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Statistical Multisource-Multitarget Information Fusion Ronald P. S. Mahler ARTECH H O U S E BOSTON LONDON artechhouse.com

Contents Preface Acknowledgments xxm xxv Chapter 1 Introduction to the Book 1 1.1 What Is the Purpose of This Book? 1 1.2 Major Challenges in Information Fusion 7 1.3 Why Random Sets or FISST? 8 1.3.1 Whylsn't Multitarget FilteringStraightforward? 9 1.3.2 Beyond Heuristics 10 1.3.3 How Do Single-Target and Multitarget Statistics Differ? 11 1.3.4 How Do Conventional and Ambiguous Data Differ? 11 1.3.5 What Is Formal Bayes Modeling? 13 1.3.6 How Is Ambiguous Information Modeled? 13 1.3.7 What Is Multisource-Multitarget Formal Modeling? 14 1.4 Random Sets in Information Fusion 15 1.4.1 Statistics of Multiobject Systems 15 1.4.2 Statistics of Expert Systems 16 1.4.3 Finite Set Statistics 17 1.5 Organization of the Book 17 1.5.1 Part I: Unified Single-Target Multisource Integration 17 1.5.2 Part II: Unified Multitarget-Multisource Integration 20 1.5.3 Part III: Approximate Multitarget Filtering 21 1.5.4 Appendixes 22 vn

Vlll Contents I Unified Single-Target Multisource Integration 23 Chapter2 Single-Target Filtering 25 2.1 Introduction to the Chapter 25 2.1.1 Summary of Major Lessons Learned 26 2.1.2 Organization of the Chapter 27 2.2 The Kaiman Filter 27 2.2.1 Kaiman Filter Initialization 28 2.2.2 Kaiman Filter Predictor 28 2.2.3 Kaiman Filter Corrector 29 2.2.4 Derivation ofthe Kaiman Filter 30 2.2.5 Measurement Fusion Using the Kaiman Filter 32 2.2.6 Constant-Gain Kaiman Filters 32 2.3 Bayes Formulation of the Kaiman Filter 33 2.3.1 Some Mathematical Preliminaries 34 2.3.2 Bayes Formulation of the KF: Predictor 35 2.3.3 Bayes Formulation of the KF: Corrector 37 2.3.4 Bayes Formulation of the KF: Estimation 40 2.4 The Single-Target Bayes Filter 42 2.4.1 Single-Target Bayes Filter: An Illustration 43 2.4.2 Relationship Between the Bayes and Kaiman Filters 45 2.4.3 Single-Target Bayes Filter: Modeling 51 2.4.4 Single-Target Bayes Filter: Formal Bayes Modeling 56 2.4.5 Single-Target Bayes Filter: Initialization 61 2.4.6 Single-Target Bayes Filter: Predictor 61 2.4.7 Single-Target Bayes Filter: Corrector 62 2.4.8 Single-Target Bayes Filter: State Estimation 63 2.4.9 Single-Target Bayes Filter: Error Estimation 64 2.4.10 Single-Target Bayes Filter: Data Fusion 67 2.4.11 Single-Target Bayes Filter: Computation 68 2.5 Single-Target Bayes Filter: Implementation 70 2.5.1 Taylor Series Approximation: The EKF 71 2.5.2 Gaussian-Mixture Approximation 72 2.5.3 Sequential Monte Carlo Approximation 79 2.6 Chapter Exercises 87 Chapter 3 General Data Modeling 89 3.1 Introduction to the Chapter 89 3.1.1 Summary of Major Lessons Learned 91

