimal Bayesian D etecto r...............................................................
19
3.3.2
Matched-ocean D e te c to r ..................................................................
21
3.3.3
Mean-ocean D e te c to r........................................................................
22
viii
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3.3.4 Energy Detector
3.4
...............................................................................
Analytical ROC Performance Expressions
4
5
23
24
3.4.2 Matched-ocean P re d ic to r.............................................................
30
.....................................................................
32
3.4.4 Energy P r e d ic to r ..........................................................................
33
3.4.5 Assumptions and Computational Com plexity.........................
35
Results and Discussions..................................................................................
36
3.5.1
Propagation Model and Environmental Configurations . . . .
36
3.5.2
Performance Predictions for the Optimal Bayesian Detector
3.5.3
Quantitative Effects of Environmental U ncertainties............
3.5.4
Comparison of Performance Predictions as a Function of Source
R a n g e ..................................................................................................
3.5.5
3.6
..............................................
3.4.1 Optimal Bayesian P r e d ic to r ......................................................
3.4.3 Mean-ocean Predictor
3.5
22
.
Effects of Wrong a priori K n o w le d g e ......................................
Summary
........................................................................................................
38
43
47
49
51
O ptim al Sonar D etectio n Perform ance Prediction: Interference
53
4.1
In tro d u c tio n ....................................................................................................
53
4.2
Interference M o d e l .......................................................................................
54
4.3
Bayesian D etecto r...........................................................................................
56
4.4
Performance Prediction in KnownE nvironm ents......................................
60
4.5
Performance Prediction inUncertain E n v iro n m e n ts................................
63
4.6
Simulation R e s u l t s .......................................................................................
67
4.7
Summary
73
.......................................................................................................
Sonar D etectio n Perform ance Prediction: SW elIEx-96 E xperim ent 75
ix
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5.1
In tro d u c tio n ....................................................................................................
75
5.2
Detection P r o b le m ........................................................................................
76
5.3
Bayesian D etecto r...........................................................................................
77
5.4
Performance P red ic tio n .................................................................................
78
5.5
Alternative D e te cto rs....................................................................................
82
5.6
Results: SWellEx-96 Experimental D a t a .................................................
83
5.6.1
Environmental and Acoustic Propagation M o d e ls ....................
84
5.6.2
Simulation R e s u l t s ...........................................................................
84
5.6.3
Experimental R e s u lts ........................................................................
88
5.7
6
Summary
........................................................................................................
97
O ptim al Sonar L ocalization Perform ance P r e d ic t i o n ...................
99
6.1
In tro d u c tio n ....................................................................................................
99
6.2
Localization P ro cessors.....................................................................................100
6.3
Formulation of the PCL E x p ressio n .............................................................. 103
6.4
6.5
6.3.1
PCL for the Matched-ocean Processor
6.3.2
PCL for the O U F P ...............................................................................105
........................................... 103
Analytical Performance Predictions with Typical Inner Product Matricesl08
6.4.1
Identity Matrix of Size L .....................................................................108
6.4.2
Equally Correlated Matrix of Size 2 ................................................. I l l
6.4.3
A Special Tridiagonal M atrix of Size L ...........................................113
6.4.4
Equally Correlated Inner Product M a trix ....................................... 117
Analytical Localization Performance Predictions in Benchmark Ocean
E n v iro n m e n ts..................................................................................................... 122
6.5.1
PCL for the Matched-ocean Processor in Known Ocean Envi­
ronments .................................................................................................. 123
X
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6.6
7
8
6.5.2
PCL for the OUFP in an Uncertain Ocean Environment
6.5.3
Effect of Environmental Uncertainty onLocalization Performance 132
Summary
. . . 127
............................................................................................................136
A R ev iew o f D ielectric P rop erties o f N orm al and M alignant B reast
T is s u e ........................................................................................................................
138
7.1
In tro d u c tio n ........................................................................................................ 138
7.2
Review of Experiment D a t a ........................................................................... 139
7.2.1
List of D ata at Low Frequency........................................................... 139
7.2.2
List of D ata at High F re q u e n c y ........................................................141
7.2.3
D ata Consistency and Inconsistency................................................. 147
7.2.4
Discussion of the Diagnostic V a lu e s ................................................. 153
7.2.5
Areas in Need of More E xperim ents................................................. 154
7.3
Mechanism: Normal vs Malignant T is s u e .................................................... 154
7.4
Summary
............................................................................................................157
B reast Tum or D iagnosis U sing M icrowave M easurem ents . . .
158
8.1
In tro d u c tio n ........................................................................................................158
8.2
Objectives and M o d e ls .....................................................................................159
8.3
8.2.1
Model of Tissue Permittivity I m a g e .................................................159
8.2.2
Forward Propagation M o d e l..............................................................164
8.2.3
R econstructions.................................................................................... 167
D etection and L ocalization A pproaches
.................................................... 168
8.3.1
Bayesian Processor for Uncertain Permittivity I m a g e ................ 170
8.3.2
Threshold Image Processor.................................................................171
XI
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8.4
8.5
9
8.3.3
Bayesian Processor for Scattered EM Field of Uncertain Per­
m ittivity I m a g e ..................................................................................... 172
8.3.4
Lower Order Approximation of the B P-S E M U PI...........................173
Simulation R e s u l t s ........................................................................................... 176
8.4.1
Basic P a ra m e te rs .................................................................................. 176
8.4.2
Performance M e tr ic s ........................................................................... 177
8.4.3
Performance Upper Bound and Tumor Characteristics
8.4.4
Convergence of the Covariance Matrix E s tim a tio n ....................... 178
8.4.5
Performance of Proposed Bayesian A pproaches..............................182
8.4.6
Performance and Array C o n fig u ratio n s...........................................188
Summary
....
177
...........................................................................................................190
C o n l u s i o n ..........................................................................................................
191
A D eflection C oefflcient I n e q u a lit ie s .........................................................
196
B
C
A .l
Deflection Coefficient after Linear T ran sfo rm atio n ................................... 197
A.2
Deflection Coefficients for Three Kinds of D ata
A.3
Inequality Relationships between Deflection Coefficients......................... 199
A.4
Two Special Cases
A.5
D iscu ssio n.......................................................................................................... 201
.......................................198
.......................................................................................... 200
Source C ode for C hapter 3 ..........................................................................
203
B.l
Analytical Performance Prediction
.............................................................203
B.2
Monte Carlo Performance E v a lu a tio n ..........................................................204
Source C ode for C hapter 4 .............................
C .l
Analytical Performance Prediction
208
.............................................................208
xii
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C.2
Monte Carlo Performance E v a lu a tio n ...........................................................210
D Source C ode for
C hapter 5 ......................................................................
214
D .l
Analytical Performance Prediction
D.2
Monte Carlo Performance E v a lu a tio n .......................................................... 215
E Source C ode for
............................................................. 214
C hapter.6 ......................................................................
218
E .l
Analytical Performance Prediction
............................................................. 218
E.2
Monte Carlo Performance E v a lu a tio n .......................................................... 222
B ib lio g r a p h y ..........................................................................................................
228
B i o g r a p h y ...............................................................................................................
237
xni
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List o f Tables
3.1 Param eters of uncertain shallow water propagation model
.................
37
3.2 Scenarios of environmental uncertainty with fixed ranges of uncertainties 38
3.3 Scenarios of environmental uncertainty with increasing ranges of un­
certainties ........................................................................................................
39
3.4 Rank estimation for various scenarios of environmental uncertainty
43
4.1 Taxonomy of results for sonar detection performance prediction
4.2 Parameters of uncertain shallow water propagation model
.
...
67
.................
68
4.3 Estimated signal-to-interference c o efficien t..............................................
72
5.1 Scenarios of source motion uncertainty for simulated d a t a ....................
85
6.1
Parameters of uncertain shallow water propagation model
.....................128
6.2 Scenarios of environmental uncertainty with increasing range of ocean
depth uncertainty in genlmis environment ..................................................132
8.1
The true and the estimated interaction coefficients.....................................182
XIV
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List o f Figures
2.1
NRL environmental m o d e l..........................................................................
11
3.1
Analytical detection performance predictions for the optimal Bayesian
p re d ic to r...........................................................................................................
30
ROC detection performance predicted by the optimal Bayesian pre­
dictor for three uncertain environment scenarios......................................
40
Detection performance predicted by the optimal Bayesian predictor,
the mean-ocean predictor, and the energy predictor in uncertain envi­
ronments.............................................................................................................
42
Detection performance prediction of the matched-ocean predictor, the
mean-ocean predictor, and the energy predictor as a function of in­
creasing environmental uncertainties............................................................
44
Analytical detection performance predictions of the optimal Bayesian
predictor illustrated using the measure of P d as a function of increasing
environmental uncertainties...........................................................................
46
Detection performance predictions illustrated using the measure of Pu
as a function of source range in the genlmis ocean...................................
48
The effect of wrong a priori range of ocean depth uncertainty on the
Bayesian detection performance....................................................................
50
Detection performance prediction in the presence of a single interferer
in a known ocean; P d versus S N R and p..................................................
62
Detection performance prediction in the presence of interference in an
uncertain ocean.................................................................................................
66
Detection performance in the presence of a single interferer in the
known benchmark ocean: P d image.............................................................
69
Detection performance prediction P d as functions of S N R in the pres­
ence an interferer in uncertain environments..............................................
70
3.2
3.3
3.4
3.5
3.6
3.7
4.1
4.2
4.3
4.4
XV
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4.5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Detection performance prediction ROC’s in the presence of an inter­
ferer with various I N R values in uncertain environments.......................
71
Shallow-water environmental configuration for the SWellEx-96 exper­
im ent...................................................................................................................
84
Comparison of analytical and Monte Carlo performance predictions
for the Bayesian detector with simulated data, by plotting P d as a
function of S N R ...............................................................................................
85
Comparison of analytical and Monte Carlo performance predictions
for the Bayesian detector with simulated data, by plotting P d as a
function of P p ...................................................................................................
86
Illustration of the effect of source motion on detection performance
with simulated d ata.........................................................................................
86
Estimated and extrapolated clean source trajectories for event S5 dur­
ing the SWellEx-96 experiment.....................................................................
89
Histograms of the 109 Hz source amplitude samples obtained from four
d ata frames of event S5 during the SWellEx-96 experiment...................
91
Illustration of the analytical performance prediction and Monte Carlo
performance prediction for the Bayesian detector with the SWellEx-96
experiment d ata................................................................................................
93
Illustration of Monte Carlo performance prediction for various detec­
tors with the SWellEx-96 experiment d ata...............................................
95
5.9
Illustration of the analytical performance prediction and Monte Carlo
performance prediction for the Bayesian detector with successive SWellEx96 d ata sets........................................................................................................ 96
6.1
Analytical optimal localization performance based on an identity inner
product m atrix of size L ..................................................................................... I l l
6.2
Analytical optimal localization performance based on a correlated in­
ner product matrix of size 2...............................................................................113
XVI
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6.3
Analytical localization performance prediction based on a tridiagonal
inner product m atrix of size L ...........................................................................118
6.4
Approximate analytical localization performance prediction and Monte
Carlo localization performance prediction based on an equally corre­
lated inner product m atrix................................................................................. 121
6.5
The absolute correlation coefficient between the signal wavefronts due
to two separated sources..................................................................................... 123
6.6
Source depth localization performance prediction for the matchedocean processor; weak correlation.................................................................... 124
6.7
Source depth localization performance prediction for the matchedocean processor: strong correlation.................................................................. 125
6.8
The performance of the matched-ocean localization processor in the
known ocean environment.................................................................................. 126
6.9
The inner product m atrix of the optimal localization signal m atrix . . 129
6.10 Verifications of the approximate analytical PCL performance predic­
tion for the OUFP................................................................................................130
6.11 The effect of environmental uncertainty on the performance of the
OUFP and the mean-ocean processor: PCL as a function of S N R . . . 133
6.12 The effect of environmental uncertainty on the performance of the
OUFP and the mean-ocean processor: PCL as a function of uncer­
tainty ranges..........................................................................................................135
7.1
Capacitance versus Resistance. Left: original data. Right: modeling
of the d ata............................................................................................................. 140
7.2 Capacitance versus Resistance........................................................................... 141
7.3 Permittivity and Conductivity versus F req u en cy ........................................141
7.4 Permittivity and Conductivity versus F req u en cy ........................................142
7.5 Permittivity and Conductivity versus Frequency[28]................................. 143
xvii
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7.6
Perm ittivity and Conductivity versus Frequency[26].................................. 143
7.7
Permittivity and Conductivity versus F req u en cy .........................................144
7.8
Conductivity versus permittivity. Left: original data. Right; modeling
of the d a t a ............................................................................................................ 145
7.9
Perm ittivity and Conductivity versus Frequency[29]..................................146
7.10 Perm ittivity and Conductivity versus Frequency[25].................................. 146
7.11 Permittivity versus frequency
7.12 Conductivity versus frequency
........................................................................ 148
..................................................................... 149
7.13 Conductivity versus Permittivity, at 900M H z.............................................. 150
7.14 Conductivity versus Permittivity, at 3.2 G H z .............................................. 151
8.1
Tissue perm ittivity image: an e x a m p le ........................................................ 164
8.2
Measurements of scattered electric fie ld (m V /m )........................................166
8.3
Reconstructed tissue perm ittivity image. Numbers 1-24 represent the
array configuration...............................................................................................167
8.4
The framework of the detection and localization approaches using the
tissue perm ittivity image data or the EM measurement d ata.................... 169
8.5
Detection and localization performance of the BP_UPI as a function of
tumor contrast, size, and local characteristics with sharp tumor mean
function.................................................................................................................. 179
8.6
Detection and localization performance of the BP_UPI tumor contrast,
size, local characteristics, with smooth tumor mean function.................... 180
8.7
The convergence of the detection performance as a function of the
number of samples used in Monte Carlo integration.................................... 182
8.8
The perm ittivity images and the probability images for tum or detector
and localization.....................................................................................................184
xviii
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8.9
Detection performance of the BP-UPI using original tissue perm ittivity
image d ata............................................................................................................. 186
8.10 Detection performance of the BP_UPI, the TIP, and the LBP_SEMUPI
for detecting a tumor with an unknown position.......................................... 186
8.11 Localization performance of the LBP_SEMUPI, the BP_UPI, and the
TIP.......................................................................................................................... 187
8.12 Deflection coefficients as a function of the tumor positions........................ 188
8.13 Effects of array configurations on detection performance............................189
XIX
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Chapter 1
Introduction
This dissertation develops and evaluates new approaches for predicting the perfor­
mance of Bayesian and sub-optimal sonar detection and localization algorithms in un­
certain ocean environments. This dissertation also investigates Bayesian approaches
for breast tumor diagnosis using microwave measurements. Sonar systems extract in­
formation about remote events by propagating acoustic waveforms, where the prop­
agation medium, the ocean, is usually uncertain. Similarly, diagnostic microwave
imaging systems examine malignant tumors by analyzing information carried with
propagating electromagnetic (EM) waveforms, where the propagation medium, i.e.,
the spatial tissues’ dielectric properties, are typically unknown.
The uncertainty
associated with the propagation medium is a challenge to both applications.
In
order to detect and locate a target, an optimal processor must correctly combine
measurements of the propagating waveform with information about the source, the
interference, the ambient noise, and particularly, the uncertain propagation medium.
The development of such optimal processors and approaches to predict quantitatively
their performance in uncertain propagation media is instructive for both applications.
In this chapter, we introduce the development of sonar detection and localization al­
gorithms and the performance evaluation approaches for those algorithms, related
work for diagnostic microwave imaging, and the quantitative performance metrics
used in this work.
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1.1
Sonar Perform ance P red iction
The acoustic field emitted from submerged targets can propagate long distances in
the ocean. A passive sonar system detects the presence of an acoustic source and
locates its position using the measurements of the propagating acoustic field, ob­
tained with an array of hydrophones distributed at distinct locations. Sonar detec­
tion performance depends upon the ability of the detection algorithm to distinguish
the propagating acoustic signal wavefronts in the measurements from ambient noise
and the interference due to other acoustic sources. The localization of a submerged
target additionally requires identifying the signal wavefront associated with a spe­
cific source position. Particular challenges to sonar detection and localization are the
uncertainties associated with the propagating signal wavefronts, which are caused by
the complicated ocean environment. A simple, fast, and realistic sonar performance
prediction approach th at incorporates information about the complicated ocean en­
vironment is im portant for the development of sonar systems. It can be used to
guide better system design as well as to optimize system parameters in operational
conditions.
The development of methodologies th at characterize and predict sonar perfor­
mance is associated with the evolution of sonar detection and localization algorithms.
Conventionally, the plane-wave propagation model is used in beamforming to differ­
entiate the direction of the propagating waveform, which suffers significant funda­
mental performance degradation. In 1976, the matched-field processing (MFP) was
formulated[l] for the localization of the range, depth, and azimuth of the source,
which correlates empirical measurements with the signal wavefronts predicted by a
full-wave propagation model for each position in a search space. W ith more exact
wave theory and state-of-the-art numerical techniques, propagating signal wavefronts
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can now be described in great detail[2]. However, the prediction of the signal wave­
fronts requires knowledge about the source position, the propagation channel, and
the receiving array configuration. Although the receiving array configuration is typ­
ically known, the knowledge of element positions is imperfect. The source position
and propagation channel characteristics are usually unknown. Of particular concern
is the uncertainty associated with the propagation channel. The spatial and tempo­
ral variations in the channel parameters, such as the ocean bottom depth and the
sound speed profile, make accurate characterization of the channel difficult. The
set of propagation channel parameters together with the source position parameters
constitute an uncertain ocean acoustic environment. The sensitivity of the perfor­
mance of conventional matched-field processing (MFP) to environmental uncertainty
is well known[3, 4, 5]. This has led to the development of the algorithms th at are
robust to environmental uncertainty, such as the minimum variance beamformer with
multiple neighboring location constraints (MV-NCL)[6], the optimal uncertain field
processor (O U FP)[7], the minimum variance beamformer with environmental pertur­
bation constraints (MV_EPC)[8], the modal space methods[4], and the reduced-rank
MFP algorithms[9], etc. Among the detection and localization algorithms th at are
robust to environmental uncertainty, the physics-based Bayesian approaches have
been widely proposed[7, 10, 11, 12]. These approaches combine statistical model­
ing of environmental uncertainty with physical modeling of the propagation medium
to achieve optimal sonar performance in the presence of environmental uncertainty.
Previous approaches include the optimal uncertain field processor (OUFP) [7] th at is
a Bayesian a posteriori probability method, full-field optimum detector in a random
wave scattering environment[11], the optimal wideband active detector[12], and the
Bayesian detectors[10], etc. The physics-based Bayesian approach enables one to in­
corporate fully the knowledge of uncertainty at the environmental parameter level.
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where the stochastic descriptions can be obtained more easily and more directly than
at the signal wavefront level.
It is of practical interest to predict the detection and localization performance
of the optimal Bayesian processors and sub-optimal processors in an uncertain en­
vironment. In many sonar systems[13], the sonar equation is used as a means for
predicting detection performance. It heuristically includes the effects of propagation
loss, acoustic source level, and ambient noise on detection performance and provides
the detection threshold needed to achieve a specific probability of detection and false
alarm. The classic sonar equation assumes th at both the signal wavefront and the
noise field directionality are known a priori. However, under practical circumstances,
the signal wavefront and the noise field directionality are usually uncertain due to
the presence of environmental uncertainty and the presence of interferers with uncer­
tain locations in the uncertain ocean environment. The classic sonar equation does
not take into account environmental uncertainty and hence is inaccurate; it is often
too optimistic for sonar detection performance prediction in a realistic circumstance.
The performance of detection algorithms for an uncertain ocean environment has
been evaluated in the past. However, few statistically valid detection performance
predictions have been reported. For example, the spatial stationarity test[14] has
been evaluated using Mediterranean vertical array data[15] to demonstrate the ro­
bustness of the algorithm. These performance analyses generate ad hoc assessments
but not general conclusions about the effects of environmental uncertainties on the
detection performance. Some performance evaluations have used a quantitative envi­
ronmental mismatch model[4, 3, 5]. For example, the degradation of the localization
performance of a matched-mo de-processor is determined as a function of increasing
bottom depth mismatch[4]. These predictions can be improved by including environ­
mental uncertainty cases other than ocean depth mismatch or sound speed profile
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mismatch. Furthermore, most detection performance evaluations have been imple­
mented using numerical approaches such as the Monte Carlo method[10], which can
be very computationally intensive.
Three performance metrics have been used for sonar localization performance
prediction: the array gain[16, 9], the mean square error (MSE)[17, 18, 19], and the
probability of correct localization (PCL)[10, 20]. The array gain is a metric of the
array performance. Although it is an im portant component of the localization perfor­
mance metric, it does not directly quantify optimal source localization performance.
The hybrid Cramer-Rao bound [17] and Barankin Bounds [18] for source localization
estimates are mean square error metrics. However, the MSE metric may not be ac­
curate when global false localizations are present due to repetitive structure of the
acoustic field. The probability of correct localization (PCL) is defined as the ratio
of the number of correct localizations to the number of total trials, which is directly
relevant to the ultim ate goal of source localization. It has been used to evaluate the
performance of the 0UFP[7] using the Monte Carlo method in a benchmark uncer­
tain environment[10]. However, the Monte Carlo method does not directly relate the
PCL to fundamental ocean environmental parameters.
In this dissertation, we derive analytical receiver operating characteristic (ROC)
detection performance predictions for the optimal Bayesian detectors[7, 10] and sev­
eral sub-optimal detectors, and derive analytical probability of correct localization
(PCL) performance predictions for the OUFP, based on a statistical model of un­
certain environments. The environmental model is presented in chapter 2. Three
different detection problems are included respectively in chapters 3, 4, and 5. For
the detection of an acoustic source whose amplitude exhibits a complex Gaussian
distribution in the presence of diffuse noise in uncertain environments, the optimal
Bayesian detector is a modified version of the 0UFP[7, 10]. The second detection
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problem includes the presence of interferers in uncertain environments. The opti­
mal Bayesian detector [21] for this detection problem is an im portant extension to
the 0UFP[7]. The third detection problem incorporates a different source amplitude
model: the signal is assumed to have unknown phase and known amplitude in an
uncertain environment. The Bayesian detection algorithm is proposed in [22]. The
optimal Bayesian detectors are designed from signal detection and Bayesian view­
points. W ith realistic assumptions about the environmental uncertainty model and
the acoustic propagation model, evaluation of the optimal Bayesian detectors provides
a practical performance upper bound. The performance predictions for sub-optimal
detectors provide useful comparisons to the optimal one. The simple analytical ROC
expressions presented in this dissertation predict sonar detection performance as a
function of three parameters: the signal-to-interference coefficient, which captures the
effect of interference in uncertain environments; the rank of the signal matrix, which
characterizes the degree of environmental uncertainty; and the signal-to-noise ratio
at the receivers, which characterizes the combined effects of sonrce level, noise level,
and propagation loss. The concept of the signal matrix is discussed in chapter 2. Sim­
ulated d ata generated using an NRL benchmark shallow-water model[14, 10, 23] are
used to verify the proposed Bayesian detection performance prediction approach for
the first two detection problems. The ROC expression for the third detection problem
is verified using simulated data as well as the SWellEx-96 experimental data. The
results demonstrate th at the simple analytical ROC expressions capture fundamen­
tal parameters of the performance of the Bayesian detectors. This provides a more
realistic detection performance prediction than does the classic sonar equation.
In contrast to previous localization performance prediction approaches which
have relied solely on the Monte Carlo method[10], in chapter 6, we present rel­
atively simple analytical approximations to the probability of correct localization
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(PCL) th a t quantifies the performance of the OUFP in uncertain environments[24].
The matched-ocean processor (i.e., B artlett processor) in known environments is
included as a special case. The analytical PCL expressions are identified and are
used to predict the performance of the optimal processors in the NRL benchmark
environments[23, 10, 14]. The agreements between the PCL results computed using
the analytical approximate approach and the Monte Carlo method verify th at the op­
timal PCL performance can be captured primarily by fundamental parameters: the
signal-to-noise ratio (S N R ) at the receivers, the number of the hypothesized source
positions, and the effective correlation coefficient between signal wavefronts th at in
part represents the level of environmental uncertainty. The results show th at the an­
alytical approximate PCL expression for the OUFP provides a practical localization
performance upper bound in uncertain environments.
1.2
B reast Tum or D iagnosis U sing M icrowave M easurem ents
Breast cancer is a significant public health problem for women. The major traditional
modality for breast cancer diagnosis is X-ray mammography. It is relatively cheap
and fast, but it exposes the body to ionized radiation. In order to operate within
safe limits, the contrast of the images tends to be low. A diagnostic microwave imag­
ing system[25] transm its and receives electromagnetic energy th at propagates through
the object using spatially distributed sensors. The propagating electromagnetic wave­
forms contain immense amount of information about the propagating medium, i.e.,
the pathologic image of the object, which is used for diagnoses. Microwave imaging is
a prom ising new m od ality for breast cancer diagnosis, because it is non-invasive and
the normal and malignant tissues of the breast have high contrast in dielectric proper­
ties in certain ranges of the electromagnetic (EM) frequencies [26, 27, 28, 29, 30, 31].
In addition, the attenuation of EM propagation in normal breast tissues is low so
7
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th a t it can also penetrate into the depth of the tissue[32][33][34][35]. A collection of
the dielectric property data on benign and malignant breast tissues from a number
of studies are presented in graphical form in chapter 7.
Most of the research in this field has focused on the experimental and theoretical
study of the dielectric properties of breast tissue at microwave frequencies[26, 27, 28,
29, 30, 32, 33, 34, 35, 31], the design of microwave imaging prototypes[25] and the
improvement of the 2D and 3D EM reconstruction algorithms[36] [37] [38]. However,
few of this research have incorporated signal detection theory directly into microwave
imaging. Bayesian theory has been applied in mammography for diagnosis, such as
[39] [40] [41]. However, [39] and [40] did not incorporate a random model of the tissue
being imaged, but assumed a simple deterministic disk object model. Most of the
applications of the Bayesian approach to diagnostic imaging focus on post-processing
the object image[41] without fully utilizing the propagation model.
In chapter 8 we present a physics-based Bayesian approach for the optimal de­
tection and localization of malignant breast tumors, which incorporate the forward
electromagnetic propagation model with a Markov random field (MRF[42]) model
of the spatially distributed uncertain propagating media[31, 43]. Chapter 8 includes
the comparison of the physics-based Bayesian approach with another Bayesian ap­
proach th at post-processes a reconstructed perm ittivity image. In the physics-based
Bayesian approach, spatial uncertainty of the tissue permitivity is included in the
forward EM propagation model and signal detection theory is applied to raw data
without prior reconstruction of an image. In the second approach, signal detection
theory is applied as a postprocessor to the reconstructed image. Simulation results in
chapter 8 present the detection and localization performance of these two Bayesian
approaches as functions of tumor contrast, size, local characteristics and shape and
show th a t the physics-based Bayesian approach achieves better detection and local8
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ization performance than the Bayesian approach th at does not utilize the forward
propagation model.
1.3
D etectio n and L ocalization Perform ance M etrics
In this dissertation, we use the receiver operating characteristic (ROC) as the metric
of detection performance and the probability of correct localization (PCL) as the
metric of localization performance, as these measures are directly relevant to the
ultim ate goal of source detection and source localization.
The ROC is a plot of the probability of detection, Pp, as a function of the prob­
ability of false alarm, Pp. From signal detection theory, Pn and Pp are defined
as
Ppif3) = P { X ' { v \ H o ) > / 3 ) ,
.
.
PD{P) = P{X' {r\ H, )>P),
^ ^^
where A'(-) is the test statistic produced by a detection algorithm and P is the thresh­
old which is determined by the performance criterion.
When the probability density functions p{X' | H q) and p{X' \ Hi) are available,
Pp and P d can be computed using
Pd W =
dX'p{X'\ Hi).
The PCL is defined as the ratio of the number of correct localizations to the
number of total trials.
The PCL can be formulated as the probability th at the
a posteriori probability of the true source position conditional to the observation
is greater than the a posteriori probability of other hypothesized source positions
conditional to the observation.
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Chapter 2
Sonar Perform ance P rediction M ethodology
This chapter presents basic models and concepts th at constitute the framework for
predicting the performance of sonar detection and localization algorithms. Uncer­
tain environmental parameters th at affect acoustic propagation are quantified using
a statistical model of environmental uncertainty. Translations from environmental
uncertainty to signal wavefront uncertainty are described by a signal model th at
combines both the physical model of acoustic propagation and the statistical model
of environmental uncertainty. Based on the signal model and Monte Carlo sampling
techniques, two new concepts, the detection signal matrix and the localization signal
set are proposed as the representations of the signal wavefronts with discrete param ­
eters in uncertain environments. The environmental uncertainty model, the signal
model, the detection signal matrix, and the localization signal set form the basis for
developing the receiver operating characteristics (ROC) and the probability of correct
localization (PCL) performance expressions.
2.1
E nvironm ental U n certain ty
The ocean environment (referred to as environment for simplicity) constitutes of
acoustic source parameters and the ocean parameters th at affect the acoustic propa­
gation. To illustrate such an environment, we utilize the general mismatch benchmark
model ( “genlmis”) proposed in the May 1993 NRL Workshop[23, 10, 14]. As shown
in Fig. 2.1, the benchmark environment model consists of a shallow water channel
with a depth of U, a linear sound speed profile, with the surface speed C q and bot­
tom speed C^, a sediment layer with density p and attenuation a, and sound speed
10
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Om
surface
tri=1500±2.5m/s
100 elements VLA
water
Crr =1480±2.5m/s
sediment
Cr,+ = 1600± 50m/s
C 1 = 1750± lOOm/s
a=0.35± 0.25dBA
p=1.75± 0.25 g/cm
Figure 2.1: NRL environmental model
profiles in the upper sediment layer and the bottom layer. This model also consists
of a single source at depth, Zg., range, fs, and a vertical line array containing
100
hydrophones spaced 1 m apart ranging from 1 to 100 m in depth. For simplicity, the
ocean environmental parameters are denoted by
and the acoustic source position
relative to the receiving array is denoted by S.
The spatial and temporal variability of the ocean and source position parameters
often cannot be sampled and modeled at scales adequate to accurately represent the
acoustic propagation environment. The limitation of the a priori knowledge of the
environmental parameters is defined in this dissertation as environmental uncertainty.
Environmental uncertainty is quantified by assigning an a priori probability density
function to both the uncertain ocean parameters and the unknown source position
param eters, over a finite range of possible values. It is assum ed th a t th e ocean param ­
eters,
and the source position parameters, S, are statistically independent a priori
with uniform probability density functions denoted by p(’F) and p(S) respectively.
The detection performance prediction can be obtained as a function of increasing
11
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range in uncertainty of the environmental parameters. The advantage of this model
compared to the mismatch environmental uncertainty model[4, 3, 5] is th at it enables
statistically valid quantitative analysis of the effects of environmental uncertainty on
sonar performance.
2.2
Signal M odel
The signal model used for source detection and localization is based on physics
model of acoustic propagation. The physics-based signal model has been used in
matched-field processing[2], the optimum uncertain field processor (OUFP) [7], and the
minimum variance beamformer with environmental perturbation constraints (MVEPC)[ 8 ] for localization, and have been used in several Bayesian detectors[ 1 2 , 44].
Here we combine physics model of acoustic propagation with the statistical model
of environmental uncertainty to relate the signal wavefront with uncertain acoustic
source and ocean acoustic parameters. This translation of uncertainties enables the
incorporation of the a priori knowledge at the ocean and source param eter level,
rather than at the signal wavefront level. The advantage of this translation is th at
it is much easier to get a stochastic description of the ocean and source parameters
than to obtain the a priori probability density function for the signal wavefronts.
The received signal is assumed to be a spatial vector in the frequency domain,
narrowband, centered at a known frequency /q. It is expressed as aoH (^, S), where
ao is a complex variable representing the amplitude and phase of the acoustic source.
Two source models are considered in this dissertation. One assumes th at ao is a
com plex G aussian distribution, which is consistent w ith th a t used in th e O U F P [7]
and the MV-EPC[ 8 ]. This model is used in chapters 3,4, and 6 . Another assumes th at
ao has a known amplitude and a random phase th at is uniformly distributed in
0
and
2tt. This assumption is used in chapter 5 for illustrating sonar detection performance
12
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with d ata collected from the SWellEx-96 experiment. The frequency-domain ocean
acoustic transfer function[45], H (^ ,S ), is sampled at an arbitrary receiving array
of N sensors, given ocean parameters ^ and source position parameters S. It is
computed using normal mode theory [2]. The ith component is given by
^
H(4', S)‘ = p„ W # „{2„
m =l
where (zs,'^s) is the source position at depth
Q-jkmTs
V
and range
{2.1)
s
referred to as S. The
parameter Zj is the depth of the ith receiver, and po is the pressure of the source
at rg=l m. The number of propagating acoustic modes is represented by K. The
terms
and
are the m th eigenvalue and eigenfunction of the Sturm-Liouville
problem, which can be calculated using M.B. Porter’s KRAKEN [46] code.
The received signal can be expressed as a o ||H (^ , S )||s(^ , S), where || • || denotes
the vector norm, and s(^ , S) is the normalized signal wavefront, given by
s('® ,S )= H (* ,S )/||H ('^ ,S )||.
(2.2)
By default, we always refer to s(^ , S) as signal wavefront, ignoring “normalized” for
simplicity.
2.3
D etectio n Signal M atrix
The signal wavefront, s (^ , S), is a function of the continuous random variables ^ and
S, which follow the probability density functions p ( ^ ) and p(S). To computationally
evaluate sonar performance in an uncertain environment, a representation of the
uncertain signal wavefront with discrete parameters is needed. A m atrix composed
of M realizations of the signal wavefront due to uncertainties of the environmental
parameters, referred to as the signal matrix 3?, is defined by applying Monte Carlo
13
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sampling techniques.
5R
where
[®l5
= [s ({ * ,S ),).s ((4 > ,S W ,...s { (< t,S )„ )|,
/q o'N
' ■'
corresponds to the ith realization of the environmental param­
eters ( ^ ,S ) , which follow the probability density function p (^ )p (S ) th at is defined
on uncertainty ranges
for ocean environmental parameters and Jls for source po­
sition parameters. Note th at although the signal matrix is used to derive the ROC
performance expressions, it is shown later th at only a few key parameters derived
from the signal m atrix are needed in the resultant approximate ROC performance
expressions.
The finite signal m atrix is a subset of the infinite set, which is composed of all
possible signal wavefront realizations following the assumed distribution. If the size
of the signal matrix, M, is large enough, the subset could be a good representation
of the whole set. Then the post-processing of the signal matrix could be considered a
good approximation to the post-processing of the original infinite set. The criterion
for determining M is problem dependent. Usually, M should be greater than the
number of the propagating mode, K , so th at independent information carried by the
propagation mode can be captured by the samples of the signal wavefronts. If the
number of elements in the array, N, is smaller than the number of the propagating
mode, the amount of independent information is limited by N, and the M should be
greater than N.
2.4
L ocalization Signal Set
In order to computationally evaluate the performance of a localization processor,
we use a localization signal set th at is composed of the realizations of the signal
wavefronts due to ocean environmental uncertainties for each hypothesized discrete
14
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source position. The localization signal set, S , is defined by applying Monte Carlo
sampling techniques [1 0 ].
Sll,
Sl2,
...
Six,
S2 1 ,
S225
•••
S2X,
Sm i ,
SM25
• • ■ SmL .
(2.4)
■ s ( ^ i , S i ),
s
(^1 ,5 2 ),
S ( ^ 2 , S i ),
s
(^2 ,5 2 ),
. S (^ M ,S i),
The first index of the signal wavefronts, i, corresponds to the ith realization of the
ocean parameters
which follow the probability density function p ( ^ ) th at is
defined on uncertainty ranges 0 ^ .
The number of the realizations of the ocean
parameters for each source position, M , should be sufficiently large. The second
index of the signal wavefront, p, is for the pth hypothesized source position, S^. The
area of the hypothesized source positions, Qs, is uniformly partitioned into L non­
overlapping rectangular regions
The pth hypothesized source position is
at the center of the pth region. This uniform probability density function of the
hypothesized source position is denoted by p(S). The number of the partitioned
region, L, is relevant to the resolution of source localization.
The localization signal set provides a representation of the signal wavefronts with
discrete parameters to facilitate the formulation and computation of the PCL for
the localization processors. It is an extension to the representation of the signal
wavefronts in the matched-field processing framework th at only considers discrete
source range and depth.
15
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Chapter 3
Sonar D etection Perform ance Prediction:
Environm ental U ncertainty
3.1
Introd u ction
Environmental uncertainty can cause severe performance degradation to sonar de­
tection algorithms th at rely on precise knowledge of the environmental parameters[4,
3, 6 ]. Sonar detection performance prediction methods th at fail to consider environ­
mental uncertainty do not capture the performance degradation it causes, leading
to overestimation of detection performance. Thus, a realistic sonar detection perfor­
mance prediction method th at incorporates the effects of environmental uncertainty,
and consequently predicts sonar detection performance more accurately, is of prac­
tical interest. Detection performance prediction typically involves both a detection
algorithm and a methodology for evaluating the algorithm’s performance. Several al­
gorithms th a t are robust to environmental uncertainty have been proposed, including
the energy detector, the spatial stationarity test [14], and generalized matched-field
processing (MFP) algorithms[ 6 , 7, 8 , 9], etc. The performance analyses of these al­
gorithms reported in the literature[14, 15, 9], however, are specific to a particular
experimental or simulated environment, and so have limited generality.
In this chapter, we derive analytical ROC performance expressions for a Bayesian
optimal detector[7, 10] and for several sub-optimal detectors, for the detection of a
submerged acoustic source in the presence of environmental uncertainty in diffuse
noise. The optimal Bayesian detector is a modified version of the OUFP [7, 10],
designed from signal detection and Bayesian viewpoints. The OUFP has been used to
study the effect of the internal waves[2 0 ] on source localization, and to solve tracking
16
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problems[47, 45]. The performance of the optimal Bayesian detector and sub-optimal
detectors can be obtained using Monte Carlo evaluation approach[10]. However, the
Monte Carlo approach can be very computationally intensive and is lack of insight
into the fundamental parameters th at impact sonar detection performance. W ith
the analytical ROC expressions developed in this chapter, the performance prediction
computation is much faster than using Monte Carlo methods[10]. Further, the simple
analytical ROC expressions characterize sonar detection performance as a function of
environmental uncertainty and other system parameters such as signal-to-noise ratio
(SNR) at the receiver. It reveals th at optimal detection performance in an uncertain
environment in the presence of diffuse noise depends primarily on the S N R and the
rank of the received signal matrix, where the rank reflects the scale of environmental
uncertainty.
3.2
D e tec tio n P rob lem
The detection problem is formulated as a hypothesis testing problem: given the
measurement data r, accept the “noise only” hypothesis Hq, or reject it to accept
the “signal present” hypothesis Hi, in the presence of environmental uncertainty.
This differs from the source localization problem th at often assumes only the Hi
hypothesis.
The two hypotheses are defined as
Hi
: r = a o H ( ^ , S ) -h Hq,
H q : r = no,
(3.1)
n o ~ N ( 0 , 2 a 2l ; v ) , a o - N ( 0 , 2
0
-
The data r are an N” x 1 spatial vector, corresponding to N spatially distributed
sensors in the receiving array. The Rh component of the data vector r isassumed
to be a narrow-band Fourier transform of the snapshot received by the sensor at the
17
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ith location. The scalar oq is assumed to be a circulant complex Gaussian random
variable th at represents the unknown source amplitude and phase in the frequency
domain, with zero mean and known variance 2(T^. The assumption about the acoustic
source is consistent with those used in the literature[7, 45].
transfer function
The ocean acoustic
S) (Eq. 2.1[2j) and the diffuse noise no are both A'’ x 1 spatial
vectors. The diffuse noise at the receiving array is defined in the frequency domain,
which is assumed to be a circulant complex Gaussian random vector. Denoting the
expectation by E(-) and the covariance m atrix by Cov(-), we assume E(no) = 0 and
Cov(no) = 2cr^Ijv, where I n is the identity m atrix of size N.
The signal-to-noise ratio at the receivers is defined as
S N R { s { ^ , S)) =
where H denotes Hermitian transpose.
Using the definition of the signal m atrix (Eq. (2.3)), the binary hypotheses and
the d ata model can be redefined as
Hi
Ho
r = a o ||H ((^ , S),)||Sj
: ao ~ N ( 0 , 2 ( 7 ^ ) ,
'■ao = 0 ,
-I -
hq, i
e 1 . .. M, no ~ N(0, 2cr^Ijv),
(3.3)
where ||H (('i', S)j)|| is the norm of the ocean acoustic function given the ith real­
ization of the environmental parameters. Environmental uncertainty is incorporated
partly through the probability, 1/ M , of the ith realization of the signal wavefront in
the received d ata and partly through the structure of the signal matrix.
