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

код для вставкиСкачатьBAYESIAN 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3177855 Copyright 2004 by Sha, Liewei All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI UMI Microform 3177855 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Copyright (c) 2004 by Liewei Sha All rights reserved Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission o f the copyright owner. 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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 Reproduced with permission of the copyright owner. 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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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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, 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 ). 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. , 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. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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)) 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. * * * 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ■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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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]}). 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 167 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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). 5 168 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 169 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 170 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 171 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 172 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 174 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). 175 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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. 176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 178 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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]. 180 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 190 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 195 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 196 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 197 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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, 198 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 . 199 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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. 200 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 % 203 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % 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. 204 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % 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 205 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 206 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % 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; 207 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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; 208 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 209 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 210 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % 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); 211 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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; 212 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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); 213 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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); 214 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % 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),:)); 216 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. %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); 217 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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; 218 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % 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); 219 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 220 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % 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) 221 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. end 227 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography [1] H. P. Bucker, “Use of calculated sound fields and matched-field detection to locate sound sources in shallow water,” J. Acoust. Soc. Am, vol. 59, no. 2, pp. 368-373, 1976. [2] F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational Ocean Acoustics, AIP series in Modern Acoustics and Signal Processing. AIP Press, New York, 1994, ISBN 1-56396-209-8. [3] A. Tolstoy, “Sensitivity of matched-field processing to sound-speed profile mis match for vertical arrays in a deep-water pacific environment,” J. Acoust. Soc. Am., vol. 85, pp. 2394-2404, 1989. [4] E. C. Shang, “Environmental mismatching effects on source localization pro cessing in mode space,” J. Acoust. Soc. Am ., vol. 89, no. 5, pp. 2285, Nov 1991. [5] G. B. Smith, H. A. Chandler, and C. Feuillade, “Performance stability of highresolution matched-field processors to sound-speed mismatch in a shallow-water environment,” J. Acoust. Soc. Am, vol. 93, no. 5, pp. 2617-2626, March 1992. [6] H. Schmidt, A.B. Baggeroer, W.A. Kuperman, and E.K. Sheer, “Environmen tally tolerant beamforming for high-resolution matched field processing: Deter ministic mismatch,” J. Aoust. Soc. Am., vol. 88, pp. 1802-1810, 1990. [7] A. M. Richardson and L. W. Nolte, “A posteriori probability source localization in an uncertain sound speed, deep ocean environment,” J. Acoust. Soc. Am., vol. 89, pp. 2280-2284, 1991. [8] Jeffrey L. Krolik, “Matched field minimum variance beamforming in a random ocean channel,” J. Aoust. Soc. Am., vol. 92, pp. 1408-1419, 1992. [9] N. Lee, L. M. Zurk, and J. Ward, “Evaluation of reduced-rank, adaptive matched field processing algorithms for passive sonar detection in a shallow-water envi ronment,” Signals, Systems, and Computers, 1999. Conference Record of the Thirty-Third Asilomar Conference on, vol. 2, pp. 876 -880, 1999. [10] J. A. Shorey, L. W. Nolte, and J. L. Krolik, “Computaionally efficient Monte Carlo estimation algorithm for matched field processing in uncertain ocean en vironments,” J. Comput. Acoust, vol. 2, pp. 285-314, 1994. 228 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [11] V. Premus, D. Alexandrou, and L. W. Nolte, “Full-field optimum detection in an uncertain, anisotropic random wave scattering environment,” J. Aoust. Soc. Am., vol. 98, no. 2, pp. 1097, August 1995. [12] M. Wazenski and D. Alexandrou, “Active, wideband detection and localization in an uncertain m ultipath environment,” J. Acoust. Soc. Am., vol. 101, no. 4, pp. 1961, Oct. 1997. [13] F.-P.A. Lam, F.P.A Benders, P. Schippers, and S.P.Beerens, “Environment adaptation for LFAS and corresponding 3d modeling of sonar performance,” UDT conference in la Spezia, 2002. [14] A. 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Machine In te l!, vol. PAMI-6, no. 6, pp. 721-741, Nov. 1984. [90] Z. Q. Zhang and Q. H. Liu, “Reconstruction of axisymmetric media with an ffht enhanced extended born approximation,” Inverse Problems,invited paper, vol. 16, no. 5, pp. 1281-1296, 2000. 235 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [91] P. M. Berg and R. E. Kleinman, “A contrast source inversion m ethod,” Inverse Problems, vol. 13, pp. 1607-1620, 1997. [92] J.S. Chipman, “On least squares with insufficient observations,” Statist. Assoc., vol. 59, pp. 1078-1111, 1964. 236 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. J. Amer. 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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