Contents ix 3.1.2 Organization of the Chapter 91 3.2 Issues in Modeling Uncertainty 92 3.3 Issues in Modeling Uncertainty in Data 94 3.4 Examples 97 3.4.1 Random, Slightly Imprecise Measurements 97 3.4.2 Imprecise, Slightly Random Measurements 101 3.4.3 Nonrandom Vague Measurements 102 3.4.4 Nonrandom Uncertain Measurements 103 3.4.5 Ambiguity Versus Randomness 106 3.5 The Core Bayesian Approach 109 3.5.1 Formal Bayes Modeling in General 109 3.5.2 The Bayes Filter in General 110 3.5.3 Bayes Combination Operators 111 3.5.4 Bayes-Invariant Measurement Conversion 113 3.6 Formal Modeling of Generalized Data 114 3.7 Chapter Exercise 117 Random Set Uncertainty Representations 119 4.1 Introduction to the Chapter 119 4.1.1 Summary of Major Lessons Learned 119 4.1.2 Organization of the Chapter 120 4.2 Universes, Events, and the Logic of Events 120 4.3 Fuzzy Set Theory 121 4.3.1 Fuzzy Logics 122 4.3.2 Random Set Representation of Fuzzy Events 123 4.3.3 Finite-Level Fuzzy Sets 126 4.3.4 Copula Fuzzy Logics 129 4.3.5 General Random Set Representations of Fuzzy Sets 131 4.4 Generalized Fuzzy Set Theory 133 4.4.1 Random Set Representation of Generalized Fuzzy Events 134 4.5 Dempster-Shafer Theory 134 4.5.1 Dempster's Combination 136 4.5.2 "Zadeh's Paradox" and Its Misinterpretation 138 4.5.3 Converting b.m.a.s to Probability Distributions 141 4.5.4 Random Set Representation of Uncertain Events 143 4.6 Fuzzy Dempster-Shafer Theory 144 4.6.1 Random Set Representation of Fuzzy DS Evidence 145

X Contents 4.7 Inference Rules 147 4.7.1 What Are Rules? 147 4.7.2 Combining Rules Using Conditional Event Algebra 148 4.7.3 Random Set Representation of First-Order Rules 150 4.7.4 Random Set Representation of Composite Rules 151 4.7.5 Random Set Representation of Second-Order Rules 152 4.8 Is Bayes Subsumed by Other Theories? 152 4.9 Chapter Exercises 154 Chapter5 UGA Measurements 157 5.1 Introduction to the Chapter 157 5.1.1 Notation 158 5.1.2 Summary of Major Lessons Learned 159 5.1.3 Organization of the Chapter 161 5.2 What Is a UGA Measurement? 162 5.2.1 Modeling UGA Measurements 162 5.2.2 Modeling the Generation of UGA Measurements 164 5.3 Likelihoods for UGA Measurements 164 5.3.1 Special Case: G Is Statistical 165 5.3.2 Special Case: 6 Is Fuzzy 166 5.3.3 Special Case: 9 Is Generalized Fuzzy 169 5.3.4 Special Case: 6 Is Discrete/Dempster-Shafer 171 5.3.5 Special Case: 0 Is Fuzzy Dempster-Shafer 173 5.3.6 Special Case: Is a First-Order Fuzzy Rule 174 5.3.7 Special Case: G Is a Composite Fuzzy Rule 179 5.3.8 Special Case: 0 Isa Second-Order Fuzzy Rule 180 5.4 Bayes Unification of UGA Fusion 181 5.4.1 Bayes Unification of UGA Fusion Using Normalized and Unnormalized Dempster's Combinations 185 5.4.2 Bayes Unification of UGA Fusion Using Normalized and Unnormalized Fuzzy Dempster's Combinations 186 5.4.3 Bayes Unification of UGA Fusion Using Copula Fuzzy Conjunctions 186 5.4.4 Bayes Unification of UGA Rule-Firing 187 5.4.5 If 3 0 Is Finite, Then Generalized Likelihoods Are Strict Likelihoods 188