In addition, since multiplying by the scalar l/\/2 (j„ on both sides of Eq. (3.3) does
not change the detection performance prediction, we further simplify the hypotheses
18
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test as
Hi
Hq
r = oa/STV^S i + n, i G 1 . . . M, n rv N(0, Ifl),
: a -N (0 ,1 ),
: a = 0,
(3.4)
where the S N R i is generated by substituting the ith realization of the environmental
parameters in Eq. 3.2. Note th at a and n are normalized so th at Cov(a) = 1, and
C ov(n) is an identity m atrix of size N.
3.3
D e te ctio n A lgorithm s
Detection algorithms can be categorized according to the level of utilizations of the a
pn'on knowledge. In this section, four detection algorithms are presented: the optimal
Bayesian detector, the matched-ocean detector, the mean-ocean detector, and an
energy detector. The optimal Bayesian detector is a modified version of the 0UFP[7,
10, 48], which fully utilizes the a priori knowledge of the environment. The matchedocean detector is not
a realistic algorithm,because a completematch between the
assumedsignal wavefront and the data ispractically
impossible.
We include it to
provide a detection performance prediction bound for an ideal situation. The meanocean detector is a modified version of the conventional, or B artlett processor[2, 49].
It represents the algorithms th at might use mismatched model parameters.
The
energy detector represents the algorithms th at do not utilize the a priori knowledge
of the environment at all.
3.3.1
O ptim al B ayesian D etector
The derivation of the optimal Bayesian detector presented here differs from th at of
the O U FP[7, 10] in th at it implements the likelihood ratio utilizing the frequency
domain data model directly. The OUFP is an algorithm intended for localization.
19
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The optimal Bayesian detector forms the likelihood ratio. Under the H q hypoth­
esis, the probability density function of the data at the receiving array is assumed to
be
Under the Hi hypothesis, the probability density function of the data conditional
to the channel and source position parameters is p(r | H i , ^ , S ) . The data, condi­
tional to the environmental parameters, is a complex Gaussian random vector. Its
probability density function depends only on the mean vector and the covariance
matrix, which are
E { r |i/i,> t ,S )
C. = C o v (r |ffi,4 ',S )
= 0,
= 2a"||H(4',S)|W 4',S)s(®,S)'' + 2aJI„.
^
Applying Woodbury’s identity[50] and the definition of the S N R (Eq. (3.2))
r -1 _
5iV B(«,S)s(4>,S)s(*,S)"
1 + S N R ( ^ , S)
The determinant of Cr is the product of all its eigenvalues. Considering th at Cr has
N - 1 identical eigenvalues 2a^ and one eigenvalue 2cr^||H(^, S )|p s (^ , S ) ^ s ( ^ , S) -H
2 (7^,
and using the definition of the SN R (E q. (3-2)),
I C r 1=
( 2 a l f { S N R { ^ , S) + 1).
(3.8)
Using Eqs. (3.7) and (3.8),
p (r I 77i,
S) = exp ( - r ^ C r “ ^r) | TrCr
=
A ( ^ ,S )
.B(^,S)
=
exp { - r ^ r / 2 a l +
S) | r ^ s ( ^ , S) p )A "^ (^ , S)(27ral)^^,
S) + 1,
= SNR{^,S)/2al{SNR{<if,S)
+ l).
(3.9)
20
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Therefore, the conditional likelihood ratio is
(3.10)
The likelihood ratio combines the probability density function of the environ­
mental parameters with the conditional likelihood ratio. Assuming ^ and S are
statistically independent a priori^ the likelihood ratio becomes
V r)
A(r I
where
and
S)
=
/„ J„ ^ rf® iS A { rig > ,S )p (4 ')p (S ),
=
exp ( B ( * , S) | r"s{ ® , S) ?)/A{<Sl, S),
(3.11)
represent the ranges of environmental uncertainties. The items
A (^ ,S ) and S ( ^ , S ) are defined in Eq. (3.9).
The optimal Bayesian detector is the likelihood ratio test
Hi
A(r) ^
^0,
(3.12)
Ho
where the threshold /5o depends on the decision criterion.
3.3.2
M atch ed -ocean D etector
When the ocean environmental parameters and the source position are known, the
probability density functions p { ^ )
and p(S) are delta functions.
This is the lim­
iting case of the optimal Bayesian detector, where there is noenvironmental un­
certainty. Substituting delta functions for p ( ^ ) and p(S) in Eq. (3.11) generates
A(r) = A(r | ^ ,S ) . Since A(’I ',S ) and 5 ( ^ , 8 ) in Eq. (3.11) are now positive con­
stants,
a monotonic function of the likelihood ratio, is also optimum.
^ matched
^
knowm
^known) \ ■
21
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(3.13)
Since the environmental parameters are assumed known, the subscript ’’known” is
used in Eq. (3.13) to label the relevant environmental parameters. The matchedocean detector is therefore
Hi
A-
(3-14)
Ho
The performance prediction based on the matched-ocean detector provides an ideal
performance upper bound. However, it can be too optimistic to use in the presence
of ocean environmental uncertainties and source position uncertainties.
3.3.3
M ean-ocean D etector
The mean-ocean detector uses a test statistic of the same form as the matched-ocean
detector. It assumes th a t the detector knows only mean values of the environmental
parameters, which are mismatched with the true environmental parameters. Sub­
script ’’mean” is used to indicate this assumption. The mean-ocean detector is ex­
pressed as
H1
A,
(3.15)
Ho
where
^ (^)mean ~l ^ ^('^mean: ^mean) j )
and
/?2
3.3.4
(3.16)
is the threshold setting.
E nergy D etecto r
The energy detection algorithm is a conventional approach to the detection of ocean
acoustic sources.Although robust to environmental uncertainties, it doesnot exploit
22
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the a priori knowledge of these uncertainties.
Using our data model defined in the frequency domain, the energy detector is
expressed as
H1
< A,
(3.17)
Ho
where
>^'i^)energy =
and
3.4
/?3
(3-18)
is the threshold setting.
A n alytical RO C Perform ance E xpressions
Analytical ROC performance expressions are derived for each of the four detectors
discussed in Section 3.3, in the presence of various amounts of environmental and
source location uncertainties. To our knowledge, the approximate closed-form ROC
expression for the optimal Bayesian detector, is a new contribution. In the past, the
ROC results have typically been obtained by Monte Carlo approaches[10,
12 ].
In
using the Monte Carlo approach to obtain the ROC’s, one needs to generate Monte
Carlo d ata samples over environmental uncertainties as well as implement Monte
Carlo integration over the uncertainties of the data, which is computationally inten­
sive. Another contribution is the derivation of precise analytical ROC expressions as
a function of environmental uncertainties for sub-optimal detectors. The analytical
ROC expression for the mean-ocean detector can also apply generally to predict the
detection performance of a beamformer, when the signal wavefront model is mis­
matched with the data.
23
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3.4.1
O ptim al B ayesian P red ictor
It is often difficult to express P d as a function of Pp in closed-form for optimal
Bayesian detectors. Typically, Monte Carlo simulation techniques are used to eval­
uate the performance numerically[10, 12], T hat approach can be computationally
intensive. In this chapter, a useful approximate analytical expression to the ROC is
developed for sonar detection performance prediction in an uncertain environment.
The development of the analytical expression for the ROC begins with the discrete
likelihood ratio. Utilizing the signal matrix, the likelihood ratio test for the optimal
Bayesian predictor is
A(r)
A,Ai
Bi
=
=
exp(P i I r^Sj
= S N R i + 1,
= S N R i / 2 a l ( S N R i + l).
.
^
^
The optimal decision rule is to decide P i if A > /5 or to decide H q if X < /3. The
optimal decision rule is approximated by the decision rule “if Aj > /5 for any i G
1 . . . M, decide Hi and if Aj < P for all f = 1 . . . M, decide Pq” • This approximation
has been studied in general[51]. However, the authors are not aware of it being used
to derive detection performance in a physics-based uncertain environment, as in this
chapter. Using this approximation, we can express the ROC as
1-P
1-P
f {P)
= P {X i< /3 ,i = l . . . M \ H o ) ,
d {P) = P { \ < P , i = l . . . M \ H i ) .
A second approximation is a constant S N R approximation, i.e., assume SNR{Si) —
S N R for z = 1 . . . M. This assumes th at the received signal energy at the receiving
array is relatively independent of the perturbations in the ocean transfer function
due to ocean environmental and source positions uncertainties. This approximation
can be satisfied if ocean environmental param eter fluctuations are limited and source
24
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range variations are not very large. For some array configurations, such as short
vertical array or horizontal array, the received signal energy is sensitive to source
range variation, which may limits the utilization of this equal energy approximation
and the resultant analytical ROC expression for the Bayesian detector.
Using this approximation, Eq. (3.20) becomes
1-P
d {^')
= P ( |r ^ s , p < ^ ' , i = l . . . M | F o ) ,
= P ( |r ^ s , p < ^ ' , i = l . . . M | i J i ) .
(3.21)
Since S N R is assumed constant, Ai and P , can also be considered constant, i.e., A
and B. The new threshold is 13' — log{(3/A)/B.
Equation (3.21) shows th at the ROC depends on the signal matrix. Assume the
rank of the signal m atrix is R. We illustrate next th at the detection performance
does not change if the d ata vector of size N in the hypotheses test is replaced by a
reduced form of size R. The derivation is similar to that in the literature[52] except
th at for the Hi hypothesis, we assume th at the signal vector s, is selected from
the signal m atrix with the probability 1/M while the reference[52] assumes th at the
signal vector is linearly transformed from a parameter vector with a known transform
matrix.
A singular value decomposition of the signal matrix
generates 3? = U S V ^ ,
where U is an A’ x A unitary matrix, V is an M x M unitary matrix, and S is
a A X M m atrix th at consists of a full diagonal m atrix A = diag[Xi, A2 , ■■■Xr] and
zeros.
S=
A- Or x (m - r )
0(N-R)xM
(3.22)
Let y = U'^r. Since U is a unitary matrix, this transformation on the data r will
not change the performance of the optimal detection algorithm. The transformation
25
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generates
y = aU ^Si + U ^ n , = a U ^ U S V f + U ^ n , = a S V f + U ^ n ,
(3.23)
where Y f is the ith column of m atrix V ^ . Let U'^ = [Ui'U'2]'^, where U i is com­
posed of the first R columns of U. This substitution produces
y =
■u f r ■
U fr
’
.
y i
ys
'
Uffn
.
(3.24)
We see th at the information (whether a = 0 or a 7^ 0) is contained in the first
R elements of y, i.e., y i. The remaining elements do not depend on the signal. We
may therefore discard y 2 in the hypothesis testing problem concerning a and using
only yi.
Replacing r, s,-, and n in Eq. 3.25 with u = U f r , v, = U f s j , and w = U f n
respectively yields
u - y /S N R a v i -f w ,i e 1 .. .M , w ~ N (0 ,lij),
Hi : a - N ( 0 , 1 ) ,
Ho : a = 0 .
(3.25)
Therefore, the dimensionality of the data vectors in the binary hypotheses testing
problem is reduced from N to R. The signal wavefronts of reduced dimensionality
preserve the property th at v f v i = 1 for i = 1 . . . M. The covariance of the diffuse
noise, w, is an identity matrix of size R. The signal-to-noise ratio, S N R , is a constant
according to the second approximation.
Since assuming Vi as the reference signal wavefront does not change the deriva­
tions for Pp and Pd, substituting Eq. (3.25) in Eq. (3.21) results in
l-P p in
= P {\^^^ri\^< /3',i = l . . . M ) ,
1-P d (/50
= P ( | { V M R a v i + w ) ^ V i p < /?', i = 1 . . . M ) .
(3.26)
Both Pd and Pp can be expressed as a multidimensional integral of a complex Gaus­
sian probability density function in a constrained space.
26
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To compute Pd, let ri = s / S N R a v i + w, where r i is a complex Gaussian random
vector with covariance m atrix Cov(r^) = 1 + S N R v i v ^ . A singular value decom­
position of Cov(r^) generates Cov(r^) = Q A Q ^, where Q is a unitary matrix and
diag{A) = [ S N R -I-1,1 ■• • 1]. Then ri can be transformed to an independent Gaus­
sian vector z with covariance m atrix A to facilitate the derivation of Pd- Substituting
ri = Qz in Eq. (3.26) and using
1 - Pd(/50
A
= S N R -I-1 yields
= P (| z ^ Q ^ S i p<
=
I zi P A -i - X :t2 I
P)7r-^A -'dz.
(3.27)
If bj = Q^Vj, then b f b j = v/^QQ'^Vj = 1. Replacing Vj by b, in both the
Pd expression (Eq. (3.27)) and the Pp expression (Eq. (3.26)), and substituting
= z ^ Q ^ in the Pp expression produces
11
Pd(P') =
- Pp(/50
( -
=
exp ( -
I z i
yR
p
1-
A
I
J2 f=2
I z *
r ) 7T
M z ,
Zi
(3.28)
Now the domain of the multidimensional integration for both Pp and Pp is the
same. This domain is represented by the symbol D for simplicity, so P = {| z-^b, p <
13'J = 1 . . . M}. Now, D is an irregular convex-body in R space. To simplify the
computation we define a regular convex-body G = {| z, p <
adjusting
7
7
, i = 1...P}.
By
, we can make C equivalent to D in the sense th at the integral of the
function
on both convex-bodies is the same, i.e.,
n
R
p
R
/ exp ( - ^ I Zj p) 7r“-^dz = / exp ( - ^ | z, \^)'K~^dz.
1=1
dn
(3.29)
Integrating the left side of Eq. (3.29) yields
1
- Pp( 7 ) =
=
(1
exp ( I Zj P)7T ^dz
- exp ( - 7 ))^.
27
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(3.30)
Next, to compute Pd, the regular domain C is used to replace the irregular domain
D of integration. Note th at in Eq. (3.28) the integral expression for Pd is similar to
th at for Pp, except for a scalar S N R + 1 along the | Zi | axis, where Zi is the first
element of vector z. When the S N R is zero, the Pd expression is the same as the Pp
expression. Under this condition, by definition of the equivalence between C and D,
Pd can be computed using C as the domain of integration without any error. When
the S N R is greater than zero, Pd computed using domain C is an approximation to
Pd computed using domain D. When the S N R reaches infinity, the Pd computed
using domain C reaches one, which is the same at th at computed using domain D.
W ith this approximation, Eq. (3.28) becomes
1 - P d (7)
~ / c e x p ( - I
(1
Z i
—exp (—7
- ^ ^ ^ 2
^ “ ^ ) )(1
I z ,-p )7 T --^ y l-id z
1
-
(1
.
—exp (—7 ))^“ U
Finally, combining Pp (Eq. (3.30)) and Pd (Eq. (3.31)) and
Pd
,
- P f ) ^ ( 1 - (1 -
(1
A
= S N R + 1 yields
- PP)i)swfer),
(3 .3 2 )
where the param eter S N R is the signal-to-noise ratio at the receivers (Eq. (3.2)),
which is assumed constant in the derivation. In real applications, the S N R can
be replaced by the average S N R over realizations of the uncertain environmental
parameters, i.e.,
SNR
1
"
i —1
This S N R computation is used in Section 3.5. The parameter R is the rank of
the signal matrix. Ideally, the rank is equal to the number of non-zero eigenvalues.
In real applications, the rank is determined by the number of significant eigenvalues
th at exceed a threshold. As shown in Eq. (3.24), the eigenvalues can be considered as
28
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weights to the signal amplitude a. Non-significant eigenvalues yield little information
(weather a =
0
or a ^
0)
for source detection, so they do not significantly impact
the optimal detection performance. Consequently, we only consider the number of
significant eigenvalues to estimate the rank in order to predict detection performance.
We use an empirical method to determine the threshold, which is 5% of the maximum
eigenvalue. This empirical method is used and discussed in Section 3.5.
Equation (3.32) is one of the key results of this chapter. The ROC expression of
Eq. (3.32) provides insight into the impact of environmental uncertainty on optimal
sonar detection performance. It avoids the computationally intensive Monte Carlo
evaluation approaches. In Section 3.5, this approximate analytical ROC form of the
optimal Bayesian predictor is checked by comparing it with Monte Carlo simulation
results using the NRL uncertain benchmark ocean environments[14, 10, 23]. It is
shown th at the optimal detection performance in an uncertain ocean environment in
diffuse noise can be predicted well by this simple analytical form, in which the only
two parameters are the S N R and the rank of the signal matrix. The rank is a good
representation of the scale of environmental uncertainty. The consistent agreements
between the analytical ROC results and Monte Carlo simulation results also verify
the empirical method to estimate the rank of the signal matrix.
Eigure 3.1 illustrates Eq. (3.32) by plotting P d as a function of the uncertainty
scale R, the S N R at the receiver, and Pp- Eigure 3.1(a) plots P d as a function of R on
a logarithmic scale, assuming Pp = 0.01 and for S N R = 0 dB, 10 dB, or 20 dB. The
scale of environmental uncertainty is represented by R. It is shown in Eig. 3.1(a) th at
the performance degradation due to environmental uncertainty is inversely proportion
to log(R) for all three S N R conditions. Figure 3.1(b) predicts the optimal detection
performance as a function of S N R in the presence of environmental uncertainty of
scale R for Pp = 0.01. The degradation in P d caused by environmental uncertainty is
29
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greater for SNR less than 20 dB. In Fig. 3.1(c) three groups of ROC curves for three
S N R conditions are plotted using normal-normal coordinates. In each group, from
top to bottom , R equals 1, 10, and 100. This plot provides a good comparison of the
impact of environmental uncertainty and the S N R on the detection performance. It
shows th at varying R from 1 to 100 or decreasing S N R about 5 dB causes comparable
detection performance degradation.
(a)
SNR=20dB
SNR=10dB
- SNR=OdB
Q
CL 0.2
500
(C)
0.999
0.99
----- SNR=20dB
SNR=10dB
- - SNR=OdB
top down
R=1,10,100
top down
R=1
R=10
R=100
Q
CL 0.2
10
20
0.01
0.1
0.5
0.9
0.99
P.
SNR(dB)
Figure 3.1: Analytical detection performance predictions for the optimal Bayesian
predictor, (a) Pd as a function of R at fixed Pj?=0.01, (b) Pd as a function of S N R at
fixed Pf=0.01, ( c ) normal-normal ROC curves with various i?’s and various S'A’P ’s.
3.4.2
M atched-ocean P red ictor
Since the test statistic for the matched-ocean detector can be considered a chi-square
random variable for both hypotheses, an accurate analytical ROC expression for the
matched-ocean detector is available. The resultant form is similar to other ROC
expressions in the literature such as the ROC for the matched filter detector[52] and
30
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the power-type ROC[53].
Using Sk as an abbreviated notation for s{^known‘>^known) in Eq. (3.13) results in
| n S/; | ,
1 (nS/; “t* n) Sj^ |
{^m atched \
matched
\
Let Xq = n^Sk and Xi = {aSk + n)^Sk- Since n is a complex Gaussian vector, scalar o
is a complexGaussian random variable, and Sk is a known vector, both
xqand
x\ are
complex Gaussian random variables. Therefore, their absolute squares are chi-square
random variables, with two degrees of freedom. The resultant conditional probability
density functions are
p { ^ matched
\
H q^Si^)
P { ^ matched \
—^exp(^),
—^ C X p ( ^ ) ,
(3 35)
where C q — Cov(a;o) = 2al^sj^Sk and Ci = Cov(a;i) = Co{SNR{sk) -h 1).
Substituting the conditional probability density functions of the test statistics in
Eq. (1.2) produces
S/;)
I
-Pd(/^
I
—
J0 dX matchedP{X
matched
\ Hq, Sfc)
— CXp (
dXffiatchedPi.X matched
\ Hi,Sk)
—CXp ( ^ )-
(3.36)
Therefore the ROG can be written as
Pjj =
Since the signal wavefront
might be any Sj in the signal m atrix with probability
1/M , the ROG expression is averaged on M realizations of the signal wavefront,
^ D m a tc h e d
_
1
^
1
\ ' pSNR(Bi) +l
/
j
•
fey qo\
(^o.ooj
j=l
This simple matched-ocean predictor is dependent only on the S N R of the known
signals.
31
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3.4.3
M ean-ocean P red ictor
To derive the mean-ocean predictor, we assume r = as< -|- n for the Hi hypothesis,
where Si represents the real signal wavefront. Replacing s(^meon) S^ean) by
simplicity in Eq. (3.16) yields the expressions for the test statistic, given
| O Sjjj | ,
m ea n \ H o ^ S f n )
m ea n
I H\., S^., Sffi)
for
and s*.
(3.39)
| (flSj-|-n) Sjji |
Let Xq = n^Sm and x% = (as* -I- n )^ s ^ . Similar to the previous subsection, the
absolute squares of both Xq and xi are chi-square random variables, with two degrees
of freedom. Their probability density functions are
Pi^mean \ Ho^Sm)
P{^ mean I Hi-, Sytii
=
(3
where D q = C o v {xq) = 2(t^s^s^ and Di = Cov(a;i) = 2al | s f
P +2(t^ | ^m
40)
12
Given the real signal wavefront St and the mean-ocean signal wavefront s^ , the
analytical ROC form for this mismatched condition can be expressed as
DO
D m ism a tch ed
+l
= Pp
1___________
Tpp{ai,8m)SNR{Bf)-\-l
~
where p{st,Sm) =
F
(3-41)
5
The above ROC expression depends on SNR{st) and
p(sti Sm) only, where p{st, Sm) is the correlation of the true signal wavefront with that
of the assumed. The param eter p characterizes the performance degradation caused
by the m ism atched environm ental param eters in th e m odel. A special case is p = 1,
for which the above ROC expression reduces to the matched-ocean case.
Considering
might be any Sj in the signal m atrix with probability 1/M , the
ROC for the mean-ocean predictor can be expressed as an average of the mismatched
32
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cases.
T)
_
^Dmean ~
1
\ ^ ^ n p(H'^-rn)S1NR(s^)+1
2-^
/n
j=l
The mean-ocean predictor predicts the average performance degradation due to en­
vironmental mismatch, which is determined primarily by the correlation coefficients
p(Sj,s^ ) , i = 1.. . M . The mean-ocean predictor also predicts possible failure of the
mean-ocean detector at longer propagation ranges, as the effects of small errors in
the exp{—jkjnrs) term (Eq. 2.1) accumulate, resulting in the mismatch between the
mean-ocean signal wavefront and the data.
3.4.4
E nergy P red ictor
The energy detection algorithm neither requires nor could it use any knowledge about
the ocean variables and source location, even if such information were available. The
performance evaluation of the energy detector provides a reasonable performance
prediction lower bound.
To derive the energy predictor, the distributions of the energy test statistic for
both hypotheses are required. One special concern in deriving the distribution func­
tion for the Hi hypothesis is th at the data depend on the reference signal wavefront
Sj, which might be any column of the signal matrix with probability 1/M. Another
special concern is th a t the conditional test statistic given the reference signal wavefront, A'(r I Si)gnergy ~
n)^(aSj -I- n), does not follow a chi-square distribution,
as commonly assumed for the energy detector [14]. The common assumption is made
by ignoring d ata spatial correlations incurred by the presence of the signal.
Since the test statistic is now the sum of correlated gamma random variables, one
can obtain analytical forms for the ROC, using the characteristic functions[54, 55].
33
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The characteristic functions of X'energy given Hi and H q conditions are
m
I Ho} = (1 - j t 2 a l ) - ^ ,
I Ho) =
=
I H u Si) =
(3.43)
I Hi, s,} =1 / - j t D
1 - 1
where D = E{{aSi + n)(aSj + n )^} = 2cr^Sjsf + 2(t^I.
| Hi,Si) is further decomposed into partial frac­
The characteristic function
tions [56](p56) to produce Fourier transforms [57](p431).
H t\H i,Si)
do
dii
=
=
=
—
{l-jtd o )-^^-^\l-itd ii)-^
Y!k= i
- jtdo)~^ + k { l - j t d i i ) ~ \
2al,
doAi,
h.
—
______ 1____
Ai
=
“
(3.44)
(1-l/Ai)^-'^ ’
S N R ( s i ) + 1.
The Fourier transforms of the characteristic functions generate the probability
density functions of the test statistics
p (X 'e n e rg y lH o )= 7 (^^,N ),
P{yenergy \ Hi) = ^ E £ l( E f = Y^ n
E ------ k) +
do
---di
1 )),
(3-45)
= W )*
Finally P d and Pp can be obtained by substituting Eq. (3.45) in Eq. (1.2)
Pp{/3)
P
d
= 1 - 7 j {P,N),
= 1 + jg Ej=l(EA:=l
{P)
Ai
= S N R ( s i ) + 1,
7j (P ,N )
=
A i{l-\/A i)r^-i'X l{l^^k) — ( ^ i _ i i \ . ^ N - i l l { ^ / A i , l ) ) ,
t^ -^ e -^ d t,
(3.46)
where
77
is the incomplete gamma function. The incomplete gamma function can be
computed by using the function “gammainc” in MATLAB. The ROC for the energy
detector depends on the number of array sensors and the S N R at the receiving array.
34
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3.4.5
A ssu m p tion s and C om p utation al C om p lexity
The ROC results computed using the Monte Carlo approach are used as a standard
to check the analytical ROC expression for the Bayesian detector. The Monte Carlo
approach makes the same assumptions as the analytical approach in regard to the
hypothesis testing model given by Eq. (3.4), which include the a priori distribution
of the signal source and the diffuse noise, and the a priori discrete signal wavefronts.
The additional approximations used for the analytical approach are summarized as
1. Decision rule approximation.
2. Constant S N R approximation.
3. Rank approximation.
4. Integration domain approximation for the derivation of P d In Section 3.5, the above four approximations are verified by consistent agreements
between the P d results obtained using Monte Carlo approach and the analytical
approach over various types and ranges of environmental uncertainties.
The computational complexity for the Monte Carlo approach is used as a com­
parison to th at of the analytical approach. We summarize the complexity for each
step of the Monte Carlo approach.
1. To compute a sample likelihood ratio given by Eq. (3.19) for one hypothesis,
the complexity is 0 { M N ) , where M is the number of the realizations of the
signal wavefronts, and N is the number of array components.
2. To generate samples of the likelihood for each hypothesis, the cost is about
0 { L M N ) ^ where L is the number of samples. Usually we use 5000 or more
samples so th at the precision of P d and Pp reaches
0 .0 0 1 .
35
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3. To obtain P d and Pp using Eq. (1.1), the complexity is about 0{L\og{L)).
The total computational complexity for Monte Carlo approach is 0 { L M N ) .
The cost for each step of the analytical approach is listed.
1. To estimate the S N R using Eq. (3.33), the cost is 0 { M ) .
2. To estimate the rank of the signal matrix, the primary cost is due to the com­
putation of the eigenvalues. According to [58], the computation of the eigen­
values of a Hermitian m atrix costs 4/3N^ using transformation methods th at
is implemented in the eig command in the MATLAB. Here N is the size of
the Hermitian matrix, i.e. the number of the array sensors. To prepare the
Hermitian m atrix s^ s, the m atrix multiplication costs 0 (A ’^®^)[59].
3. To compute Eq. (3.32), the cost is 4 multiplication and 3 exponential operations,
which can be ignored.
The total computational complexity for the analytical approach is 0{N^).
Typically, the number of the realizations of the signal wavefronts is comparable
to or greater than the number of the array sensors. The number of the samples of the
likelihood ratio is 10 times greater than the number of the array sensors. Therefore,
the analytical approach speedups more than an order of magnitude.
3.5
R esu lts and D iscussions
3.5.1 P ropagation M odel and E nvironm ental C onfigurations
The propagation model used here to check the analytical ROC expressions is a mod­
ified version of the general mismatch benchmark model (“genlmis” ) proposed in the
May 1993 NRL Workshop on Acoustic Models in Signal Processing[23, 10, 14]. This
36
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model is reviewed in Fig. 2.1. The environmental configurations used in this chapter
are extensions of the “genlmis” model[14, 10, 23].
They are summarized in Ta­
ble 3.1-3.3. Table 3.1 defines the environmental parameters, their mean values and
their ranges of uncertainty. Since the quantitative effects of environmental uncer­
tainty are the main subject, we denote the ranges of the environmental uncertainties
(D, Coi
Q , p, a, Zs) using A variables instead of constants. The source range
Vs and the upper sediment thickness t were assumed fixed at their mean values in the
simulations. The units for each parameter have been listed in Table 3.1.
Table 3.1: Parameters of uncertain shallow water propagation model
Environmental parameter
mean value ± range of uncertainty
D-bottom depth [m]
lO O iA O
Co-surface sound speed [m/s]
1500±ACo
-bottom sound speed [m/s]
1480±ACo
-upper sediment sound speed [m/s]
1600±AC^
Q-lower sediment sound speed [m/s]
1750±ACi
p-sediment density [g/cm^]
1.7±Ap
Q-sediment attenuation [dB/A]
0.35±A a
t-sediment thickness [m]
10 0
Zj-source depth [m]
50 ±A.Zs
fs-source range [km]
6
Table 3.2 presents a group of three simulation scenarios used to check the optimal
Bayesian predictor. Scenario A considers a single water depth uncertain case with 5
m uncertainty. Scenario B defines a general uncertain case th at includes seven un­
certain parameters (£), (7o,
(7^, Q , p, a) with fixed known uncertain ranges. Sce­
nario C includes one more uncertain parameter
in addition to the seven uncertain
param eters in scenario B. T h ese scenarios represent different levels of environm en­
tal uncertainty. Since source depth and water depth are two of the most sensitive
parameters, they are considered separately. The presence of other environmental un­
certainties appears to be relatively less sensitive to the detection performance. There37
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fore, we use A 6*to represent (ACq, A(7^,
AC;, Ap, A a) for simplicity and use
A 6*o = (2.5, 2.5,50,100,0.25,0.25) as a typical set of values for A9.
Table 3.2; Scenarios of environmental uncertainty with fixed ranges of uncertainties
Scenario
A
B
C
Uncertainty configuration
AD = 5
A D = 5 ,A 9 = A9o
A D = 5, A^, = 40, A9 - A^o
Define A9 = (ACo, A C ^ , A C ^ , A Q , Ap, A a ) and A 6»o = (2.5,2.5,50,100,0.25,0.25)
for simplicity. The units are consistent with those in Table 3.1.
Table 3.3 presents another group of simulation scenarios to study the detection
performance predictions as a function of increasing environmental uncertainties and
to further check the optimal Bayesian predictor. In Table 3.3, we also use A9 and A^o
for a simple representation. Six scenarios were included. Scenario A is a single source
depth uncertain case with Azg increasing from 0 to 50 m. Scenario B represents a
single water depth uncertainty case with A D increasing from 0 to 5m. Scenario
C considers source depth and water depth uncertainties increasing together from
{Azg, A D ) = (0, 0) to (50,5) m. Scenario D is scenario A plus water depth uncertainty
(A D =5 m) and the general uncertainty {A9 = A9 q). Scenario E (or F) is scenario B
(or C) combined with the presence of the general uncertainty case. These scenarios
reflect various amounts of environmental uncertainties.
3.5.2
Perform ance P red iction s for th e O ptim al B ayesian D etector
Three simulation scenarios summarized in Table 3.2 were used to check the opti­
mal Bayesian predictor (Eq. (3.32)) by comparing its prediction results with those
obtained using Monte Carlo evaluations. The resultant ROC’s are plotted in Figs.
3.2(a)-3.2(c) corresponding to scenarios A-C in Table 3.2. In each plot, the dash38
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Table 3.3: Scenarios of environm ental uncertainty w ith increasing ranges of uncer­
tainties
Scenario
A
B
C
D
F
F
Uncertainty configuration
A^, e [0, 50]
A D e [0, 5]
(A D ,A .^ ,) g [(0,0),(5,50)]
A.2, e [0,50], AD = 5, A0 = A^o
A D e [0,5],A^ = A^o
(AD, A^,) e [(0,0), (5,50)], A0 = A^o
Define M = (ACq,
AC+, A G , Ap, A a) and A^o = (2.5, 2.5, 50,100,0.25,0.25)
for simplicity. The units are consistent with those in Table 3.1.
dotted line is the ROC’s computed using Eq. (3.32) and the solid line is the ROC’s
obtained using the Monte Carlo approach.
For each uncertain environment, 80 realizations of the environmental parameters
were generated (i.e., M =80) and used for both analytical and numerical prediction
approaches. The analytical form for the optimal prediction (Eq. (3.32)) contains
two parameters, the rank of the signal matrix and the S N R . Figures 3.2(d)- 3.2(f)
illustrate the procedure of choosing an approximate rank by counting the number of
the eigenvalues th at exceed a threshold. The eigenvalues come from the matrix s^s.
This result is fundamentally the same as using the eigenvalues of the signal matrix
s. The threshold distinguishes the eigenvalues with significant energy, and therefore
determines the approximation to the dimension of the signal wavefront structure. It
is difficult to theoretically solve the best threshold for the rank estimation. However,
we show in this Section and next Section th at an empirical threshold: 5% of the
maximum eigenvalue, gives consistent agreement between Monte Carlo simulation
results and analytical prediction results for various types of environmental uncertain­
ties. We show next th at the variation in P d due to changes in the threshold from
5% to either 10% or 1% is not significant. Figures 3.2(d)- 3.2(f) illustrate three rank
39
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(c)
0.5
0-0.5
Y
If
0.5
X
8 It
10
X 10 -4 (e)
-4 (d )
,R = 6
'w
X
^ 4 ;,R = 12
O) 4tl 1
LU
X
8
,R = 10
6
4 ;,R = 13
4
6
2
50
;,R = 23
^ ,R
II
=
21
2
--------
0
100
(f)
iR = 21
^ II
2 4 11
10’
8
0
50
100
0
k
ll,R = 25
.................
50
100
F ig u re 3.2: Analytical detection performance predictions for the optimal Bayesian
predictor, using the measure of ROC curves at three uncertain environment scenarios,
which have been defined in Table 3.2. The S N R at the receivers is about 10 dB. (a)(d)
the case of single unknown ocean depth with 5m uncertain range A D = b m, (b)(e) the
case of general uncertain ocean environment with A D = 5 m and A9 = A^O) and (c)(f)
the case of 40m uncertain source depth plus general uncertain ocean environment with
Azs=iO m, A D = 5 m, and A 9 = A^o- (a)-(c) ROC curves of analytical predictions
for optimal Bayesian predictor (dashdot, Eq. (3.32)), ROC curves for the optimal
detector using Monte Carlo evaluation techniques (solid-using 5% threshold, vertical
lines-using 1 %-1 0 % threshold ). (d)-(f) estimations of the rank in the analytical
prediction expression (Eq. (3.32)) for each scenario, using 10%,5% and 1% thresholds
respectively from left to right.
40
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estimations th a t are computed using 10%, 5%, and 1% of the maximum eigenvalue.
In the corresponding analytical ROC prediction plots in Figs. 3.2(a)- 3.2(c) in the
upper row, the dash-dotted line is computed using the 5% threshold, and the short
vertical lines illustrate changes in P d th at are due to the threshold variation from
10% to 1%. In Fig. 3.2(a), the maximum difference between P ^ s computed from 10%
or 1 % thresholds is smaller than 0.054. In Figs. 3.2(b) and 3.2(c), the P d variations
due to threshold variation are even smaller. The results show th at limited threshold
variation does not significantly affect the P d computation.
Figures 3.3(a)-3.3(c) compare the ROC’s predicted by the optimal Bayesian pre­
dictor, the mean-ocean predictor, and the energy predictor in uncertain environments
defined in Table 3.2. The S N R at the receiver is about 10 dB. The dotted lines are
for the mean-ocean predictor and the dashed lines are for the energy predictor. These
results show the importance of fully incorporating the environmental uncertainty into
the sonar detection performance prediction algorithm.
Figures 3.3(a)-3.3(c) compare the ROC’s predicted by the optimal Bayesian pre­
dictor, the mean-ocean predictor, and the energy predictor in uncertain environments
defined in Table 3.2. The ROC’s predicted by the optimal Bayesian predictor are the
same as those in Figures 3.2(a)-3.2(c) except th at they are now plotted in normalnormal coordinates in order to expand the ranges of Pd and Pp- The dotted lines
are for the mean-ocean predictor and the dashed lines are for the energy predictor.
The S N R at the receivers is about 10 dB. These results show th at the difference in
P d predicted by the Monte Carlo approach and the Optimal Bayesian predictor is
small for Pp beyond
0 .1 ,
and is modest (< 0.06) for Pp in the range of
41
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0 .0 0 1
—0 . 1 .
(a)
.99
.01
,01
.1
5
9 .99
(C)
(b)
.99
.99
.01
.01
01
.1
,5
.9 .99
,01
.1
,5
9 .99
Figure 3.3: Detection performance predictions for the optimal Bayesian predic­
tor (solid-Monte Carlo, dashdot-analytical, Eq. (3.32)), the mean-ocean predictor
(dotted-Eq. (3.42) ), and the energy predictor(dashed-Eq. (3.46)). The uncertain
environments for (a)-(c) have been defined in Table 3.2. The S N R at the receivers
is about 10 dB.
42
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3.5.3
Q uantitative Effects of Environmental Uncertainties
Quantitative effects of environmental uncertainty on various detection performance
predictions are first presented using the estimations of the rank of the signal m atrix for
various scenarios of environmental uncertainty, then illustrated in Figs. 3.4 and 3.5,
by plotting the probability of detection as a function of the range of environmental
uncertainties for the scenarios defined in Table 3.3.
Table 3.4: Rank estimation for
Rank of the
1a d I
signal m atrix
< 2.5 m
Benchmark ocean
8
Mediterranean Sea* 6
various scenarios of environmental
A^s
1 ACol
ACB
< 2.5 m /s < 2.5 m /s < 5 m
3
3
3
3
3
3
uncertainty
A e = A9o
13
9
Define A9 = {AC q, A C ^ , A C ^ , AC/, Ap, A a) and A^o = (2.5,2.5,50,100,0.25,0.25)
for simplicity. The units are consistent with those in Table 3.1. Mediterranean Sea
environment is introduced in [60].
The effect of environmental uncertainty on optimal sonar detection performance
is captured by the rank parameter in the proposed analytical ROC expression for
the Bayesian detector. We summarized the rank of the signal m atrix for variance
uncertain scenarios in Table 3.4 for two ocean environments, one is the benchmark
ocean (Fig. 2.1, Table 3.1), another is the Mediterranean Sea environment[60]. The
signal m atrix is composed of 1 0 0 realizations of the signal wavefront for each scenario,
where the realizations of the signal wavefront are predicted using KRAKEN code[46],
assuming a single acoustic source at
6
km in range and 50 m in depth, and a full
spanning vertical array of 1 0 0 hydrophones uniformly spaced from surface to bottom.
A 5% of the maximum eigenvalue of the signal matrix is used as the threshold to
estimate the rank. It is verified th at the ROC curves predicted using these rank
estimations and the analytical expression agree with those computed using the Monte
Carlo method. The results in Table 3.4 show th at the rank param eter can be used to
43
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represent the scale of environmental uncertainty. These rank parameters provide fair
comparisons across various types of uncertain environmental parameters in terms of
their effects on optimal detection performance.
(a)
1
1
0.8
0.8
0.8
o
d 0.6
II
LL
CL
0.6*
\
\
® 0.4 \
Q
CL
(c)
(b)
1
0.4
\
0.2 ..... 'v ; ...................
0
0
\
A Zg [m]
50
0.6* i .............................
\
0.4 \
\
V,
\
_______
0.2
0
A D [m]
\
'
0.2
0
A z [x10m] and A D [m]
(e)
(d)
(f)
1
1
1
0.8
0.8
0.8
o
d 0.6(<>
0.6*
^0.4
>
' K,
0.6*
0.4
0.4
0.2
0.2
0
0
>
Q
CL
0.2
0
0
50
A Zg [m]
A D [m]
, A..............................................
---------------------
A Zg [x1 Om] and A D [m]
Figure 3.4:
Detection performance predictions of matched-ocean (circle),
mean-ocean (dashed line) and energy (solid line) predictors using the measure of
probability of correct detection ( P d ) as a function of increasing environmental uncer­
tainties at given probability of false alarm (Pp—0.01) and at fixed noise level { S N R
about lOdB), for six uncertain environment scenarios, which have been defined in
Table 3.3. (a) single source depth unknown, (b) single ocean depth unknown, (c)
joint source depth and ocean depth unknown, (d) source depth unknown plus ocean
general uncertain, (e) ocean depth unknown plus ocean general uncertain, and (f)
joint source depth and ocean depth unknown plus ocean general uncertain.
