Statistical Multisource-Multitarget Information Fusion

Transcription

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

2 Contents Preface Acknowledgments xxm xxv Chapter 1 Introduction to the Book What Is the Purpose of This Book? Major Challenges in Information Fusion Why Random Sets or FISST? Whylsn't Multitarget FilteringStraightforward? Beyond Heuristics How Do Single-Target and Multitarget Statistics Differ? How Do Conventional and Ambiguous Data Differ? What Is Formal Bayes Modeling? How Is Ambiguous Information Modeled? What Is Multisource-Multitarget Formal Modeling? Random Sets in Information Fusion Statistics of Multiobject Systems Statistics of Expert Systems Finite Set Statistics Organization of the Book Part I: Unified Single-Target Multisource Integration Part II: Unified Multitarget-Multisource Integration Part III: Approximate Multitarget Filtering Appendixes 22 vn

3 Vlll Contents I Unified Single-Target Multisource Integration 23 Chapter2 Single-Target Filtering Introduction to the Chapter Summary of Major Lessons Learned Organization of the Chapter The Kaiman Filter Kaiman Filter Initialization Kaiman Filter Predictor Kaiman Filter Corrector Derivation ofthe Kaiman Filter Measurement Fusion Using the Kaiman Filter Constant-Gain Kaiman Filters Bayes Formulation of the Kaiman Filter Some Mathematical Preliminaries Bayes Formulation of the KF: Predictor Bayes Formulation of the KF: Corrector Bayes Formulation of the KF: Estimation The Single-Target Bayes Filter Single-Target Bayes Filter: An Illustration Relationship Between the Bayes and Kaiman Filters Single-Target Bayes Filter: Modeling Single-Target Bayes Filter: Formal Bayes Modeling Single-Target Bayes Filter: Initialization Single-Target Bayes Filter: Predictor Single-Target Bayes Filter: Corrector Single-Target Bayes Filter: State Estimation Single-Target Bayes Filter: Error Estimation Single-Target Bayes Filter: Data Fusion Single-Target Bayes Filter: Computation Single-Target Bayes Filter: Implementation Taylor Series Approximation: The EKF Gaussian-Mixture Approximation Sequential Monte Carlo Approximation Chapter Exercises 87 Chapter 3 General Data Modeling Introduction to the Chapter Summary of Major Lessons Learned 91

4 Contents ix Organization of the Chapter Issues in Modeling Uncertainty Issues in Modeling Uncertainty in Data Examples Random, Slightly Imprecise Measurements Imprecise, Slightly Random Measurements Nonrandom Vague Measurements Nonrandom Uncertain Measurements Ambiguity Versus Randomness The Core Bayesian Approach Formal Bayes Modeling in General The Bayes Filter in General Bayes Combination Operators Bayes-Invariant Measurement Conversion Formal Modeling of Generalized Data Chapter Exercise 117 Random Set Uncertainty Representations Introduction to the Chapter Summary of Major Lessons Learned Organization of the Chapter Universes, Events, and the Logic of Events Fuzzy Set Theory Fuzzy Logics Random Set Representation of Fuzzy Events Finite-Level Fuzzy Sets Copula Fuzzy Logics General Random Set Representations of Fuzzy Sets Generalized Fuzzy Set Theory Random Set Representation of Generalized Fuzzy Events Dempster-Shafer Theory Dempster's Combination "Zadeh's Paradox" and Its Misinterpretation Converting b.m.a.s to Probability Distributions Random Set Representation of Uncertain Events Fuzzy Dempster-Shafer Theory Random Set Representation of Fuzzy DS Evidence 145