Contents XI 5.4.6 Bayes-Invariant Conversions Between UGA Measurements 189 5.5 Modeling Other Kinds of Uncertainty 194 5.5.1 Modeling Unknown Statistical Dependencies 195 5.5.2 Modeling Unknown Target Types 196 5.6 The Kaiman Evidential Filter (KEF) 199 5.6.1 Definitions 204 5.6.2 KEFPredictor 205 5.6.3 KEF Corrector (Fuzzy DS Measurements) 205 5.6.4 KEF Corrector (Conventional Measurements) 207 5.6.5 KEF State Estimation 208 5.6.6 KEF Compared to Gaussian-Mixture and Kaiman Filters 208 5.7 Chapter Exercises 209 Chapter6 AGA Measurements 211 6.1 Introduction to the Chapter 211 6.1.1 Summary of Major Lessons Learned 212 6.1.2 Organization of the Chapter 213 6.2 AGA Measurements Defined 213 6.3 Likelihoods for AGA Measurements 214 6.3.1 Special Case: 9 and E x Are Fuzzy 215 6.3.2 Special Case: 9 and E x Are Generalized Fuzzy 219 6.3.3 Special Case: 9 and E x Are Dempster-Shafer 219 6.3.4 Special Case: 9andE x Are Fuzzy DS 220 6.4 Filtering with Fuzzy AGA Measurements 221 6.5 Example: Filtering with Poor Data 222 6.5.1 A Robust-Bayes Classifier 223 6.5.2 Simulation 1: More Imprecise, More Random 225 6.5.3 Simulation 2: Less Imprecise, Less Random 225 6.5.4 Interpretation of the Results 232 6.6 Unmodeled Target Types 232 6.7 Example: Target ID Using Link INT Data 238 6.7.1 Robust-Bayes Classifier 240 6.7.2 "Pseudodata" Simulation Results 243 6.7.3 "LONEWOLF-98" Simulation Results 243 6.8 Example: Unmodeled Target Types 244 6.9 Chapter Exercises 245

Xll Contents Chapter7 AGU Measurements 249 7.1 Introduction to the Chapter 249 7.1.1 Summary of Major Lessons Learned 250 7.1.2 Why Not Robust Statistics? 250 7.1.3 Organization of the Chapter 251 7.2 Random Set Models of UGA Measurements 252 7.2.1 Random Error Bars 252 7.2.2 Random Error Bars: Joint Likelihoods 252 7.3 Likelihoods for AGU Measurements 254 7.4 Fuzzy Models of AGU Measurements 255 7.5 Robust ATR Using SAR Data 260 7.5.1 Summary of Methodology 264 7.5.2 Experimental Ground Rules 266 7.5.3 Summary of Experimental Results 268 Chapter 8 Generalized State-Estimates 271 8.1 Introduction to the Chapter 271 8.1.1 Summary of Major Lessons Learned 273 8.1.2 Organization of the Chapter 274 8.2 What Is a Generalized State-Estimate? 274 8.3 What Is a UGA DS State-Estimate? 275 8.4 Posterior Distributions and State-Estimates 277 8.4.1 The Likelihood of a DS State-Estimate 278 8.4.2 Posterior Distribution Conditioned on a DS State- Estimate 278 8.4.3 Posterior Distributions and Pignistic Probability 279 8.5 Unification of State-Estimate Fusion Using Modified Dempster's Combination 280 8.6 Bayes-Invariant Transformation 280 8.7 Extension to Fuzzy DS State-Estimates 281 8.8 Chapter Exercises 285 Chapter 9 Finite-Set Measurements 287 9.1 Introduction to the Chapter 287 9.1.1 Summary of Major Lessons Learned 287 9.1.2 Organization of the Chapter 288 9.2 Examples of Finite-Set Measurements 288 9.2.1 Ground-to-Air Radar Detection Measurements 288 9.2.2 Air-to-Ground Doppler Detection Measurements 291