Figure 3.4 compares the effects of environmental uncertainty on the matchedocean predictor, the mean-ocean predictor, and the energy predictor, which are all
precise analytical calculations. Uncertain environment scenarios A-F defined in Table
3.3 were used to generate Figs. 3.4(a)-3.4(f). The Ppi’s in each plot were computed
44
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using Eqs. (3.38), (3.42), and (3.46) respectively, at a fixed Pf= 0.01, and for an S N R
of about 10 dB at the receivers.
In Fig. 3.4(a) for example, the circle is the P d predicted by the matched-ocean
predictor at Azg=0 m, corresponding to no environmental uncertainty. The P d for
the matched-ocean predictor is determined only by the S N R of the mean-ocean signal
and Pp. In each of Figs. 3.4(d)-3.4(f), the circle is P d averaged over the fluctuating
S N R s in uncertain environments. The ROC’s of the matched-ocean detector in all
scenarios provide performance prediction upper bounds.
In each plot of Fig. 3.4, the dotted line is for the mean-ocean predictor and the
solid line is for the energy predictor. All curves are analytical results. The curves in
Fig. 3.4(a) show th at without environmental uncertainty the mean-ocean predictor
predicts the same detection performance as th at of the matched-ocean predictor.
However, the mean-ocean predictions (Pd ) fall off rapidly with the increasing range
of uncertain source depth. These results indicate th at the a priori information used
by the mean-ocean detector is so limited th a t it is useful only when the uncertain
environment is close to a known deterministic environment. The mean-ocean detector
is very sensitive to increasing environmental uncertainties.
For the energy predictor, P d does not change with environmental uncertainties,
but remains a small number. This is because the energy detection algorithm does not
incorporate the a priori knowledge of the environment at all. It nevertheless provides
a useful performance lower bound.
Figures 3.4(a)-3.4(f) compare the mean-ocean predictor and the energy predictor
for different scales o f environm ental uncertainty. B y using th e energy predictor as a
performance lower bound, the plots indicate at what scale of environmental uncer­
tainty one could use the mean-ocean detector. For example, Fig.
6 (a)
shows th at
for a single param eter uncertainty in source depth, ± 18 m, the performance of the
45
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mean-ocean detector degrades dramatically. Figure 6 (b) shows th at for the single pa­
rameter of water depth uncertainty, ± 5 m, the mean-ocean detector performs better
than the energy detector. For more general ocean uncertainty, corresponding to the
scenarios D-F in Table 3.3, the mean-ocean detector’s performance is poor, as shown
in Figs. 3.4(d)-3.4(f).
(a)
(c)
(b)
1
0.8
0.8
0.8
0.6
0.6
0.6
(§) 0.4
0.4
0.4
0.2
0.2
°
LL
Q.
Q
^ 0.2
0
A Zg [m]
50
A D [m]
^
A^g [x1 Om] and A D [^ ]
(e)
(d )
(f)
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
(§) 0.4
0.4
0.4
0.2 -----------
0.2
CL
Q
^ 0.2
50
A Zg [m]
0
A D [m]
_____
..................
0
0
5
A z [x10m] and A D [m]
F ig u re 3.5: Analytical detection performance predictions of the optimal Bayesian
predictor, as compared with the prediction using Monte Carlo evaluation techniques
and the performance predictions of the mean predictor and the energy predictor,
using the measure of P d as a function of increasing environmental uncertainties at
given Pp = 0.01 and at fixed noise level { S N R about lOdB) for six uncertain envi­
ronment scenarios, which have been defined in Table 3.3. Four prediction curves are
plotted: analytical prediction for the optimal Bayesian predictor (dashdot), Monte
Carlo evaluation for the optimal Bayesian predictor (solid), analytical prediction for
the mean predictor (dotted) and the energy predictor (dashed).
Figure 3.5 illustrates the effects of environmental uncertainty on the optimal
Bayesian predictor. The P d 's for the optimal Bayesian predictor were computed
46
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using Eq. (3.32), at a fixed Pp=OM, for about 10 dB S N R at the receiver. They
were plotted using solid curves. The P ^ ’s obtained by the Monte Carlo approach were
plotted using dash-dot lines. It is shown th a t the predictions made by the optimal
Bayesian predictor agree with those obtained by the Monte Carlo approach. In this
case, the performance falls off gradually with increasing environmental uncertainty.
The results confirm th at the optimal detection performance can be predicted, at least
for the situations considered so far, if one has the rank of the signal m atrix as a rep­
resentation of environmental uncertainty, and the mean S N R . Note th at the Monte
Carlo approach used 5000 d ata samples for each hypothesis and each uncertainty
scale to get a single P d point. This is considerably more computationally intensive
than computing the analytical expression for the optimal Bayesian predictor.
In addition, for most uncertain environments considered, the optimal performance
is much better than the performance predicted by the energy predictor, as shown in
Figs. 3.5(a)-3.5(f). The result is reasonable because we assume th at the optimal
Bayesian detector fully utilizes the a priori knowledge of the uncertain environ­
ment. The optimal Bayesian predictor provides a more practical sonar detection
performance upper bound than the mean-ocean predictor, or the conventional sonar
equation. It directly relates the environmental uncertainty to optimal performance
through a simple but im portant parameter i?, in addition to the parameter S N R ,
which in turn incorporates the effects of propagation loss and diffuse noise.
3.5.4
C om parison o f Perform ance P red iction s as a Function o f Source
R ange
The detection probability as a function of source range is shown in Fig. 3.6 for
scenario B defined in Table 3.3. From top to bottom, five approaches are illustrated:
the matched-ocean predictor (circle), the optimal Bayesian predictor (dashdot), the
47
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Monte Carlo performance evaluations for the optimal Bayesian detector (solid), the
energy predictor (plus), and the mean-ocean predictor (dotted).
1
o.a
° ° ° O O n ,
0 0 .6
°°OO o
"4
5
6
7
Source Range(km), Source Depth = 41m
8
F ig u re 3.6: Detection performance predictions using the measure of P d as a func­
tion of source range at a fixed source depth =41m, in the genlmis ocean, at given
Pp=0.01 and at a fixed noise level ( S N R about lOdB) for five performance predic­
tion approaches: analytical predictions for the matched-ocean predictor (circle) and
the optimal Bayesian predictor (dashdot), Monte Carlo performance evaluation for
the optimal Bayesian predictor (solid), analytical predictions for the energy predictor
(plus) and the mean-ocean predictor (dotted).
The difference between the Pd predicted by the matched-ocean predictor and the
optimal Bayesian predictor reflects the performance degradation due to environmen­
tal uncertainty. The performance degradation in source range can be illustrated by
fixing Pd = 0.6 and obtaining the ranges of detection in the plot. The matched-ocean
predictor predicts
6 .8
km range of detection and the optimal Bayesian predictor pre­
dicts 4.9 km range of detection. The 1.9 km difference is due to the presence of
environmental uncertainty. On the other hand, the performance degradation can
also be illustrated by fixing the range of detection at 4.9 km and to compare
P d 's .
The matched-ocean predictor predicts Pd — 0.72 and the optimal Bayesian predictor
predicts Pd = 0.6. The 0.12 difference in Pd is due to the presence of environmental
uncertainty.
48
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The performance degradation due to environmental uncertainty can be compara­
ble to th at due to propagation loss. Suppose a target is moving from 4.9 km to
6 .8
km in this uncertain environment, the optimal predictor predicts th at P d decreases
from 0.6 to 0.48. This decrease is mainly due to propagation loss. Therefore, neither
propagation loss nor environmental uncertainty can be neglected. Simulation results
indicate th at the general pattern of P d as a function of source range is relatively
independent of source depth.
3.5.5
E ffects o f W rong a p r io r i K now ledge
The a priori knowledge is im portant to the performance of the Bayesian detector. If
the a priori distribution of model parameters or the a priori ranges of environmental
uncertainties are wrong, the detection performance degrades. Figure 3.7 illustrates
the effects of overestimating or underestimating ranges of ocean depth uncertainty
on the detection performance of the Bayesian detector, using Monte Carlo simulation
approach. The number of signal wavefronts is 80 and the simulated data samples for
each hypothesis is
20000.
The ROC curves are plotted using normal-normal coordinates, where the dashed
line denotes the case using correct
±1
m range of ocean depth uncertainty; the dash-
dotted line is for the scenario assuming ± 5 m ocean depth uncertainty while the tru th
is ± 1 m; the solid line denotes the scenario using correct ± 5 m range of ocean depth
uncertainty; and the dotted one denotes the scenario using an underestimated range of
±
1
m ocean depth uncertainty, while the tru th is ±5 m. The maximum degradation
from the dashed curve to the dash-dotted curve is 0.03, which shows th at if the
range of ocean depth uncertainty is overestimated, the performance degradation is
modest. However, if the range of uncertainty is underestimated, the P d performance
degradation is as large as 0 . 1 , which is significant.
49
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0.999
0.99
0.9
0.5
0.01
0.001 h............
0.001 0.01
0.1
0.5
0.9
0.99
0.999
F ig u re 3.7: The effect of wrong a priori range of ocean depth uncertainty on
the Bayesian detection performance, illustrated by the ROC curves computed us­
ing Monte Carlo approach, using normal-normal coordinates. From top to bottom,
four cases are plotted: correct ±1 m range (dashed), assuming ± 5 m range while
the truth is ± 1 m (dash-dotted), using correct ± 5 m range (solid), and using ± 1
m range while the tru th is ±5 m (dotted).
Note th at the ROC expression (Eq. 3.32) can be applied if the range of uncer­
tainty is overestimated, because the derivations of the ROC expression can be applied
to this case without any modification. The rank is estimated from the signal matrix
th a t consists of the signal wavefronts due to environmental uncertainty in the overes­
tim ated uncertainty range. For example, if the true range of ocean depth uncertainty
is smaller than ±5 m, and if we assumed ±5 m, the Bayesian detection performance
is the same as if the true uncertainty range is ±5 m. The overlapping of the dashdotted curve and the solid curve in Fig. 3.7 verifies this point. The overestimation
of the range of uncertainty causes the increase of the rank of the signal matrix and
therefore causes smooth (logarithmical) decrease of the detection performance. Un­
derestimating the range of uncertainty means th at the signal wavefronts in the signal
matrix are mismatched with the data. Although an analytical ROC formulation for
50
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the underestimate case is not available yet, we know from the ROC expression for
the mean-ocean detector th at mismatched signal wavefronts cause the S N R decreas­
ing by a factor th a t is determined by the similarity between the mismatched signal
wavefronts and the true signal wavefronts. The decrease in S N R may degrades the
detection performance significantly.
In summary, the results indicate th at we would rather overestimate the range
of environmental uncertainty than underestimate the range of environmental uncer­
tainty.
3.6
Sum m ary
Analytical approximate ROC expressions, th at can be computed rapidly, have been
developed for sonar performance prediction. Using a statistical decision theory frame­
work, detection performance prediction algorithms were derived th a t incorporate the
uncertainty of the physics of the acoustic propagation channel as well as the uncer­
tainty of source position in an optimal manner. These analytical expressions for the
ROC enable one to compute sonar performance prediction in a much simpler manner
than is usually impossible using Monte Carlo methods.
Analytical forms were obtained for the ROC for the matched-ocean detector, the
mean-ocean detector, and the energy detector. The matched-ocean predictor provides
an upper limiting performance bound for the case where there is no uncertainty
in the ocean environment. The mean-ocean predictor, which uses only the meanocean signal wavefront rather than the signal matrix, illustrates the degradation in
perform ance due to m ism atched m odel param eters.
The optimal Bayesian predictor incorporates the uncertainty of the physics of
the acoustic propagation channel. An algebraic expression was obtained th at was
shown to be an excellent approximation for the ROC for several examples. This
51
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expression indicates th at the optimal sonar detection performance (ROC) in diffuse
noise depends primarily on the ocean environmental uncertainty, which is captured by
the signal matrix, and the mean signal-to-noise ratio at the receivers. These results
let one obtain the ROC’s in a simple way, as contrasted to computationally intensive
Monte Carlo approaches. These results also provide a more meaningful and realistic
performance prediction since it incorporates the environmental uncertainty, a feature
th at is lacking in the classic sonar equation.
52
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Chapter 4
Optim al Sonar D etection Perform ance Prediction:
Interference
4.1
Introd u ction
In addition to environmental uncertainty, the presence of interference is another lim­
iting factor to sonar detection performance. To detect an object in the presence of
interference, an optimal detection algorithm must incorporate the a priori knowledge
of the interference. A realistic performance prediction approach must accordingly in­
corporate the effect of the interference on sonar detection performance. Some detec­
tion algorithms assume a known plane-wave interference model[61, 16]. The resultant
performance predictions can be too optimistic. Other detection algorithms[62, 63]
assume th a t the noise field is completely unknown, requiring either in situ measure­
ments or adaptive estimations from noise-only data, which do not take advantage of
available information about the environment and the interferers.
Here we develop an analytical ROC expression for the Bayesian detector in the
presence of interference in uncertain environments[2 1 ]. This work is a significant
extension to the work illustrated in chapter 3, which only considers diffuse noise
field. The principal cause of the interference modeled here is the acoustic sources
other than the target. The 0UFP[7] is modified in order to incorporate the a priori
information of the interferers. The resultant ROC expression is an algebraic function
of a few parameters: the signal-to-interference coefficient, which captures the effect
of the interference in uncertain environments; the rank of the signal matrix, which
characterizes the scale of environmental uncertainty; and the S N R at the receivers
th a t represents the combined effects of source level, noise level, and propagation
53
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loss. Simulated data generated using an NRL benchmark shallow water model[14,
10,
23] were used to verify the proposed detection performance prediction approach,
by comparing the analytical results with the Monte Carlo performance evaluation
results. Good agreement of those results demonstrates th at the simple analytical
ROC expression derived here captures fundamental parameters of the performance
for the Bayesian detectors. This provides a more realistic detection performance
prediction than does the classic sonar equation. The simulation results also indicate
th a t detection performance degradation due to the interference is magnified by the
presence of environmental uncertainty.
4.2
Interference M odel
The interference model presented here is different from conventional plane-wave
model for detection or parameter estimation problems[61, 16], or the Gaussian dis­
tribution model commonly used in beamforming[64]. This model incorporates the
acoustic interferers and utilizes a new concept; the interference matrix.
The interference are assumed to be spatial vectors in the frequency domain, nar­
rowband, centered at a known frequency /q. The A:th interference consists of three
components,
Sfe)cr^, hk, and 4 ( ^ ,8 * ) . The term ^ J I N R k { ^ ,
rep­
resents the strength of the ^th interferer, where the interference-to-noise ratio (INR)
is defined at the receivers.
INRk = 4 h ( ^ ,
(7
where
S ,),
(4.1)
is the variance of the A;th interferer. A complex Gaussain random variable,
hk ~ N (0 ,1), represents the normalized amplitude and phase of the interferer. The
interference wavefront, t k { ^ ,S k )i is the normalized ocean transfer function sampled
at the receiving array of N sensors, given the ocean environmental parameters
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and
the interferer position parameters S^.
Sk) = H (^, S,)/||H(^, S,)||.
(4.2)
Note th at i k h = 1- The kth. interference matrix is defined similarly to the signal
matrix.
=
[ f k l 5
f k 2
/A
5 ■ • • f k n ]
-[fk((^,sOi),fk((^,Sk)2),...fk((^,Sk)J,
o \
^
^
where L is the number of Monte Carlo samples.
The detection problem is formulated as a doubly composite binary hypotheses
testing problem. For an arbitrary array of N sensors, the measurement data are
an N X 1 spatial vector in the frequency domain, obtained using a narrow-band
Fourier transform of the snapshots. The ith data element is transformed from the
snapshot received by the sensor at the zth location. Given the measurement data,
we must decide between two hypotheses; the “null” hypothesis H q and the “signal
present” hypothesis Hi, in the presence of interference and environmental uncertainty.
The null hypothesis assumes th at the data consist of diffuse noise and uncertain
interference. The Hi hypothesis assumes th at the data consist of an uncertain signal
at the receivers, in addition to diffuse noise and the interference.
Hi
: r
= y ^ S N R { ^ , S ) a l a s { ^ , S) + ui,
Ho
: r
= n i,
ni = E f= i
Sk)albkfk{^, Sk) + a„no,
no ~ N (0 ,Ijv ),
^ - p ( ^ ) ,S ~ p ( S ) ,S ,- p ( S ,) ,
(4.4)
where the normalized acoustic source, a, is assumed a complex Gaussian with zero
mean and unit variance. The signal-to-noise ratio, S N R { ^ , S), and the signal wavefront, s ( ^ ,S ) , have been defined in chapter 3. Diffuse noise cr„no is modeled by
a spatial complex Gaussian vector in the frequency domain, with zero mean and
identity covariance m atrix of size N.
55
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Since multiplying by a scalar l/cr„ on both sides of Eq. (4.4) does not affect the
detection performance prediction, the hypothesis testing problem can be simplified
to be
Hi
: r
= . y S N R ( ^ , S ) a s ( ^ , S ) + i i i,
Ho
: r
= n i,
—J 2 k = l
^ k)bk^k{'^-, ^ k ) +
no
~ N ( 0 , Ijv),
^
- p ( ^ ) ,S ~ p ( S ) ,S f e ~ p ( S ,) .
(4-5)
Equation (4.5) is the formulation of the detection problem in a continuous form.
By applying the concept of the signal m atrix and the interference matrices, the
detection problem can also be formulated in a discrete form.
Hi
: r
= ^/SNRiaSi + n i , i G 1 . . . M,
Ho
: r
= ni,
no
_______
= Y^k=i V^HRkibkfki + no, / e . . . L,
Ijv),
where S N R i = S N R { ( ^ , S ) i ) , I N R k i — I H R k ( ( ^ , S k ) i ) , Sj is an arbitrary column
of the signal m atrix with probability 1/M , and fki is an arbitrary column of the A:th
interference m atrix with probability 1/L.
In the detection problem, diffuse noise no, interference source bk, and signal source
a are assumed statistically independent. The S N R ' s and the I N R ' s are assumed
known a priori.
4.3
B ayesian D etecto r
From signal detection theory, the optimal detector is the Bayesian detector th at fully
incorporates the a priori knowledge of the uncertain parameters. The derivation of
the Bayesian detector begins with the likelihood ratio [65]
A(r) =
P ( r \ Ho )
56
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(4.7)
For the detection problem defined in Eq. (4.5), the above likelihood ratio is a
doubly composite hypotheses test, requiring integration over the uncertainties under
each hypothesis, yielding
U!sffsd'^dSfdSp{r\Hu<l>,§f,S)pi<Sf)p{Ef)p(S)
The data vector under both hypotheses, given the model parameters, is assumed to
be a complex Gaussian distribution.
(4.9)
=
p (l* I
|^M o('5',S /)|
’
where
M i(«,S/,S)
M„(*, S,)
= S A f f l ( ® ,S ) s ( ® ,S ) s » ( * ,S )+ M „ { ® ,S ; ) ,
= E L INRt{9,St)U’^, S»)ff ( * . Si) + 1„.
(4.10)
Rather than using the likelihood ratio defined by Eqs. (4.8), (4.9), and (4.10), we
are going to derive a simplified test statistic by making the second order assumption
for the noise field. It is assumed th at the noise field, i.e., the sum of the interference
plus diffuse noise, is a multivariate complex Gaussian distribution. This assumption
ignores high order statistical characteristics of the noise field. It only uses the second
order statistic, i.e., the covariance matrix M qi to characterize the noise field.
Mo
= /* 4 < i* d S ,r t« ')p (S /)E f.i^ A 'fli(4 ',S » )ft(> l> ,S t)f« (> t.S t) + 1.
(4.11)
However, the covariance m atrix Mo incorporates the effect of environmental uncer­
tainty in the interference through integrations over uncertain ocean environmental
parameters and interferer position parameters. This approach is different from the
approach th at assumes a known covariance matrix, which requires too much a priori
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
information. It also differs from the approach th at uses the estimation of the covari­
ance m atrix from the collection of noise-only data, which relies too much on the data,
but does not take advantage of the available a priori information about the interferer
positions and the environmental parameters.
The concept of the interference matrices is applied to numerically compute the
covariance matrix, yielding
Mo = r V ^ - . V f - . I N R u S A ^ + l .
(4.12)
Applying the second order assumption of the noise field, the distributions of the
d ata conditional to both hypotheses become
(4.13)
where M , = SN R ( < S , S ) s ( 9 , S )s (* , S )" + Mo.
The likelihood ratio can be approximated by
Mr)
=f^fsd^dSMvl^,S)p(i')p(S),
(4.14)
where the conditional likelihood ratio is given by
=
(4.15)
Using Woodbury’s identity[50],
M -i = M - i 1
0
m ;;^s n r (%s )s(^, s )s(%s )«m ;;^
.
l+ 5 iV i? (’*',S)s('5i,S)^M o U ('5',S)^ ■
Using properties of the m atrix determinant,
I M l 1=1 M o I (1
+ S N R ( ^ , S )s (^ , S ) ^ M o ^ s(^ , S)).
58
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(4.17)
Substituting Eqs. (4.13), (4.16), and (4.17) in Eq. (4.15) results in
A(r|^,s)
— l-^.sNR('^,S)G{'^,S)^
G ( ^ ,S )
= s ^ ( ^ , S ) M o 's ( ^ „ S ) ,
i+SNR{<s>,s)a(^s,s)
^
(4.18)
where the term G (^ , S) is referred to as the signal-to-interference coefRcient th at
represents the effect of the interference. The product of the S N R and G can be
referred to as the ”signal-to-interference-plus-noise ratio ( S I N E ) " , which is compa­
rable to the S I N E defined in the literature [50, 16]. The differences between our
definition of the S I N E and others are th at the product of the S N E and G in Eq.
(4.18) is derived from Bayesian detection view points. The S N E represents the effect
of diffuse noise and G reflects the effect of the presence of interference in uncertain
environments. Other definitions of the S I N E are defined from beamforming perspec­
tives, concentrating on the total effects of the noise field, usually used in a known
environment.
Finally, the Bayesian detector th at incorporates the a priori information of envi­
ronmental uncertainties and the interferers is given by Eqs. (4.14), (4.18), and (4.11).
To numerically compute the likelihood ratio, the concept of the signal matrix is
applied to generate a discrete version of the approximate likelihood ratio.
A(r)
Ai
G,
SNEi
= i E £ i a.,
USNRi*Gi^
^
= sfM o^ s,,
= SN E{{^,S)i).
59
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(4-19)
4.4
Perform ance P red iction in K now n E nvironm ents
In a known environment, it is assumed th at the ocean and the source position param ­
eters are known a priori. Consequently, the signal and interference wavefronts are
assumed known. In this case, the ROC for the proposed Bayesian detector provides
a performance benchmark for an ideal situation.
A known environment assumption is equivalent to assuming th at the a priori
distributions p ( ^ ) , p(S), and p{Sk) are delta functions.Substituting these conditions
in Eq. (4.18) results in the likelihood ratio specific to the known environment case.
.
G
where Mq = E f= i
W
(4.20)
= s ^ M o ^s,
+ I^v, S N R , G, s, I N R k , and
denote S N R { ^ , S ) ,
G (^ ,S ), s (^ ,S ), INRif{'^,Sh), and 4 (^ ,S a ;) respectively for simplicity.
Since S N R and G are constants under the assumption of a known environment,
A', a monotonic function of the likelihood ratio, is also optimal.
A'(r) =1 r^M o
f .
(4.21)
Since A' can be considered a chi-square random variable with two degrees of freedom
for both the H\ and H q hypotheses, an accurate analytical ROC expression for the
optimal Bayesian detector is available using similar derivations as described in [44].
First, the probability density functions of X' to both hypotheses is given by
p{X'\Ho)
=Jexp(^),
P(X
—
\
Hi)
S N R * G ^
+
G
(4.22)
( s N
R
* G
^ +
a y
From signal detection theory
P d (P) =
dX'p(X' I Ho) = exp ( - | ) ,
dX'piX' I Hi) = exp
60
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Combining P d and Pp in Eq. (4.23) yields the ROC expression for the Bayesian
detector in the presence of interference in known environments.
1
Po = pSNR,a+i ^Q ^ s^M o ^s.
When a single interferer is present, substituting M q^ = I -
(4.24)
in Eqs.
(4.20) and (4.24) yields the detector statistic
A'(r) =1 r«{I -
|^
(4.25)
I p P),
(4.26)
and the performance expression
=
where
P f ^ , G
= (1 - ^
f denotes the known interference wavefront, and p = s-^f is the correlation
coefficient between the signal wavefront and the interference wavefront.
With the simple analytical ROC expressions, the effect of interference on detection
performance is captured by the role of the key parameter G = s^M q^s.
Since
the covariance m atrix Mq is positive definite, G is a positive number. Since the
eigenvalues of Mq are equal or greater than one, the eigenvalues of Mg ^ are in the
range of zero to one. A singular value decomposition of Mg ^ generates Mg ^ =
UATJ-^, where U is a unitary matrix and A is a diagonal m atrix composed of the
eigenvalues of Mg^. Since the eigenvalues are in the range of zero and one, G =
s^UATJ-^s < s ^ U U ^ s = s^ s = 1. Therefore, G is in the range of zero and one.
Considering G is a weighting factor to the S N R in the ROC expression in Eq. (4.24),
this range of G means th at the presence of interference always decreases the S N R
and consequently decreases the detection performance. The stronger the interference,
the closer the G approaches zero, and the greater the performance degrades. On the
other hand, if there is no interference, G is one, which does not impact the detection
performance.
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(b)
(a)
1
1
0 .9
0 .9
0.8
0.8
CO
S 0.7
§ O .r
cc 0 . 6
cc 0 .6
CO
CO
<S>
. cjO.3
0.1
O
' \
' \
"uO.4
CO.3
0 .4
0.2
s
.^-O .5
---r° 5
O-
^ Vs
..... ........... ..
,
|p |= 0
|p |= 0 .4
|p |= 0 .6
|p|=O.B
-------- I N R = - 1 0 d B
^
IN R=O dB
-------------IN R = 1 0 d B
------IN R = 2 0 d B
IPl- 1
0 .5
p .c o r r e ia tio n c o e f f e c ie n t
-1 0
O
10
20
30
In te rfe re n c e —
t o —n o i s e ra tio (dB )
F ig u re 4.1: Detection performance prediction in the presence of a single interferer in
a known ocean (Eq. 4.25) (a) P d as a function of I N R at fixed Pp = 0.1 for various
p’s, (h)PD as a function of p at fixed Pp = 0.1 for various I N K ’S.
In the presence of a single interferer, G is determined by the I N R and | p |, as
shown in Eq. (4.26). The I N R represents the energy of the interference relative to
the energy of diffuse noise at the receivers. The | p | characterizes spatial similarity
between the interference wavefront and the signal wavefront. This case is illustrated
in Fig. 4.1. In Fig. 4.1(a), P d is plotted as a function of I N R for a set of | p |’s.
It shows th at the stronger the interference, the worse the detection performance.
However, performance degradation is modest if the correlation coefficient between
the interference wavefront and the signal wavefront is small. In Fig. 4.1(b), P d is
plotted as a function of | p | for a set of I N R ’s. It shows th at when | p | is less
th a t 0.5, a single interferer has little impact on the detection performance, even if its
energy is very strong.
The effect of the interference on detection performance in known environments has
been studied previously using different performance metrics[61, 16]. The performance
metric used in [61] is a distance measure, and is array gain used in [16]. Here we use
the ROC metric, which is directly relevant to the ultim ate goal of target detection.
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 6.11 in [16] shows th at the stronger the interference, the better the optimum
array gain performance, which is different from our ROC performance results shown
in Fig. 4.1. Optimum array gain is an empirical metric defined as the array output
to input signal-to-noise ratio. When the noise field is diffuse noise only, the array
gain metric is consistent with the ROC metric. In the presence of interference, the
array gain metric cannot completely capture the effect of interference on detection
performance. Utilization of array gain as the detection performance metric in the
presence of interference can result in misleading conclusions.
4.5
Perform ance P red iction in U ncertain E nvironm ents
Detection performance predictions become more difficult in the presence of both in­
terference and environmental uncertainty. Both factors can greatly limit the detection
performance. Q uantitative descriptions about the role of each factor on the detection
performance are of practical interest. In this section, the analytical ROC expression
is derived for the Bayesian detector in the presence of interference in an uncertain
environment, which is a significant extension to the work included in chapter 3. De­
tection performance predictions can be performed much faster with this analytical
ROC expression than using Monte Carlo methods[10j.
In chapter 3, where the noise field is assumed to be diffuse noise only, the resultant
analytical ROC expression for the Bayesian detector in uncertain environments is
given by Eq. (3.32), where R is the rank of the signal matrix, characterizing the scale
of environmental uncertainty. The S N R is the signal-to-noise ratio at the receivers,
assum ed in th e derivation to be constan t over environm ental uncertainty.
Here we transform the problem to the detection problem defined in chapter 3
and exploit the results obtained in chapter 3. First, a pre-whitening procedure is
performed on the noise field. A singular value decomposition of the covariance matrix
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
of the noise field generates Mq = U / A / U f , where U ; is a unitary m atrix and A/
is a diagonal m atrix composed of the eigenvalues of
M
q.
If Q =
then Q
is full rank. Since multiplying a full rank matrix on both side of Eq. (4.6) does not
change the detection performance, the detection problem becomes
i?i
:r'
Ho
:
= ^ /S N R iaQ si + n ,i E 1 . . . M,
r'
=n,
(4.27)
~N(0,Iiv),
n
where the noisefield n is diffuse noise only, and the data vector r' = Q r .
Then the portion of the signal, y/SN RiaQsi, is rewritten to fit into the framework
of the detection problem defined in chapter 3. The new signal wavefront vector is
defined as
s'
= Q s j / V ' ( Q s j ) ^ Q s^,
= Q s ,/i/s f M o 's ,-,
(4.28)
Q sj/
The new signal wavefront vector preserves the property
= 1.
Substituting Eq. (4.28) in Eq. (4.27) results in
Hj
Ho
: r' = ^ S N R i *
: r' = n,
n ~ N(0, I at).
+ n ,i E 1 ... M,
(4.29)
Now the detection problem th at considers the presence of interference is translated
to the detection problem th a t assumes diffuse noise only,
SNRi
with the product of the
and Gi replacing S N R i , and with the new signalm atrix
3?'= [s(, S2 , . ..
replacing the original signal matrix 3?. The S N R i is the signal-to-noise ratio at the
receivers, which is assumed constant in the derivations in chapter 3. The parameter
Gi is the signal-to-interference coefficient. An additional assumption used here is th at
Gi is constant over environmental uncertainty, i.e., Gi = G for i = 1 . . . M . Since
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
= Q 3^/a/G and the transformation matrix, Q/ ^/G , is full rank, the rank of the
new signal m atrix is equal to the rank of the original signal matrix, which is R.
Using similar derivation procedures as used in chapter 3, an analytical ROC ex­
pression for the Bayesian detector in the presence of interference in uncertain environ­
ments is obtained. This ROC expression differs from Eq. (3.32) in th at the product
of the S N R and G replaces the S N R .
P d = 1 - ( 1 - Pf ) ^ { 1 - (1 - (1 - P F ) ^ ) snr\ g+i ),
(4.30)
In real applications, the S N R and G could be replaced by the average S N R and the
average G over realizations of the uncertain environmental parameters, i.e.,
SNR
=
r
-
jsT .t,SN F U ,
1
n
These computations are used in Section 4.6.
Equation (4.30) is the key result in this chapter. This simple analytical ROC
expression captures the effects of the presence of interference and environmental
uncertainty on detection performance. The effect of the interference on detection
performance is represented by the param eter G. In Section 4.4, we have shown that
G is in the range of zero and one for a known environment. It can also be shown
th a t G is in the range of zero and one for an uncertain environment. This means
th a t the presence of the interferers always decreases detection performance, in either
a known or an uncertain environment. However, in an uncertain environment, the
computation of G is affected by the presence of environmental uncertainty, through
the covariance m atrix M q- Simulation results In Section 4.6 showed th at the value of
G decreases with Increased scale of environmental uncertainty. This means th at the
detection performance degradation due to the presence of Interferers Is magnified by
the presence of environmental uncertainty.
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(a)
1
0 .9
0 .9
0.8
CO
_ 0 .7
^ o.e
CO
.^"0.5
nr- 0 . 6
^
"
0 .5
“^0.4
(§
>
^czO.3
o..
^^
\
V
0.2
—
R=1
\
^.
0 .4
\
\
'">s \ \
G=1
G = 0 .6 7
0.1 ------------ R = 1 0
OO
0=0.42
------R=1 OO
0 .5
1
1 - G , G: s ig n a l—
t o —i n te r f e r e n c e c o e ffic ie n t
10
10
R, r a n k of t h e s ig n a l m atrix
1O
F ig u re 4.2: Detection performance prediction in the presence of interference in an
uncertain ocean (Eq. (4.30)) (a) P d as a function of G at fixed Pp = 0-1, S N R —10
dB for various R's. (b) P d as a function of R at fixed Pp = 0.1, S N R ^ I O dB for
various G ’s.
Figure 4.2 illustrates Eq. (4.30) by plotting P d as a function of the key parameters:
the signal-to-interference coefficient G and the uncertainty scale R. Figure 4.2(a)
plots P d as a function of 1 —G, assuming Pp=0.1, S N R —10 dB, for i? = l, 10, and
100. It shows th at for i? = 1, i.e., a known environment, P d starts at 0.81 for
G = 1, then quickly drops to 0.1 for G = 0. While for R = 100, i.e., a greatly
uncertain environment, P d drops from 0.59 to 0.1 for the same range of G. The plots
indicate th a t when G is close to zero, detection performance is primarily determined
by G rather th a t R; when G is close to one, detection performance is independently
determined by G and R. Figure 4.2(b) plots P d as a function of i? on a logarithmic
scale, assuming Pi^=0.1 and for G =1, 0.67, and 0.42. It shows th at the performance
degradation is inversely proportional to log{R) for all three G conditions.
Equation (4.30) is a general result th at is consistent with the results for many
special cases. For example, substituting i? = 1 in Eq. (4.30) yields the ROC per­
formance prediction expression for known environments and substituting G = 1 in
Eq. (4.30) generates the results for diffuse noise only. Generality is an advantage of
66
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T ab le 4.1: Taxonomy of results for sonar detection performance prediction
Problem
Known ocean
Detector statistic
ROC expression
multiple interferers
-Fdiffuse noise
A'(r) = r^M g
Eq. (4.21)
Eq. (4.24)
single interferer
-|-diffuse noise
A (r) =1 r ^ ( I
Eq. (4.25)
diffuse noise only
A'(r) =1 r ^ s p
Eq. (3.13)
Uncertain ocean
multiple interferers
p
Eq. (3.37)
A.,
-l-diffuse noise
~ l+SNRi*Gi
Eq. (4.18)
diffuse noise only
A(r) = ^ E £ i A „
A- =
1+SNRi
P
p5iViS(l
— Up
Eq. (4.26)
1
A(rt) = ^
1 .... -
p
p
l+SNRi*Gi
1+SNR,
Eq. (3.19)
Pp = l - ( l - P p ) ^ X
- (1 - (1 - P F ) ^ ) snr*g+i )
Eq. (4.30)
Pp = 1 - ( l - P p ) ^ X
(1
(1 — (1 — (1 — PP)fl)sJVfl+l)
Eq. (3.32)
this detection performance prediction approach in addition to simplicity and com­
putational feasibility. The developed detector statistics and the ROC performance
predictions along with the expressions for special cases are summarized in Table I.
4.6
S im u la tio n R e s u lts
A modified version of the benchmark propagation model (Fig. 2.1) is used to check
the analytical ROC expressions and to study the effect of the interference on detection
performance in uncertain environments. The modification includes a single interferer
fixed at 7 km range and 11 m depth, in addition to the target th at is fixed at
6
km
range and 50 m depth. Table 4.2 summarizes the ocean environmental parameters,
their mean values and their ranges of uncertainty. Three environmental configura-
67
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tion scenarios are used. The known ocean scenario assumes th a t the environmental
parameters are known a priori, which have the mean values given by Table 4.2. The
uncertain water depth scenario assumes th at the water depth parameter is uncertain
as defined in Table 4.2, and other parameters take the mean values. The general
uncertain scenario considers seven uncertain parameters, D, C q,
(7^, Ci, p, and a,
which are defined in Table 4.2.
Table 4.2: Parameters of uncertain shallow water propagation model
mean value ± range of uncertainty
Environmental parameter
D-bottom depth [m]
102.5±2.5
Co-surface sound speed [m/s]
1500±2.5
C^-bottom sound speed [m/s]
1480±2.5
C^-upper sediment sound speed [m/s]
1600T50
Crlower sediment sound speed [m/s]
1750±100
p-sediment density [g/cm^]
1.75±0.25
a-sediment attenuation [dB/A]
0.35±0.25
t-sediment thickness [m]
100
The acoustic transfer function is computed using normal mode theory [2 ], defined
by Eq. (2.1).
The developed analytical ROC expression is verified by comparing its prediction
results with those computed using Monte Carlo evaluations. Figure 4.3 illustrates
the ROC performance for the known ocean scenario, computed using Eq. (4.26),
which can be used as a benchmark performance. In Fig. 4.3, P d at fixed Pp = 0.1
is plotted as functions of the range and depth of the interferer, for about 10 dB
S N R at the receivers and for the target fixed at 5950 m range and 55 m depth.
In Figs. 4.3(a) and 4.3(b), the interferer levels are the same as, or 30 dB more
than, the target source level. The plots show th at in the known benchmark ocean,
the presence of a single interferer does not impact the detection performance, even
though the interferer level is very high. This is determined by the spatial correlation
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10.83
0 .8 3
O.T’9
0 .7 9
0 .7 4
0 .7 4
0.68
0 .6 8
0 .5 8
0 .5 8
0.31
0.31
4000
6000
8000
in te rfe re r r a n g e (m)
4000
6000
8000
in te rfe re r r a n g e (m)
F ig u re 4.3; Detection performance in the presence of a single interferer in the known
benchmark ocean. The P d image is plotted at fixed Pp = 0.1 for S N R — lOdB at
receivers. The coordinates of the image are the range and depth of the interferer.
Target source is located at 5950m in range and 55m in depth, (a)
(b)
^2 = I 0 0 0 a 2
property of the benchmark ocean. In the known benchmark ocean, the absolute value
of the correlation coefficient between the wavefronts th at come from two arbitrarily
separated sources is very low (| p |<
0 .2 ),
if the source separation is greater than
about 30 m in range and 4 m in depth.
Figures 4.4 illustrates P d as a function of the S N R and various interferer levels
for a fixed Pp = 0.1 in uncertain environments. Figure 4.4(a) is for the uncertain
water depth scenario and Fig. 4.4(b) is for the general uncertain scenario. In each
plot, the solid curve is the benchmark performance, computed using Eq. (4.26) for the
known ocean scenario. The dashed curves are analytical prediction results obtained
using Eq. (4.30) and the circles are Monte Carlo evaluation results for the Bayesian
detector defined by Eq. (4.19). Each of the dashed curves corresponds to an interferer
level: cr^/cr^ =
0
means the interferer is absent; ol/a'l =
is the same as the source level; and o l / o l =
10
1
means the interferer level
means the interferer level is
10
times
stronger than the target source level. Note th a t these three interferer levels do not
69
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(b)
(a)
0 .9
0 .9
//Q
O.B
0.8
0 .7
0 .7
0.6
a- 0 .4
a- 0 .4
0 .3
0 .3
0.2
0.2
10
20
10
30
S N R (dB )
20
30
S N R (dB )
F ig u re 4.4: Detection performance prediction P d as functions of S N R in the pres­
ence an interferer in uncertain environments(Eq. (4.26)), comparing with Monte Carlo
performance evaluation of the Bayesian Detector (Eq. (4.19)) (a) Uncertain ocean
depth scenario | A D |=2.5 m. (b) General uncertain scenario.
affect the performance prediction for the known ocean scenario, since the correlation
coefficient between the interference wavefront and the signal wavefront is low. The
plots show th at the P d results generated using the analytical Bayesian performance
prediction approach agree well with those obtained using the Monte Carlo evaluation
approach for both scenarios, and for various S N R ’s and various interferer levels. The
agreement between the analytical and Monte Carlo approaches are further illustrated
in Fig. 4.5 by plotting ROC curves for both uncertain scenarios for S N R = 1 0 dB for
three interferer levels. The computations are the same as those performed for Fig. 4.4.
The results indicate th a t the effects of the presence of interference and environmental
uncertainty are captured by the signal-to-interference coefficient and the rank of the
signal m atrix in the proposed Bayesian performance prediction ROC expression.