5 X Contents 4.7 Inference Rules What Are Rules? Combining Rules Using Conditional Event Algebra Random Set Representation of First-Order Rules Random Set Representation of Composite Rules Random Set Representation of Second-Order Rules Is Bayes Subsumed by Other Theories? Chapter Exercises 154 Chapter5 UGA Measurements Introduction to the Chapter Notation Summary of Major Lessons Learned Organization of the Chapter What Is a UGA Measurement? Modeling UGA Measurements Modeling the Generation of UGA Measurements Likelihoods for UGA Measurements Special Case: G Is Statistical Special Case: 6 Is Fuzzy Special Case: 9 Is Generalized Fuzzy Special Case: 6 Is Discrete/Dempster-Shafer Special Case: 0 Is Fuzzy Dempster-Shafer Special Case: Is a First-Order Fuzzy Rule Special Case: G Is a Composite Fuzzy Rule Special Case: 0 Isa Second-Order Fuzzy Rule Bayes Unification of UGA Fusion Bayes Unification of UGA Fusion Using Normalized and Unnormalized Dempster's Combinations Bayes Unification of UGA Fusion Using Normalized and Unnormalized Fuzzy Dempster's Combinations Bayes Unification of UGA Fusion Using Copula Fuzzy Conjunctions Bayes Unification of UGA Rule-Firing If 3 0 Is Finite, Then Generalized Likelihoods Are Strict Likelihoods 188

6 Contents XI Bayes-Invariant Conversions Between UGA Measurements Modeling Other Kinds of Uncertainty Modeling Unknown Statistical Dependencies Modeling Unknown Target Types The Kaiman Evidential Filter (KEF) Definitions KEFPredictor KEF Corrector (Fuzzy DS Measurements) KEF Corrector (Conventional Measurements) KEF State Estimation KEF Compared to Gaussian-Mixture and Kaiman Filters Chapter Exercises 209 Chapter6 AGA Measurements Introduction to the Chapter Summary of Major Lessons Learned Organization of the Chapter AGA Measurements Defined Likelihoods for AGA Measurements Special Case: 9 and E x Are Fuzzy Special Case: 9 and E x Are Generalized Fuzzy Special Case: 9 and E x Are Dempster-Shafer Special Case: 9andE x Are Fuzzy DS Filtering with Fuzzy AGA Measurements Example: Filtering with Poor Data A Robust-Bayes Classifier Simulation 1: More Imprecise, More Random Simulation 2: Less Imprecise, Less Random Interpretation of the Results Unmodeled Target Types Example: Target ID Using Link INT Data Robust-Bayes Classifier "Pseudodata" Simulation Results "LONEWOLF-98" Simulation Results Example: Unmodeled Target Types Chapter Exercises 245

7 Xll Contents Chapter7 AGU Measurements Introduction to the Chapter Summary of Major Lessons Learned Why Not Robust Statistics? Organization of the Chapter Random Set Models of UGA Measurements Random Error Bars Random Error Bars: Joint Likelihoods Likelihoods for AGU Measurements Fuzzy Models of AGU Measurements Robust ATR Using SAR Data Summary of Methodology Experimental Ground Rules Summary of Experimental Results 268 Chapter 8 Generalized State-Estimates Introduction to the Chapter Summary of Major Lessons Learned Organization of the Chapter What Is a Generalized State-Estimate? What Is a UGA DS State-Estimate? Posterior Distributions and State-Estimates The Likelihood of a DS State-Estimate Posterior Distribution Conditioned on a DS State- Estimate Posterior Distributions and Pignistic Probability Unification of State-Estimate Fusion Using Modified Dempster's Combination Bayes-Invariant Transformation Extension to Fuzzy DS State-Estimates Chapter Exercises 285 Chapter 9 Finite-Set Measurements Introduction to the Chapter Summary of Major Lessons Learned Organization of the Chapter Examples of Finite-Set Measurements Ground-to-Air Radar Detection Measurements Air-to-Ground Doppler Detection Measurements 291

8 Contents Xlll Extended-Target Detection Measurements Features Extracted from Images Human-Mediated Features General Finite-Set Measurements Modeling Finite-Set Measurements? Formal Modeling of Finite-Set Measurements Multiobject Integrals Finite-Set Measurement Models True Likelihoods for Finite-Set Measurements Constructive Likelihood Functions Chapter Exercises 303 II Unified Multitarget-Multisource Integration 305 Chapter 10 Conventional Multitarget Filtering Introduction to the Chapter Summary of Major Lessons Learned Organization of the Chapter Standard Multitarget Models Standard Multitarget Measurement Model Standard Multitarget Motion Model Measurement-to-Track Association Distance Between Measurements and Tracks Single-Hypothesis Correlation (SHC) SHC: No Missed Detections, No False Alarms SHC: Missed Detections and False Alarms Multihypothesis Correlation (MHC) Elements of MHC MHC: No Missed Detections or False Alarms MHC: False Alarms, No Missed Detections MHC: Missed Detections and False Alarms Composite-Hypothesis Correlation (CHC) Elements of CHC CHC: No Missed Detections or False Alarms CHC: Probabilistic Data Association (PDA) CHC: Missed Detections, False Alarms Conventional Filtering: Limitations Real-Time Performance 338