Contents Xlll 9.2.3 Extended-Target Detection Measurements 292 9.2.4 Features Extracted from Images 292 9.2.5 Human-Mediated Features 292 9.2.6 General Finite-Set Measurements 293 9.3 Modeling Finite-Set Measurements? 293 9.3.1 Formal Modeling of Finite-Set Measurements 293 9.3.2 Multiobject Integrals 297 9.3.3 Finite-Set Measurement Models 299 9.3.4 True Likelihoods for Finite-Set Measurements 302 9.3.5 Constructive Likelihood Functions 302 9.4 Chapter Exercises 303 II Unified Multitarget-Multisource Integration 305 Chapter 10 Conventional Multitarget Filtering 307 10.1 Introduction to the Chapter 307 10.1.1 Summary of Major Lessons Learned 308 10.1.2 Organization of the Chapter 311 10.2 Standard Multitarget Models 311 10.2.1 Standard Multitarget Measurement Model 311 10.2.2 Standard Multitarget Motion Model 313 10.3 Measurement-to-Track Association 315 10.3.1 Distance Between Measurements and Tracks 315 10.4 Single-Hypothesis Correlation (SHC) 319 10.4.1 SHC: No Missed Detections, No False Alarms 319 10.4.2 SHC: Missed Detections and False Alarms 320 10.5 Multihypothesis Correlation (MHC) 321 10.5.1 Elements of MHC 323 10.5.2 MHC: No Missed Detections or False Alarms 326 10.5.3 MHC: False Alarms, No Missed Detections 329 10.5.4 MHC: Missed Detections and False Alarms 332 10.6 Composite-Hypothesis Correlation (CHC) 335 10.6.1 Elements of CHC 335 10.6.2 CHC: No Missed Detections or False Alarms 337 10.6.3 CHC: Probabilistic Data Association (PDA) 337 10.6.4 CHC: Missed Detections, False Alarms 338 10.7 Conventional Filtering: Limitations 338 10.7.1 Real-Time Performance 338

XIV Contents 10.7.2 Is a Hypothesis Actually a State Variable? 340 10.8 MHC with Fuzzy DS Measurements 341 Chapter 11 Multitarget Calculus 343 11.1 Introduction to the Chapter 343 11.1.1 Transform Methods in Conventional Statistics 344 11.1.2 Transform Methods in Multitarget Statistics 345 11.1.3 Summaryof Major Lessons Learned 346 11.1.4 Organization of the Chapter 348 11.2 Random Finite Sets 348 11.3 Fundamental Statistical Descriptors 356 11.3.1 Multitarget Calculus Why? 357 11.3.2 Belief-Mass Functions 359 11.3.3 Multiobject Density Functions and Set Integrals 360 11.3.4 Important Multiobject Probability Distributions 364 11.3.5 Probability-Generating Functionals (p.g.fl.s) 370 11.4 Functional Derivatives and Set Derivatives 375 11.4.1 Functional Derivatives 375 11.4.2 Set Derivatives 380 11.5 Key Multiobject-Calculus Formulas 383 11.5.1 Fundamental Theorem of Multiobject Calculus 384 11.5.2 Radon-Nikodym Theorems 385 11.5.3 Fundamental Convolution Formula 385 11.6 Basic Differentiation Rules 386 11.7 Chapter Exercises 394 Chapter 12 Multitarget Likelihood Functions 399 12.1 Introduction to the Chapter 399 12.1.1 Summary of Major Lessons Learned 401 12.1.2 Organization of the Chapter 402 12.2 Multitarget State and Measurement Spaces 403 12.2.1 Multitarget State Spaces 403 12.2.2 Multisensor State Spaces 406 12.2.3 Single-Sensor, Multitarget Measurement Spaces 407 12.2.4 Multisensor-Multitarget Measurement Spaces 408 12.3 The Standard Measurement Model 408 12.3.1 Measurement Equation for the Standard Model 411 12.3.2 CaseI:No Target IsPresent 412 12.3.3 Case II: One Target Is Present 414

Contents xv 12.3.4 Case III: No Missed Detections or False Alarms 416 12.3.5 Case IV: Missed Detections, No False Alarms 418 12.3.6 Case V: Missed Detections and False Alarms 420 12.3.7 p.g.fl.s for the Standard Measurement Model 421 12.4 Relationship with MHC 422 12.5 State-Dependent False Alarms 424 12.5.1 p.g.fl. for State-Dependent False Alarms 426 12.6 Transmission Drop-Outs 426 12.6.1 p.g.fl. for Transmission Drop-Outs 427 12.7 Extended Targets 427 12.7.1 Single Extended Target 428 12.7.2 Multiple Extended Targets 430 12.7.3 Poisson Approximation 431 12.8 Unresolved Targets 432 12.8.1 Point Target Clusters 434 12.8.2 Single-Cluster Likelihoods 435 12.8.3 MultiCluster Likelihoods 442 12.8.4 Continuityof Multicluster Likelihoods 444 12.9 Multisource Measurement Models 445 12.9.1 Conventional Measurements 445 12.9.2 Generalized Measurements 447 12.10 A Model for Bearing-Only Measurements 448 12.10.1 Multitarget Measurement Model 450 12.10.2Belief-Mass Function 451 12.10.3 Multitarget Likelihood Function 452 12.11 A Model for Data-Cluster Extraction 452 12.11.1 Finite-Mixture Models 453 12.11.2 A Likelihood for Finite-Mixture Modeling 456 12.11.3 Extraction of Soft Data Classes 457 12.12 Chapter Exercises 458 Chapterl3 Multitarget Markov Densities 461 13.1 Introduction to the Chapter 461 13.1.1 Summary of Major Lessons Learned 465 13.1.2 Organization of the Chapter 466 13.2 "Standard" Multitarget Motion Model 466 13.2.1 Case I: At Most One Target Is Present 469 13.2.2 Case II: No Target Death or Birth 470