In Figs. 4.4 and 4.5, to obtain th e analytical results, th e signal m atrix and th e
interference m atrix are constructed using
100
realizations of the signal wavefronts and
the interference wavefronts respectively for eaeh uncertain scenario. The estimated
rank of the signal m atrix is 5 for the uncertain water depth scenario and is 13 for the
70
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(a)
(b)
0.9
0 .9
0.8
0.8
0.7
0 .7
0.6
0.6
0.4
0.3
1b p
0 .4
i o p O ow n
=0
0.2
0.2
0.1
0.1
O ow n
=o
0 .3
=io
o.s
Pp
F ig u re 4.5: Detection performance prediction ROC’s in the presence of an interferer
with various I N R values in uncertain environments(Eq. (4.26)), comparing with
Monte Carlo performance evaluation ROC’s of the Bayesian Detector (Eq. (4.19))
(a) Uncertain ocean depth scenario | A D |=2.5 m. (b) General uncertain scenario.
general uncertain scenario. The signal-to-interference coefficients are computed for
each of the S N R and interferer level conditions using Eqs. (4.19), (4.12), and (4.31),
which are summarized in Table 4.3. The numerical Monte Carlo results are generated
using 5000 trials for each hypothesis and for each of the S N R ' ’s and interferer levels.
In each trial, 500 realizations of the signal wavefronts and interference wavefronts
are used in the Bayesian detector to incorporate environmental uncertainty. The
computation of the analytical performance prediction results is considerably faster
than th at of the Monte Carlo performance evaluation results.
The effect of interference on detection performance in uncertain environments
is captured by the signal-to-interference coefficient. Table 4.3 lists the signal-tointerference coefficient values for three scenarios and various SNR^s and interferer
levels. It show s th a t G decreases significantly w ith increased environm ental uncer­
tainty, although the interferer position and the interferer level are the same. For
example, for the entries of S N R = 1 0 dB and o \ j o \ = 10, G decreases from 0.9662
for the known environment scenario to 0.6927 for the uncertain water depth scenario
71
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and then to 0.5977 for the general uncertain scenario. The degrees of environmental
uncertainty, i.e., the rank of the signal matrix, are 1, 5, and 13 respectively for these
three scenarios. The result shows th at G decreases with increased environmental
uncertainty, which means th at the degradation effect of the interference on detection
performance is strengthened by environmental uncertainty. This is because the rank
of the interference matrix also increases with environmental uncertainty, resulting in
an increase in the number of effective eigenvalues th at inversely impact the computa­
tion of G. The higher the interferer level, the higher the eigenvalues for the covariance
matrix, and the greater the inverse impact on the G values.
T a b le 4.3: Estimated signal-to-interference coefficient
SNR
Interferer level
30dB
Scenario
OdB
lOdB
20dB
Known
0.9859 0.9700 0.9662 0.9658
Environment
0.9700 0.9662 0.9658 0.9657
Uncertain
0.9667 0.8470 0.6927 0.5969
Water Depth
0.8470 0.6927 0.5969 0.5398
oH ol=\^
General
0.9683 0.8316 0.5977 0.4017
oHol=\
Uncertain
0.8316 0.5977 0.4017 0.2428
Figure 4.4 illustrates the combined effects of the interference and environmental
uncertainty on detection performance. In each plot, the benchmark
performance
predictions, denoted by solid lines, are computed for the known environment scenario,
for three interferer levels:
= 0,1,10. Since the absolute value of the correlation
coefficient between known signal and interference wavefronts is very small, three
benchmark P d curves are nearly indistinguishable.
The degradation of detection
performance due to interference and environmental uncertainty can be illustrated
by an increased S N R threshold in order to achieve a fixed P d - For example, to
achieve a fixed P d —^ S , Fig. 4.4(a) shows th at the S N R threshold is 10 dB for the
72
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known environment scenario and are about 12, 13, and 14 dB for the uncertain water
depth scenario for three increased interferer levels. The uncertainty of the uncertain
water scenario (-R=5) results in a 2-4 dB S N E increase; Fig. 4.4(b) shows th at the
uncertainty of the general uncertain scenario (E=13) results in a 2.5-7 dB S N E
increase. Thus, detection performance degradation due to interference is magnified
by the presence of environmental uncertainty. On the other hand, the results also
indicate th a t in the presence of interference, environmental uncertainty plays a more
im portant role on detection performance. To achieve a fixed Pn = 0.8, the S N E
threshold increases from 12 dB for the uncertain water depth scenario to 12.5 dB for
the general uncertain scenario when the interferer is absent (cr|/cr^=0). This agrees
with the result in [44] th at the ROC detection performance degrades gradually with
increased degree of environmental uncertainty in diffuse noise. Figure 4.4 further
shows th at the S N E threshold increases from 14 dB for the uncertain water depth
scenario to 16.5 dB for the general uncertain scenario when the interferer is present
(cTj/af=10). This means th at in the presence of interference, detection performance
degrades much faster with increased degree of environmental uncertainty.
4.7
Sum m ary
Analytical ROC performance prediction expressions, which are computationally fast,
are developed for the Bayesian detector in the presence of interference in uncertain
environments. This is a significant extension to previous detection performance pre­
diction approach proposed for diffuse noise only circumstance in chapter 3[44j. The
ROC expression is developed w ithin a B ayesian decision fram ework th a t incorporates
uncertainties in ocean environmental parameters and source (target and interferers)
position parameters.
The analytical ROC expression is verified using several uncertain environment
73
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scenarios extended from benchmark propagation models. The results demonstrate
th a t the ROC expression is a good approximation to the ROC obtained using Monte
Carlo methods, and can be computed much faster. The simple ROC expression cap­
tures fundamental parameters th at impact sonar detection performance: the signalto-noise ratio, the rank of the signal matrix, and the signal-to-interference coefRcient.
It provides a more realistic performance prediction than the classic sonar equation
th at fails to incorporate the effects of the interference and environmental uncertainty.
The signal-to-interference coefRcient in the ROC expression characterizes the ef­
fect of the interference on detection performance in uncertain environments. In com­
puting the signal-to-interference coefRcient, environmental uncertainty is incorpo­
rated through the covariance m atrix of the noise Reid, which is obtained based on the
a priori information of uncertain environmental parameters and uncertain interferer
position parameters. The values of the signal-to-interference coefRcient in different
simulation scenarios show th at environmental uncertainty magniRes degrading effect
of the interference on detection performance.
74
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Chapter 5
Sonar D etection Perform ance Prediction:
SW ellEx-96 Experim ent
5.1
Introduction
In chapters 3 and 4, we developed approximate analjdical ROC expressions th at
incorporate the effects of environmental uncertainty and interference on the perfor­
mance of the Bayesian detectors, assuming a complex Gaussian signal[44, 21], Here
we develop an approximate analytical ROC expression for a new Bayesian algorithm
for the detection of signal with known amplitude and unknown phase in an uncertain
environment.
It is im portant to test the proposed analytical ROC expression against real data,
in controlled operating conditions. The simulated data can be used to test the ap­
proximations particularly used by the proposed analytical approach, including the
constant SNR assumption, the decision rule approximation, the rank approximation,
and the regular integration domain approximation (chapter 3). However, only with
the real d ata can we truly verify the data model and the acoustic propagation model
used by both the analytical approach and the Monte Carlo method.
The ROC’s for the Bayesian detector are computed using both the analytical
approach and the Monte Carlo method and are verified using both the simulated data
and the d ata collected from the SWellEx-96 experiment. The agreement between
these ROC results showed th at the analytical ROC expression captures the effect
of source motion uncertainty by the rank of the signal m atrix and captures the
combined effects of source level, noise level, and propagation loss by the signal-tonoise ratio at the receivers.
The analytical ROC expression provides a fast and
75
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realistic performance prediction for the Bayesian detector.
5.2
D etectio n Problem
The signal model has been defined in chapter 2. In order to investigate the loss of
detection performance due to environmental uncertainty using the SWellEx-96 real
data set, the signal source is assumed to have known amplitude and unknown phase
which is uniformly distributed from
0
to 27t.
The detection problem is formulated as a binary hypotheses testing problem. For
an arbitrary array of N sensors, the observation is an
x 1 spatial vector in the
frequency domain, obtained using a narrow-band Fourier transform of the snapshot.
Given the observation, we must decide between two hypotheses: the “null” hypoth­
esis Ho and the “signal present” hypothesis Hi, in the presence of environmental
uncertainty.
Hi : r
=
S) + no,
Ho : r
no,
no ~ N ( 0 , 2 a 2 j^ ) ,
^ - p ( ^ ) , S ~ p (S ) ,
c; i
where the norm of the acoustic transfer function, ||H (^ ,S )||, and the source am­
plitude, I cio I) are merged and their product is denoted by yU, because the effects
of these two components on the detection performance are indistinguishable. The
signal wavefront s ( ^ , S) is defined in chapter 2. The noise field, no, is modeled by
an A" X 1 spatial complex Gaussian vector in the frequency domain, with zero mean
and covariance m atrix 2a^lN- The symbol Ijv means the size of the identity matrix
is N and the symbol ~ means “distributed as” .
Equation (5.1) is the formulation of the detection problem in a continuous form.
By applying the concept of the signal matrix, the detection problem can be formulated
76
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in a discrete form.
Hi
: r
= iie^^Si + no, i € 1 . . . M ,
H q \ V = no,
no ~ N(0,2(j^Iiv),
(5.2)
where r, Sj, and no denotes the data, the signal wavefront, and diffuse noise. The
probability of selecting s, among M column of the signal m atrix is 1/M .
5.3
B ay e sia n D e te c to r
Here we consider wavefront uncertainty caused by uncertain environmental parame­
ters. We derive the Bayesian detector th at incorporates the a priori information of
the environmental parameters, and develop an approximate analytical ROC expres­
sion for the performance of the Bayesian detector.
The derivation of the Bayesian detector begins with the likelihood ratio, which
requires integration over the uncertain environmental parameters for the H\ hypoth­
esis, and is given by
=
p(r|Ho)
’
Js d ^ d S \ { i I
(5.3)
S )p (^ )p (S ),
where A(r | ^^, 8 ) is the conditional likelihood ratio, which can be derived from the
probability density functions (pdf) of the data conditional to the H q hypothesis and
the Hi hypothesis given known environmental parameters ^ and S.
p ( r |M i , ^ , S )
=
^J^^d 9 p {r\e,H u ^,S ),
2n fo
1
r^r
SJVH(*,S)
.
(27ro-2)
/ o ( ^ | r « s ( * , S ) I),
77
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(5.5)
where I q{-) is the modified zero-order Bessel function of the first kind. The signalto-noise ratio at the receivers, S N R { ^ , S), is defined as
S N R { ^ , S) =1 a |2
S)\\yal
= | fi / a l
(5.6)
Combining Eqs. (5.4) and (5.5) yields the conditional likelihood ratio,
A(r I
S) =
(J
| r ^ s ( ^ , S) |).
(5.7)
The conditional likelihood ratio, which is a kernel in the likelihood ratio for incorpo­
rating signal wavefront uncertainty, is consistent with the optimal Bayes test for the
detection of signal with unknown phase but known signal wavefront in the literature
(Eq. (13) in [6 6 ] or Eq. (33.11) in [51]).
Finally, the Bayesian detector for the detection of signal with knownamplitude
and unknown phase in an uncertain environment in diffuse noiseisgiven by Eqs.
(5.3), (5.7), and (5.6).
To facilitate the computation of the likelihood ratio and the derivation of the
ROC expression, a discrete likelihood ratio is obtained by applying the concept of
the signal matrix.
5.4
A(r)
A,
=
— ep
SNRi
=
Tni
I rT^S'
n
s, \),
SN R {{^,S)i).
^
^
P e rfo rm a n c e P r e d ic tio n
The detection problem defined in Eq. (5.2) can be placed into a more general frame­
work, the detection of M-correlated signals in Gaussian noise. An accurate analytical
ROC expression for the Bayesian detector for M-correlated signal detection problem
is formidable. Here, we develop an approximate analytical ROC expression for the
78
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M-correlated signal detection problem, where the correlated signals consist of realiza­
tions of the signal wavefront due to uncertain environmental parameters. A similar
general derivation technique [44] was used in chapter 3 for the case where the source
amplitude was also unknown, so we present here the major steps.
The derivation of the ROC involves the following steps:
1.
Reduce the dimensionality of the signaland noise veetors from N to R, where
N is the number of the receiving array components and R is the rank of the
signal matrix. As shown in chapter 3, the reduction of the dimensionality can
be achieved without affecting the detection performance. A similar technique
for signal dimensionality reduction can be found in [52].
2.
Approximate the optimal decision rule to be “if Aj > /3 for any i e 1 . . . M ,
decide Hi and if A, < /5 for all i —
decide H q" . W ith this approx­
imation, the probability of false alarm (Pp) and the probability of detection
(Pd)
can be expressed as integrations of the available joint probability density
function of A, rather than the unknown probability density function of A. This
approximation was suggested in a different context in [51].
3. Assume the signal-to-noise ratio, SN R^ is constant over environmental uncer­
tainty. W ith this assumption. A* can be replaced by its monotonic function,
I r^Sj I, in the decision rule.
4. Approximate the integration domain for the computation of the probability of
detection.
Since the first three steps and the resultant Pp and Pp expressions are the same
as those in chapter 3, we directly utilize those results.
1-Pp{^')
l-P fl(/5 0
=
=
P{\r^Sif<P',i = l . . .M\ Ho),
Pi\r^Si\^< P ',i = l . . . M \ H i ) ,
79
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(5.9)
where 15' is the decision threshold. The data r is a random vector of size R x l .
Both P d and Pp can be expressed as a R-dimensional integral of the pdf of the
data conditional to the Hi or Ho hypotheses.
I-P
1-P
f W)
=
d {I5') =
.
s
The pdf’s of the d ata are given by Eqs. (5.4) and (5.5). Replacing s ( # ,S ) with
Si in Eq. (5.5) yielding
p(r I H u S i )
=
( 2 7 r a l ) - ^ e x p { - r ^ r / 2 a l - S N R / 2 ) I o { ^ \ r^S i |),
(5.11)
where the first column of the signal matrix, Si, is the reference signal wavefront.
Utilizing Si does not change the derivation for Pp and PpLet P = [si T] be an i? X R m atrix whose columns form a complete orthonormal
basis. The d ata vector r can be transformed to z through P to facilitate the deriva­
tions of Pp and Pd - Since P is orthonormal, substituting z = rP in Eqs. (5.4) and
(5.11) yields the pdf’s of z.
I Ho)
=
(5.12)
SNR
p ( z I Hi)
=
^
’ 7o(^ I
Zi
I),
where Zi is the first component of z and Zj_j=2...fl are the rest components of z.
Substituting Eq. (5.12) and r = z P “^ in Eq. (5.10) yields
1
1 -
Pp{(5') - ijzHp-Hg.|
2<^<_,-=i...n^dz^2^^2 ^^e
|z « P - ^ S j|2 < j5 ',i= l...M ^ ■ ^ ( 2 7 1 0 - 2 J "
P d (^ 0
=
i j z f f p - H s i|2 < ^ ',j= l.. .M ^^(27ra2)fl6
^
1
(5 13)
^
. f o ( ^ | Z l | ) -
Similar to chapter 3[44], we define a regular convex-body C = {| Zj p <
7
,i =
1 . .. R} to replace the irregular integral domain R = {| z-^P^-^Sj p < P',i = 1 . . . M}.
80
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By adjusting
7
, we can make C equivalent to D in the sense th at the integral of
p (z I Ho) on both convex-bodies are the same, i.e.,
r
1
I -----------e
r
I
1
e
z^z
.
(5.14)
Integrate the right side of Eq. (5.13) using domain C for both Pp and P d generates
1 --P r(T )
=
(1
-e
(5.15)
where Q{a,(3) is the Q function defined by Marcum and Swerling[67],
Q{a,P)= f
h
ve~'’ 2“ Io[av)dv.
(5.16)
Finally, combining Pp and Pp in Eq. (5.15) yields
P d = \ - { \ - Q { ^ M R , - ^ J 2 l o g { l - (1 - P f)i/'')))(1 -
(5.17)
where R is the rank of the signal m atrix and the S N R is assumed constant in the
derivation. In real applications, we estimate the rank of the signal m atrix by counting
the number of significant eigenvalues of the signal matrix th a t exceed a threshold.
Empirical results suggest th at using 5% of the maximum eigenvalue as the threshold
gives consistent agreement between the Monte Carlo simulation results and the an­
alytical ROC prediction results. Further, the optimal detection performance is not
highly sensitive to the exact setting of this threshold. The S N R can be replaced by
the average S N R over the realizations of the uncertain environmental parameters.
M
(5-18)
i=l
81
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5.5
A lternative D etectors
A special case of the Bayesian detector is the matched-ocean detector, when the
environmental parameters ^ and S are assumed known a priori. The matched-ocean
detector is implemented by the monotonic function of the likelihood ratio, which is
also optimal.
A '=1 r ^ s ( ^ ,S ) I .
(5.19)
The performance of the matched-ocean detector can be obtained by substituting
R = 1 in Eq. (5.17), since the signal matrix for the matched-ocean scenario only has
one column, the known signal wavefront s ( ^ ,S ) .
= Q {VSN R,-^2log{PF)).
(5.20)
The mathematical expressions for the matched-ocean detector (Eq. (5.19)) and its
performance (Eq. (5.20)) are similar to the expressions for the optimal Bayes detector
for the detection of signal with unknown phase but known wavefront in Gaussian noise
in [51,
6 6 ].
For comparison, the performances of an energy detector and a mean-ocean detec­
tor is computed using the Monte Carlo evaluation method discussed in Section 5.6.
The energy detector is a simple detector th at does not exploit the signal wavefront
uncertainty. The test statistic of the energy detector is given by
A' =1 r ^ r I .
(5.21)
The mean-ocean detector assumes a deterministic ocean, using only the mean
values of the uncertain environmental parameters. The test statistic of the meanocean detector is given by
A'
=1
r^Smean I,
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(5.22)
where the subscript “mean” indicates th at the a priori signal wavefront is computed
from the mean values of the environmental parameters.
5.6
Results: SW ellEx-96 E xperim ental D a ta
The data collected from event S5 of the SWellEx-96 experiment were used to demon­
strate the approximate analytical ROC expression derived in this chapter for the
Bayesian detector. The SWellEx-96 experiment was conducted in May 1996, west of
Point Loma. This site is a shallow water channel approximately 200 m in depth with
a relatively flat bottom and a downward refracting sound speed profile. The uncer­
tainties in the channel parameters are insignificant. However, the acoustic source was
moving, which caused variations of the signal wavefronts at the receiving array. In an
ideal matched scenario, we could catch up with source motion to accurately predict
the varying signal wavefront. Consequently, the Bayesian detector can be considered
as the matched-ocean detector th at provides a performance upper bound. However,
in a realistic circumstance, we often have limited amount of information about source
motion, which causes signal wavefront uncertainty. Since we know the source track
during the SWellEx-96 experiment from the GPS data, both ideal and realistic sce­
narios can be implemented for the verification of the analytical ROC expression and
the analysis of the effect of source motion on detection performance.
Before presenting results using the experimental data, the accuracy of the ap­
proximate analytical ROC expression is verified using simulated data with the same
SWellEx-96 environmental and acoustic propagation models. It is instructive to pro­
vide both the simulation results and the experimental results. The simulated results
check the approximations used for deriving the analytical ROC expressions. The
experimental d ata results test the signal and noise model.
83
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Om surface
Q) = 1 5 2 2 m/s
water
21
D=2 1 3 m
elements VLA
Crr = 1 4 8 8 m/s
Cn+ = 1 5 7 2 m/s
T=2 3 .5 m sediment
C , = 1 5 9 3 m/s
C u= 1 8 8 1 m/s
a =0 .2 dB/KmHz
P =1 .7 6 g/cm?
Figure 5.1: Shallow-water environmental configuration for the SWellEx-96 experi­
ment.
5.6.1
Environm ental and A cou stic P ropagation M odels
The shallow water environmental model is presented in Fig. 5.1. This model also
consists of a source at about 15 m deep, traveling the nearby area at 2.5 m /s speed,
and a 21-element vertical line array (VLA) suspended about 122m above the ocean
floor.
The acoustic transfer function is computed using normal mode theory [2], which
is given by Eq. (2.1) in chapter 2.
5.6.2
Sim ulation R esults
Using the simulated data, the approximate analytical ROC expression is verified by
comparing its prediction results with those obtained using the Monte Carlo method.
The resultant Pd is plotted as a function of the S N R , Pp, and source motion un­
certainty in Figs. 5.2-5.4. Four scenarios denoted by A, B, C, and D are considered,
which are listed in Table 5.1, corresponding to four pieces of event S5 source trajec­
tory with increasing uncertainty in ranges of source motion. For both analytical and
Monte Carlo approaches, the range and depth of source motion for each scenario is
84
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0.9
0.8
0.7
a.‘^.4
0.3
0.2
20
SNR a t receivers (dB)
F ig u re 5.2: Comparison of analytical and Monte Carlo performance predictions for
the Bayesian detector with simulated data, by plotting Pp as a function of S N R , for
Pp = 0.1. Solid: analytical approach. Star: Monte Carlo approach. Left: scenario
A. Right: scenario C (Table 5.1).
assumed known a priori and the signal m atrix consists of realizations of the signal
wavefronts due to source motion within th a t area. For the analytical approach, the
S N R was computed using Eqs. (5.6) and (5.18), assuming the signal amplitude to
noise variance ratio, | a | /a ^ , is known a priori. The rank R was estimated from the
signal m atrix using a threshold of 5% maximum eigenvalue. The Monte Carlo method
used 5000 Hi trials and 5000 H q trials to obtain the distribution of the likelihood
ratio and then calculated Pd and Pp.
T ab le 5.1: Scenarios of
scenarios source motion
A
3300 m
B
from 3300 m to
from 3300 m to
C
D
from 3300 m to
source motion uncertainty for simulated data
range of source motion source depth
0 m
15±2 m
15±2
m
3044 m 256 m
15±2 m
2750 m 550 m
15±2 m
2276 m 1034 m
85
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0.9
0.8
0.7
o0.6
0.4
0.3i
0 .2 ?
0.4
0.2
0.8
0.6
P,
F
F ig u re 5.3: Comparison of analytical and Monte Carlo performance predictions for
the Bayesian detector with simulated data, by plotting P d as a function of Pp, for
about 10 dB S N R at the receivers. Solid: analytical approach. Star: Monte Carlo
approach. Upper: scenario A. Lower: scenario C (Table 5.1).
0.9,
0.8
0.7
R=(
R=
R=)
0.6
0.4
0.3
0.2
200
400
600
800
1000
R a n g e of so u rc e motion (m)
F ig u re 5.4: Illustration of the effect of source motion on detection performance with
simulated data, by plotting P d as a function of increasing uncertainty in the range
of source motion, for Pp = 0.1 and for about 10 dB S N R at the receivers. X-axis
is the range of source motion defined in Table 5.1. Solid: analytical approach. Star:
Monte Carlo approach.
86
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Figure 5.2 illustrates Pq as a function of S N R for scenarios A and C. The solid
curves are the analytical ROC’s computed using Eq. (5.17) and the stars are the
ROC’s obtained using Monte Carlo evaluation approach[10] for the Bayesian detector
defined in Eq. (5.8). Figure 5.2 shows th at the analytical ROC predictions agree with
the Monte Carlo predictions for both the known source range scenario A and the
uncertain source range scenario C. The uncertainty in source range causes about 2-4
dB increase in S N R to achieve the same P d performance as if the source range is
known. Figure 5.3 compares the analytical and Monte Carlo approaches by plotting
P d as a function of Pp for about 10 dB S N R at the receivers for scenarios A and C.
Again, the analytical and the Monte Carlo approaches agree well for both scenarios.
The agreements between the analytical and Monte Carlo approaches demonstrate
th at rapid Bayesian detection performance prediction can be achieved by capturing
two fundamental parameters in the proposed analytical ROC expression: the rank
of the signal m atrix th a t represents the scale of environmental uncertainty, and the
signal-to-noise ratio at the receivers th at represents the combined effect of the source
level, the noise level, and the propagation loss.
Figure 5.4 illustrate the effect of source motion by plotting P d as a function of
increasing source range uncertainty for Pp = 0.1 and for about 10 dB S N R at the
receivers. The solid curve denotes the analytical results. The estimated rank of the
signal m atrix is illustrated in the plot. The stars denote Monte Carlo computations.
It shows th at the performance of the Bayesian detector gradually degrades with in­
creasing source range uncertainty. From scenarios C to D, where the uncertain ranges
of source range are 550 m and 1034 m respectively, the performance degradation of
the Bayesian detector is hardly noticeable. Figure 5.4 further illustrates th at the
source motion effect is well predicted by the analytical ROC expression, which can
be computed much faster than using the Monte Carlo approach.
87
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5.6.3
Experim ental R esults
The analytical ROC expression is further verified using the real data collected from
event S5 during the SWellEx-96 experiment. We first select signal and noise fre­
quencies for the Hi and H q hypotheses, then compare data characteristics with those
assumed in the detection model, and finally perform analytical and Monte Carlo per­
formance predictions with the data. In the 75 minutes data of event S5, a total of
13,500 time-domain snapshots were generated for each of 21 array components. Each
time-domain snapshot has 1000 points, occupying 2/3 seconds, with 50% overlap
between successive snapshots. An 8192 length fast Eourier transform (EET) is per­
formed on each snapshot to generate 13,500 frequency-domain spatial data vectors
for each frequency bin under consideration. A data frame consists of 900 successive
spatial vectors, occupying 5 minutes. A single ROC result is computed using a frame
of the Hi d ata and a frame of the corresponding
Hq
data. W ith a series of the data
frames, more ROC results can be obtained.
The 109 Hz frequency data is selected as the signal data because of its high signalto-noise ratio. One im portant characteristic of the 109 Hz data is the match between
the data and the wavefronts predicted using the acoustic transfer function Eq. (2.1)
and the a pn'on shallow water environmental parameters (Fig. 5.1). Figure 5.5 plots
the time-evolving source trajectory in range and depth, which was estimated using a
B artlett processor with the 109 Hz data. The source trajectory between 35 minutes
and 75 minutes agrees with th at computed from the GPS data, indicating th at the
wavefront prediction is reliable, which matches with real signal wavefronts in the
data. D etails o f th e source trajectory estim ation using th e B a rtlett processor can be
found in [6 8 ]. A clean source trajectory th at was extrapolated from the estimated
trajectory is also plotted in Fig. 5.5. The clean trajectory is used next as the a priori
source positions.
88
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(a)
10
—
—
clean
estimated
O)
5
0.
0
10
20
30
40
50
60
70
80
(b)
—
clean
estimated
Time (min)
F ig u re 5.5: Estimated and extrapolated clean source trajectories in range and
depth for event S5 during the SWellEx-96 experiment, using a B artlett processor
with SWellEx-96 environmental parameters shown in Fig. 5.1. Dashed; estimated.
Solid; extrapolated clean source track. The origin of the x-axis is the beginning of
event S5.
89
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Another im portant characteristic of the 109 Hz data is th at the source ampli­
tude is a constant, as was assumed in the detection model. Figure 5.6 plots four
histograms of the 109 Hz source amplitude, corresponding to four data frames starts
at 30, 40, 50, and 60 minutes respectively. The sample of the source amplitude is
obtained as the root of the energy of the data vector divided by the energy of the
acoustic transfer function given the a priori ocean and source position parameters,
i.e., I a I = y ^ r ^ r /H ( ^ ,
S). The standard deviation estimated from source
amplitude samples is used to generate the Rayleigh probability density function. Fig­
ure 5.6 shows th a t the histograms of the source amplitude are much narrower than
the Rayleigh probability density functions, indicating th at the source amplitude can
be approximated as a constant. The match between data characteristics and the de­
tection model enables the application of the proposed Bayesian detector (Eq. (5.8))
and the analytical performance prediction approach (Eq. (5.17)) to the experimental
data.
Since the 109 Hz acoustic source was always on during event 85, we don’t have
the 109 Hz noise-only data. The 99 Hz frequency data is selected to represent the
noise for the H q hypothesis, because it does not overlap with any source frequency
and it is close to 109 Hz. A weighed sum of the 109 Hz and 99 Hz data is used as
the signal plus noise d ata for the Hi hypothesis.
rio9 + fli-rgg,
H\
:
Yi
=
H q
:
ro
= 6 irg 9 ,
/r 23')
^
where riog and rgg means 109 Hz and 99 Hz spatial data vectors. The coefficient,
flj, is assigned to adjust the noise level of the rth data frame so th at the detection
performance can be evaluated at a wide range of the SN R 's . The coefficient,
is
used to prevent the effect of noise level mismatch. It is calculated by equalizing the
total noise energy in a frame of ri and the total noise energy in the corresponding
90
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(b)
(a)
0.12
0.08
0.06
■« 0.08
■(U
a
^0.06
o 0.04
a.
0.02
0.02
0
20
40
60
signal amplitude
80
0
100
40
60
signal amplitude
80
100
(d)
(c)
0.06
20
0.08
0.05
^0.03
2 0.02
Q.
0.02
0.01
20
40
60
signal amplitude
100
20
40
60
signal amplitude
100
F ig u re 5.6: Histograms of the 109Hz source amplitude samples obtained from four
data frames of event S5, comparing with the Rayleigh probability density functions
(pdf) generated using the same standard deviation of the source amplitude samples.
Solid: histograms, dashed: Rayleigh pd f’s, (a) The data frame starts at 30 minutes,
(b) the d ata frame starts at 40 minutes, (c) the data frame starts at 50 minutes, and
(d) the d ata frame starts at 60 minutes.
91
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frame of
tq.
where r±^j denotes the portion of the j t h spatial vector in the ith data frame (either
r i or rgg) th at is orthogonal to the predicted signal wavefront, s ( ^ , 8 ,^). The source
position Sij corresponds to the j t h spatial vector in the zth data frame.
Although this is a synthetic way of obtaining the Hi and Ho data, it preserves
the wavefront uncertainty in the data and preserves source motion as the primary
cause of wavefront uncertainty, which is the focus of this work.
The analytical performance prediction approach is verified using a series of ri and
ro data frames along the entire event S5 trajectory. In constructing the synthesis data,
Gi is first assigned to be
1 .6
for all data frames and 6, is calculated for each data frame
using Eq. (5.24). Other Oj values can also be used to construct the data, which is
discussed later. The verification is performed by comparing the analytical prediction
results with Monte Carlo prediction results, by comparing the results across various
data frames, and by comparing the performance prediction for the Bayesian detector
with those for the energy detector and the mean-ocean detector. Two scenarios are
considered. In scenario I, the Bayesian detector has accurate source position for each
snapshot, obtained from the source trajectory estimation (Fig. 5.5). The performance
prediction for the Bayesian detector for this scenario provides a performance upper
bound. In scenario II, the Bayesian detector has the a priori range of source motion
for each Hi d ata frame. The analytical performance prediction approach captures
the performance degradation caused by the limited a priori knowledge about source
position, and provides a rapid and realistic P b prediction.
Figure 5.7 plots the P d prediction results for the Bayesian detectors as a function
of the center of the source range of the data frames, given Pp = 0.1. In Fig. 5.7(a),
92
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(a)
+ +
+ + + -H -.
(b)
XX
^
X
^%
Jf H^ x_^5x»(>
Source Range (km)
F ig u re 5.7: Illustration of the analytical performance prediction and Monte Carlo
performance prediction for the Bayesian detector with the data collected during event
S5 in the SWellEx-96 experiment. The P d result is plotted as a function of the center
of the source range across d ata frames, (a) Scenario I. Solid: Monte Carlo results
and plus: analytical PCL prediction results, (b) Scenario II. Dashed: Monte Carlo
results and times: analytical PCL prediction results.
93
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scenario I is considered. The solid curve represents Monte Carlo evaluation results for
the matched-ocean detector, obtained using Eq. (5.19) to generate the distribution
of the likelihood ratio for each data frame. The signal wavefront is predicted using
Eq. (2.1) and the known source position and ocean parameters for each spatial vector
in the d ata frame. The plus denotes the analytical P d results computed using Eq.
(5 .2 0 ) with param eter S N R i estimated from the ith data frame using the following
empirical formula,
SNRi =
'900 II
||2
fiH = E S l |r » , I P ,
E«, =
l|ro«IP,
(5.25)
where ri^- and ro^ are the j th spatial vector in the ith data frame of ri and ro
respectively. In Fig. 5.7(b), scenario II is used. The dashed curve represents Monte
Carlo Pd result for the Bayesian detector, obtained using Eq. (5.8). The signal matrix
used in Eq. (5.8) consists of 120 realizations of the signal wavefront due to source
motion. The S N R param eter is the same as th at used for the matched-ocean detector
for scenario I. The times denote analytical Pd results computed using Eq. (5.17). The
analytical approach uses the same S N R parameters and the signal m atrix th at are
used in Monte Carlo method. The parameter R is estimated from the signal matrix
using a threshold th a t equals 5% of the maximum eigenvalue. Since the variation of
R across data frames is small, an averaged R = 7 is used for all data frames.
Figure 5.7 shows th a t the analytical prediction Pd results agree with the Monte
Carlo evaluation results over most of the event 85 track, for both scenarios. The
consistency of the agreement is a good demonstration of the analytical prediction
approach.
Some minor disagreements might be due to inaccurate shallow water
environment parameters used by the detector. The performance degradation from
Pd
result for scenario I to th at for scenario II is caused by the limited amount
94
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0.9
0.8
0.7
0.6
0.4
0.3
0.2
s»
Source Range (km)
F ig u re 5.8; Illustration of Monte Carlo performance prediction for various detectors
with the d ata collected during event S5 in the SWellEx-96 experiment. The Pd result
is plotted as a function of the center of the source range across data frames. Solid,
dashed, and dash-dotted curves denote the P d results for the Bayesian detector, the
mean-ocean detector, and the energy detector for scenario II respectively.
of information about the source position, which is correctly captured by the rank
parameter in the analytical ROC expression.
The P d performance of the Bayesian detector is compared with th at of the meanocean detector and the energy detector in Fig. 5.8, denoted by the solid curve, the
dashed curve, and the dash-dotted curve respectively. The results for the Bayesian
detector are the same as those illustrated in Fig. 5.7(b). The results for the meanocean detector and the energy detector are computed using Eqs. (5.22) and (5.21)
for scenario II. The mean-ocean signal wavefront is predicted using the center source
position for each d ata frame and the known ocean parameters. The results presented
95
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(b)
(a)
0.8
0.4
0.2
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
)00 0
)00 o
20
)00 0
20
(d )
1
(c)
(f)
(e)
—..... ............
1
--------- —
0.8
0.8
0.6
0.6
0.4
0.4
0.4
0.2
0.2
LL
0.2
0
^ ^ o c o o o o '^
0
10
20
S N R at receivers (dB)
0
0
10
20
S N R at receivers (dB)
0
7V :
J+
/+/ ;
:...........
0
10
20
S N R at receivers (dB)
Figure 5.9: Illustration of the analytical performance prediction and Monte Carlo
performance prediction for the Bayesian detector with SWellEx-96, event S5 data.
The Pd result is plotted as a function of S N R for Pp = 0.1. Solid: analytical
results, star: Monte Carlo results. Left: scenario I, known signal wavefronts. Right:
scenario II, uncertain signal wavefronts. Circle: Monte Carlo performance results for
the energy detector. D ata frames start at (a) 30 min, (b) 35 min, (c) 40 min, (d) 45
min, (e) 65 min, and (f) 70 min.
in Fig. 5.8 show th a t the performance of the Bayesian detector is better than those
of the mean-ocean detector and the energy detector, because the Bayesian detector
incorporates the a priori source position information in an optimal way.
To test the consistency of predicted Pd results as a function of S N R across data
frames, the coefficient a, is varied from 0.4 to 16 to generate additional synthesis
data frames. Coefficient 5,- is changed accordingly using Eq. (5.24). Figure 5.9 plots
Pd
as a function of the S N R for the data frames start at 30, 35, 40, 45, 65, and 70
96
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minutes respectively, given Pp = 0.1. The solid curves are the analytical prediction
results obtained using Eq. (5.17) for scenario I and Eq. (5.20) for scenario II. The
stars are Monte Carlo evaluation results, with the x-axis to be the S N R estimated
from the data frame using Eq. (5.25), and the y-axis to be the P ^ ’s computed using
Eq. (5.19) and Eq. (5.8) for scenarios I and II respectively. The circles denote Monte
Carlo evaluation results for the energy detector for scenario II. Figure 5.9 shows th at
the analytical and Monte Carlo Pp versus S N R curves agree with each other. The
agreement is consistent across various data frames. It indicates th at the detection
model is feasible in characterizing the data in a real circumstance. It further indicates
th a t the performance of the Bayesian detector in the presence of source motion can
be rapidly predicted by the proposed analytical ROC expression. The difference
between analytical and Monte Carlo results might due to model parameter mismatch,
the S N R estimation error, and limited number of trials for Monte Carlo evaluation.
5.7
Sum m ary
If the environmental parameters are precisely measured and the acoustic signal wavefront is accurately modeled and predicted, environmental uncertainty can be avoided.
However, it is usually difficult to get an accurate description of the target position
and the ocean parameters. W hat can often be obtained are the ranges of possible
values of uncertain environmental parameters, such as “water depth are 100 ± 5 m” .
Our approach translates the a priori ranges of the uncertain environmental param­
eters to uncertainty in the signal wavefront, and then to analytical ROC expression
for the B ayesian d etection perform ance.
The analytical Bayesian ROC expression developed in this paper predicts the
detection performance degradation caused by environmental uncertainty. W ith the
analytical ROC expression, the effect of environmental uncertainty on detection per97
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formance can be quantitatively described and compared with the effect of other fac­
tors such as noise level and propagation loss. Further, the effects of various types of
environmental uncertainties can be compared quantitatively. The limiting factor to
the detection performance can be identified by those quantitative comparisons and
can be used to guide better sonar system design.
Although the simulation and experimental verifications in this chapter only con­
sider the uncertainty caused by source motion, the application of the analytical ROC
expression is not limited to these cases. It can be extended to various other environ­
mental uncertainty scenarios[44]. Furthermore, the developed ROC expression can
be a solution to a general M-correlated signal detection problem, where the correlated
signals are realizations of the propagation transfer function due to perturbations of
the environmental parameters.
In summary, we have developed an approach for translating environmental uncer­
tainty into detection performance prediction. In particular, the detection model and
the ROC expressions are verified by both simulated data and real experimental data
in which there is signal wavefront uncertainty caused by source motion. The sim­
ulation results demonstrate th at the analytical ROC expression yields accurate and
faster performance prediction as compared to conventional Monte Carlo performance
evaluation approach, as least for the test cases. The experimental results with the
real SWellEx-96 d ata show th at the signal source model and the acoustic propagation
model are practical. The analytical ROC expression for the Bayesian detector can be
applied in a realistic circumstance to provide fast detection performance prediction
in an uncertain environment.
98
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Chapter 6
Optim al Sonar Localization Perform ance
Prediction
6.1
Introd u ction
The performance of passive source localization is limited by inaccurate knowledge
of the parameters th at describe the acoustic environment, the repetitive structure of
the acoustic field, and the noise from the background. The sensitivity of conventional
matched-field processing to environmental uncertainty[3, 4, 5] motivates the develop­
ment of the algorithms th at are robust to environmental uncertainty[6 , 7, 8 , 4]. It is
instructive to characterize the performance of those robust algorithms in an uncertain
environment. Among the performance metrics used in the literature, the array gain
does not directly quantify optimal localization performance prediction. The MSE
metric[17, 18] may not be accurate when global false localizations are present due
to the repetitive structure of the acoustic field. The probability of correct localiza­
tion (PCL) is directly relevant to the ultim ate goal of source localization. It has
been used to evaluate the performance of the 0UFP[7] in a benchmark uncertain
environment[10]. However, the PCL is computed using Monte Carlo method in [10],
which is computationally intensive.
In this chapter, we develop relatively simple analytical approximations to the
PCL th at quantifies the performance of the matched-ocean processor in known en­
vironments and the performance of the OUFP in uncertain environments. The PCL
for the matched-ocean processor and the OUFP is formulated as a constrained mul­
tidimensional integral of the probability density function of the ambiguity vector (a
discrete extension to the ambiguity function defined in the matched-field processing
99
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framework), whose covariance matrix is determined by the S N R at the receivers
and the inner product m atrix of the signal wavefronts. Given several typical inner
product matrices, the analytical PCL expressions are derived and are used to predict
the performance of the optimal processors in benchmark environments[23, 10, 14].
The results show th a t the optimal PCL performance can be captured primarily by
fundamental parameters using the analytical PCL expressions; the signal-to-noise
ratio ( S N R ) at the receivers, the number of the hypothesized source positions, and
the effective correlation coefficient between signal wavefronts th at in part represents
the level of environmental uncertainty.