9 XIV Contents Is a Hypothesis Actually a State Variable? MHC with Fuzzy DS Measurements 341 Chapter 11 Multitarget Calculus Introduction to the Chapter Transform Methods in Conventional Statistics Transform Methods in Multitarget Statistics Summaryof Major Lessons Learned Organization of the Chapter Random Finite Sets Fundamental Statistical Descriptors Multitarget Calculus Why? Belief-Mass Functions Multiobject Density Functions and Set Integrals Important Multiobject Probability Distributions Probability-Generating Functionals (p.g.fl.s) Functional Derivatives and Set Derivatives Functional Derivatives Set Derivatives Key Multiobject-Calculus Formulas Fundamental Theorem of Multiobject Calculus Radon-Nikodym Theorems Fundamental Convolution Formula Basic Differentiation Rules Chapter Exercises 394 Chapter 12 Multitarget Likelihood Functions Introduction to the Chapter Summary of Major Lessons Learned Organization of the Chapter Multitarget State and Measurement Spaces Multitarget State Spaces Multisensor State Spaces Single-Sensor, Multitarget Measurement Spaces Multisensor-Multitarget Measurement Spaces The Standard Measurement Model Measurement Equation for the Standard Model CaseI:No Target IsPresent Case II: One Target Is Present 414

10 Contents xv Case III: No Missed Detections or False Alarms Case IV: Missed Detections, No False Alarms Case V: Missed Detections and False Alarms p.g.fl.s for the Standard Measurement Model Relationship with MHC State-Dependent False Alarms p.g.fl. for State-Dependent False Alarms Transmission Drop-Outs p.g.fl. for Transmission Drop-Outs Extended Targets Single Extended Target Multiple Extended Targets Poisson Approximation Unresolved Targets Point Target Clusters Single-Cluster Likelihoods MultiCluster Likelihoods Continuityof Multicluster Likelihoods Multisource Measurement Models Conventional Measurements Generalized Measurements A Model for Bearing-Only Measurements Multitarget Measurement Model Belief-Mass Function Multitarget Likelihood Function A Model for Data-Cluster Extraction Finite-Mixture Models A Likelihood for Finite-Mixture Modeling Extraction of Soft Data Classes Chapter Exercises 458 Chapterl3 Multitarget Markov Densities Introduction to the Chapter Summary of Major Lessons Learned Organization of the Chapter "Standard" Multitarget Motion Model Case I: At Most One Target Is Present Case II: No Target Death or Birth 470

11 xvi Contents Case III: Target Death, No Birth Case IV: Target Death and Birth Case V: Target Death and Birth with Spawning p.g.fl.s for the Standard Motion Model Extended Targets Unresolved Targets Intuitive Dynamic Behavior of Point Clusters Markov Densities for Single Point Clusters Markov Densities for Multiple Point Clusters Coordinated Multitarget Motion Simple Virtual Leader-Follower General Virtual Leader-Follower Chapter Exercises 482 Chapterl4 The Multitarget Bayes Filter Introduction to the Chapter Summary of Major Lessons Learned Organization of the Chapter Multitarget Bayes Filter: Initialization Initialization: Multitarget Poisson Process Initialization: Target NumberKnown Multitarget Bayes Filter: Predictor Predictor: No Target Birth or Death Multitarget Bayes Filter: Corrector Conventional Measurements Generalized Measurements Unified Multitarget-Multisource Integration Multitarget Bayes Filter: State Estimation The Failure of the Classical State Estimators Marginal Multitarget (MaM) Estimator Joint Multitarget (JoM) Estimator JoM and MaM Estimators Compared Computational Issues State Estimation and Track Labeling Multitarget Bayes Filter: Error Estimation Target Number RMS Deviation Track Covariances Global Mean Deviation 510