xvi Contents 13.2.3 Case III: Target Death, No Birth 471 13.2.4 Case IV: Target Death and Birth 471 13.2.5 Case V: Target Death and Birth with Spawning 472 13.2.6 p.g.fl.s for the Standard Motion Model 473 13.3 Extended Targets 474 13.4 Unresolved Targets 475 13.4.1 Intuitive Dynamic Behavior of Point Clusters 475 13.4.2 Markov Densities for Single Point Clusters 476 13.4.3 Markov Densities for Multiple Point Clusters 477 13.5 Coordinated Multitarget Motion 478 13.5.1 Simple Virtual Leader-Follower 478 13.5.2 General Virtual Leader-Follower 481 13.6 Chapter Exercises 482 Chapterl4 The Multitarget Bayes Filter 483 14.1 Introduction to the Chapter 483 14.1.1 Summary of Major Lessons Learned 484 14.1.2 Organization of the Chapter 486 14.2 Multitarget Bayes Filter: Initialization 486 14.2.1 Initialization: Multitarget Poisson Process 486 14.2.2 Initialization: Target NumberKnown 487 14.3 Multitarget Bayes Filter: Predictor 487 14.3.1 Predictor: No Target Birth or Death 489 14.4 Multitarget Bayes Filter: Corrector 490 14.4.1 Conventional Measurements 490 14.4.2 Generalized Measurements 493 14.4.3 Unified Multitarget-Multisource Integration 493 14.5 Multitarget Bayes Filter: State Estimation 494 14.5.1 The Failure of the Classical State Estimators 494 14.5.2 Marginal Multitarget (MaM) Estimator 497 14.5.3 Joint Multitarget (JoM) Estimator 498 14.5.4 JoM and MaM Estimators Compared 501 14.5.5 Computational Issues 504 14.5.6 State Estimation and Track Labeling 505 14.6 Multitarget Bayes Filter: Error Estimation 509 14.6.1 Target Number RMS Deviation 509 14.6.2 Track Covariances 509 14.6.3 Global Mean Deviation 510

Contents xvii 14.6.4 Information Measures of Multitarget Dispersion 512 14.7 The JoTT Filter 514 14.7.1 JoTT Filter: Models 516 14.7.2 JoTT Filter: Initialization 518 14.7.3 JoTT Filter: Predictor 519 14.7.4 JoTT Filter: Corrector 520 14.7.5 JoTT Filter: Estimation 520 14.7.6 JoTT Filter: Error Estimation 523 14.7.7 SMC Implementation of JoTT Filter 523 14.8 The p.g.fl. Multitarget Bayes Filter 528 14.8.1 The p.g.fl. Multitarget Predictor 528 14.8.2 The p.g.fl. Multitarget Corrector 530 14.9 Target Prioritization 531 14.9.1 Tactical Importance Functions (TIFs) 533 14.9.2 The p.g.fl. for a TIF 533 14.9.3 The Multitarget Posterior for a TIF 535 14.10 Chapter Exercises 537 Approximate Multitarget Filtering 539 er 15 Multitarget Patticle Approximation 541 15.1 Introduction to the Chapter 541 15.1.1 Summary of Major Lessons Learned 542 15.1.2 Organization of the Chapter 543 15.2 The Multitarget Filter: Computation 543 15.2.1 Fixed-Grid Approximation 544 15.2.2 SMC Approximation 545 15.2.3 When Is the Multitarget Filter Appropnate? 546 15.2.4 Implementations of the Multitarget Filter 547 15.3 Multitarget Particle Systems 551 15.4 M-SMC Filter Initialization 554 15.4.1 Target Number is Known 554 15.4.2 Null Multitarget Prior 555 15.4.3 Poisson Multitarget Prior 555 15.5 M-SMC Filter Predictor 556 15.5.1 Persisting and Disappearing Targets 557 15.5.2 Appearing Targets 558 15.6 M-SMC Filter Corrector 560