6.2
L ocalization P rocessors
The observation is assumed to be the sum of the received signal and diffuse noise,
and is given by
r
=
ao||H (^ r,S )||s(^ ,S)+ n o,
(6.1)
where no is the diffuse noise from the environment and the receivers. The spatial
covariance m atrix of the diffuse noise is assumed to be 2 a-^Ijv.
Multiplying both sides of Eq. (6.1) by a scalar 1 /\/2 ct„ does not affect the local­
ization performance prediction and yields
r
=
^S N R {^,S )a s{'^,S )+ n ,
(6.2)
where the scalar a and the vector n are normalized so th a t variance of a is one
and the covariance m atrix of n is an identity matrix of size N. The signal-to-noise
ratio at the receivers, for a given source position and a givenrealization of the ocean
environmental parameters, is defined as
S N R {^,S ) =
\ ^
’ C
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(6 . 3 )
Note th at the S N R includes array gain.
Applying the concept of the localization signal set generates the discrete observa­
tion model,
r = s jS N R ip aS ip -f n , z = 1
. . . M, p = 1 . . . L,
(6.4)
where Sjp is the realization of the signal wavefront due to the ith realization of the
ocean environmental parameters and the pth hypothesized source position.
The
SNRip is the corresponding signal-to-noise ratio at the receivers. The probability
of selecting s,p as the reference signal wavefront is
The 0UFP[7] computes the a posteriori probability density function of the source
position given the observation. Using the observation model defined by Eq. (6.2) and
using the S N R defined by Eq. (6.3), we derive the a posteriori probability density
function of the source position conditional to the observation data [7],
p(S|r)
=
|r"s(*.S)
( 6 .6 )
where C is a normalization constant, and ^ and S are assumed statistically indepen­
dent a priori, whose probability density functions are p ( ^ ) and p(S).
Utilizing the concept of the localization signal set yields the OUFP in discrete
form.
max
o fp'>P
Sp
S
f
Jp
_
Z^i=l
1 • • • L,
••• ’
1____ nvn ( r1^ r 1 I- r
(SNRip+l)7T^ t:x p i^
( 6 .6 )
II A
IFiJ ,
where fp is a monotonic function of the a posteriori function of the source position,
conditional to th e observation d a ta r.
If the ocean parameters are assumed known a priori, the optimal localization
processor is the matched-ocean localization processor. The localization signal set
consists of the signal wavefronts due to known ocean environmental parameters and
101
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total of L hypothesized source positions. We refer to this signal set as the matched
localization signal matrix, which is denoted by Smatch — [sii • • • Sil].
Substituting M — l i n Eq. (6 .6 ) generates the matched-ocean localization proces­
sor.
r
— a^ySNRlpSlp +
s
=
(6.7)
fm atched,p.P = l - - - L ,
Op
_ _
f
J matched,p
n,p = 1. . .L,
1
/
H
I
^ ^ i i T
~
I
SNRip
SNRip+1 I
g
|2N
^
I
b
where Sip is the signal wavefront due to known ocean environmental parameters
and the pth hypothesized source position. The subscript “matched” means th at the
reference ocean environmental parameters used in the processor is matched with the
true environmental parameters.
The mean-ocean localization processor is considered for comparison purposes. It
knows only mean values of the ocean environmental parameters, which are in general
mismatched with the true environmental parameters. The mean-ocean localization
processor is defined as
-I- n, i = 1 . . . M, p = 1 . . . L,
r
=
_
®
max r
.
T
— C J m e a n , p - i P — 1 . . . T,
f
= _____ 1_____ cxn
J m e a n ,p
S N R ip S ip
/X
(6 .8 )
Op
(S7ViJ„p+l)7r^
(
r^ r I
I ^ ^^
SNRmp+l
I
'
I b
where s^j, is the signal wavefront for the pth hypothesized source position given mean
values of the ocean environmental parameters and the
SN Rm p
is the corresponding
signal-to-noise ratio. Note th at the form of the mean-ocean processor is the same as
th at of the matched-ocean processor, except th at the mean-ocean signal wavefronts
are mismatched with the data.
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6.3
Form ulation o f th e PC L E xpression
The PCL is formulated as the probability th at the a posteriori probability of the
true source position conditional to the observation is greater than the a posteriori
probability of other hypothesized source positions conditional to the observation. We
show th at the PCL for the matched-ocean localization processor is a function of the
S N R at the receivers and the inner product matrix of the matched localization signal
matrix. The PCL for the OUFP can be approximately expressed as a function of
the S N R at the receivers and the inner product matrix of a special subset of the
localization signal set, referred to as the optimal localization signal matrix. Both
PCL expressions can be formulated as a constrained multidimensional integral of the
probability density function of the ambiguity vector th at is a discrete extension to
the ambiguity function defined in the matched-field processing framework.
6.3.1
PC L for th e M atched-ocean P rocessor
It is assumed th at the matched-ocean processor uses accurate ocean environmental
parameters to compute the reference signal wavefronts.
The matched-ocean pro­
cessor is not a realistic algorithm, because a complete match between the reference
signal wavefront and the d ata is practically impossible. We include it to provide a
localization performance upper bound for an ideal situation.
If the received signal is due to known ocean environmental parameters and the
pth hypothesized source position, the performance of the matched-ocean localization
processor is given by
PCLip{SNR,Smatch)
~ P^fmatchq ^ fmatchpi
/g gx
|r = U-y/S N R\pS\p •+■n).
By assuming th at the S N R is a constant over realizations of ocean environmental
103
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parameters and hypothesized source positions, fmatchp and fmatchq can be replaced by
their monotonic function | r^Sip p and | r^Si^
to generate
PC Lip[SNR^^jnatch)
tfi in'!
= P{\ r^Siq p<| r^Sip \'^,q = l . . . L , q ^ p \ r = aVSNRsip + n).
If Xq = r^Siq = (y/SNRaSip + n)^Si^, Xi,X 2, ■■■, x l constitute a random vector,
denoted by x, and referred to as the ambiguity vector. Since the ambiguity vector
is a linear combination of the complex Gaussian random variable a and the complex
Gaussian vector n, it is a complex Gaussian vector. Substituting x in Eq. (6.10)
yields
P C L \p{SN R , 'T^match) — I\xq\^<\xp\'^,q=l. . . L, q^p dxG{lC, 0, Cp),
(6.11)
where G(x, 0, Cp) denotes the complex Gaussian distribution of the ambiguity vector,
with zero mean and covariance m atrix Cp. The subscript p is the index for the true
source position.
The element of Cp is given by
C p(^, I) — SNRpq^pPp^i + pq,u q.,1 —1 . . . L,
(6.12)
where Pq^p = s^Sip is the correlation coefficient of the signal wavefronts Si^ and Sip
th at are due to the ^th and pth hypothesized source positions given known ocean
environmental parameters. The m atrix of Pq^p,q,p — 1 . . . L is the inner
matrix
product
of the matched localization signal matrix, denoted byOimatch-Note th at
^match — '-‘match^rnatchGiven the inner product m atrix a , the covariance m atrix of the ambiguity vector
can be expressed as
Cp = SNRapUp + a .
Here, ap is the pth column of a.
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(6.13)
Substituting Eq. (6.13) in Eq. (6.11) and replacing
a
with
amatch
yields
PCLip(^SNRf OLmatch^
/g
^ ^ ^ ^ ‘rr>-atch,pOi^atch,p + (^match)-
~
The final formulation of the PCL expression is the averaged PCL over the variety of
the true source positions.
PC L{ S NR , amatch) =
I
I\xq\^<\xp\^,q=l...L,q^p
.g
dxG{'X.^ 0, S N Ramat ch, p<^mat ch, p +
(^match)-
Equation (6.15) shows th at the performance of the matched-ocean localization proces­
sor is a multidimensional integral of the probability density function of the ambiguity
vector, whose covariance m atrix is determined by the S N R at the receivers and the
inner product m atrix of the matched localization signal matrix.
6.3.2
PC L for th e O U F P
The OUFP incorporates the a priori knowledge of the uncertain ocean environment.
It is difficult to formulate an exact PCL expression for the OUFP. Here, we use three
approximations to express the PCL expression as a function of the S N R and the
inner product m atrix of the optimal localization signal matrix, which is a special
subset of the localization signal set (Eq. (2.4)).
If the signal is due to the ith realization of the ocean parameters and the pth
hypothesized source position, the PCL for the OUFP is given by
PCLip{SNR, S) = P{fg < fp,q ^ 1. . . L, q
p \ r = ^/SNRaSip -h n),
(6.16)
where H is the localization signal set. The monotonic function of the a posteriori
probability of the source position conditional to the observation, fp, is defined by Eq.
( 6 . 6 ).
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To simplify the PCL expression, we approximate the sum of the exponentials in
the
by the maximum of the exponentials.
fp
^0
=
iy v
I
I').
(6-17)
This approximation has been used to simplify the optimal detection[51] and localization[69]
algorithms. Here, it is specially used for performance prediction.
The second approximation is to assume th at the S N R at the receivers is constant
over the realizations of the ocean environmental parameters and the hypothesized
source positions, so th a t
and
Vp,
fq
and
fp
can be replaced by their monotonic function,
Vq
to generate
P C L ip (S N R ,E ) = P{vq < Vp,q = 1.. . L , q ^ p \ r — V SNRaSip + n ) ,
(6.18)
where
Vq = maXj=i,„M I
P•
(6.19)
The first subscript of the signal wavefront Sjp is the index of the realizations of the
ocean environmental parameters. The one th at maximizes | v^Sjq p is referred to as
the “optimal ocean index” , denoted by
rriq.
Substituting
in Eq. (6.19) yields
Vq =1 r^Sm^q P •
(6.20)
In order to simplify the PCL expression, the optimal ocean index is approximately re­
placed by the index th a t maximizes the covariance of the
rather than maximizes
the absolute square of
ruq
= a r g m a X jC o v { r ^ S jq | r = ^ / S N R a S i p -f- n ),
= argmaXjCov{-\/SNRapip^jq -|- n^Sj^),
= a r g m a x j { S N R | Pip,jq P -1-1),
= argmaxj | pipjq p.
106
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,
where pip^iq = sf^Sjq. W ith this approximation, the optimal ocean index is a known
param eter th at is determined by the m atrix of p%p,jq and the indexes of the true signal
wavefront, ip. The optimal ocean index maximizes the correlation of the candidate
signal wavefronts Sjq with the true signal wavefront Sip. A comparable idea is used in
the Widrow-Hoff LMS algorithm, which uses the realization of the error rather than
the mean value of the error to control the convergence of the adaptive filter.
The signal wavefronts s^gq, q = 1 . . . L constitute a matrix, Sjp = [ 3 ^ 11 - • -Smq-iq-i
Srugq Srug+iq+i- ■-SmiL], which is a subsct of thc localization signal set defined in Eq.
(2.4), and is referred to as the optimal localization signal matrix. The inner product
matrix of the optimal localization signal matrix is given by aip — SipSf^.
Similar to the formulation of the PCL for the matched-ocean localization proces­
sor, the PCL for the optimal localization processor can be expressed as a constrained
multidimensional integral of the probability density function of the ambiguity vector
for the optimal localization processor, whose covariance matrix is determined by the
S N R at the receivers and the inner product matrix of the optimal localization signal
matrix.
The components of the ambiguity vector for the optimal localization processor is
defined as Xg — r ^ s ^ g q = V S N Rapip^rngq + n^s^gq^ q = 1 . . . L . Since the index nip
th a t maximizes Pip,mpp is equal to i (note th at Pip^ip = 1), replacing i with rrip yields
Xq — a/ S N Rapmpp,mgq +
^rrigq- The covariaucc m atrix of the ambiguity vector can
be expressed as
Cip — S N RckippCx^pp -|- Q!jp,
( 6 .2 2 )
where aip^p is the pth column of the inner product matrix of the optimal localization
signal matrix.
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utilizing the inner product m atrix a,j„ the PCL expression is formulated as
P C L ip { S N R ,a ip ) = [
dxG(:ic,0,SNRaip,pafp^p + aip).
(6.23)
The final PCL expression for the OUFP is the averaged PCL over the variety of
the realizations of the ocean environmental parameters and the hypothesized source
positions.
P C L ( S N R , 3) = jJ j
6.4
Y . U P C U , { S N R , at,).
(6.24)
A n alytical Perform ance P red iction s w ith T ypical Inner P rod u ct M a­
trices
It has been illustrated in section 6.3 th at the PCL performance for the matched-oeean
localization processor and the OUFP are constrained multidimensional integrals of
the probability density function of the ambiguity vector, whose covariance matrix is
determined by the signal-to-noise ratio at the receivers and the inner product matrix
of the localization signal matrix. Before studying various scenarios of the real-world
inner product m atrix corresponding to different ocean environments and localization
problems, we derive analytical PCL expressions based on several typical inner product
matrices, which only take into account the following fundamental parameters; the
SN R
at the receivers, the number of the hypothesized source positions, denoted by
I/, and the correlation coefficient between the signal wavefronts, denoted by p.
6.4.1
Id en tity M atrix o f Size L
Assume th a t the inner product m atrix is an identity m atrix of size L and the first
column of the signal m atrix is the reference signal wavefront. Substituting a = 7^ in
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Eq. (6.13) yields the covariance m atrix of the ambiguity vector.
SNR + \
0
...
0
C
10
0
...
...
0
0
0
(6.25)
1
The determinant of the covariance matrix is given by
C \= S N R + l,
(6.26)
and the inverse of the covariance m atrix is
C
-1
_
SNR+l
^
0
1
0
0
.
0
0
0
(6.27)
1
Using Eqs. (6.25)-(6.27), the probability density function of the ambiguity vector
is expressed as
p(x)
=
G(x, 0, C)
1 „—x^C
7T^(5ViJ+l)
g
(6.28)
1^1 I
S iV f l+ 1
Replacing the components of the ambiguity vector with Xi =
yields the PCL
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expression
P C L { S N R , a)
= U<,ui=2...LdM^)
d9L /o°° qidqi
= lo^ d9i...
= /o °° qidqi
= r
J o ' q2dq2
■■• f S '
dqf ^ e - ^ i n t 2
= SNm r
d y ie ^(l -
= l ^ k =0
fc
^
q2dq2 •- -
qLdqi^i^sNR+i)
q L d q ig A fl^
r
kSNR+k+1
k(SNR+l)
nL-1
k=l k(SNR+l)+l-
(6.29)
The PCL is completely determined by two parameters: the signal-to-noise ratio at
the receivers ( S N R ) and the number of the hypothesized source locations (L).
Figure 6.1 illustrates the analytical PCL performance computed using Eq. (6.29).
In Fig. 6.1(a), the PCL is plotted as a function of the S N R for various numbers of the
hypothesized source positions. As expected, the PCL increases with the S N R . When
the S N R approaches —oo, the PCL approaches 1/L. When the S N R approaches oo,
the PCL approaches one. Fig. 6.1(b) illustrates the PCL as a function of the number
of hypothesized source positions using a logarithmic scalefor various S N R ’s. It is
shown th a t the PCL decreases linearly as a
function oflogL for S N R >lOdB. For
S N R < lOdB, the PCL decreases much faster.
110
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N
ca
o
o
—I
0.5
20
CL
40
SNR(dB)
Figure 6.1; Analytical optimal localization performance based on an identity inner
product m atrix of size L, computed using Eq. (6.29). (a) PCL as a function of the
S N R , from left to right, L=2, 4, 16, 256, and 4096. (b) PCL as a function of the
number of possible source positions, from top to bottom, SNR =20dB , lOdB, OdB
and -lOdB.
6.4.2
E qually C orrelated M atrix o f Size 2
The correlations between signal wavefronts from different hypothesized source posi­
tions often affect localization performance. Here, we consider an equally correlated
inner product m atrix of size 2 .
a
1
P*
p
1
(6.30)
where p is the correlation coefficient of the two signal wavefronts. Assuming the
first column of the signal m atrix is the reference signal wavefront, the elements of
the covariance m atrix of the ambiguity vector are given by C n = S N R -I-1, C'12 =
{ S N R + 1 ) / , C21 = { S N R + l)p, and C 22 = S N R | p P +1.
Using z =\ Xi \^ — \ X2
x ^ D x , where x = [X1X2] and D = diag{[l,—l]),
yields P C L = P (| X2 P < | x\ p) = P{z > 0). The characteristic function of 2: can be
111
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expressed as
\-kC\
=
=
Sx'
11 - i w c
(1
—itAi)
(6.31)
^(1
—i^Aa) ^
~
^ “ Ai-^2
where Ai and A2 are the eigenvalues of the m atrix DC.
Solving
Ai + As = Cii - C22 = S N R { 1 - I p p),
(6.32)
A1A2 =1 C'12 P -CX 1C 22 = - ( S N R + 1 )(1 - I p p),
results in
Ai = ^ { S N R ( 1 - I p n + y / S N R ^ ( l - I p |2)2 + 4(S'iVi? + 1 )(1 - | p H ),
(6.33)
A2 = l ( S N R ( l - I p n - y / S N R ^ ( l - I p |2)2 + 4 ( S N R + 1 )(1 - | p H ).
Considering Ai > 0 and A2 < 0 yields the probability density function of
p(z) =
Ai — As
e ^^^- ^C U ( z) —
e^i-^C f/(—^),
Ai —As
2
(6.34)
where U(t) is the step function. Finally, the PCL is given by
PCL
— /q°° dzp(z)
Ai
(6.35)
SNRy/l-\p\'^
2-yy{SNR+2P-{SNR\p\P
Figures 6.2 illustrate the PCL as functions of the S N R and the correlation coeffi­
cient using Eq. (6.35). The plots in Fig. 6.2(a) show th at the effect of the correlation
112
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c:
_o
(b)
(a)
CO
0.5
0.5
>>
0.5
SNR(clB)
p 1 (correlation coeficient)
Figure 6.2: Analytical optimal localization performance based on a correlated inner
product m atrix of size 2, computed using Eq. (6.35). (a) PCL as a function of SNR^
from left to right, | p | = 0, 0.7, 0.9, and 0.95. (b) PCL as a function of correlation
coefficient, from top to bottom, S N R =30dB, 20dB, lOdB, OdB, and -lOdB.
coefficient is reflected by a shift of the PCL along the axis of S N R { d B ) . The higher
the I p I, the higher the S N R is required to achieve the same localization perfor­
mance (PCL value). Figure 6.2(b) shows th a t for all the S N R ’s^ the PCL degrada­
tion caused by a small | p | can be ignored (| p |< 0.5). When | p p approaches 1,
the PCL approaches 1/L for any fixed S N R .
6.4.3
A Special Tridiagonal M atrix o f Size L
Here we consider a special correlation pattern of the signal wavefronts; in the lo­
calization signal matrix, there exists a pair of signal wavefronts th at are strongly
correlated, but all other signal wavefronts are orthogonal to each other. Assuming
th at the first column of the signal matrix is the reference signal wavefront and the
second is the one strongly correlated to the first, the inner product matrix is given
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by
p
1
a =
0
1
0...
...
0
0
(6.36)
where p is the correlation coefficient.
Using Eq. (6.13), the covariance m atrix of the ambiguity vector is expressed as
SNR +1
SNRp + p
SNRp* + p* S N R \ p \ ' ^ + l
0
0
0
0
1 0 ...
0
...0
1
0
0
0
(6.37)
The determinant and the inverse of the covariance matrix are identified as
c h
(6.38)
(5 iV i? + i ) ( i - I p H ,
and
( 1 - | p |2 )(5 V H + 1 )
\-W
p*
1
1- I p P
0
0
. ..
0
...
0
0
...
0 ..
0
0
1
1
0
(6.39)
Using Eqs. (6.37)-(6.39), the probability density function of the ambiguity vector is
expressed as
p{x)
'kC
(6.40)
Replacing the components of the ambiguity vector with Xi = ri{cos{9i) -\-jsin{9i))
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generates
PCL
— F (|
x i p)
X 2 p < | a:i p , . . . I x l P < |
. . . dXLp{x)
= / x i i | x , P < | x i | ^ ■ • ■i | x x P < | x i j ^
_
l o ^ d S i - f o ' ^ d e L f ^ n d r i . . . f p ri,dri, e x p ( - ( y 3 ^ [ '^g^i^l^ + --rf+ r^ + 2 |p |r ir 2 c o s ( g i - 6 2 ) ] ~ E j ^ 3 r f )
“
7 r i(5 T V B + l)(l-|/j|2 )
_
4 /o ° ° ^ ir fn /o ’'^»-2 dr 2 e x p ( - j ^ ^ | £ S ^ ! ^ J ^ r f - Y : ^ ) ( l - e x p ( - r 2 ) ) ^ - 2 j ^ ( ^ - ^ r i r 2 )
( 5 iV J l+ l) ( l- lp |2 )
(6.41)
Since (1 - e ' i ) ‘-
= J 2 t j ( ^ j; ^ ) ® ‘’’■{-I)*,
(6.42)
.«2
= / o ° ° / J ' " r2dr2e
i^ /o (i^ rir2 ),
where
— (5 Arij+i)(i-|p|
+i 2) +i ^
wnere F —
Using ri = rcos{6) and r 2 = rsin{9)^ the item
=
5
/
o
°
°
rdr
2
r
d9e
d
r
e
2
is given by
2(rrifj% ^/Q (_ ^
^ ^e
2a-ipiVVo(j3 ^
r ^ 5 m ( 0 ))r^ 5 m ( 0 ).
(6.43)
Using X = cos(0) and y = sin{9) = \ / l P,k
4^ Ji or
=
Q!
r' ^^d' r Jo
j^dxe
yields
* '=■
''■' '-'O V i-Ip I
i /o ° ° ^ ^ ^ / o c ? 3 ;e -® ’' ^ i + ^ ) e " ^ J o ( j | ^ r 2 / )
= ( ^ + F a;)/( 1 - I p |2 ),
= C y/(l-|pn.
where A = ( ^ ^ ^ S u f + 1) +
I P P). -B = ( 7 ^ +
1^
~ 1) + ^(1“ I P
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and
C =
2
I p |.
Using Eq. (6.623.2) in [56],
xe~^^h{-jPx)dx =
yields
(i~l/^P)^ „
p
»
= f ' *
•’+“
If i? = a + 5a: + ca:^, A - 4ac —5^, where a =
—C*^, 5 = 2 ^ 5 and c —
according to Eqs. (2.264.5) and (2.264.6) in [56],
dx
2{2cx + b).i
I0 \ / ^ - \ f R
xdx _
I T ^ ~
Then
^
_
_
^
2(2c + 5)
A-y^a + 5 + c
25
Ay/a
2(2a + 5 a ; ) n _
2(2a + 5)
4a
A V S “ '" “ “ A v/a + 6 + c '^ V S '
(6.46)
(a
'■
■'
can be expressed as
_
2 ( 2 c + 6 ) ^ - 2 ( 2 a + 6 ) B , 4 qB - 2 6 ^
A v T j-6 + c
1 6 { A + B ) \ p \ ^ _ 1 6 |p p B
A (j4 + B )
Aa
S N R + l _____________________________________ ( S N R - k ( S N R + l ) ) ( S N R + l )
4 ( l- |p |^ ) ( l+ A ;( 5 7 V a + l) )
4 ( l- |p |2 ) i/2 ( i+ A ;( S iV iJ + l) ) y /( A :( 5 W iJ + l) + 5 W i? + 2 ) 2 - |p i2 ( i? - A ;( 5 W « + l) ) 2 ‘
(6.48)
Finally, the PCL for the special tri-diagonal inner product m atrix of size L can
be expressed as
Ak = k S N R +
Bk = k S N R +
Ck = k S N R +
k + \,
k - SNR,
k + S N R + 2.
N ote th a t su b stitu tin g p = 0 in Eq. (6.49) yields Eq. (6.29) and su b stitu tin g
(6.49)
L =2
in Eq. (6.49) generates Eq. (6.35).
Figure 6.3 illustrates the PCL computed using Eq. (6.49). Figure 6.3(a) plots the
PCL’s as a function of the S N R for various | p |’s, given L = 50. The left most PCL
116
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curve is corresponding to | p |=
0,
which is exactly the same as th at predicted using
Eq. (6.29). Prom the left to the right of Fig. 6.3(a), the PCL is computed using an
increased correlation coefficient. All of the PCL curves approach 1/L when the S N R
approaches —oo, and reach one when the S N R approaches oo. The pattern of the
PCL as a function of | p
for given L = 50 is illustrated in Fig. 6.3(b). It is shown
th a t the decreasing of the PCL with the increasing | p | is modest when | p |< 0.8,
but is significant when | p |> 0.98. And the PCL approaches 1/L when the | p
approachs one for any S N R . Figure 6.3(c) illustrates the PCL as a function of L
using a logarithmic scale, for combinations of various S N R ’s and | p |^’s. It is shown
th a t the PCL decreases with the increasing logL as well as the increasing | p p. Both
factors result in a shift of the PCL along the axis of S N R ( d B ) . Note th at when L is
small, the correlation coefficient has a strong impact, and when L is large, the impact
of the correlation coefficient is fading. This result is reasonable because only one pair
of correlated signal wavefronts is considered in this inner product m atrix model.
6.4.4
E qually C orrelated Inner P rod u ct M atrix
Usually, the signal wavefronts due to various hypothesized source positions are corre­
lated with each other. One of the limiting scenarios is th at all the signal wavefronts
are equally correlated and the inner product matrix is given by
a =
1
p
1
...
p...
p
p
p
p
...
p
1
(6.50)
where p is a non-negative real number.
Assume th a t the first column of the localization signal matrix is the reference
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(a)
0.8
0.2
-10
SNR(dB), L=50
p I ,L=50
F ig u re 6.3: Analytical Localization Performance prediction based on a tridiagonal
inner product m atrix of size A, computed using Eq. 6.49. (a) PCL as functions of
the S N R given L = 50, for | p | = 0, 0.7, 0.9, and 0.95 for left to right, b) PCL as
a function of the correlation coefficient | p \ given L — 50, for SNR=SOdB, 20dB,
lOdB, OdB, and -lOdB from top to bottom, c) PCL as a function of M for various
S N R ’s and | p |’s. Solid: 20dB, dashed: lOdB, dash-dotted: OdB, dotted: -lOdB.
For each S N R configuration, | p |=0, 0.7, 0.9, and 0.95 from top to bottom.
118
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signal wavefront, the covariance matrix of the ambiguity vector is
SN R + 1 SNRp + p
S N R p + p SNR p^ + 1 S N R p “
^ + p . ..
S N R p + p SN R p ^ + p S N R p ‘^ + 1 S N R fP + p.
C
SNRf? + p
S N R p + p S N R p^ + p
SNRp + p
SNRfP^ + p
SNRp^ + p
S N R fP + 1
(6.51)
The inverse of the covariance matrix is identified as
where A =
A
B
B
D
B
...
B
...
...
B
B
B
D
(6.52)
B = - f , D = !£z2£±l, and A = ((L - l ) p + 1)(1 - p).
The determinant of the covariance matrix is
(6.53)
C 1= (l-p )^ -^ (5 A T i? + l ) ( ( L - l ) p + l ) .
Using the method of Section II in [70], the PCL is derived as a double integral.
PCL
where Q{a,l3) —
^
/o°° fo°° dwdrwr exp (-
2
SNRp+l
2{l-p){SNR+l)
r%
(6.54)
xexp{—\{x^ + a^))lQ{ax) is the Marcum-Q function[70]. It is
difficult to calculate the PCL in Eq. (6.54) using brute-force integration. To tabulate
the PCL function, we use the following Monte Carlo simulation method.
1. Generate a signal m atrix [sn . .. Sil] from the inner product matrix a (Eq. 6.50)
using Cholesky factorization.
2.
Generate samples of the ambiguity vector x. The elements of the ambiguity
vector are given by x^ = {V SNRaSip + n)^Siq, q = 1 . . .L, where p and q are
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the indexes to the column of the localization signal matrix. The scalar a and
the vector n follow the data model (Eq. (6.4)).
3. The PCL is computed as the ratio of the number of the samples of the ambiguity
vectors th a t satisfies | Xp |< | Xq |,^ = 1.. . L , q ^ p over the total number of
the samples.
Monte Carlo simulations are computationally intensive. We introduce an approx­
imation PCL expression to accelerate the PCL computation, which is given by
Pr^TfQ ATR n
r L y L ,[P I \n , p,ly)
R ( S N R , p, L)
__ T -r r /-l k ( R ( S N R , p , L ) +l )
— 1 1 ^ = 1 k{ Ri SNR, p , L) +l ) +l - >
^^
SNRy/l-\p\0W
)
S N R ^ l- \p \ 0W
=
_
(g 55)
■\/(SNR+2p-SNR'^\l,fW
I3{L)
+ 1.17.
The PCL expression in Eq. (6.55) combines the factors of the SNR., L, and p such
th at it can be reduced to the PCL expression for the identity inner product matrix
of size L by substituting | p |= 0 or reduced to the PCL expression for the correlated
inner product m atrix of size 2 by substituting L — 2. The function /5(L) reflects the
degradation of the PCL as a function of increased number of the hypothesized source
positions. The parameters of the (3{L) are estimated using data fitting method to fit
the approximate PCL’s to the PCL’s computed using Monte Carlo simulations over
the parameter ranges of S N R = 0 —20dB, L = 2 — 50, and p = 0 —0.9.
The function ^{L) is plotted in Fig. 6.4(a). Figure 6.4(b)-(c) illustrates the PCL as
functions of S N R , L and p computed using the approximation approach (Eq. (6.55)),
denoted by dashed lines, and computed using the Monte Carlo simulation approach,
denoted by stars. Each Monte Carlo PCL computation uses 10000 samples of the
ambiguity vectors. Figure 6.4(b) plots the PCL as a function of L using a logarithmic
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(a)
2
1.8
1.4
10
1
,3
,2
10 '
10 '
L
(C)
r
0.8
0 .6 '
V
*****
O
QL
*i*4l
0.4
*
1
■
..*,
0.2
*****
* *
f i *
f f 4 P...............
* * ' *
f4 /* *- '
.
*...
*
0
0.5
20
SNR(dB)
Ipl
40
F ig u re 6.4: Approximate (dashed, Eq. (6.54)) and Monte Carlo (stars) localization
performance prediction based on the equally correlated inner product matrix, (a)
The function 'y(L), (b) The PCL as functions of L, given S N R = lOdB, for | p | = 0,
0.7, 0.9, and 0.95 from top to bottom, (c) The PCL as a function of p, given L=100,
for S'Ai?=20dB, lOdB, and OdB from top to bottom, and (d) the PCL as a function
of S N R {d B ) , given L=100, for p=0, 0.7,0.9,and 0.95 from top to bottom.
121
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scale for various | p |’s, for S N R = lOdB. The PCL decrease with logL and | p |
obviously. Figure 6.4(c) illustrates the PCL as a function of | p | for three S N R ' s at
L = 100, which reflects the decreasing effect of | p | on the localization performance.
Figure 6.4(d) shows th at the PCL as a function of the S N R is similar to the PCL
function given identity inner product matrix, except th at the cross correlation causes
a shift of the PCL along the axis of S N R ( d B ) to the right.
6.5
A n alytical L ocalization Perform ance P red iction s in B enchm ark O cean
Environm ents
In this section, we verify th a t the analytical approximate PCL expressions can be
used to predict the performance of the OUFP and the matched-ocean processor in
benchmark environments, by comparing the analytical results with the PCL’s com­
puted using the Monte Carlo performance evaluation method[10]. The assumptions
used by the Monte Carlo method include the data model defined by Eq. (6.4), the
localization signal set defined by Eq. (2.4), and the normal mode acoustic propaga­
tion model[2]. In using the Monte Carlo method, the source localization results are
obtained by processing the sample data sets using the localization processors (e.g.,
the matched-ocean processor, the OUFP, and the mean-ocean processor). Only when
the localization result is the same as the true source position is the trial considered
a correct trial. The PCL is calculated as a ratio of the number of correct trials over
the number of total trials.
Figure 2.1 illustrates benchmark environmental model, proposed in the May 1993
NRL Workshop on Acoustic Models in Signal Processing[23].
a known ocean environment, those mean values are used.
When we refer to
When we refer to an
uncertain ocean environment, the ranges of environmental uncertainties are provided
additionally. This model also consists of a single source at depth, Zg, range, r^, and
122
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a vertical line array containing
100
100
hydrophones spaced
1
m apart ranging from
1
to
m in depth.
6.5.1
PC L for th e M atched-ocean P rocessor in K now n O cean E nviron­
m ents
In known ocean environments, the optimal localization processor is the matchedocean processor. The PCL performance for the matched-ocean processor provides a
performance upper bound for ideal situations.
0.9
0.8
0.7
0.6
clO.5
0.4
0.3
0.2
source depth(m ) relative to ttie center of ttie w ater
Figure 6.5: The absolute correlation coefficient between the signal wavefronts due
to a hypothesized source at an arbitrary depth and a source at the center of the water
in depth as a function of the arbitrary hypothesis source depth relative to the center.
We first consider source depth localization problems. The analytical PCL predic­
tion begins with the analysis of the correlation pattern between the signal wavefronts.
The absolute correlation coefficient of two signal wavefronts, one from a hypothesized
source at an arbitrary depth and another from a source at the center of the water
in depth, is plotted as a function of the separation of the hypothesized source depth
relative to the center in Fig. 6.5. The plot shows th at the signal wavefronts are or123
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thogonal to each other if the depth separation is greater than about
6 m,
which is
about twice the wavelength of the acoustic source (f=450Hz). Assuming a 6 m depth
separation, the inner m atrix of the signal wavefronts due to L hypothesized source
depth positions can be approximated by an identity matrix of size L.
c
o
0.8
0.6
0.5
0.4
Ana.
0.2
5
10
15
source depth index
SNR(dB)
F ig u re 6 . 6 : Source depth localization performance prediction for the matched-ocean
localization processor, (a) Inner product matrix of the signal wavefronts due to
hypothesis source positions evenly located from 7.5m to 97.5m in depth, 6 km in
range, with 5m depth resolution, (b) The PCL as a function of the 577i?(dB) at
the receivers. Solid line; analytical results computed using Eq. (6.29). stars; Monte
Carlo performance evaluation results.
Consider a depth localization problem; locate the source from 18 hypothesized
source positions th a t are evenly distributed from 7.5m to 97.5m in depth and
6 km
in range in the known benchmark ocean environment. Figure 6 .6 (a) plots the inner
product m atrix of the matched localization signal m atrix due to hypothesized source
positions given a known ocean environment. It is shown th at the inner product m atrix
is close to the identity m atrix of size 18. Figure 6 .6 (b) demonstrates th at the PCL’s
for the matched-ocean localization processor (Eq. (6.7)) computed analytically using
Eq. (6.29) agree well with the PCL’s obtained using the Monte Carlo method.
If the hypothesized source depth separation decreases, the correlation coefficient
124
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c
o
(a)
N
1
1 401
CC
c
0.8
£30
0.6
Q.
CD
O
O
—I
O
0
go
-0 20
0
o
0.4
0.2
o
o
- - Ana. I
— Ana.I
CO
10
20
30
0
40
o
source depth index
SNR(dB)
F ig u re 6.7; Source depth localization performance prediction for the matched-ocean
localization processor. (a)Inner product matrix of the signal wavefronts due to 46
hypothesis source positions evenly located from 6 m to 96 m in depth and 6 km in
range, in the known benchmark ocean environment, (b) The PCL as a function of
the S N R at the receivers. Dotted line: analytical result computed using Eq. (6.49),
assuming p = 0. Solid line; analytical result computed using Eq. (6.49), assuming
p = 0.825. Stars; Monte Carlo performance evaluation results.
of the signal wavefronts due to hypothesized source positions increases. Consider
another source depth localization problem; locate the source from 46 hypothesized
source positions th a t are evenly distributed from 6 m to 96m in depth and
6
km in
range, in the known benchmark ocean environment. Figure 6.7(a) illustrates the
inner product m atrix of the matched localization signal matrix. By computing the
mean of the subdiagonal and superdiagnal values of the inner product matrix, the
correlation coelRcient is estimated as p = 0.83. This value is substituted in Eq. (6.49)
to compute the analytical PCL results, which is plotted as a function of the S N R
at the receivers in Fig. 6.7(b), denoted by the solid line. It agrees with the PCL
results computed using the Monte Carlo method, denoted by stars. The dashed line
in Fig. 6.7(b) is computed using Eq. (6.49) assuming p = 0, which is used to illustrate
the PCL degradation caused by the correlation between the signal wavefronts due to
hypothesized source positions.
125
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* * * p
/
0.9
1 0.8
toN
5 0.7
•■■/..........
/
/
„o 0.6
(U
so .5
O
o 0..
1
/*
/
/
/
_/*
SO.;
JD
|o . ;
0.1
/
/
i
/*
*
-10
10
20
30
SNR(dB)
40
- ana.
MC
50
60
F ig u re 6.8; The performance of the matched-ocean localization processor in the
known ocean environment, illustrated by plotting the PCL as a function of the S N R
at the receivers. Assume 738 hypothesis sources positions th a t are uniformly dis­
tributed from 4km to 8km with 50 m separation in range and from 10m to 95m with
5 m separation in depth. Dashed line: analytical PCL results computed using Eq.
6.29. Stars: PCL results computed using Monte Carlo approach.
126
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The analytical PCL expression for the identity inner product m atrix can also be
used to predict the range and depth localization performance for the matched-ocean
processor. Consider the problem of localizing a source from 738 hypothesized source
positions th at are uniformly distributed from 4km to 8km with 50 m separation in
range and from 10m to 95m with 5 m separation in depth. The correlation between
the signal wavefronts due to hypothesized source positions is moderate and can be
ignored. The dashed line in Fig. 6.8 shows th at the analytical prediction computed
using the identity inner product m atrix model (Eq. (6.29)), using L = 738 and the
S N R . This agrees with the Monte Carlo simulation results, denoted by stars.
6.5.2
PC L for th e O U F P in an U n certain O cean E nvironm ent
The PCL for the OUFP is formulated as an integral of the probability density function
of the ambiguity vector, whose covariance m atrix is determined by the S N R at the
receivers and the inner product matrix of the optimal localization signal matrix. Here
we use an equally correlated m atrix of size L to approximate the inner product matrix
of the optimal localization signal matrix and use the approximate PCL expression
for the equally correlated inner product matrix (Eq. (6.55)) to predict the PCL for
the OUFP.
A “genlmis” uncertain ocean environment, defined in the May 1993 NRL Work­
shop on Acoustic Models in Signal Processing[23], is used to check the approximate
analytical PCL expression for the OUFP. Table 6.1 summaries the ranges of environ­
mental uncertainties. We consider the problem of localizing a source in the genlmis
environment from 738 hypothesized source positions th at are uniformly distributed
from 4 km to 8 km with 50 m separation in range and from 10 m to 95 m with 5 m
separation in depth. The PCL results computed using the Monte Carlo simulation
approach are used as comparisons. In using the Monte Carlo approach, 5000 sim127
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ulated observation d ata vectors are generated using the data model defined by Eq.
(6.4) for each PCL calculation. The localization signal set used in the data model
(Eq. (6.4)) and used by the OUFP (Eq. (6.6)) is generated from 40 realizations of
the genlmis ocean environmental parameters for each of the 738 hypothesized source
positions.
T a b le 6.1: Parameters of uncertain shallow water propagation model
Environmental param eter
mean value ± range of uncertainty
dbottom depth [m]
IGOiAd, A d = 5
Co-surface sound speed [m/s]
1500±AC'o, AC q = 2.5
-bottom sound speed [m/s]
1480±AC'^,AC^ = 2.5
C^-upper sediment sound speed [m/s]
1600±AC'^,AC'^ = 50
Crlower sediment sound speed [m/s]
1750d=AC'/,AQ = 100
p-sediment density \g/cm^]
1.7±Ap,Ap = 0.25,
Q-sediment attenuation [dB/A]
0.35±Aa,Ao; = 0.25
t-sediment thickness [m]
100
In using the approximate analytical approach, the same localization signal set used
for the Monte Carlo performance evaluation approach is used to generate the optimal
localization signal m atrix for the formulation of the PCL for the OUFP. Assuming
the true source is located at 6 km in range and 50 m in depth for simplicity, the inner
product m atrix of the optimal localization signal matrix is plotted in Fig. 6.9. The
correlation coefficients between the signal wavefronts in the optimal localization signal
matrix are varied across the hypothesized source positions. It is difficult to take into
account all the different correlation coefficients in the analytical PCL expression. We
define the primary correlation level of the signal wavefronts using a single parameter
p, referred to as th e effective correlation coefficient, and we use th e equally correlated
matrix with the param eter p to approximate the inner product m atrix of the optimal
localization signal matrix.