12 Contents xvii Information Measures of Multitarget Dispersion The JoTT Filter JoTT Filter: Models JoTT Filter: Initialization JoTT Filter: Predictor JoTT Filter: Corrector JoTT Filter: Estimation JoTT Filter: Error Estimation SMC Implementation of JoTT Filter The p.g.fl. Multitarget Bayes Filter The p.g.fl. Multitarget Predictor The p.g.fl. Multitarget Corrector Target Prioritization Tactical Importance Functions (TIFs) The p.g.fl. for a TIF The Multitarget Posterior for a TIF Chapter Exercises 537 Approximate Multitarget Filtering 539 er 15 Multitarget Patticle Approximation Introduction to the Chapter Summary of Major Lessons Learned Organization of the Chapter The Multitarget Filter: Computation Fixed-Grid Approximation SMC Approximation When Is the Multitarget Filter Appropnate? Implementations of the Multitarget Filter Multitarget Particle Systems M-SMC Filter Initialization Target Number is Known Null Multitarget Prior Poisson Multitarget Prior M-SMC Filter Predictor Persisting and Disappearing Targets Appearing Targets M-SMC Filter Corrector 560

13 XV111 Contents 15.7 M-SMC Filter State and Error Estimation PHD-Based State and Error Estimation Global Mean Deviation Track Labeling for the Multitarget SMC Filter 563 Chapter 16 Multitarget-Moment Approximation Introduction to the Chapter Single-Target Moment-Statistic Filters First-Order Multitarget-Moment Filtering Second-Order Multitarget-Moment Filtering Summary of Major Lessons Learned Organization of the Chapter The Probability Hypothesis Density (PHD) First-Order Multitarget Moments PHD as a Continuous Fuzzy Membership Function PHDs and Multitarget Calculus ExamplesofPHDs Higher-Order Multitarget Moments The PHD Filter PHD Filter Initialization PHD Filter Predictor PHD Filter Corrector PHD Filter State and Error Estimation Target ID and the PHD Filter Physical Interpretation of PHD Filter Physical Interpretation of PHD Predictor Physical Interpretation of PHD Corrector Implementing the PHD Filter Surveyof PHD Filter Implementations SMC-PHD Approximation GM-PHD Approximation Limitations of the PHD Filter The Cardinalized PHD (CPHD) Filter CPHD Filter Initialization CPHD Filter Predictor CPHD Filter Single-Sensor Corrector CPHD Filter State and Error Estimation Computational Complexity of the CPHD Filter 640

14 Contents xix CPHD and JoTT Filters Compared Physical Interpretation of CPHD Filter Implementing the CPHD Filter Surveyof CPHD Filter Implementations Particle Approximation (SMC-CPHD) Gaussian-Mixture Approximation (GM-CPHD) Deriving the PHD and CPHD Filters Derivation of PHD and CPHD Predictors Derivation of PHD and CPHD Correctors Partial Second-Order Filters? Chapter Exercise 653 Chapter 17 Multi-Bernoulli Approximation Introduction to the Chapter p.g.fl.-based Multitarget Approximation Why Multitarget Multi-Bernoulli Processes? The Multitarget Multi-Bernoulli Filter The Para-Gaussian Filter Summary of Major Lessons Learned Organization of the Chapter Multitarget Multi-Bernoulli Filter MeMBer Filter Initialization MeMBer Filter Predictor MeMBer Filter Corrector MeMBer Filter Pruning and Merging MeMBer Filter State and Error Estimation Relationship with the Moreland-Challa Filter Para-Gaussian Filter Para-Gaussian Filter Initialization Para-Gaussian Filter Predictor Para-Gaussian Filter Corrector Para-Gaussian Filter Pruning and Merging Para-Gaussian Filter State and Error Estimation MeMBer Filter Derivation Derivation of the MeMBer Filter Predictor Derivation of the MeMBer Filter Corrector Chapter Exercise 682 Appendix A Glossary of Notation 683

15 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

16 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