XV111 Contents 15.7 M-SMC Filter State and Error Estimation 561 15.7.1 PHD-Based State and Error Estimation 561 15.7.2 Global Mean Deviation 562 15.7.3 Track Labeling for the Multitarget SMC Filter 563 Chapter 16 Multitarget-Moment Approximation 565 16.1 Introduction to the Chapter 565 16.1.1 Single-Target Moment-Statistic Filters 566 16.1.2 First-Order Multitarget-Moment Filtering 568 16.1.3 Second-Order Multitarget-Moment Filtering 572 16.1.4 Summary of Major Lessons Learned 574 16.1.5 Organization of the Chapter 575 16.2 The Probability Hypothesis Density (PHD) 576 16.2.1 First-Order Multitarget Moments 576 16.2.2 PHD as a Continuous Fuzzy Membership Function 579 16.2.3 PHDs and Multitarget Calculus 580 16.2.4 ExamplesofPHDs 583 16.2.5 Higher-Order Multitarget Moments 586 16.3 The PHD Filter 587 16.3.1 PHD Filter Initialization 587 16.3.2 PHD Filter Predictor 587 16.3.3 PHD Filter Corrector 590 16.3.4 PHD Filter State and Error Estimation 595 16.3.5 Target ID and the PHD Filter 599 16.4 Physical Interpretation of PHD Filter 599 16.4.1 Physical Interpretation of PHD Predictor 600 16.4.2 Physical Interpretation of PHD Corrector 603 16.5 Implementing the PHD Filter 609 16.5.1 Surveyof PHD Filter Implementations 610 16.5.2 SMC-PHD Approximation 615 16.5.3 GM-PHD Approximation 623 16.6 Limitations of the PHD Filter 631 16.7 The Cardinalized PHD (CPHD) Filter 632 16.7.1 CPHD Filter Initialization 633 16.7.2 CPHD Filter Predictor 634 16.7.3 CPHD Filter Single-Sensor Corrector 636 16.7.4 CPHD Filter State and Error Estimation 639 16.7.5 Computational Complexity of the CPHD Filter 640

Contents xix 16.7.6 CPHD and JoTT Filters Compared 641 16.8 Physical Interpretation of CPHD Filter 642 16.9 Implementing the CPHD Filter 642 16.9.1 Surveyof CPHD Filter Implementations 643 16.9.2 Particle Approximation (SMC-CPHD) 644 16.9.3 Gaussian-Mixture Approximation (GM-CPHD) 646 16.10 Deriving the PHD and CPHD Filters 649 16.10.1 Derivation of PHD and CPHD Predictors 650 16.10.2 Derivation of PHD and CPHD Correctors 651 16.11 Partial Second-Order Filters? 652 16.12 Chapter Exercise 653 Chapter 17 Multi-Bernoulli Approximation 655 17.1 Introduction to the Chapter 655 17.1.1 p.g.fl.-based Multitarget Approximation 655 17.1.2 Why Multitarget Multi-Bernoulli Processes? 657 17.1.3 The Multitarget Multi-Bernoulli Filter 657 17.1.4 The Para-Gaussian Filter 658 17.1.5 Summary of Major Lessons Learned 659 17.1.6 Organization of the Chapter 660 17.2 Multitarget Multi-Bernoulli Filter 660 17.2.1 MeMBer Filter Initialization 661 17.2.2 MeMBer Filter Predictor 661 17.2.3 MeMBer Filter Corrector 662 17.2.4 MeMBer Filter Pruning and Merging 665 17.2.5 MeMBer Filter State and Error Estimation 666 17.2.6 Relationship with the Moreland-Challa Filter 667 17.3 Para-Gaussian Filter 668 17.3.1 Para-Gaussian Filter Initialization 669 17.3.2 Para-Gaussian Filter Predictor 669 17.3.3 Para-Gaussian Filter Corrector 671 17.3.4 Para-Gaussian Filter Pruning and Merging 673 17.3.5 Para-Gaussian Filter State and Error Estimation 675 17.4 MeMBer Filter Derivation 675 17.4.1 Derivation of the MeMBer Filter Predictor 675 17.4.2 Derivation of the MeMBer Filter Corrector 677 17.5 Chapter Exercise 682 Appendix A Glossary of Notation 683