The effective correlation coefficient p is estimated from the inner product matrix
128
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■M
Figure 6.9: The inner product m atrix of the optimal localization signal matrix
of the optimal localization signal matrix by selecting the R largest non-diagonal
components from each column of the inner product m atrix and calculating the mean
of the R X L components selected from the inner product matrix. Here L is the
number of the hypothesized source positions and R is rank of the localization signal
matrix, which is determined by the number of propagating modes. Note th at the
equally correlated inner product m atrix corresponds to an ideal signal matrix th at is
a full rank L-simplexes, the correlation coefficients are the same for every pair of the
signal wavefronts. However, the optimal signal matrix is i?-dimensions, where R is
much less than L. Using the first column of the optimal signal m atrix as the reference,
the R columns of the optimal signal m atrix th at generate the R largest correlation
coefficients to the reference constitute a special subset of the signal wavefronts th at
are most similar to the reference. The localization performance is primarily limited
by the ability of the processor to differentiate the most similar signal wavefronts
from the reference. Other signal wavefronts with lower correlation coefficients to the
reference do not significantly affect the localization performance. This estimation
129
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method empirically takes into account the most im portant correlation coefficient
values rather than using the average of all cross correlation coefficients.
0.9
.2 0.8
=S0.7
0.5
0.1
-1 0
20
SNR(dB)
40
F ig u re 6.10: Verify the approximate analytical PCL performance prediction for the
OUFP using the scenario defined in Table 6.1, by comparing the PCL results with th at
computed using the Monte Carlo approach. Assume 738 hypothesis sources positions
th at are uniformly distributed from 4km to 8km with 50 m separation in range and
from 10m to 95m with 5 m separation in depth. DashedrPCL for the matched-ocean
processor; Solid:PCL for the OUFP computed using Eq. (6.55); Stars:PCL for the
OUFP computed using Monte Carlo approach; Dash-dotted:PCL for the mean-ocean
processor computed using Monte Carlo approach.
Using 1% of the maximum eigenvalue of the localization signal matrix as the
threshold to count the number of the eigenvalues th at exceed the threshold, the rank
is estimated to be 26. The effective correlation coefficient p is estimated to be 0.64.
The approximate analytical PCL is computed by substituting L = 738, p = 0.64, and
the S N R in Eq. (6.55). Figure 6.10 illustrates the approximate PCL as a function of
the S N R at the receivers using a solid line. The PCL’s computed using the Monte
Carlo approach are denoted by stars. Figure 6.10 shows th at the analytical PCL
results agree well with the Monte Carlo PCL results.
The PCL results computed with Monte Carlo method are used to check additional
130
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assumptions made by the approximate analytical approach, including
1. The approximations used to formulate the PCL for the OUFP in Sec. 6.3:
replacing the sum of the exponentials with the maximum of the exponentials;
assuming equal S N R over hypothesized source positions and realizations of
ocean environmental parameters; approximate the optimal ocean index using
the maximum covariance criterion rather than using the maximum absolute
square criterion.
2. The analytical approximate PCL expression given the equally correlated inner
product matrix.
3. The approximation to the inner product matrix of the optimal signal m atrix us­
ing equally correlated matrix and empirically estimate the effective correlation
coefficient.
The agreement between PCL results obtained using the Monte Carlo method and the
approximate analytical approach shows th at the optimal localization performance is
captured primarily by the analytical PCL expression using three fundamental pa­
rameters: the effective correlation coefficient, the number of hypothesized source
positions, and the S N R at the receivers.
For comparisons, the performance of the matched-ocean processor and the meanocean processor computed using the Monte Carlo method are illustrated in Fig. 6.10
by plotting the PCL as a function of the S N R at the receivers, denoted by the dashed
line and the dash-dotted line respectively. The difference between the dashed line and
the solid line indicates th at the optimal PCL performance degrade in the presence of
genlmis environmental uncertainty. In order to achieve the same PCL performance
in the genlmis environment as th at in the matched-ocean environment, a 3-5 dB in­
crease of the S N R at the receivers is required. The dash-dotted line shows th at the
131
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performance of the mean-ocean processor is poor in the genlmis environment. Even
if the S N R is very high, the PCL for the mean-ocean processor is lower than 0.1.
This is because the signal wavefront computed using the mean-ocean environmental
parameters is mostly mismatched with the data. In the m ultipath acoustic propaga­
tion channel, the signal wavefront computed using mismatched ocean environmental
parameters may match with the signal wavefronts from false hypothesized source
positions and cause false localizations.
6.5.3
Effect o f E nvironm ental U n certain ty on L ocalization Perform ance
The effect of environmental uncertainty on the performance of various processors are
illustrated in Fig. 6.11 by plotting the PCL as a function of the S N R for various
scenarios defined in Table 6.2 with increased range of ocean depth uncertainty in
genlmis environment. The hypothesized source positions are uniformly distributed
from 4 km to 8 km with 50 m separation in range and from 10 m to 95 m with 5 m
separation in depth.
Table 6.2: Scenarios of environmental uncertainty with increasing range of ocean
depth uncertainty in genlmis environment
Scenario
A
B
C
D
F
F
Uncertainty configuration
A d e [0,0], A^ = A^o
A d e [0,0.5], A9 = A9o
A d e [0,1], A0 = A9o
A d E [0,1.5], A 9 = A9o
A d e [0,2.5], A9 = A9o
A d e [0,5], A 9 = A9o
Define A9 = (ACq, A C ^, A C ^, A Q , Ap, A a ) and A^o = (2.5, 2.5, 50,100, 0.25, 0.25)
for simplicity. The units are consistent with those in Table 6.1.
In each plot of Fig. 6.11, the solid line is the approximate analytical PCL for the
OUFP, computed by substituting L = 738, the S N R values, and the effective corre132
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(a)
0.8
/
/
/ /
/f
....................... / . .
_ l 0.6
o
CL 0.4
0.2
0
.
f
0.8
1
/
/•
0.4
/ /
20
40
/ 1
/ T
/' /fI
'j /
/ 7 /'
/d.y'
0
0
0.2
..
(d)
/
20
(c)
/
0.8 .................r .
/ /
/ /
0.6
/ /
/ 7
0.4
/ /
/ y
0.2 ,. /.. / ........
/ j
//
0.6
1J
//
'A
!f '
!
(b)
1
1
40
0
(e)
20
40
(f)
0.8
0.8
0.8
_ l 0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
o
20
SNR(dB)
40
40
SNR(dB)
20
SNR(dB)
Figure 6.11: Illustrate the effect of environmental uncertainty on the performance
of the OUFP and the mean-ocean processor by plotting the PCL as a function of the
S N R at the receivers scenarios with increased environmental uncertainty. Assume
738 hypothesis sources positions th at are uniformly distributed from 4 km to 8 km
with 50 m separation in range and from 10 m to 95 m with 5 m separation in depth.
(a)-(f) are generated using the scenarios defined in Table II. In each of the plots,
Dashed-PCL for the matched-ocean processor computed using Eq. (6.29); Solid-PCL
for the OUFP computed using Eq. (6.55); Stars-PCL for the OUEP computed using
Monte Carlo approach; Dash-dotted-PCL for the mean-ocean processor computed
using Monte Carlo approach.
133
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lation coefficient value in Eq. (6.55). Using the same estimation method as used in
previous section, and use the localization signal set th at is composed of 40 realizations
of the environmental parameters for each of the 738 hypothesized source positions
for each scenario, the effective correlation coefficients are p =0.57, 0.58, 0.67, 0.68,
0.68, and 0.68 for scenarios A-F. The stars in Figs. 6.11(a)-6.11(f) are the PCL for
the OUFP computed using the Monte Carlo method, using 5000 realizations of the
simulated d ata vectors for each of the PCL computation, and using the same localiza­
tion signal set for the data model as used by the analytical approach. The agreement
between the analytical PCL results and the Monte Carlo PCL results indicate th at
the effect of environment uncertainty on the optimal localization performance is cap­
tured primarily by the effective correlation coefficient in the approximate analytical
PCL expression.
In Figs. 6.11(a)-6.11(f), the dashed line is the analytical PCL for the matchedocean processor, obtained by substituting L — 738 and the S N R values in Fq. (6.29),
which provide a performance upper bound for the scenario without environmental
uncertainty. The difference between the dashed line and the solid line is small, which
indicates th at the performance degradation due to increased ocean depth uncertainty
in the genlmis environment is modest for the OUFP. The dash-dotted lines is the PCL
as a function of the S N R at the receivers for the mean-ocean processor. From Fig.
6.11(a) to Fig. 6.11(f), the PCL for the mean-ocean processor for high S N R values
{ S N R > 20dB) degrades dramatically, due to increased ocean depth uncertainty
in the genlmis environment. The PCL for the mean-ocean processor is sensitive to
environmental uncertainty. In the presence of a worst case environmental uncertainty,
the mean-ocean processor generates a uniformly random localization result among L
hypothesized source positions and it’s PCL performance approaches 1/L.
Using previous PCL results for the OUFP, the matched-ocean processor, and the
134
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(b)
(a)
—H
i—^
—-
C)
A
\
_L
0.5
\
\
\
\
N
s
0
0
0
(d)
(c )
C)
O Q .5
OL
★
0.5
*
*
C)
*
0
0
2
,
0
4
A d I (m)
0
2
,
4
A d | (m)
F ig u re 6.12: Illustrate the effect of environmental uncertainty on the performance
of the OUFP and the mean-ocean processor by plotting the PCL as a function of
the range of ocean depth uncertainty for a fixed S N R ’s at the receivers. Circle-PCL
for the matched-ocean processor, computed using Eq. 6.29 Solid-PCL for the OUFP
computed using Eq. 6.55; Stars-PCL for the OUEP computed using Monte Carlo ap­
proach; Dash-dotted-PCL for the mean-ocean processor computed using Monte Carlo
approach, (a) SNR=30.3dB (b) SNR=2Q.3 dB (c) SNR =13.3 dB (d) SNR=7.3dB.
135
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mean-ocean processor, Fig. 6.12 illustrates the effect of environmental uncertainty on
the performance of various processors by plotting the PCL as a function of the range
of ocean depth uncertainty. Figures 6.12(a)-6.12(d) correspond to S N R =30.3dB,
20.3dB, 13.3 dB, and 7.3 dB respectively. The PCL for the OUFP computed using
the approximate analytical approach and the Monte Carlo approach are denoted by
solid lines and stars. The PCL for the matched-ocean processor is denoted by a circle
in each plot, and the PCL for the mean-ocean processor is denoted by dashed lines.
Figure 6.12 shows th a t the performance of the OUFP degrades gradually with the
increase in the range of ocean depth uncertainty in the genlmis environment, while
the performance of the mean-ocean processor degrades dramatically.
6.6
Sum m ary
Analytical approximate probability of correct localization (PCL) expressions for the
performance prediction of the matched-ocean processor and the OUFP have been
developed. Incorporating both source position uncertainty and ocean environmental
uncertainty, the optimal PCL performance is formulated as a constrained integral
of the probability density function of the ambiguity vector, whose covariance matrix
is determined by the S N R at the receivers and the inner product matrix of the
localization signal matrix. The analytical PCL expressions were obtained by solving
the constrained multidimensional integrals based on some typical types of the inner
product matrices.
The analytical PCL expressions for the matched-ocean processor were verified
using known benchm ark ocean environm ents, by com paring th e PC L results w ith
those computed using Monte Carlo method. The results show th at the performance of
the matched-ocean processor can be captured primarily by the S N R at the receivers,
the number of the hypothesized source positions, and spatial correlation of the signal
136
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wavefronts. The analytical PCL expression for the matched-ocean processor provides
a performance upper bound for an ideal case where there is no ocean environmental
uncertainty, which is simple and fast for computation.
The analytical approximate PCL expression for the OUFP is checked using sce­
narios extended from the uncertain benchmark environments, using the PCL results
generated with Monte Carlo simulations as comparisons. The agreement between two
PCL results demonstrates additional assumptions used for the development of the
approximate analytical PCL expression as summarized in Section 6.5. It also shows
th a t the effect of environmental uncertainty on the optimal localization performance
is primary captured by the effective correlation coefficient in the PCL expression. The
performance degradation of the OUFP due to environmental uncertainty is modest
compared to the large degradation in localization performance of the mean-ocean
localization processor in an uncertain ocean environment.
137
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Chapter 7
A R eview o f D ielectric P roperties of Norm al and
M alignant Breast Tissue
7.1
Introd u ction
The contrast in the dielectric properties between normal and malignant tissues is a
basis for diagnostic applications using microwave devices. The study of normal tissues
has been widely reviewed. In this chapter, we collect together dielectric property data
on both benign and malignant breast tissues from a number of studies and present
them in graphical form so th at this information is convenient for general reference. It
should be emphasized th a t the data shown have been interpolated, extrapolated or
computed from the graphs and tables, so it is not necessarily precise. This chapter
also reviews the mechanisms behind the differences in dielectric properties of normal
and malignant breast tissues.
Most d ata are represented in terms of conductivity a and relative perm ittivity e',
since a and e' of biological materials are practically independent of frequency up to
the microwave range[71]. For the two low frequency cases with no a and e' available,
the data are represented in terms of a parallel combination of a conductance G and
a capacitance C. The two pairs of terms are equivalent in th at
F* = G + j u C = {A/d){a + jueQs'),
(7.1)
where Y* is the complex admittance of the equivalent circuit of an idealized parallel
plate capacitor filled with the tissue of a and e'. A /d is the geometry factor. We
assume (e', a) follow a bi-variate normal distribution. Therefore, the modeling of
the data can be represented by the specific cross section of the distribution function,
138
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which satisfies
+ (y - m y f l o \ - 2p{x - m^){y - my)la^Oy)l{\ - p^) = 1,
((a; -
(7.2)
where, x is the relative permittivity, y is the conductivity, and rrix, my and a^, Oy are
the marginal mean and variance, p is the correlation coefficient of e' and a. When
only the mean and variance values are available but not the original data pairs, we
assume an independent distribution of (e', o) and assume p=0. All data are from
human breast tissue, except for one case from rats. The category of breast tissues in
the literature is ambiguous. In this chapter, we define the following major categories
of breast tissue:
• fat,
• normal, includes glands tissue (lobules th at produce milk), and connective tissue
(fibrous tissue th at surrounds the lobules and ducts),
• benign, includes fibroadenoma and mastitis,
• malignant, i.e. breast carcinomas.
We first display the d ata in the low and high frequency regions, in the order of their
publication date. Then we discuss the consistency and inconsistency in the data as
well as the diagnostic value of the dielectric properties from the data. Finally the
mechanisms are reviewed.
7.2
7.2.1
R ev iew o f E xp erim en t D ata
List o f D a ta at Low Frequency
l.Fricke et.al.[27] (1926, 20 kHz, 24‘’C), measured the parallel capacitance and re­
sistance (i? = l/G ) of excised samples from 55 patients, using a wheatstone bridge.
139
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Several types of tissue were studied: fat, gland, mastitis, fibroadenoma and carci­
noma. D ata is displayed in Fig. 7.1. Only one sample of fat is measured, no variance
is available for this type.
900
1000
800
900
700
800
w ' 700
E600
>
O
&500
o
to
>
cc
.
eg
> >
>
«
>
+
0
>>
o
200
O
*
+
++
0 4>
>
>
200
>
>
.......
500
1000
1500
2000
2500
400
'(/)
oC
> 300
D
>
100
600
(O
D 500
c
3000
3500
100
0
500
1000
1500
2000
2500
3000
Capacitance(pf)
Capacitance(pf)
F ig u re 7.1: Capacitance versus Resistance. Left: original data. Right: modeling of
the data.
2. Morimoto et.al.[72, 73] (1990, 10 kHz, 37°C) obtained in vivo measurements
of breast cancer, fibroadenoma, normal breast tissue and fatty tissue using a threeelectrode method. The proposed equivalent circuit is composed of Rg parallel with the
series of Ri and Cm- We transformed it to parallel R and C, using R = RiRg/[Ri -\Re), C — Cm- W ith no original data and correlation coefficient available, Fig. 7.2
displays the modeling of the data, assuming p=0.
3. Jossinet et.al. [74, 75] (1996, 488Hz-lMHz, 21°C) measured 120 samples from
64 patients, using impedance probe sensors connected with a microcomputer system.
In Fig. 7.3, d ata of six types of tissue is displayed, gland (o), connective (*), fat (•),
mastopathy (square), fibroadenoma (-f), carcinoma (i>). We calculated the relative
perm ittivity and conductivity from the original complex impedance data p* (not the
characteristic impedance) using
e' = / m ( l / p * ) / ( a ; e o ) , cr = R e ( l / p * ) , p* = 1 / ( ( t -f ju e o e ').
140
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(7.3)
0
normal
4000
> maligna It
3000
t>
2000
4000
6000
8000
10000
Capacltance(pf)
Figure 7,2: Capacitance versus Resistance.
10
10
10
10 ‘
frequency (Hz)
lO'’
frequency (Hz)
Figure 7.3: Perm ittivity and Conductivity versus Frequency
7.2.2
List o f D a ta at H igh Frequency
4. T. S. England et.al. [32, 33] (1949-50, 3-24 GHz, 37"C) measured the attenuation
a nepers/cm, and phase constant ^ radians/cm of the standing wave pattern of
the excised human breast fat and carcinoma tissue samples in the wave-guide. We
computed the relative perm ittivity e' and conductivity a using Eq. (4.28) in [76].
£ = - (o ^ - 0^) /up'jioeo-u = 2a!3/ujjj,Q
141
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(7.4)
The variance due to these measurements were computed and presented with the light
marker in Fig. 7.4. Our extrapolations are shown with dashed lines.
300
earcinoma
80
250
> 60
P200
150
0)40
cloo
breast fat
frequency (Hz)
frequency (Hz)
F ig u re 7.4: Permittivity and Conductivity versus Frequency
5. W. T. Joines et.al. [28, 77, 78] (1980, 30 MHz-2 GHz, 37°C) obtained in vivo
measurements of SMT-2A tum or and mammary gland tissue samples from 22 rats.
The nondestructive method uses an open-ended coaxial probe to produce a fringing
field in the termination tissue and a directional coupler and an oscilloscope to detect
the fringing pattern. The dielectric properties are then computed. The data is shown
in Fig. 7.5, in which skin effect is not corrected, and the data cannot be compared
with in vitro data directly.
6.
S. S. Chaudhary et.al.[26] (1984, 3 MHz-3 GHz, 24'’C), measured excised
normal and malignant breast tissues from 15 patients, using the time domain spec­
troscopy system of HP. We use the total spread over the mean value (0.8%) to compute
the variance of the data, which is shown in Fig. 7.6
7. A. J. Surowiec at.el. [30] (1988, 0.02MHz-100MHz, 37"C) measured the in­
put reflection coefficient of 28 samples from 7 patients, using a coaxial line sensor
connected to an HP3577 network analyzer. Tissue types include ductal carcinoma,
lobular carcinoma, and surrounding tissues. The measured dielectric values are avail142
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
SMTi-2A tumor
^SM T^2A tumor
■> 70
P12
0.60
normal; gland
normal gland
frequency (Hz)
frequency (Hz)
F ig u re 7.5; Perm ittivity and Conductivity versus Frequency[28]
450
400
350
P10
>300
E250
Q.
<D
malignant
mafignant
0 200
•^150
100
50
normal
normal
frequency (Hz)
frequency (Hz)
F ig u re 7.6: Perm ittivity and Conductivity versus Frequency[26]
143
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
able only at 100 kHz and 100 MHz, and we use square symbols to represent the mean
and m e a n istd values of those data in Fig. 7.7. The authors provided the parameters
S(xn^s^T-,(Js and a , by fitting the data with the Cole-Cole equations[71]
e * = £ m / i » + {£> -
£ c o )/(l +
Uf/fc)
“ ) - jff./w e o ,
(7.5)
where, fc = lf2'KT, r is the relaxation time, fc is the relaxation frequency, a. is the
distribution param eter th a t reflects the range of r. s represents the low frequencies
/ <C /c, oo represents the high frequencies / ^ fc- When O!=0, Eq. (7.5) is the same
as Debye equation. The curves in Fig. 7.7 are computed from the fitted Cole-Cole
5000
4000
Q.,
■3000
0)2000
earcindi
1000
surrounding tissue
frequency (Hz)
frequency (Hz)
F ig u re 7.7: Permittivity and Conductivity versus Frequency
model in [30]. The dark lines are the mean values, the light lines are the m e a n istd
values.
Extrapolations are shown in light color.
Comparing the mean from the
measurements and the model, the conductivity values agree well, but the relative
perm ittivity values show some inconsistency.
8.
A. M. C am pbell e t.a l.[35] (1992, 3.2 G Hz, 24'’C) m easured 39 sam ples of
normal breast fat, 18 samples of benign tumors, 22 samples of glandular connective
tissue and 20 samples of cancer from 37 patients, using a resonant cavity perturbation
method. Dielectric properties were measured using the observation of the changes in
144
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
resonant frequency. In Fig. 7.8, the left plot displays the original data. The right
plot illustrates the modeling of the data with Eq. (7.2).
45
50
45
40
>>
40
'
E30
— > frlaiignar
>>
■■■.' fat;..................... .........
0 normal ^
-S
£30 + be nign ;
CO
E . > meliignant.......
^25
35
> >>
+ b enign
I 35
E
>
t520
■13o
o
; |2 5
c3
*c=aJ20
O 15
O
10
.
°.o..... ............
0<S>
10
5
5
0
0.
10
20
30
40
50
60
70
0.
10
20
30
40
50
60
70
80
relative permiticity
relative permiticity
F ig u re 7.8: Conductivity versus permittivity. Left: original data. Right: modeling
of the data
9. W. T. Joines et al.[29] (1994, 50MHz-900MHz, 24'’C) measured adm ittance of
12 normal mammary samples and 12 malignant mammary samples from 12 patients,
using a flat-ended coaxial probe connected to a network analyzer HP 8754A. e' and a
are then computed from the admittance with the knowledge of the geometry factor.
In Fig. 7.9, the mean values are presented with the solid lines, the standard error
on the mean is presented with the dashed lines. Extrapolations are represented with
light lines.
10. P.M. Meaney et.al.[25] (2000 900MHz,37‘’C) obtained the in vivo breast mi­
crowave imaging of 5 patients, all non-malignant, at 900MHz. We display the mod­
eling of the d ata of individuals in Fig. 7.10 to represent the heterogeneity within and
across patients. The correlation coefficient is not available and assumed to be 0.
145
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
’> 60
fnalignant
D.
CD40
d20
■a
015
malignant
normal
frequency (Hz]
frequency (Hz)
F ig u re 7.9; Permittivity and Conductivity versus Prequency[29]
12
10
E
-y
8
CO
E
>>
%®
o
|4
o
10
20
30
relative permiticity
40
50
F ig u re 7.10: Permittivity and Conductivity versus Frequency[25]
146
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7.2.3
D a ta C on sisten cy and Inconsisten cy
Low Frequencies
At low frequency ranges, the dielectric values of the four types of tissue are all
available for the first three cases, as shown in Figs. 7.1, 7.2, and 7.3. We cannot
compare them directly, because of the unknown geometry factors in cases 1 and 2.
Yet we can still make comparisons according to the relative distribution of the data
for the same tissue types. The consistencies are listed below
• The conductivity of the malignant tissue falls between the fat (plus connective
tissue) and the normal gland tissue (plus the benign fibroadenoma and mastitis
tissues).
• The benign and normal tissues can be grouped together relative to the malig­
nant tissues on the s'-a plane.
• The relative position of the fat tissue on the e'-a plane compared to the other
types is the same.
This information of d ata consistency provides a basis for identifying breast cancer,
benign tum or and normal breast tissue using the tissue conductivity at 1 kHz-1
MHz. The inconsistency in the first three cases is th at the malignant tissue has lower
capacitance (or permittivity) than th at of the normal and benign tissues in case 2 and
in the lower frequency region of case 3, which is not true in cases 1 and in the higher
frequency region of case 3. One of the possible reasons for the inconsistency is the
frequency difference. It is 10 kHz in case 2, 20 kHz in case 1 and 488 Hz-1 MHz in case
3. Therefore, the inconsistency can be related with a turning frequency point in tens
of kHz, above which the capacitance value of cancerous tissues became greater than
th a t of normal and benign tissues. Other reasons for the inconsistency are the intrinsic
147
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heterogeneity and the tem perature difference of tissue samples. This information of
inconsistency suggests th at in the range 1 kHz to 1 MHz, the capacitance of breast
tissues may not be used to diagnose breast cancer. More experiments and analyses
on the capacitance properties of normal and malignant breast tissues are needed in
this frequency range.
H ig h F req u en cies
In the high frequency range, we can compare the dielectric data directly.
Figs. 7.11
300
250
^200
;>
1
8.150
0
>
........... >■■■
2 100
>
>
" l > .........
50
o
JO
OGo
10°
10
o
o
o
° ° OOOOCG
O O OO o
10®
>
>
o
o
10®
V
.•
10^°
frequency (H z)
F ig u re 7.11: Permittivity versus frequency
and 7.12 illustrate the dielectric data together in the range 500KHz to 20 GHz. One
more case of breast fat [79] is included. Four types of breast tissue: fat (•), normal
(o), benigh (+), malignant (>) are displayed. For a clearer view. Figs. 7.13 and 7.14
illustrate the modeling of the data for multiple cases on the e'-a plane at 900MHz
148
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
>
2 5
E 20
O
£
■■i'is
o
■O
■>
§10
>
•
%
10®
o
''
,o
O
Ci
10^
o
O ,cy
jn
O o oooccJ£>
O OOOCCCf^
•
*
10®
10‘,9
O
«
• •
•
10
10
frequency (H z)
F ig u re 7.12: Conductivity versus frequency
149
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and 3.2 GHz. Again, fat (•), normal (o), benigh (+), malignant (>) are displayed.
20
I
1
1
,
...
!
18
16
su ro w ie c
:
e n g la n d
e
14
jo in e s
|12
jo in e s - ra
■•|’1Q
o
^ 8
c
o
“ 6
Chau lh ary
o
4
2
Met a n e y
o
ch au d h ary
. . ° .......... j o i n e s
, .........O' ■ '
e n g la n d
0
0
°
jo in e s -
rat
su ro w ie c
.......... ■i"
20
40
,
60
80
100
120
140
re la tiv e p e rm itiv lty
F ig u re 7.13: Conductivity versus Permittivity, at 900MHz
From Figs. 7.11-7.14, we observe the data inconsistency
• The mean dielectric values of normal and malignant breast tissues have obvious
variability.
• The mean conductivity values of normal breast tissues in Joines’ rat data,
Campbell’s data, and Meaney’s data are more than twice th at of other normal
breast tissue cases at the corresponding frequencies. The mean permittivity
values of normal breast tissue in Joines’ rat data and Campbell’s data are more
than twice th at in England’s data and Chaudhary’s data at 3.2 CHz frequency.
The possible two reasons for the inconsistency are listed below, which may help
explain the results and improve the future experiment designs.
150
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45
40
35
E
■i^SO
....... >■
: cam pbe I \ en gland
CO
E
>>25
■>
■■^20
H—
*
Campbell
> jo n es - rat
■o
•o
i'=
cam piD.e.ll
■■■■'........ o
O
10
chauc ary
..‘^..ChelUdary
jo in e s - rat
5
0
"0
england
Campbell
10
20
30
40
50
60
relative perm iticity
F ig u re 7.14; Conductivity versus Permittivity, at 3.2 GHz
151
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70
Experiment method
1. Limitations in experiments. Chaudhary’s samples were collected in physi­
ological saline, which will affect the accuracy of the data. Campbell’s ma­
lignant samples are frozen and defrosted before the measurement, which
may affect the accuracy of the data of this type.
2. In vivo vs in vitro. Joines’ rat data is from in vivo measurements with un­
corrected skin effects. Meaney’s data come from reconstruction of in vivo
microwave imaging. Others are from excised samples. In vivo methods
seem to have higher dielectric values.
3. Sample tem perature differences. Lower sample tem perature will make the
dielectric value a little bit lower, when the frequency is below 2 GHz.
Intrinsic heterogeneity
1. Normal breast tissues are composed of breast fat, connective tissue and
gland tissue, etc. In the literature, the composition of normal breast tissues
from case to case may differ.
2. Different stages of tumor development will change the tum or’s dielectric
property and introduce variability [27, 30]. Some samples of malignant
tissues were actually composed of small parts of malignant cells infiltrating
within a large part of normal cells, which may decrease the mean value of
the malignant tissue samples.
3. Across patients. The breast tissue samples from patients with different wa­
ter content or fat content and in different stage of menstruation, pregnancy
or lactation will have obvious differences in dielectric values. Campbell’s
d ata came from a relatively larger patient group, which may introduce
152
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wider variability.
This inconsistency information indicates the importance of using proper sample stor­
age method before experiments and suggests a standardization of the experiment
conditions like the sample and environment temperatures, as well as the record of
patients’ information for later analysis.
Although there are so many conditions out of control, we still observe the data
consistency from Figs. 7.11-7.14
• The mean conductivity of the normal tissue is less than 15ms/cm up to 3.2
GHz.
• Malignant tissues have higher mean perm ittivity and conductivity values than
those of normal breast tissues
• Fat tissues have the lowest mean perm ittivity and conductivity values
This data consistency information provides the basis for breast cancer diagnosis using
the dielectric properties in the microwave frequency range.
7.2.4
D iscu ssion o f th e D iagn ostic Values
It is misleading to use only the contrast of the mean values to judge the diagnostic
value of the dielectric properties. Since the intrinsic heterogeneity in malignant tissue
is large, this will decrease the mean contrast. The mean values from samples across
patients will decrease the contrast as compared to an individual patient. Therefore,
the diagnostic value of the dielectric properties seems to be underestimated, as in
case 8. A better concept might be to use the contrast of the maximum value of the
malignant tissue with the mean of the neighborhood normal tissue samples[27, 30].
Better criterion can be defined using the probability of detection and false alarm, in
153
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which the random model of the point dielectric values and the spatially distributed
dielectric values are incorporated. In summary, we observed the diagnostic value of
the dielectric properties from the data, as
• The low conductivity values of the normal breast tissue enable penetration
of microwave frequencies up to the low GHz range, which coincides with the
simulation results in [34].
• At 100 MHz-1 GHz, dielectric properties can significantly help classify normal
and malignant tissues.
• At frequency ranges of 1 GHz-3 GHz, dielectric properties can help classify
normal and malignant tissues.
• At 10 kHz-1 MHz, dielectric property can help classify normal, benign and
malignant tissues, yet mainly depends on the conductivity.
7.2.5
A reas in N eed o f M ore E xperim en ts
• The dielectric properties of benign tissues compared with th at of the malignant
tissues and normal tissues in the frequency range of 100 MHz-3 GHz.
• The spatial distribution of the dielectric properties of normal, benign and ma­
lignant breast tissues.
• The dielectric properties of human breast cancer in different development stages.
7.3
Mechanism: Normal vs Malignant Tissue
We first review the mechanism of the dielectric properties of biological tissues in
general.
154
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The frequency dependence of dielectric properties of biological tissues is related
to the polarization of molecules and structural interfaces in response to the applied
electric field[28]. D ata from Schwan and Foster on high water content muscle tissue
suggest the presence of three dispersion regions; alpha, beta, and gamma, with the
relaxation frequencies to be kHz, hundreds of kHz, and GHz[80]. The delta dispersion,
located in half way between beta and gamma regions, has also been identified[71, 81,
82].
For engineering applications, the alpha dispersion has little significance[71]. Beta
dispersion occurs at radio frequencies, and arises principally from the charging of
cellular membranes, with smaller contributions from the protein constituents and
ionic diffusion along surfaces in the tissue[71, 81, 82].
Tissues typically exhibit a small dispersion between 0.1 and 3 GHz, which have
been termed the delta dispersion[71, 82] or ”UHF relaxation” [81]. A combination of
mechanisms are suggested for this region: bipolar relaxation of the water of hydration
’’bound” to proteins, a Maxwell-Wager effect due to ions in the cytopalsm collection
against relative nonconductive protein surface and rotation of polar side-chains on
the protein surface [71, 28, 81]. The relaxation frequency is dominant mostly by
bound water {fr of 100-1000 MHz[71], protein molecules {fr of 40-300 MHz) and free
water (/r= 2 5 GHz).
The gamma dispersion occurs with a center frequency near 25 GHz at body tem­
perature, due to the dipolar relaxation of the free and bound water and ionic con­
ductivity. Campbell and Land[35] attribute higher than expected conductivity at 3.2
GHz to the ’’tail end” of /3-dispersion effects. In recent studies, a variety of factors
have been explored, which lead to pronounced difference in dielectric properties in
normal compared with malignant tissues, as listed below;
• Necrosis.
155
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Inflammation and necrosis are commonly found in malignant breast tissues.
Presence of necrosis leads to breakdown of cell membranes and thus a larger
fraction of the tissue th at can carry current at low frequencies[83], which de­
creases the capacitance of the tumor[27].
• Charging of the cell membrane.
In breast carcinoma, there is a progressive replacement of fat lobules with fi­
broblastic proliferation and epithelial cells, which also accompanied by a vari­
ety of alterations at the transformed cell surface[26]. Cancer cells have reduced
membrane potentials and tend to have altered ability to absorb positive ions[82].
They have a higher negative surface charge on their membranes[28, 82]. Ac­
cording to Joines et.al., conductivity of the malignant tissues is increased with
this mobile charge being displaced and rotated by the microwave field [28].
• Relaxation times.
The relaxation times in malignant tissues are larger than those in normal tis­
sue, indicating th at a significant increase in the motional freedom of water has
occurred[82]. Surowiec et.al.[30] reported th at cancerous breast tissues had av­
erage dielectric relaxation times between 0.6 us and 1.4 us and the surrounding
normal tissues had shorter relaxation times of 0.3 us.
• Sodium concentration and water content.
The sodium concentration in tum or cells is higher than th at in normal cells[82].
The excessive sodium concentrations not only affect the cell membrane poten­
tials [77], but causes malignant tissue to retain more fluid. According to Joines
et.al, the excess sodium fluid alone would yield greater conductivity and per­
m ittivity values in malignant tissue than those in normal tissue. In addition,
156
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the fluid is retained in the form of bound water, which has larger values of
a and £ than free water [77]. Malignant tissues have significantly higher wa­
ter contents than normal tissues have[81, 82]. The data from Campbell and
Land[35] illustrates the dielectric properties related with the water content at
3.2 GHz of the breast tissues. The relationship between relative perm ittivity
and water content is strikingly similar to the relationship between conductivity
and water content. This leads to the conclusion th at the same mechanism is
responsible for the change in both dielectric properties. Malignant breast tissue
has a higher ratio of water content compared with th a t of the normal tissue,
which coincides its higher values of perm ittivity and conductivity than normal
breast tissue at the same microwave frequency. However, in this data taken at
3.2 GHz, there is no marked difference in the water content of benign breast
tissue and malignant tumor.
7.4
Sum m ary
This chapter presents an initial review and consolidation of the dielectric properties
of normal, benign and malignant tissues in the range of 10 kHz-20 GHz. A brief
explanation of the experiment methods is presented as well as the mechanisms that
explain the difference in the dielectric properties of normal and malignant tissues.
The consistency and inconsistency of the data are discussed as well as suggestions
for the possible inconsistency. It is observed th a t the dielectric properties of breast
tissue, even though containing uncertainty, have good diagnostic value in the range
of 100 MHz-3 GHz.
157
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Chapter 8
Breast Tumor D iagnosis U sing Microwave
M easurem ents
8.1
Introd u ction
The normal and malignant tissues of the breast have high contrast in dielectric prop­
erties in certain ranges of the electromagnetic (EM) frequencies [26, 27, 28, 29, 30, 31].
This forms the basis for breast tumor diagnoses using microwave measurements. Here,
we use tum or to represent the category of malignant tissues. At this time, microwave
measurements are not intended to replace X-rays, but to provide additional informa­
tion to the radiologist to improve the performance of breast tum or diagnoses.
In this chapter, breast tumor diagnoses are formulated as a detection and localiza­
tion problem. As compared to previous Bayesian approaches th at have relied solely
on the post-processed image data for the detection and localization of the tumor, we
develop a physics-based Bayesian approach th at directly processes the measurements
of the propagating EM field. The physics-based Bayesian approach incorporates the
measurements of the propagating EM field with a Markov Random Field (MRF)
model th a t characterizes the spatial uncertainty of the tissue perm ittivity and a for­
ward propagation model th a t predicts the propagating EM field. The physics-based
Bayesian approach enables a fully utilization of the a priori knowledge about the un­
certain propagation medium. For comparison, another Bayesian approach is proposed
th a t op tim ally p ost-processes a reconstructed tissue p erm ittivity im age.
The objectives, the probability model, the forward EM propagation model, and
the reconstruction algorithm are introduced first. Secondly, the physics-based Bayesian
approach and alternative post-processing approaches are developed. Finally, simu158
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lation results are presented for: 1) the detection and localization performance as a
function of tum or contrast, size, local characteristics and shape, 2) the convergence
of the model covariance matrix estimation as support for the number of Monte Carlo
trials for the Bayesian approaches, 3) eomparisons of the physics-based Bayesian ap­
proaches th at incorporate uncertainty in the electromagnetic propagation with the
Bayesian approach th a t post-processes a reconstructed image, and 4) the effect of the
sensor array configuration on the performance of the Bayesian processors.
8.2
O b jectives and M odels
The problem of breast tumor diagnoses is placed within the framework of physicsbased signal detection theory. The decisions are first the detection problem: is a
tumor present or not in the tissue? Secondly, the localization problem: if a tumor
is present, where is its location? We are particularly interested in improving the
early detection and localization performance for the more difficult and less obvious
situations due to uncertainties in tissue characteristics. The detection problem can
be expressed by the binary hypotheses
Hi'. Tumor present, loeated at an unknown position S with area A in a 2D perm it­
tivity cross section of the tissue,
H q\ No tum or present.
The basic objective is to make optimal decisions about whether Hi or H q is true
and if Hi is true, what is the best estimate of the size, location, and perm ittivity
characteristics of the tissue.
8.2.1
M od el o f T issu e P e r m ittiv ity Im age
Many different types of uncertainties can be incorporated into the processing of the
microwave measurements. Here, the principal uncertainties considered are the spatial
159
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uncertainty of normal and malignant breast tissue permittivity, tumor position, its
size and shape, and the noise at the sensors.
We utilize the Markov Random Field (MRF [42]) to model the spatial uncertain­
ties of the breast tissue permittivity. The MRF model implements the idea th at the
perm ittivity of adjacent tissues is similar. The Markov property captures the local
characteristics of the perm ittivity distribution and thereby reduces the uncertainty
space. A MRF satisfies[84]
p{xij\xki e Q \
= p{xij\xki e ^ i j ) ,
(8.1)
where (i,j) denotes a 2 dimensional position, Xij is the tissue perm ittivity at the
position (i, j), and
is the neighborhood system of (i,j) [42].
There are several MRF models, such as Auto-Models, Multi-Level Logistic Mod­
els. The Gaussian MRF (GMRF[85][84], one of Auto-Models) is ehosen because it
can be easily ineorporated in the Bayesian approaches. The GMRF is defined as
p{xij\xki e f l \ { { i j ) } ) =
where
and af- are the mean and variance at (i, j) , and
are the interaction
coefficients, which describe the local characteristics of the tissue permittivity. The
m atrix form of
is
/5 =
.
where
^0,0
/5i,o
/5o,i
A ,i
•■■ Po,p
•■■ Pi,p
I^pfi
I3p,i
■
has been simplified to
■■
(8.3)
I3p,p
assuming stationary and symmetric
properties of the interaction coefficients. The (3 matrix provides two pieces of infor­
mation: one is the neighborhood area of (i, j) , which is a (2 P -f 1) x (2P-h 1) square.
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This implies th a t the perm ittivity at {i,j) only depends on the perm ittivity in this
neighborhood area. Another piece of information is the impact of the neighbors on
the point (i, j), as represented by the value of the interaction coefficients. The (3
matrix is equivalent to the neighborhood system
notations,
in the GMRF. Using conventional
in this model is composed of one-site and two-site cliques of P th order.
A clique is a set of points th at are neighbors of one another.
Eq. (8.2) only defines the conditional distribution of the tissue permittivity. In
order to develop the Bayesian approaches, we must formulate the joint distribution
of the tissue permittivity. According to the Hammersley-Clifford theorem[84], Eq.
(8.2) is equivalent to the joint distribution of a Gaussian Field[86], expressed as
r,(X\
=
|g|V^
^ ’
X {k )
where X is a
(8.4)
= e r ( i j ) , k ^ ( i - 1) X N + j , l < i j < N,
x l vector, a re-ordered mapping of the perm ittivity image
on an N X N lattice Q. The sample space of
is
is
defined
The size of the mean vector yU
X 1 and the size of the inverse covariance matrix Q is N'^ x N'^. Q is symmetric
and positive definite.
To describe the equivalence, the relationship of the parameters in Eq. (8.2) and
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those in Eq. (8.4) are summarized.
Eq. (8.4)
Eq. (8.2)
k=^(i — l ) y . N + j
Q = IIBU
^
^
=
B(kl,k2)
, k l — k2 = {i — 1) * N + j
for k l ^ k 2
(8.5)
=
k l — (i — 1) X N + j
k2 = {k — 1) X N + 1.
When the random field is assumed stationary and isotropic, IT is simplified to ^7,
and i? is a Toeplitz block circulant matrix.
In many studies, normal and malignant tissues have been modeled as homoge­
neous. Here, we model inhomogeneous tissue characteristics using means, variances,
and interaction coefficients to designate both the normal tissue and the malignant
tumor. The mean values of tumor and background are denoted by jit and fib respec­
tively, which are scalar variables. The interaction coefficients of tum or and back­
ground are denoted by m atrix j5t and Pb respectively. An algorithm proposed by
Rue[87] is used to fit the interaction coefficients to the Gaussian field with proposed
correlation length.
Translating the local characteristics to the global characteristics generates a uni­
form mean vector fi^° —
:i e
for H q condition, and a non-uniform mean
vector fi^^ = {fit — fib) x V + fi^° for Hi condition, where V is the tumor mean func­
tion usedto describe the position, the shape, and the area of
the tumor. Weconsider
both sharp and smooth tum or mean functions. The sharp tumor mean function
given by
E(:r)
V(x)
= 1, o: g A(5),
= 0,x^A {S).
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is
where S is the tum or position and A is the tumor area centered at S.
The smooth tumor mean function is given by
V{x)
D
C
V{x)
= ^ { x - S y - § i x - S f + l,xe A {S ),
= (7/0.6041,
= max( x — S ) , x E A(S),
= 0,a: ^ yl(<S'),
where (7 is a constant. Both the sharp and smooth mean functions are assumptions
since real data of spatially distributed tissue perm ittivity are not available yet. The
data in [30] suggest a sharp mean function, i.e. the permitivity at the tum or boundary
is higher than the perm ittivity at the tumor center.
Prom the local characteristics /?6 and variance <7^, the inverse covariance matrix
under H q condition is derived.
Q
( ^ 0 )
~
(QQ \
^
=
0, (kl) ^
Under Hi condition, the inverse covariance m atrix
, is derived from both
and
/3t, as well as cr^, which is a function of tum or position S.
^
=
^
e )^ij)
(8.9)
0, [kl) ^
In summary, for normal tissues, the Hq condition, the distribution of
multivariate complex Gaussian with mean vector
and inverse covariance m atrix
qFo; fQj. malignant tissues, the Hi condition, the distribution of
complex Gaussian with mean vector
is a
is a multivariate
and inverse covariance m atrix
. In the
follow ing sections, we suppress th e superscript Hi and H q if th e suppression does not
cause any ambiguity.
The possible parameter space is limited in order to have a definite positive covariance m atrix of the Gaussian Markov Random field[88]. However, we still have
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flexible choices for the /3’s, a ’s and /i’s in order to capture tissue variances in in­
dividuals as well as background structures across patients. The model is simple so
th at computational costs are tractable. Since the Gaussian field is a class of Gibbs
distribution (Gibbs [89][84]), it is possible to extend the GMRF to a wide range of
Gibbs distributions to model more complex microwave tissue characteristics.
An example realization of a 2D cross section of the tissue perm ittivity using the
MRF model, and a particular sensor array configuration, is illustrated in Fig. 8.1. In
this case, 24 sensors are arranged in a rectangle surrounding the tissue. It is im portant
to realize th at our characterization is statistical so th at the model does not convey
detailed deterministic anatomical features. The conductivity data is assumed to be
a known constant for simplicity. The numbers 1-24 denote one of the configurations
of sensor positions used in this chapter. All 24 sensors can both transm it and receive
signals at microwave frequencies.
Figure 8.1: Tissue perm ittivity image: an example. Numbers 1-24 label the sensor
positions.
8.2.2
Forward P ropagation M odel
The propagating EM signal is predicted by computing the scattered electric field
through the uncertain tissue medium, using Extended Born Approximation (EBA)
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implemented with improved CG FFT method[37][90]. The electric field can be ex­
pressed as
E, =
where
(8.10)
is the incident field from an infinite line current source
^ r^ (p ) = - W o / * dqG{p,q)J,{q).
JD
(8.11)
is the scattered field
EI^^\pR,qT)
JDdqG(pR,q)Al{q)E^{q,qT) ,
=
(8.12)
where qr and Pr are the transm itter and receiver positions. A k ‘^{q) = k^{q) — kl
and kl = uP'piQeh — ju>p,o<^b, where k denotes wave number and subscript b indicates
background. D is the inhomogeneous object domain. G is the Green function for a
homogeneous background media
G(p,q) = ^ H S ^ \ k b \ p - q \ ) .
(8.13)
The EBA method exploits the fact th at G(p,q) is close to 6(p — q) to approximate
Eq. (8.12) as
E^z‘" '\ p r ^Qt ) = E^ipR.qr) [ dqG{pR,q)Al(q).
Jd
(8.14)
Substituting Eq. (8.14) in Eq. (8.10) yields the solution of the electric field
E,{p) = { l - [ d q G { p , q ) A e ( q ) ) - ^ E r .
Jd
(8.15)
The approximation of the scattered field is obtained by substituting Eq. (8.15) in Eq.
(8.12). W ith the improved CG FFT method ([37][90]), the approximated scattered
electric field can be computed using
Escat
J(^e)(p)
^
=
F -H F [G ]F [j(^ ^ )F ;r]} ,
AA;2(1 - F-HF[G]F[Aifc2]}).
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1000
E
500
0
<0
oaj
LU
-500
- 1000 .
100
200
300
400
500
100
200
300
400
500
1000
E
>
o
CO
CO
oT
DC
-1000
0
Figure 8.2: Measurements of scattered electric field(mV/m)
The microwave signals at the sensors are represented by a reordered spatial vector
s = [sii, Si2 ...SA:i^] of size K'^ x 1 in frequency domain, where Ski represents a single
measurement relevant to the kth transm itter and the Ith receiver. An illustration
of one realization of the signal vector s, real and imaginary components versus their
reordered index, is shown in Fig. 8.2, with K ^‘ = 576. The signals are results of the
scattered electromagnetic field th at has propagated through the uncertain perm ittiv­
ity medium. A realization of the uncertain tissue perm ittivity is illustrated in Fig.
8 . 1.
More specifically, under the Hi hypothesis, the data vector r can be expressed as
r
=
s -I- n,
s,
=
Er\pk,qi\k = l...KJ = l...K,
SNR{dB)
=
^ = M!
(8.17)
where the subscript k and I are the indices to the transm itter and the receiver re­
spectively, and the subscript i is the reordered index to the signal and data vectors.
The data r, the signal s, and the additive noise n are all
x 1 complex vectors.
It is assumed th at the additive noise is a multivariate complex Gaussian vector with
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zero mean and covariance m atrix 7/^2, an identity matrix with size K'^ x K'^.
8.2.3
R e c o n s tru c tio n s
18
17
16
15
14
13
F ig u re 8.3: Reconstructed tissue perm ittivity image. Numbers 1-24 represent the
array configuration.
A reconstructed image example, shown in Fig. 8.3, is obtained using the backpropagation as initial solution to the measurements illustrated in Fig. 8.2, followed
by the contrast source inversion(CSI Berg[91]) method, which is proposed by Zhang
and Liu[37]. A summary of CSI is given below. First define the contrast source uj(p)
as
= x(p)Ez{p), x{p) = ^l{p)/kl-
(8-18)
The scattered field can be expressed as
E r ‘(P) = t i f dpG(p,q)x(q)B.(q).
JD
(8.19)
The scattered field is measured in the domain S (outside D) where x vanishes. As­
sume no measurement error, the measurement data r satisfy
’(P) = kl [ dpG{p,q)x{q)E^(q),p
JD
e
S.
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(8 .20)
The state equation and the data equation are
u{p)
r(p)
=
=
xET'" + x G duJ , P ^ D,
G s U ),p e S ,
where G d (^ and Gsco are defined as
G d ,s (^ = kb /d dpG{p, q)x{(l)Ez{q), P ^ D
or p e S.
(8.22)
The contrast source can be obtained by minimizing the cost function F(o;, y), which
is the summation of the error in the data equation and the state equation.
x) = (E f=i Ik,-111)-^ E f= i Ik,- -
(E,^=i WxErjWl)-^ E,ti llx^^lr -
+ xGoi^All.
^
^
The Polak — Ribiere conjugate gradient procedure is used to update both to and x
alternately.
8.3
D etec tio n and L ocalization A pproaches
According to signal detection theory, the optimal detector for binary hypotheses is
the likelihood ratio (A) of the data vector X followed by a threshold whose value is
determined by the optimum criterion (T. C. Birdsail).
The optimal localization processor computes the a posteriori probability of the tumor
location, given the d ata vector r and the Hi condition. We use the maximum a
posteriori probability of the tum or position in order to obtain quantitative localization
performance in terms of the probability of correct localization (PCL).
p{S\X).
5
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(8.25)
Based on signal detection theory, we obtain the optimal physics-based Bayesian image
processor using two kinds of data: the data of the tissue perm ittivity image as well as
the data of the scattered EM field measurements. Two Bayesian results are presented
in the form of likelihood ratio and the a posteriori probability image. We also present
non-Bayesian approaches for comparisons. For all those approaches, we assume th at
the tumor position is a random variable with uniform distribution on the 2D lattice.
Original tissue
: permittivity image
forward propagation
Scattered EM
Measurements
Reconstruction of tissue
permittivity image
inverse algorithm
C) Bayesian Processor for
Scattered EM field of
Uncertain Permitivity Image
(BP_SEUPI)
D) Low order approximation of
BP_SEUPI
(LBP_SEUPI)
A) Bayesian Processor for
Uncertain Permittivity Image
i) Known parameters
(BP_UPI)
ii) Estimated parameters
(EBP_UPI)
B) Threshold Image Processor
Tumor detection and localization
Tumor detection and localization
F ig u re 8.4: The framework of the detection and localization approaches using the
tissue perm ittivity image d ata or the EM measurement data.
Figure 8.4 is a flow chart of various detection and localization approaches. The
left part in Fig. 8.4 illustrates the first Bayesian approach for tum or detection and
localization, which optimally post-process the reconstructed image. The Bayesian
processor is derived using th e m ean, variance, and interaction coefficients o f the
tissue perm ittivity image, and is referred to as the Bayesian Processor for Uncertain
Permittivity Image(BP_UPI).
The performance upper bounds are obtained based on the BP_UPI algorithm th at
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optimally processes the original perm ittivity image data, with its prescribed uncer­
tainties. If accurate a priori knowledge of the GMRF is not available, a sub-optimal
approach could use the estimated parameters. We refer to the output for such a case
the Estimated Bayesian Processor for Uncertain Permittivity Image (EBP_UPI). One
of the performance lower bounds is obtained by defining the Threshold Image Processor(TIP) of the reconstructed tissue perm ittivity image, which is perhaps similar
to what a human observer might do in looking at such an image.
A second approach is presented in the right part of Fig. 8.4. Here, the likelihood
ratio and the a posteriori probability image are formed directly from the measure­
ment data, incorporating the a priori knowledge of the uncertainties of the original
tissue characteristics as projected through the nonlinear propagation. The output of
this processor is called the Bayesian Processor for Scattered EM field of Uncertain
Permittivity Image (BP-SEMUPI). The Monte Carlo Integration is used to make the
computation of the BP-SEMUPI tractable. Since the scale of the uncertainty is very
large, the Monte Carlo Integration, although useful, takes a long time to converge.
To reduce the computational complexity, we propose a lower order approximation
of the BP-SEMUPI, referred to as the LBP-SEMUPI. It is based on projecting the
uncertainties at the measurement domain onto a multivariate Gaussian field.
8.3.1
B ayesian P rocessor for U ncertain P e rm ittiv ity Im age
B P -U P I D etecto r
When the tum or position S is unknown, the likelihood ratio is
-
^-p(e;|Ho)------- ■
Substituting the a priori distribution of the permittivity image (Section 8.2) yields
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the likelihood ratio for the BP.UPI
-
/
X
V — ^
I
^
I
( e r - M J
T r \t
A(er) OCE se n I <55 I e
Q ( e r “
, . N _ / _____
^\T
M ) - ( € r - j U / ? )
( q
c %i- 7 \
^
B P _U P I L ocalization Processor
The a posteriori probability of the tissue permittivity, e^, given th at the tum or is
expressed as
PS\eriS\€r)
=
eS S S w sJ-
Substituting the a priori distribution of the perm ittivity image for Hi hypothesis
(Section 8.2) in Eq. (8.28) generates
p{S
I 6r) a {Cr - j l f Q { e r ~ yu)+
X
log{\ Q s \ ) - { ^ r - P s f Q s i ^ r ~ Ps)-
(8.29)
The computations of the Bayesian processor are proportional to the tumor area
8.3.2
A.
T hreshold Im age P rocessor
The threshold processors mimic a routine visual examination of the maximum bright­
ness of the object image.
T IP D etecto r
The threshold detector is defined by
A = m ax
where
is th e perm itivity im age data.
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(8.30)
T IP L ocalization Processor
The threshold localization processor is defined by
5=
(8.31)
where S is the tumor position.
8.3.3
B ayesian P rocessor for S cattered E M Field o f U n certain P erm it­
tiv ity Im age
B P _S E M U P I D etecto r
In the measurement domain, the likelihood ratio is defined by
X/
N
_
p{r\Hi,€r,S)p{er\Hi,S)p{S)d€r
f^^p{r\€r,Ho)p(€r\Ho)d€r
’
E g g n /e „
I '
,
.
V•
1
where S is the unknown tum or position.
Incorporating the a priori pdf of the additive noise, the conditional probability
density function of r given T{cr \ C) is
p(r I T{er I C)) = (27rcr„)
e
^
(g.SB)
where T is the mapping function from the perm ittivity image to the measurement
data, T(cr) is the received signal s. And C represents the condition Hq or (Hi,S ).
The complete likelihood ratio is given by
4 e
2-n
P {^r\HuS )d€r
----------------------- •
p{e, I Ho)d€r
(8-34)
Because T{er) is a complex nonlinear function, we cannot obtain a closed form for
the likelihood ratio. However, Monte Carlo Integration[10] can be used to compute
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A(r) in a computationally efficient manner. This can be expressed as
l^i=l,erij~p(.er\Sj) ---------
X ( j - ) OC
--------------------(r-T(6ri.|gn))^(r-T(erfe|Hn))
2cr,^
A^er}c'^p{er\f^0)
(8.35)
whereM is the number of all possible tumor positions and L is a number large enough
to make the Monte Carlo Integration a good approximation to the likelihood ratio.
B P _ S E M U P I lo c a liz a tio n p ro c e sso r
In the measurement domain, we derive the a posteriori probability of the tumor
position S given the received measurements r.
^(^Ir)
= J^^p{r \ er,S)p{€r \ S)p{S)/p{r)d€r.
(8.36)
Substituting Eq. (8.33) in Eq. (8.36), and ignoring the uniform p{S) and the constant
p(r) for simplicity, yields
p(S'|r)
p{er\Hi,S)der.
a
(8.37)
Monte Carlo integration can be used to make the computation tractable
Jp(S'|r) OC
( r - T ( e r t ) )^ ( r - T ( e , .,) )
,
e
(8.38)
k=l
where erk{S) is the multivariate real Gaussian distribution N r {pls^Q s ) i given known
tumor position S. L is a large number to make the Monte Carlo approximation
converge to a good approximation of the a posteriori probability.
8.3.4
Lower Order A p p roxim ation o f th e B P _S E M U P I
To reduce the huge computational cost for the BP_SEMUP1, a lower order distribu­
tion is developed to approximate the true distribution of the measurement data. The
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resultant processor is referred to as the LBP-SEMUPI. The multivariate Gaussian
model is chosen which demonstrates good performance in the simulations. From a
generated large database, we estimate the mean and covariance m atrix of the mea­
surement d ata under both
Hi
and iLo, which are denoted by U and D for
Hq
and Us
and Ds for (Hi^S). The size of the Us and Ds matrices are the same as the size of
the possible tumor positions. These estimations are then used in the LBP_SEMUPI
processor to make detection and localization decisions.
L B P -S E M U P I D etecto r
Using a multivariate complex Gaussian model,
p(Ter I c) =
where subscript C represents
( H q)
-Uc)^
(8 .3 9 )
or (Hi,S). K is the number of sensors.
Applying the theorem of Multiplication,
P(r \ C ) = [
P(r I T,^\c)p{T,,ic)dT,^.
(8.40)
Substituting Eq. (8.39) and Eq. (8.33) in Eq. (8.40) generates
I Cl
I
~
+ Dc,
D ci
= 4
(8-41)
Vci
= D cl(^+ D cV c).
T h e likelihood ratio can b e expressed as
“
where
p(r I i/„)
’
is the set of all possible tumor positions.
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Substituting Eq. (8.41) in Eq. (8.42), with C replaced by (i?i,S) and { H q) respec­
tively, yields the simplified form of the LBP_SEMUPI detector
• ' W
«
E s t n
I - O s
!l - O s i
l“ ‘
(8.43)
^Usi"DsxUsi-Us^DsUs-Ui»DiUi^
where f/5 1 , Dsi are given by Eq. (8.41) with C replaced by S. U\ and Di are given
by Eq. (8.41) with C ignored.
When (7^ approaches zero, i.e. there is no measurement error, the expressions are
simplified to
rx(r
I
n\
-
i^ol,
Dn{r-Ur)
L B P_SE M U PI L ocalization P rocessor
The localization problem has already been
variate Gaussian model
defined in Eq. (8.36).W ith the multi­
approximation, the conditional probability density function
of r given Hi and S is defined in Eq. (8.41), with C replaced by (ffi,S). Then the a
posteriori probability expression is given by
Dsr = ^ ^ + D s ,
(8-45)
Usi=Dsi{x^^+ DsUs).
W hen there is no additive noise, th e LB P_SE M U PI localization processor is sim plified
to
p{S Ir)
OC
log{\ Ds |) - (r - U s )^ D s (r - Us).
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(8.46)
8.4
8.4.1
Sim ulation R esu lts
B asic Param eters
We assume th at the tissue perm ittivity image has size 9.2 cm x 9.2 cm and the unit
area is 4 m m x 4 mm. The number of pixels in the image is 23x23. For the first
three simulation scenarios, we use 24 sensors evenly spaced outside the object area,
7 at each side and 1.6 cm between the neighborhood sensors (Fig. 8.1). The fre­
quency used for the forward propagation and the reconstruction is 1 GHz. The forth
simulation scenario discusses the effect of different array configurations on detection
performance.
The mean value of normal background tissue was set to
= 30, which is within
the range of real normal breast perm ittivity data reported experimentally [31]. The
mean value of malignant tumor
is assumed to be a variable in the first simulation
to study the effect of the tumor contrast, and it takes constant value of 40 in the
later simulations. The variance for both the background and the tumor is 1. The
interaction coefficients of tum or and background, denoted by /3f and
respectively,
have been identified to fit the Gaussian field with 3 and 30 pixel correlation length.
I^b =
1.0000 -0.1879 0.0191 ■
-0.1879 -0.1724 0.0453
0.0191
0.0453 0.0007
11 0.1 ■
0.1 0.1
(8.47)
Two tumor types, one with sharp boundaries and the other with smooth boundaries
are considered, as shown in Figs. 8.5(a) and 8.6(a). We consider two categories of
the interaction coefficients. In one category, both tumor and normal tissues have the
same interaction coefficient jif,. In another, tumor and normal tissues have different
interaction coefficients, denoted by Pt and
respectively.
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8.4.2
Perform ance M etrics
To evaluate the detection performance, we use the receiver operating characteristic
(ROC), which is defined by Eqs. (1.1) and (1.2). We also use the detectability in­
dex, d, i.e., the defiection coefficient. The scalar variable d completely characterize
the ROC curves for the case where the underlying probability density functions are
Gaussian. In other cases, d can be estimated from the ROC curve through Eq. (8.48)
using a d ata fitting method and used as an approximation to the ROC’s.
Pd = 1
—Gau2cdf{Gau2inv{l — Pp) — d),
(8.48)
where Gau2cdf and Gau2inv are normalized Gaussian cumulative density function
(CDF) and inverse CDF functions.
To evaluate the localization performance, we use the probability of correct local­
ization (PCL), which is introduced in chapter 1.
A database of the original and reconstructed tissue perm ittivity image and mea­
surement d ata is generated for testing the proposed detection and localization algo­
rithms, including 400 realizations for the H\ hypothesis, assuming a tum or located
at random positions and 400 realizations for the H q hypothesis, assuming no tumor
present. The test database is independent of a large database generated to study the
statistics of the measurement data.
8.4.3
Perform ance U pper B ou nd and Tum or C haracteristics
The detection performances of the BP_UPI are illustrated in Figs. 8.5(b), 8.5(c),
8.6(b), and 8.6(c). In those figures, the detectability is plotted as a function of tumor
area, tumor contrast
sharp or smooth tumor mean function, and weather
or not the interaction coefficients for malignant and background tissue are the same
or different. The computation follows Eq. (8.27).
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The localization performances are quantified by the PCL curved plotted in Figs.
8.5(d), 8.5(e), 8.6(d), and 8.6(e), computed using Eq. (8.29).
The simulation results demonstrate th at
1. The tumor detection and localization performance increases with the tumor
contrast
which is shown in all 8 plots.
2. Figs. 8.5 and 8.6 verifie th at it is easier to detect and locate the tum or with a
sharp weight function than th at with a smooth weight function.
3. By comparing (b) and (c), (d) and (e) in both Figs. 8.5 and 8.6, one can
see th at capturing different interaction coefficients for the background and the
tumor improves the detection and localization performance.
4. The performance as a function of tumor area is not a simple scaling relationship
since it is affected by the tum or mean function and the interaction coefficients
together. When the tumor and the background have the same interaction coef­
ficients and the tum or has a smooth shape function, the larger tum or area may
cause stronger similarity between the tumor and the background such th at the
detection performance may decrease, as shown in Figs. 8.6(c) and 8.6(e).
8.4.4
C onvergence o f th e Covariance M atrix E stim ation
The estimation of the mean and covariance m atrix is im portant for the performance
of the EBP-UPI and the LBP_SEMUPI. The large uncertainty scale of the uncer­
tain perm ittivity image motivates two approaches to study the convergence of the
estimates,
1. Determine the performance of the processor versus the number of samples used
to form the estimates. The ROC curve should reach some stable state as the
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20
20
y axis
xaxis
b
o4
20
30
40
20
30
40
50
0.9
0.7
0.6
0.5
0.4
0.3
20.2
0.2
20
30
40
tumor area ( pixel num ber)
0.1
50
20
30
tumor area ( pixel num ber)
F ig u re 8.5: Detection and localization performance of the BP_UPI as a function
of tumor contrast, size, local characteristics, with sharp tum or mean function, (a)
an example of sharp tum or mean function, tumor at (11,7), occupying 25 pixels.
(b)(c) ROC, (d)(e) PCL curves. Interaction coefficients for background and tumor:
left-differ, right-same. From top to bottom, contrast
; Left-f5 2.5 2 1.5 1 01,
Right-[5 2.5 2 1.5 Ij.
179
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20
y axis
Xaxis
b
3.5
2.5
24
0.5
30
40
20
30
40
50
0.8
0 0.4
0.4
20 .2
0.2
20
30
tumor area ( pixel num ber}
20
30
40
tumor area { pixel num ber)
F ig u re 8.6: Detection and localization performance of the BP_UPI as a function
of tumor contrast, size, local characteristics, with smooth tum or mean function, (a)
an example of smooth tumor mean function, tumor at (11,17), occupying 25 pixels,
(b)(c) ROC, (d)(e) PCL curves. Interaction coefficients for background and tumor:
left-differ, right-same. From top to bottom, contrast
Left-[5 2.5 2 1.5 1 0],
Right-[5 2.5 2 1.5 1].
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number of the samples increases.
2. Estimate the mean and covariance m atrix of the original tissue perm ittivity
image d ata from the database, and compare them with the true values. The
difference should be within a small threshold. The number of unknowns is 529
for tissue perm ittivity image and 600 for the measurement data. Since they are
comparable, we can use the convergence of the estimates for the image data to
predict the convergence of the estimates for the measurement data.
Figure 8.7 illustrates the detection performance of the Bayesian processors versus
the number of samples used to estimate the mean vector and the covariance matrix.
Figure 8.7(a) plots the ROC of the BP_UPI th at processes the original perm ittivity
image data.
The ROC curves become stable near the perfect upper left corner.
The corresponding deflection coefficient converges to around 6.4 after 1400 samples.
Figure 8.7(b) plots the ROC of the BP-SEMUPI th at processes the scattered EM
measurement data. The ROC curves converge with some degradation compared to
the ROC for the BP_UPI th a t processes the original perm ittivity image data. The
corresponding deflection coefficient converges around 3.6 after 1400 samples. The
performance degradation due to limited sample size also reflects the effect of nonaccurate a priori knowledge of the mean and covariance m atrix on the processor’s
performance.
Table 8.1 lists the value of the estimated coefficients from 2000 samples and the
true values using the original tissue perm ittivity image data. There are 35%-50%
increases in the absolute estimation values. However, the ROC performance curves
demonstrate th at this level of accuracy in interaction coefficients is adequate.
181
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0.6
0.4
5 0 .4
0.2
0.2
lower to upper, sample#(x100)
9 11 13 15 17 19
10 12 14 16 1820
0.2
0.4
0.6
Probobliity of False Alarm
lower to upper, sample#(x100)
9 11 13 15 17 19
10 12 14 16 1 8 2 0 _____
0.2
0.8
0.4
0.6
Probobliity of False Alarm
Figure 8.7: The convergence of the detection performance as a function of the
number of samples used to estimate the mean and covariance matrix. Assume the
tumor position unknown, (a) ROC curves of the BP_UPI detector using the original
tissue perm ittivity image d ata b) ROC curves of the LBP-SEMUPI detector using
the measurement data
Table 8.1: The true and the estimated interaction coefficients
l^bll
^bl2
A 22
/^613
ture
1
-0.1879 0.0191 -0.1724
estimate 1.3557 -0.2547 0.0255 -0.2337
I^b33
A ll
A 23
A i 2,22
1
ture
0.0453 0.00067
0.1
estimate 0.0607 0.0007 1.3613 0.1459
8.4.5
Perform ance o f P rop osed B ayesian Approaches
A n E xam ple
Figure 8.8(a) is an example of a stochastic background perm ittivity image of the
tissue, along with a simulated tumor, modeled by the GMRF. Figs. 8.8(b), 8.8(c),
and 8.8(d) are the reconstructed perm ittivity images from the clean measurement
d ata or from 60 dB and 50 dB noisy measurement data.
The signal detection approach th at processes the perm ittivity image data is to
compute the a posteriori probability of the tum or location given either the original
perm ittivity image data, or the reconstructed image data, with Fq. (8.29). Figure
182
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8.8(e) illustrates an upper bound on tum or localization by plotting the a posteriori
probability of tumor location using the tissue data of Fig. 8.8(a). Figure 8.8(i) pro­
vides a sub-optimal performance comparison for tumor localization by plotting the a
posteriori probability of tumor location using the tissue data of Fig. 8.8(a) and the
estimation of the mean and covariance matrix. Figures 8.8(f)-8.8(h) is the a poste­
riori plot of tum or location based on post-processing the reconstructed tissue data
shown in Figs. 8.8(b)-8.8(d).
Using the measurement data, the a posteriori probability of tum or location is
computed with Eq. (8.45). Figures 8.8(j)-8.8(l) show the a posteriori plot of tumor
location based on the same measurements to get the reconstruction in Figs. 8.8(b)8.8(d).
Plots 8.8(g)-8.8(h) show th at at 60 dB and 50 dB additive noise condition, the
BP_UPI using the reconstruction tissue data misses the correct tum or localization,
and Figs. 8.8(k)-8.8(l) show th at the BP-SEMUPI using the measurement data gets
the correct location of the tumor. This is a specific example where the BP_SEMUPI
works better.
In the following Sections, it is demonstrated statistically th at the
processors based directly on the likelihood ratio and a posteriori probabilities of the
measurement d ata have better performance.
D etectio n Perform ance
Figure 8.9 illustrates the detection performance of the proposed processors assuming
clean microwave measurement data. In Fig. 8.9, the ROC for the threshold detector
provides a perform ance lower bound.
T h e T IP m im ics a routine visu al exam ina­
tion of the image. Although it is not sophisticated, its ROC reflects the problem of
high positive predictive value(PPV) of conventional mammography. Using the recon­
structed perm ittivity image data, the BP-UPI detector is much better than the TIP
183
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e-100
e-25G
e-100
e-250
0-500
u
P(^Y'(r))
0-250
8-500
0-250
0-500
P(S| r'(r))
18-250
■©-500
Figure 8.8: (a) Original tissue perm ittivity image; (b)-(d) Reconstructed perm it­
tivity image from the measurement data, (b) without additive noise, (c) with ad­
ditive noise, SNR=60 dB, (d) with additive noise, SNR=50 dBj (e)-(h) the a pos­
teriori probability of the tum or position given the perm ittivity image data and i?i
condition-p(S' | er,Hi), d ata
comes from (a)-(d); (i) the p{S \ er,i?i) computed
using the estimation of the mean and covariance m atrix and the data from (a); (j)-(I)
the a posteriori probability of the tum or position given the measurement data and
the Hi condition, p{S | r, Hi).
184
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detector, especially when the probability of false alarm is low, because the BP_UPI
utilizes the a priori knowledge of different characteristics of normal and malignant
tissues to improve the detection performance.
Figure 8.9 also shows th at the BP_SEMUPI detector th at directly processes the
microwave measurement d ata is better than the BP_UPI detector th at post-processes
the reconstructed perm ittivity image data, yet worse than the BP_UPI detector th at
processes the original perm ittivity image data. Note th at in reality, we cannot get
the original tissue perm ittivity image directly but the EM measurements. The per­
formance of the BP_UPI th at processes the original perm ittivity image data provides
an upper bound for an ideal situation. The BP_UPI th at processes the original per­
mittivity image d ata is better than the BP_SEMUPI because the forward EM field
maps the variables from the original perm ittivity domain to the measurement domain,
which reduces the random variable space and decreases the detectablity. Proofs in
Appendix A support this explanation. In addition, the multivariate Gaussain ap­
proximation used to compute Eq. (8.36) may cause performance degradation. The
LBP-SEMUPI is better than the BP_UPI using reconstructed perm ittivity data is
because of the inherent algorithm limitations in the reconstruction procedure.
Figure 8.10 illustrates the performance of the detectors when noise is present.
Additive noise degrades the performance of all three detectors. Figure 8.10(a) shows
th at for the three SNR conditions, the BP_UPI detector is better than the TIP
detector. The ROC curve at 60 dB SNR condition is so close to the ROC curve for
a very large SNR th at we consider 60 dB as a threshold for the BP_UPI detector
using reconstruction data. Figure 8.10(b) shows th at for the three SNR conditions,
the LBP-SEMUPI detector using the measurement data is better than the BP_UPI
detector using the reconstructed tissue perm ittivity image data.
185
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1
0.8
i
0.6
(
f
1
0.4
0.2
/
/
r'' . ,
.......... 1
___ ___________ 1
______
* BP UPl.orig.
----- LBP_SEMUPI,meas.
BP_UPi,recon.
-----TIP.recon.
0.4
0.2
0.6
0.8
F ig u re 8.9: Detection performance of the BP_UPI th at processes original perm it­
tivity image data, detection performance of the BP_UPI th at processes reconstructed
perm ittivity image data, and detection performance of the TIP th a t processes the
reconstructed perm ittivity image data, and the BP_SEMUPI th at processes the mi­
crowave measurement data. No additive noise is present at sensors, tumor position
unknown.
'/
0.6
0.6
0.4
1
0.4
//
jt'
0.2
f
Y
)
0.2
upper to lower (Inf,60,50) dB
BP_UPI,recon.
-----TIP,recon.
/ ''
0.2
0.4
0.6
0.8
1
upper to lower: (Inf,60,50) dB
LBP_SEMUPI,meas.
BP_UPI,recon._____________
0.2
0.4
0.8
P,F
Pp
F ig u re 8.10: Detection performance comparisons. Tumor position unknown.
SNR=[Inf, 60,50]dB. (a) the BP_UPI and the T IP detectors using the reconstructed
tissue perm ittivity image, (b) the BP-UPI detector using the reconstructed tissue
perm ittivity image and the LBP-SEMUPI detector using the measurement data.
186
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L ocalization Perform ance
Figure 8.11 illustrates the localization performance of the LBP-SEMUPI th at pro­
cesses the microwave measurement data, and the BP_UPI and the TIP th at post­
process the reconstructed tissue perm ittivity image data. The localization perforb) window diameter =1
a) window diameter =0
=5 0.8
0.8
0.6
^0.4
0.4
0.2
0.2
40
60
20
d) window diameter =2.8284
c) window diameter =2
•■ =
0.8
0.8
- •
0.6
0.6
^0.4
0.4
0.2
0.2
jQ
20
SNRdB
40
SNRdB
Figure 8.11: The PCL performance of the proposed processors. Solid line, the
LBP-SEMUPI th at processes the microwave measurement data; dotted line, the
BP-UPI th at post-processes the reconstructed tissue perm ittivity image data; dashed
line, the TIP th at post-processes the reconstructed tissue perm ittivity image data.
mance is illustrated by the probability of correct localization (PCL) curves. The
localization is correct if the estimated position is located within the test window.
If it is required th at the estimated position is the same as the real position to be
true, the window diameter is zero. For the other values of the window diameter: 1,
2, and 2.8284, the window areas are 9, 13, and 25. Figure 8.11 demonstrates th at
at 50 dB or 60 dB SNR condition, both the BP-UPI and the LBP_SEMUPI have
187
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better performance than 0.9 PCL. The Bayesian processors are much better than the
threshold method for all the tested SNR conditions. The LBP_SEMUPI localization
using the measurement data is the best of the three. The detection and localization
performance results show th at the LBP_SEMUPI, which processes the measurement
data directly, provides a potential way to help the doctor make a diagnosis by using
the measurement d ata directly.
8.4.6
Perform ance and Array C onfigurations
In this Section, we study the effect of sensor array configurations on detection perfor­
mance. Figure 8.12 displays the deflection coefficient indexed by 2D source positions,
given th at the possible tumor position is known to the detectors.
18
17
16
16
14
J
13
15
■ ■ “
14
1
I
Figure 8.12: Deflection coefficients as a function of the tum or positions, assume
tumor position known for the detector, (a) the BP_UPI th at processes the original
tissue perm ittivity image data, (b) the EBP_UPI th at processes the original per­
mittivity data, (c)-(f) the LBP_SEMUPI th at processes the microwave measurement
d ata directly, (c) configuration 1:1:24, total of 24 sensors, (d) configuration 1:2:24,
total of 12 sensors, e) configuration 2:2:24, total of 12 sensors, (f) configuration 1:1:12,
total of 12 sensors.
Figure 8.12(a) provides the upper bound by computing the deflection coefficient
188
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of the BP-UPI using the original perm ittivity data. Figure 8.12(b) is a sub-optimal
comparison by computing the deflection coefficient of the EBP-UPI using the original
perm ittivity data. The degradation in Fig. 8.12(b) compared with th at in Fig. 8.12(a)
reflects the effect of the estimation procedure, which is independent of the tumor
positions. Figures 8.12(c)- 8.12(f) is the performances of the LBP_SEMUPI detector
for four different array configurations. The four plots indicate a strong relationship
between the detection performance and the distance of the tumor to the sensors. The
deeper the tumor is located, the more difficult it is to detect. Figure 8.13 illustrates
1
0.8
0.6
0.4
•
LBP_SEMUPI,meas.
LBP_SEMUPI,meas.
- - LBP_SEMUPI,meas.
- - - LBP_SEMUPI,meas.
BP_UPI,recon. array
0.2
0.2
0.4
0.6
array 1:1:24
array 1:2:24
array 2:2:24
array 1:1:12
1:1:24_____
0.8
1
F ig u re 8.13: Detection performance using the measurement data and the recon­
structed perm ittivity image data. Assume tumor position unknown. Comparison of
different array configurations.
the ROC detection performance when the tumor position is unknown. By comparing
the ROC curves of the first configuration with those of three other configurations
in Fig. 8.13, one sees th a t the performance degrades due to the limited number of
sensors. The ratio of the d ata length to the number of the uncertain variables is
600:529 for the first configuration and 156:529 for three other configurations. The
189
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results also indicate th a t if we lower the performance requirements, the number of the
sensors can be reduced. The plots also show th at the configuration of sensors affects
the processor’s performance. For the case where the tumor position follows a uniform
a priori distribution, it appears th at the evenly distributed array configuration yields
better performance. But if the a priori tumor position is not a uniform distribution,
we could improve the performance by having more sensors close to the possible tumor
positions.
8.5
Sum m ary
This chapter develops two Bayesian approaches for breast tumor diagnoses. One post­
processes the reconstructed tissue perm ittivity image data. The other is a physicsbased Bayesian approach th at incorporates the scattered EM measurement data into
an optimal likelihood ratio detector.
The breast perm ittivity cross section prop­
agation media is modeled by a Gaussian Markov Random Field. The simulation
results demonstrate th a t the physics-based Bayesian approach achieved better de­
tection and localization performance. The simulation results also provide an upper
bound on early detection and localization of malignant tissue, as a function of un­
certain and variable tissue perm ittivity characteristics, tumor contrast, tum or size,
and local characteristics and shape. The effectiveness of the multivariate Gaussian
distribution approximation for the measurement data to reduce the computational
complexity was illustrated. Finally, the effect of array configurations on detection
performance was discussed.
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Chapter 9
Conlusion
Realistic sonar performance prediction approaches are im portant because they not
only guide better algorithm design but also can be used to optimize system param^
eters in operational conditions. In order to accurately predict the performance of
sonar detection and localization algorithms in complicated ocean environments, the
presence of environmental uncertainty must be considered. Although a number of
detection and localization algorithms th at are robust to environmental uncertainty
have been proposed, few statistically valid performance prediction approaches have
been developed. Conventional sonar equations are often too optimistic for perfor­
mance prediction because they assume a deterministic ocean environment. Previous
sonar detection and localization performance evaluation approaches th at incorporate
environmental uncertainty often rely on Monte Carlo simulation methods, which do
not provide insights into fundamental parameters of sonar performance. In this work,
analytical approximate ROC expressions and PCL expressions have been derived to
predict optimal detection performance, sub-optimal detection performance, and opti­
mal localization performance in uncertain environments. The analytical approaches
characterize sonar detection and localization performance as a function of fundamen­
tal parameters, which can be computed much faster than algorithms th at rely on
Monte Carlo methods.
In chapter 3, an algebraic expression was derived to approximate the ROC for a
Bayesian detector th a t is a modified version of the OUFP in diffuse noise in uncertain
environments. The analytical ROC expression was verified using several benchmark
environments.
The results showed th at the optimal sonar detection performance
191
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(ROC) in diffuse noise depends primarily on the ocean environmental uncertainty,
which is captured by the rank of the signal matrix, and the mean signal-to-noise
ratio at the receivers. Accurate analytical ROC expressions were also derived for a
matched-ocean detector, a mean-ocean detector, and an energy detector. The analyt­
ical performance prediction for the mean-ocean detector illustrates the degradation
in performance due to mismatched model parameters, characterized by a correlation
coefficient between the signal wavefront due to the mean-value of the environmental
parameters and the reference signal wavefront.
The presence of interference is another im portant limiting factor for sonar detec­
tion performance. In chapter 4, an optimal Bayesian detector was developed th at
incorporates the a proiri knowledge about the uncertain positions of the interferers
and the a priori knowledge of the uncertain environment, which is a significant ex­
tension to the OUFR The analytical approximate ROC expression was derived th at
predicts the optimal detection performance in the presence of interferences in uncer­
tain environments. Several NRL benchmark environments were used to check the
proposed ROC expression. The results illustrate th at the degradation on detection
performance due to interferences is greatly magnified by the presence of environmen­
tal uncertainty and th at the Bayesian sonar detection performance depends on the
following fundamental parameters: the signal-to-noise ratio at the receivers, the rank
of the signal matrix, and the signal-to-interference coefficient.