XX Contents A.l Transparent Notational System 683 A.2 General Mathematics 684 A.3 SetTheory 685 A.4 Fuzzy Logic and Dempster-Shafer Theory 686 A.5 Probability and Statistics 687 A.6 Random Sets 689 A.7 Multitarget Calculus 690 A.8 Finite-Set Statistics 691 A.9 Generalized Measurements 692 Appendix B Dirac Delta Functions 693 Appendix C Gradient Derivatives 695 C.l Relationship with Partial Derivatives 696 C.2 Multidimensional Taylor Series 696 C.3 Multidimensional Extrema 696 Appendix D Fundamental Gaussian Identity 699 Appendix E Finite Point Processes 705 E.l Mathematical Representations of Multiplicity 705 E.2 Random Point Processes 707 E.3 Point Processes Versus Random Finite Sets 708 Appendix F FISST and Probability Theory 711 F.l Multiobject Probability Theory 711 F.2 Belief-Mass Functions Versus Probability Measures 713 F.3 Set Integrals Versus Measure Theoretic Integrals 714 F.4 Set Derivatives Versus Radon-Nikodym Derivatives 715 Appendix G Mathematical Proofs 717 G. 1 Likelihoods for First-Order Fuzzy Rules 717 G.2 Likelihoods for Composite Rules 718 G.3 Likelihoods for Second-Order Fuzzy Rules 720 G.4 UnificationofDSCombinations 721 G.5 UnificationofRule-Firing 722 G.6 Generalized Likelihoods: 3o Is Finite 723 G.7 NOTA for Fuzzy DS Measurements 724 G.8 KEFPredictor 726

Contents xxi G.9 KEF Corrector (Fuzzy DS Measurements) 729 G. 10 Likehhoods for AGA Fuzzy Measurements 732 G. 11 Likehhoods for AGA Generalized Fuzzy Measurements 733 G. 12 Likehhoods for AGA Fuzzy DS Measurements 734 G. 13 Interval Argsup Formula 735 G. 14 Consonance of the Random State Set r z 736 G.15 Sufficient Statistics and Modified Combination 737 G. 16 Transformation Invariance 738 G. 17 MHT Hypothesis Probabilities 739 G. 18 Likelihood for Standard Measurement Model 742 G. 19 p.g.fl. for Standard Measurement Model 745 G.20 Multisensor Multitarget Likehhoods 747 G.21 Continuity of Likehhoods for Unresolved Targets 749 G.22 Association for Fuzzy Dempster-Shafer 751 G.23 JoTT Filter Predictor 753 G.24 JoTT Filter Corrector 755 G.25 p.g.fl. Form of the Multitarget Corrector 757 G.26 Induced Particle Approximation of PHD 758 G.27 PHD Counting Property 760 G.28 GM-PHD Filter Predictor 761 G.29 GM-PHD Filter Corrector 763 G.30 Exact PHD Corrector 765 G.31 GM-CPHD Filter Predictor 767 G.32 GM-CPHD Filter Corrector 768 G.33 MeMBer Filter Target Number 771 G.34 Para-Gaussian Filter Predictor 773 G.35 Para-Gaussian Filter Corrector 774 Appendix H Solutions to Exercises 777 References 821 About the Author 837 Index 839

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