Considering a signal model th at assumes a random phase distributed uniformly
from 0 to 27t and a fixed amplitude, which is different from the complex Guassian
signal model used in chapters 3 and 4, we developed a new Bayesian detector and
its analytical approximate ROC performance prediction expressions for uncertain en­
vironments in chapter 5. The performance prediction approach was verified using
the SWellEx-96 experimental data and simulated data using the SWellEx-96 envi192
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ronment, where source motion is the primary source of wavefront uncertainty. The
experimental results showed th at the signal source model and the acoustic propaga­
tion model are practical. The analytical ROC expression for the Bayesian detector
can be applied in a realistic circumstance to provide fast detection performance pre­
diction in which there is environmental uncertainty.
In chapter 6, analytical approximate PCL expressions for the OUFP in uncertain
environments and the matched-ocean processor in known environments were derived.
The optimal PCL expressions were formulated as a constrained integral of the proba­
bility density function of the ambiguity vector, whose covariance m atrix is determined
by the S N R at the receivers and the inner product m atrix of the localization sig­
nal matrix. The analytical PCL expressions were obtained by solving the constrained
multidimensional integrals based on some typical types of the inner product matrices.
The scenarios extended from the known and uncertain benchmark shallow water en­
vironments were used to check the approximations made by the analytical approach,
by comparing the PCL results with th a t computed using the Monte Carlo method.
The results verified th at the optimal localization performance is captured primarily
by the analytical PCL expression using three fundamental parameters: the effective
correlation coefficient, the number of hypothesized source positions, and the S N R at
the receivers.
The analytical sonar performance prediction approaches translate the range of
uncertain environmental parameters quantitatively to the uncertainty in the signal
wavefronts and then incorporate it in the analytical ROC and PCL performance ex­
pressions. W ith the analytical ROC and PCL expressions, the effect of environmental
uncertainty on sonar detection and localization performance can be quantitatively
described and compared with the effect of other factors such as noise level and prop­
agation loss. Further, the effects of various types of environmental uncertainties can
193
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be compared quantitatively. The limiting factor in the detection performance can
be identified by these quantitative comparisons, and can be used to improve sonar
system. The analytical ROC expressions derived in this dissertation can be solutions
to a general M-correlated signal detection problem, where the correlated signals are
realizations of the propagation transfer function due to perturbations of the environ­
mental parameters. And the PCL expressions derived in chapter 6 can be solutions to
the classification problems th at have similar signal wavefront structures whose inner
product m atrix belongs to one of those categories; the identity matrix, the correlated
matrix of size 2, the tri-diagonal matrix, and the equally correlated matrix.
The physics-based Bayesian detection approach was applied to diagnostic mi­
crowave imaging for detecting and localizing breast cancer. The basis for the de­
velopment of diagnostic microwave imaging techniques for breast cancer is the high
contrast between the dielectric properties of normal and malignant breast tissues
for radio through microwave frequencies. In chapter 7, we illustrated a collection of
dielectric property data on normal and malignant tissues from a number of studies
using graphics and provided a brief summary of the experiment methods and the
mechanisms th at explain the difference in the dielectric properties of normal and
malignant breast tissue. In chapter 8, we developed two Bayesian approaches. One
directly processes the raw measurements of the EM field for tum or detection and
localization by incorporating the forward propagation model of the electromagnetic
field with the Markov random fields model th at characterizes the spatial variety of
the propagation medium. Another post-processes the reconstructed tissue perm ittiv­
ity image. The ROC and PCL results computed using a large simulated data sets
showed th at the decision-aided breast tumor diagnostic algorithms based on signal
detection theory and microwave energy, which has no radiation danger at low power
levels, have the potential of providing additional information for radiologists so as
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to improve the probability of detecting breast tumors as well as their correct posi­
tions. The LBP-SEUPI gains better performance than the BP_UPI by modeling the
microwave measurements as a function of a random propagating medium, and mak­
ing decisions before the reconstruction procedure. Tumor detection and localization
performance upper bounds, in terms of the ROC and PCL, are given as a function
of tumor characteristics, including the malignant normal perm ittivity contrast and
tumor area.
In the future, sonar performance prediction approaches can be improved by mod­
eling and incorporating other realistic factors such as the limited number of support
Monte Carlo samples used by the processor. This information is im portant for real­
time sonar applications. Also, the methods for estimating the im portant performance
parameters, such as the SNR at the receivers, are im portant and need further investi­
gation. The physics-based Bayesian approach to microwave imaging diagnosis is still
in a preliminary stage. The Gaussian Markov Random Field Model can be extended
to other types of Markov Random field model to enable more flexible descriptions
of breast tissue. In addition, the Monte Carlo integration method can be further
improved to reduce the huge computational cost. The physics-based Bayesian ap­
proach can also be used together with a data fusion method so th a t the measurements
from different modalities can be correctly combined to improve the performance of
computer-aided diagnoses.
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A ppendix A
D eflection CoefRcient Inequalities
For linear projection and reconstruction medical system, the performance of the op­
timal detectors to distinguish the tumor of some known position from the normal
tissue background can be precisely represented by the deflection coefhcient if the
object image and the measurement error can be modeled by multivariate normal
distributions. In this Appendix, we derive the deflection coefficients based on the
assumptions about the known tumor position, the linear projection and reconstruc­
tion, and the multivariate normal distribution of the data. We prove inequalities
between the deflection coefficients for the optimal detectors th at respectively pro­
cess the original image data, the measurement data, and the reconstructed image
data. The effects of the forward propagation, the reconstruction procedure, and the
measurement error on optimal detection performance are discussed.
T h e o re m 1 vFC~^u > v
i
F
i
f
C is a N x N real symmetric positive
definite matrix, u is a N x 1 vector, B is a M x N matrix, and B C B ^ is non singular.
The matrix version of the well-known Cauchy-Schwarz inequality[92] is
X ^ X > lX ^ Y { Y '^ Y )-^ Y ^ X ,
(A.l)
where, X is an n x m matrix, Y is an n x g matrix, and Y ^ Y is nonsingular. > l is
the Loewner ordering relationship.
C onsidering C is a real sym m etric p ositive definite m atrix yields C = L L '^ and
where L is an Y x Y simple matrix.
Substituting X =
and Y = L XB ^ into Eq. A .l generates C ^^> lBX {B CB X)~^B , i.e.
fi^B'^[BCBX)-^Bu.
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u^C~^u >
A .l
D eflection CoefRcient after Linear Transform ation
Any linear transformation can be represented by a matrix A. Assume x is a A x 1
vector of multivariate Gaussian distribution, y is a M x 1 vector transformed from
X
by M
X
A m atrix A
y = Ax.
(A-2)
HI :
AO: a;~ N (0 o ,C '),
.
^ ‘ ^
Assume the binary hypotheses for x
where
is the A x 1 mean vector, C is the A x A covariance matrix, which is
real, symmetric and positive definite. The symbols N{a,P ) denote a multivariate
Gaussian distribution with mean vector a and covariance m atrix /5.
Using Eq. (A.2) and Eq. (A.3)
E { y I H I } = A9i
E { y I AO} = A 6o
COV{y} = ACA^.
(A.4)
We derive the deflection coefficient no m atter whether the COV{y) be singular or
non-singular. Let Ra = rank{A). There are Ra independent rows in A. we assume
A\ is composed of Ra independent rows of A, and A 2 is composed of the rest rows
of A. A2 can be represented by a linear combination of rows in Ai, i.e. A 2 = GA\ to
generate
y
yi =
j/2 =
Equation (A.5) results in
Py{y)
'T il ■
' Vi '
X
. 2/2 .
. ^2
Aix
A 2X — G A ix — Gyi-
(A.5)
— 6 {y2 — Gyi)py^{yi). Since vy 2 is determined by
uj/i, which is independent of the hypotheses, we find th at the likelihood ratio of y
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is the same as the likelihood of y i, so is the deflection coefficient. It can be proved
th at COV{yi) = A i C A ( is a real symmetric positive definite matrix. Then the
post-transformation deflection coefficient becomes
dF = A e ^ A ^ { A ^ C A j ) - ^ A i A 9 .
A .2
(A.6)
D eflection C oefflcients for T hree K inds o f D ata
1. Original image data
The binary hypotheses have been defined in Eq. (A.3), at which Hi represents
the normal tissue background with a tumor present, Hq represents the normal
tissue background only, and x in this problem is the perm ittivity data
used
for detection. The deflection coefficient with the original object data becomes
A r
=
2. Measurement data
If we consider the measurement error, and use the additive independent Gaus­
sian vector to represent it, the measurement data vr can be represented by
r
— Acr + n = \_ A
n ~
I m ~\
n
N(0,Jm<7^),
where A is a M x A matrix, I m is a M x M Identity matrix,
vector,
n
is a M
X
(A.8)
is an A x 1
1 vector. Using Eqs. (A.5) and (A.6), with A replaced by
[A I m ], X replaced by
n
, and y replaced by r, we obtain the deflection
coefficient using the measurement data
4
=
A e ^ A ^ { A i C A j + lM ,al)- ^A iA 9,
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(A.9)
where, [Ai I m i ] are composed of
independent rows of m atrix [A I m ]: R a =
rank([A I m ])3. Reconstructed image data
After reconstruction, the estimated object data ^ can be represented by
^
=
ri
T2
Br = B
= Dn,
D =
B
BG
(A.IO)
Where, r i , r 2 are correspond to yi-,y2 in Eq. (A.5). G is some matrix satisfied
T2 = Gri. Using Eqs. (A.5) and (A.6) again, with A replaced by D, x replaced
by r\ and y replaced by
we obtain the deflection coefficient of the optimal
detector using reconstruction data
=
A 0^(S iA i)^(R iA iC '(R iA i)^
+B^BlGl)-^B^A,^^,
where Bi is composed of R b independent rows of D, R b = rank{D).
A .3
Inequality R elationsh ip s b etw een D eflection C oefficients
We have
= A9'^G~^A9. Assume
Apply theorem 1, we get dl >
= 0, we have
(A iC'A f)“ ^ i A d .
It can be proved th at d^ is inversely proportional
to (j2, or d^ < d^^, so we get
dl > d^.
Let A 0 = AiAd and E = A i G A j +
(A.12)
and substitute them into Eq. A.9 and
Eq. A .ll, we have
dl
d^
=
AQ '^E-^A Q
A 0 ^ (R i)'^ (R iE R f)-iR iA 0 .
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(A.13)
Since E is real symmetric positive definite, B E B ^ is non singular, apply theorem 1,
we get
dl, > dl-
A .4
(A. 14)
Tw o S pecial Cases
1. Case 1: Assume the measurement error is not present, and the only uncertainty
is the normal and malignant tissue properties.
Let al = 0. Using Eq. A.9 and Eq. A .ll,
ct
= A 0^A f(A ,C A l)-'A ,A 0
4
= AO‘^ (BiAif(BiAiC{BiAif)-^BiAiAe.
^
’
The inequality relationships can be inherited from Eq. (A.14) and Eq. (A.12)
to get
d l > d l , > dl2. Case 2: Assume signal
(A.16)
under binary hypotheses are determined, in stead of
random. The uncertainty in this problem is the additive measurement noise
only.
Let C be Zero m atrix and let A replace Ai. Using Eq. A.9 and Eq. A .ll,
dl
A0
=
=
=
A Q '^ A Q /a l
A e '^B f(B iB l)-^B iA e/al
AA9.
(A.17)
The inequality relationship can be inherited from Eq. (A. 14) to get
dl > dl
The
two
forms ofthe deflection coefficients and their inequality relationships
(A.18)
for the
special cases can also be derived directly from the corresponding special binary
hypotheses.
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A.5
Discussion
The form of the defiection coefficients for special case 1, and their inequality rela­
tionships demonstrate th a t it is the reduction of the random variable space during
the forward and reconstruction procedures th at causes the decrease of the detection
performance. The detection performance of the optimal detector using the recon­
struction d ata cannot be better than th at using the measurement data, even without
any measurement noise.
The form of the deflection coefficients for special case 2 demonstrates th at dm and
dr are inversely proportional to cr^, which reflects the direct effect of the measurement
noise on the detection performance.
In addition, the forms and their inequality
relationships also demonstrate th at the inadequate rank of the reconstruction matrix
5 , which is less than the length the measurement data, reduces the random variable
space in dealing with the uncertainty of the measurement noise, and causes the
decrease of the detection performance at the reconstruction domain. The effect of the
measurement error on the detection performance has been negatively strengthened
by the reconstruction procedure. This conclusion agrees well with the simulation
result in [40].
The more general form of dm and dr in Eq. (A.9) and Eq. (A .ll) clarifies the
effects of the combination of the projection and reconstruction procedures as well
as the measurement noise. The inequalities in Eqs. (A.14) and (A.12) illustrate the
degradation of the detection performance during the processing procedures. This
general conclusion is consistent with our simulation results for the performance of
the microwave imaging system.
Clarifying the factors th at cause the degradation of the detection performance
could help us design a better system. It suggests th at good sampling of the mea-
201
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
surement data, which maximize the random variable space in dealing with the uncer­
tainties (i.e. increase the rank of A), will improve the detection performance. If the
measurement error cannot be ignored, the length of the measurement data should be
limited (i.e. decrease the difference in rank of B and I m )
reduce the degradation
of post-processing the reconstructed image. It also suggests th a t making optimal
decisions before reconstruction will improve the detection performance.
202
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A ppendix B
Source Code for Chapter 3
B .l
A n alytical Perform ance P red iction
%---------------------------------------------------------------------
% R_orth predict the probability of detection as a function of
% the probability of false alarm, the signal to noise ratio
% at the receivers, and the rank of the signal matrix
%
% input PF probability of false alarm
% SNR signal to noise ratio at the receivers
% R rank of the signal matrix
%
% output PD probability of detection
%
% 06/11/02 Liewei Sha
%
-
function PD = M_orth(PF,SNR,R)
X = (1 -P F )/(1 ./R );
PD = l-(l-(l-x )N (l./(S N R + l))).* x /(R -l);
% Est_det estimate the signal-to-noise ratio at the receivers and
% the rank of the signal m atrix
%
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% input s a m atrix of size M x N, representing an ensemble of
% the original signal wavefronts where
%
is the number of the array components and M is the
% number of the signal wavefront samples.
% SIGMAN2 the variance of the diffuse noise.
% th a percentage value in the range of [0.01 0.1], used for
% the rank estimation
% output SNR the estimated signal-to-noise ratio at the receivers
% R the estimated rank of the signal matrix
% 06/11/02 Liewei Sha
-
%
function [SNR, R] = Est_det(s,SIGMAN2,th)
% covariance matrix
C = s’*s;
C C = abs(diag(C));
% eigenvalues
e = -sort(-real(eig(C)));
SNR = mean(CC)/SIGMAN2;
R =length(find(e>e(l)*th));
B .2
M on te Carlo Perform ance E valuation
% ROC-opt use the detection model th at assumes the signal amplitude is a
% complex Gaussian random variable and the diffuse noise is a complex
% Gaussian random vector.
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% The simulated d ata is used to compute samples of the likelihood
% ratio.
% And the ROC performance is obtained from the statistics of the
% likelihood ratio for Hi and H q hypotheses.
%
% input
% PF probability of false alarm
% ss(NxM): N number of the array elements, M number of the samples,
% reference signal wavefronts used by the processor
% ssr(NxM): N number of the array elements, M number of the samples,
% the signal wavefronts used to generate the simulated data
% SIGMAN2; diffuse noise variance
% RepeatN: the number of Monte Carlo samples
%
% output
% PD probability of detection
% 06/11/02 Liewei Sha
-
%
function PD = ROC_opt(PF,ssr,ss,SIGMAN2,RepeatN)
randn(’state’,sum(100*clock));
rand (’state’,sum (100*clock));
N = size(ss,l);
M=size(ssr,2);
Ess = abs(diag(ss’*ss)’);
ssl = ss./sqrt(Ess(ones(N,l),:));
%normalized so th a t s s l’ssl= L M
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SNR_s = Ess/SIGMAN2;
SNR_s2 = sqrt(SNR-s);
Tsub = l+SNR_s;
LTsub =log(Tsub);
seeds = fix(rand(l,R6peatN )*M )+l;
Essr = abs(diag(ssr’*ssr)’);
SNR_r2 = sqrt(Essr/SIGMAN2);
ssrl = ssr./sqrt(Essr(ones(N,l),0);
for i=l:R epeatN
a = (randn(l)+sqrt(-l)*randn(l))*SNR_r2(seeds(i));
rO = ran d n (N ,l)+ sqrt(-l)*randn(N ,l);
r l = a.*ssrl(;,seeds(i))+rO;
L0(i)=addexp(abs(r0’*ssl)A2./Tsub-LTsub);
Ll(i)=addexp(abs(rr*ssl)A 2./T sub-LT sub);
end
PD = ROC-PD(L1,LO,PF);
%----------------------------------------------------------------------------
% ROC-PD compute the ROC curve using samples of the likelihood
% ratio for Hi and H q hypotheses.
%
% input
% PF the probability of false alarm
% LLl N samples of the likelihood ratio for the Hi hypothesis
% size Ix N
% LLO N samples of the likelihood ratio for the H q hypothesis
% size Ix N
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% output
% 01/01/02 Liewei Sha
%----------------------------------
function PD = ROC_PD(PF,LL1,LLO)
RepeatN = length(LLl);
[beta I] = sort([LLl LLO]);
beta = [ones(size(LLl)) zeros(size(LLO))];
clear c;
c = beta(I);
pdO = cumsum (c(size(c,2);-l:l),2)/RepeatN;
pfO = [l:size(c,2)]/RepeatN-pdO;
for kk= l:length(P F )-l
[a b]=min(abs(pfO-PF(kk)),[],2);
PD(l,kk)=pdO(b);
end
P D (l,length(pft))= l;
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A ppendix C
Source Code for Chapter 4
C .l
A n alytical Perform ance P red iction
% ROC-infe predict detection performance PD based on the detection model
% defined in chapter 4 th a t assume
% known signal wavefront
% known interference wavefront
% known variance of signal source
% known variance of interference source
% known variance of diffuse noise
%
% input
% PF the probability of false alarm
% SNRs signal to noise ratio at the receivers
% SNRi interference to noise ratio
% rho correlation coefficient between the signal wavefront and the
% interference wavefront
% output
% PD the probability of detection
% G the signal to interference coefficient
% 07/01/03 Liewei Sha
%
-
function [PD,G]=ROC_infe(PF,SNRs,SNRi,rho)
k = SN Ri./(l+SN Ri);
rho2 = abs(rho)N2;
G = l-k.*rho2;
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PD = P F .(l./(S N R s.* (l-k *rho2)+l));
% ROC-UnJnfe
% predict detection performance PD based on the detection model
% defined in chapter 4 th at assumes an uncertain
% environment.
%
% input
% ss a m atrix of size M x N , representing an ensemble of
% the original signal wavefronts where N is the number of the
% array components and M is the number of the signal wavefront
% realizations.
% sf a m atrix of size M x N , representing an ensemble of
% the original interference wavefronts where N is the number of the
% array components and M is the number of the interference wavefront
% realizations.
% SIGMAN2 the variance of the diffuse noise.
% SIGMAF2_SIGMAA2 the ratio of the interference variance to signal
% variance.
% t a percentage value in the range of [0.01 0.1], used for
% the rank estimation.
% PF the probability of false alarm.
%
% output
% PD the probability of detection
% R estimation of the rank of the signal matrix
% SNR estimation of the signal-to-noise ratio at the receivers
% G estimation of the signal to interference coefficient
%
% 07/01/03 Liewei Sha
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function [PD,R,SNR,G]=ROC-Un_infe(ss,sf,SIGMAN2,SIGMAF2_SIGMAA2,t,PF)
N = size(sf,l);
M = size(sf,2);
Ess = abs(diag(ss’*ss)’);
ssl = ss./sqrt(Ess(ones(N,l),:));
%normalized so th at ssrssl= I_M
SNR_s = Ess/SIGMAN2;
SNR_s2 = sqrt(SNR_s);
Esf = abs(diag(sf’*sf)’);
sfl = sf./sqrt(Esf(ones(N ,l),;));
SN RJ = Esf/SIGMAN2.*SIGMAF2_SIGMAA2;
SNR_12 = sqrt(SN R j);
SNR = mean(SNR-s);
e = -sort(-real(eig(ss’*ss)));
R = length(find(e>e(l)*t));
MO = sfl*diag(SNR_i)*sfr/M +eye(N);
IMG = inv(MO);
for i= l:M
efs(i) — abs(ssl(:,i)’*IMO*ssl(:,i));
end
G = mean(efs);
PD=M_orth(PF,SNR*G,R);
C .2
M on te Carlo Perform ance E valuation
ROC-UnJnfe_MG
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% predict the detection performance of the Bayesian detector in the
% presence of interference in uncertain environments using Monte
% Carlo method. The interference model for uncertain environments is
% defined in chapter 4.
% input
% ssr(NxM): an ensemble of the signal wavefronts used to generate the
% simulated data, N is the number of array elements, M is the number of
% sample wavefronts.
% sfr(NxM); an ensemble of the interference wavefronts used to generate
% the simulated data, N is the number of array elements, M is the number
% of sample wavefronts.
% ss(NxM); an ensemble of the signal wavefronts used by the processor,
% N is the number of array elements, M is the number of sample wavefronts.
% sf(NxM): an ensemble of the interference wavefronts used by the processor,
% N is the number of array elements, M is the number of sample
wavefronts.
% SN2: diffuse noise variance
% SFA2: the ratio of the interference source variance to the
% signal source variance
%
%output
% PD the probability of detection
%
% 03/06/03 Liewei Sha
function P D = R O C -U n_infe_M C (ssr,sfr,ss,sf,SN 2,SFA 2,R epeatN )
randn(’state’,sum(100*clock));
rand(’state’,sum(100*clock));
N = size(ss,l);
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M=size(ss,2);
K=size(sf,2)/M;
Ess = abs(diag(ss’*ss)’);
ssl = ss./sqrt(Ess(ones(N,l),:));
%normalized so th at s s l’ssl=I_M
SNR_s = Ess/SN2;
SNR_s2 = sqrt(SNR_s);
Esf = abs(diag(sf’*sf)’);
sfl = sf./sqrt(Esf(ones(N,l),:));
SNRJ = Esf/SN2.*SFA2;
SNRJ2 = sqrt(SN RJ);
MO = sfl*diag(SN R J)*sfr/M +eye(N );
%NxN m atrix
IMG = inv(MO);
Tsub = l+SNR_s.*abs(diag(ssr*IM O*ssl))’;
LTsub =log(Tsub);
Gsub = SNR_s2(ones(N,l),0-*(IMO*ssl);
seeds = fix(rand(l,RepeatN )*M )+l;
Essr = abs(diag(ssr’*ssr)’);
ssrl = ssr./sqrt(Essr(ones(N ,l),:));
Esfr = abs(diag(sfr’*sfr)’);
sfrl = sfr./sqrt(Esfr(ones(N ,l),0);
for l=l:R epeatN
b = (randn(l,K )+sqrt(-l)*randn(l,K )).*SN RJ2(seeds(i):M ;end);
a = (randn(l)+sqrt(-l)*randn(l))*SNR_s2(seeds(i));
n = randn(N ,l)+ sqrt(-l)*randn(N ,l);
rO = sum(b(ones(N,l),:).*sfrl(:,seeds(i):M ;end),2)+n;
%size Nxl
r l = a.*ssrl(;,seeds(i))+rO;
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LO(i) = addexp{abs(rO’*Gsub).''2./Tsub-LTsub);
Ll(i) = ad d ex p (ab s(rr* G su b )/2./Tsub-LTsub);
end
PD = ROC-PD(L1,LO,PF);
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A ppendix D
Source Code for Chapter 5
D .l
A n alytical Perform ance P red iction
% R_orth_RMB
% predict the probability of detection as a function of
% the probability of false alarm, the signal to noise ratio
% at the receivers, and the rank of the signal matrix, assume
% the signal is with known amplitude and unknown phase uniformly
% distributed from 0-27T. The detection model is defined in
% chapter 5
%
% input
% PF probability of false alarm
% SNR signal to noise ratio at the receivers
% R rank of the signal matrix
% note SNR and PF have the same dimension, or one is scalar, the other is % matrix
or vector, marcumq is a function in MATLAB.
% output
% PD probability of detection
%
% 07/31/03 Liewei Sha
_
%
function PD = M_orth_RMB(PF,SNR,R)
x = ( l- P F ) .^ ( l./R ) ;
PDO = m arcum q(sqrt(SN R),sqrt(2*log(l./(l-x))));
PD = l-(l-PD O).*x.^(R-l);
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% est_SNR
% estimate the signal-to-noise ratio at the receivers using
% Hi d ata samples and H q d ata samples, assume th at the
% interference is absent.
%
% input
% rrl A N xL matrix, the H i data in frequency domain, obtained by
% processing the successive snapshots using 8192 FFT. N is the
% number of array elements. L is the number of snapshots.
% rrO A N xL matrix, the H q data in frequency domain, obtained by
% processing the successive snapshots using 8192 FFT. N is the
% number of array elements. L is the number of snapshots.
% output
% SNR the signal-to-noise ratio at the receivers.
%
% 03/06/03 Liewei Sha
-
%
function SNR = est_SNR(rrl,rrO)
Lei = sum (abs(rrl)."2,l);
LeO = sum (abs(rr0)."2,l);
SNR = (sum(Lel)-sum(LeO))/sum(LeO)*size(rrl,l);
D .2
M on te Carlo Perform ance E valuation
%----------------------------------------------------------------------------------
% ROC_Un_exp_RMB
% predict the detection performance for the Bayesian detector
% in uncertain environments using experimental data. The detection
215
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% model is defined in chapter 5
%
% input
% r l A N xL matrix, the Hi data in frequency domain, obtained by
% processing the successive snapshots using 8192 FFT. N is the
% number of array elements. L is the number of snapshots.
% rO A N xL matrix, the H q data in frequency domain, obtained by
% processing the successive snapshots using 8192 FFT. N is the
% number of array elements. L is the number of snapshots.
% ss a m atrix of size M x N, representing an ensemble of
% the original signal wavefronts where N is the number of the
% array components and M is the number of the signal wavefront
% realizations.
%
% SNR the signal-to-noise ratio at the receives
% RepeatN the number used for Monte Carlo integration
% PF the probability of false alarm
% output
% PD the probability of detection
%
% 03/06/03 Liewei Sha
-
%
function PD = ROC_Un_exp_RMB(rl,rO,ss,SNR,RepeatN,PF)
randn(’state’,sum(100*clock));
rand(’state’,sum(100*clock));
N = size(rl,l);
L=size(rl,2);
Ess = abs(diag(ss’*ss)’);
ssl = ss./sqrt(Ess(ones(N,l),:));
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%normalized so th at s srss l= L M
SNR_s = SNR;
SNR_s2 = sqrt(SNR_s);
seeds = fix(rand(l,R epeatN )*L)+l;
j= sq rt(-l);
for i=l:R epeatN
LO(i) = addexp(log(besseli(0,SNR_s2.*abs(r0(:,seeds(i))’*ssl)))-SNR_s/2);
Ll(i) = addexp(log(besseli(0,SNR_s2.*abs(rl{:,seeds(i))’*ssl)))-SNR_s/2);
end
PD = ROC-PD(L1,LO,PF);
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A ppendix E
Source Code for Chapter 6
E .l
A n alytical Perform ance P red iction
% gamma_orth
% predict the localization performance of identifying one of the M
% orthogonal signal wavefronts in complex Gaussian noise.
% The localization model is defined in chapter 6
% input
% SNR signal-to-noise ratio at the receivers
% M number of signal wavefronts
% output
% PCL the probability of correct localization
% 08/07/02 Liewei Sha
-
%
function PCL = gamma_orth(SNR,M)
PCL=ones(size(SNR));
A = SNR+1;
for i= l:M -l
PCL = PC L.*(i*A )./(i*A +l);
end
%---------------------------------------------------------------------------------
% gamma2a
% predict the PCL of identifying one of two correlated signal wavefronts
% in complex Gaussian noise, two gamma-type correlated variable
%
% input;
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% SNR the signal-to-noise ratio at the receivers
% rho2 absolute square of the correlation coefficient between the two
% signal wavefronts.
%
% 08/07/02 Liewei Sha
-
%
function PCL = gamma2a(SNR,rho2)
SNR =SNR(:);
PCL = SN R*sqrt(l-rho)./sqrt((SNR+2).^2-SNR.^2*rho)/2+0.5;
%-------------------------------------------------------------------% PCL_sl
% predict the PCL of identifying one of M signal wavefronts
% in complex Gaussian noise, where the first and the second signal
% wavefronts have strong correlation, and all others are orthogonal
% to each other.
%
% input:
% SNR the signal-to-noise ratio at the receivers
% M the number of candidate signal wavefronts
% rho2 absolute square of the correlation coefficient between the
% first two signal wavefronts.
%
% output
% PCL the probability of correct localization
%
% 10/18/02 Liewei Sha
-
%
function PCL=PCL_sl(SNR,M,rho2)
LR = length(SNR);
Lrho = length(rho2);
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for k=0;M-2
a (k + l) = nchoosek(M-2,k)*(-l)k;
end
if Lrho = = 1
rhol = sqrt(l-rho2);
SNR=SNR(:);
K = 0:M-2;
A = 1+(SNR+1)*K;
B = SNR*ones(l,length(K))-(SNR+l)*K;
C = (SNR+l)*K +(SNR+2)*ones(l,length(K ));
D = sq rt(C /2 -rh o 2 * B /2 );
P C L = sum ((ones(L R ,l)*a).*(l+ B *rhol./D )./A ,2)/2;
else
rho2 = rho2(;);
rhol = sqrt(l-rho2);
K = 0:M-2;
A = ones(L rho,l)*(l+(SN R +l)*K );
B = (SNR-(SNR+1)*K);
C = ones(Lrho,l)*((SN R +l)*K +(SN R +2));
D = sqrt(CA 2-(rho2*B)/2);
PC L =sum ((ones(L rho,l)*a).*(l+(rhol*B )./D )./A ,2)/2;
end
%----------------------------------------------------------------------------------
% PCL_eq
% predict the approximate PCL of identifying one of M equal correlated
% signal wavefronts in complex Gaussian noise.
%
% input;
% SNR the signal-to-noise ratio at the receivers
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% M the number of candidate signal wavefronts
% rho the correlation coefficient between the signal wavefronts.
%
% 10/18/02 Liewei Sha
%------------------------------------------------------------function PCL_eq(SNR,R,rho)
gamma = 1.16*R."(-0.49)+1.17;
rhol = rho. "gamma;
SNR =SNR(;);
PCLO = SNR*sqrt(l-rho)./sqrt((SNR+2)."2-SNR."2*rho)/2+0.5;
A = PCL0./(1-PCL0);
PCL = gamma_orth(A-l,M);
%----------------------------------------------------------------------------
% Est-det estimate the signal-to-noise ratio at the receivers and
% the rank of the signal m atrix
%
% input s a m atrix of size M xN, representing an ensemble of
% the original signal wavefronts where
% N is the number of the array components and M is the
% number of the signal wavefront samples.
% SIGMAN2 the variance of the diffuse noise.
% th a percentage value in the range of [0.01 0.1], used for
% the rank estimation
% output SNR the estimated signal-to-noise ratio at the receivers
% R the estimated rank of the signal matrix
%
% 06/11/02 Liewei Sha
%
.
function rho = est_rho(s,th)
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cl = s*s’;
el = sort(abs(eig(cl)));
R = length(find(el>el(end)*th));
Esl = diag(s’*s);
for k=l:size(s,2)
rhol(k,:) = abs(s(:,k)’* s)./sq rt(E sr)/sq rt(E sl(k ));
[temp,I]=sort(-rhol(k,:));
m rhol(k)=m ean(rhol(k,I(2:R ))/2);
end
rho = mean(m rhol);
E.2
M onte Carlo Perform ance E valuation
%-------------------------------------------------------------------------
% PCL_MC
% predict the localization performance of the matched-ocean
% processor using Monte Carlo method.
%
% input
% s a m atrix of size M xN, representing an ensemble of
% the original signal wavefronts where N is the number
% of the array components and M is the number of source
% positions.
% S1GNAM2
% the variance of the diffuse noise.
% output
% PCL the probability of correct localization, of size 1x3
% each value corresponds to a specific window. If the estimated
% location is inside the window centered at the true source position
% the localization is considered correct.
222
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06/11/02 Liewei Sha
function PCL = PCL_MC(s,SIGMAN2)
randn(’state’,sum(100*clock));
rand(’state’,sum(100*clock));
SIGMAN = sqrt(SIGMAN2);
sigmaa2 = 1;
sigmaa = sqrt(sigmaa2);
F = sigmaa2/SIGMAN2;
RepeatN=2000;
Es = abs(diag(s’*s)’);
A = F*E s+l;
sa = 2*A*sigmaa2;
nsl = size(s,l);
Es2=Es*0.5;
LA = log(A);
M = size(s,2);
Org = fix(rand(l,RepeatN )*M +l);
for repeat = l:R epeatN
n = (randn(nsl,l)+sqrt(-l)*randn(nsl,l))*SIG M A N ;
a = (randn(l)+sqrt(-l)*randn(l))*sigm aa;
r = a*s(:,Org(repeat))+n;
temp = abs(E*r’*s)A2./sa-LA;
[amp,pos] =m ax(tem p);
Pos (repeat)=pos;
end
PG L(l) = count (Org,Pos,0)/RepeatN;
PGL(2) = count (Org,Pos, l)/R epeatN ;
PGL(3) = count(Org,Pos,2)/RepeatN;
223
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
% PCL_opt
% predict the localization performance of the OUFP in uncertain
% environments using Monte Carlo method.
%
% input
% s a m atrix of size M l x M2 x M3,
% representing an ensemble of the original signal wavefronts used
% by the OUFP, where M l is the number of the array components,
% M2 is the number of candidate source positions and M3 is the
% number of the realizations of ocean environmental parameters.
% st a m atrix of size M lx M t, representing an ensemble of the
% original signal wavefronts used to generate the data. Mt is
% number of different realizations of the signal wavefronts.
% PosO PosO is vector of size Mt. Each PosO(i) is the source position
% index to the ith column signal wavefront in st.
% SIGMAN2
% the variance of the diffuse noise.
% output
% PCL the probability of correct localization, of size 1x3
% each value corresponds to a specific window. If the estimated
% location is inside the window centered at the true source positio
% the localization is considered correct.
% 10/20/02 Liewei Sha
-
%
function [PCL,Pos] = PCL_opt(s,st,PosO,SIGMAN2)
randn(’state’,sum(100*clock));
rand(’state’,sum(100*clock));
SIGMAN = sqrt(SIGMAN2);
224
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
sigmaa2 = 1;
sigmaa = sqrt(sigmaa2);
F = sigmaa2/SIGMAN2;
M l = size(s,l);
M2 = size(s,2);
M3 = size(s,3);
for 1=1 :M3
Es(:,i) = abs(diag(s(:,;,i)’*s(:,:,i))’);
end
A = F *E s+ l;
sa = 2*A*sigmaa2;
Es2 = Es*0.5;
LA = log(A);
Mt=size(st,2);
Rep = fix(2000/Mt);
RepeatN = Mt*Rep;
Org = reshape(ones(Rep,l)*[l:M t],l,RepeatN);
PosO = reshape(ones(Rep,l)*PosO,l,RepeatN);
for repeat = l:R epeatN
n = (randn(M l,l)+sqrt(-l)*randn(M l,l))*SIG M A N ;
a = (randn(l)+sqrt(-l)*randn(l))*sigm aa;
r = a*st(:,O rg(repeat))+n;
for 13 = 1;M3
temp(13,:) = abs(F*r’*s(:,:,13))A2./sa(;,13)’-LA(:,13)’;
end
[amp,pos] =m ax(m ax(tem p, 0,1));
Pos (repeat) = pos;
end %repeat
PG L(l) = count(PosO,Pos,0)/RepeatN;
225
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PCL(2) = count(PosO,Pos,l)/RepeatN;
PCL(3) = count(PosO,Pos,2)/RepeatN;
%--------------------------------------------------------------------------
% count
% computed the probability th at th at I and J are
% within the window win*win
%
% input
% 1(2 X N) is 2D coordinates for true source position
% J(2x N) is 2D coordinates for the locations under test
% N is the number of trials
% output
% 1 the probability of correlate localization in both coordinates
% 11 the probability of correlate localization in the first coordinates
% 12 the probability of correlate localization in the second coordinates
% 07/05/01 Liewei Sha
-
%
function [l,ll,12]=count(I,J,win)
if siz e (I,l)= = l
len = length(find(l(l,;)>0));
1 = length(find((abs(I(l,;)-J(l,:))<=w in) & I(1,;)>0 & J(1,;)>0));
11= 0 ;
12 = 0 ;
else
len = length (find (I (2,:) >0 & I(1,:)>0));
I = length(find((abs(I(l,:)-J(l,:))<=w in) & (abs(I(2,:)-J(2,:))<=win) ..
& I(2,:)>0 & I(1,:)>0 & J(2,:)>0 & J(1,;)>0));
II = length(find((abs(I(l,:)-J(l,:))<=w in) & I(1,:)>0 & J(1,:)>0 ));
12 = length(find((abs(I(2,:)-J(2,:))<=win) & I(2,:)>0 & J(2,:)>0 ));
226
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end
227
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Biography
Liewei Sha was born in Shanghai, China on March 28, 1972. Liewei received her
Bachelor of Science in Electronic Engineering from Tsinghua University in July, 1995.
During her undergraduate study, she received first class fellowship from Tsinghua
University in 1993 and 1994 and she was a member of Tsinghua military band.
She received her Master of Engineering in signal and information processing from
Institute of Acoustics, Chinese Academy of Sciences in August 1998, and her Master
of Science in Electrical Engineering from Duke University in May 2000. She received
full scholarship from Duke University in September 1998-August 2003. She has been
a Firmware Engineer for GE Medical Systems since August 2003. Her publications
include:
REFERRED JOURNAL PUBLICATIONS
[1] Liewei Sha and L. W. Nolte, “Effects of environmental uncertainties on sonar
detection performance prediction,” J. Acoust. Soc. Am. (Under Review, Sub­
m itted May, 2003).
[2] Liewei Sha and L. W. Nolte, “Bayesian sonar detection performance prediction
in the presence of interferences in uncertain environments,” J. Acoust. Soc.
Am. (Submitted Oct., 2003).
[3] Liewei Sha and L. W. Nolte, “Bayesian sonar detection performance prediction
with environmental uncertainty using SWellEx-96 vertical array data,” IEEE
J. Oceanic Eng. (Submitted Feb., 2004).
[4] Liewei Sha and L. W. Nolte, “Analytical approximations of Bayesian optimal lo­
calization performance prediction in uncertain environments,” IEEE J. Oceanic
Eng. (Submitted Feb., 2004).
[5] Liewei Sha, L. W. Nolte, Z. Q. Zhang and Q. H. Liu, “Incorporating statistical
models of tissue perm ittivity into an optimal signal detection theory frame237
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
work for the detection of breast cancer at microwave frequencies,” IEEE trans.
Medical Imaging (Submitted Dec., 2003).
[6] Liewei Sha, H. Guo and A. Song “ An Improved Gridding Method for Spiral
MRI Using Nonuniform Fast Fourier Transform,” Journal of Magnetic Reso­
nance 162, 250-258 (2003)
REFERRED CONFERENCE PROCEEDINGS AND ABSTRACTS
[1] Liewei Sha, L. W. Nolte, Z. Q. Zhang and Q. H. Liu, “Performance analysis for
Bayesian microwave imaging in decision aided breast tum or diagnosis,” IEEE
International Symposium on Biomedical Imaging. Proceedings 1039 -1042 (July
2002)
[2] Liewei Sha, Erika Ward and Brandon Stroy “A review of dielectric properties
of normal and malignant breast tissue,” IEEE SoutheastCon 2002 Proceedings
,457 -462(2002)
[3] L. W. Nolte, S. L. Tantum, and Liewei Sha, “Incorporating uncertainty in ocean
acoustics for optimum signal detection,” J. Acoust. Soc. Am. 109 2382 (2001)
[4] Renhe Zhang, Zhenge Sun and Liewei Sha etc, “ Normal-mode analysis for
signal fluctuations in the Yellow Sea,” J. Acoust. Soc. Am. 101 3180 (1997)
PRESENTATIONS
[1] Liewei Sha and L. W. Nolte, “ Incorporating environmental uncertainty into
Bayesian sonar detection performance prediction,” Underwater Acoustic Signal
Processing Workshop (Sep.,2003)
[2] Liewei Sha and L. W. Nolte “ A computer-aided algorithm for breast tumor
diagnosis using microwave diffraction measurements,” Era of Hope Conference
Concept Award (Sep., 2002)
238
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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