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E-raamat: Statistical Multisource-Multitarget Information Fusion

  • Formaat: 888 pages
  • Ilmumisaeg: 31-Jan-2007
  • Kirjastus: Artech House Publishers
  • ISBN-13: 9781596930933
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Statistical Multisource-Multitarget Information FusionSuurem pilt
  • Formaat: 888 pages
  • Ilmumisaeg: 31-Jan-2007
  • Kirjastus: Artech House Publishers
  • ISBN-13: 9781596930933
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Mahler applies his decades of experience in tactical systems along with his expertise in theory to this all-inclusive resource on finite-set statistics (FSST), a new method that unites much of information fusion under a single Bayesian paradigm. He focuses on the needs of practitioners for complete information on unified single-target multisource integration, including single-target filtering, general data modeling, random set uncertainty representation, measurements of UGA and AGA as well as AGU, generalized state estimates and finite-set measurements, then covers unified multitarget multisource integration in terms of conventional filtering, calculus, likelihood functions, Markov densities, and the Bayes filter. He closes with approximate multitarget filtering including particle and moment approximation, and Bernoulli approximation. Each chapter includes exercises, and Mahler supplies support information for such complex topics as Dirac delta functions, mathematical proofs, probability theory, gradient derivatives and Gaussian identity. Annotation ©2007 Book News, Inc., Portland, OR (booknews.com)
Preface xxiii
Acknowledgments 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 Why Isn't Multitarget Filtering Straightforward?
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
I Unified Single-Target Multisource Integration 23
Chapter 2 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 Kalman Filter
27
2.2.1 Kalman Filter Initialization
28
2.2.2 Kalman Filter Predictor
28
2.2.3 Kalman Filter Corrector
29
2.2.4 Derivation of the Kalman Filter
30
2.2.5 Measurement Fusion Using the Kalman Filter
32
2.2.6 Constant-Gain Kalman Filters
32
2.3 Bayes Formulation of the Kalman 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 Kalman 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
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
Chapter 4 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
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
Chapter 5 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: Θ Is Statistical
165
5.3.2 Special Case: Θ Is Fuzzy
166
5.3.3 Special Case: Θ Is Generalized Fuzzy
169
5.3.4 Special Case: Θ Is Discrete/Dempster-Shafer
171
5.3.5 Special Case: Θ Is Fuzzy Dempster-Shafer
173
5.3.6 Special Case: Θ Is a First-Order Fuzzy Rule
174
5.3.7 Special Case: Θ Is a Composite Fuzzy Rule
179
5.3.8 Special Case: Θ Is a 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 30 Is Finite, Then Generalized Likelihoods Are Strict Likelihoods
188
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 Kalman Evidential Filter (KEF)
199
5.6.1 Definitions
204
5.6.2 KEF Predictor
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 Kalman Filters
208
5.7
Chapter Exercises
209
Chapter 6 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: Θ and Σx Are Fuzzy
215
6.3.2 Special Case: Θ and Σx Are Generalized Fuzzy
219
6.3.3 Special Case: Θ and Σx Are Dempster-Shafer
219
6.3.4 Special Case: Θ and Σx 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
Chapter 7 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
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
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 Summary of 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-Nikod$rm 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 Case I: No Target Is Present
412
12.3.3 Case II: One Target Is Present
414
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 Continuity of 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
12.10.2 Belief-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
Chapter 13 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
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
Chapter 14 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 Number Known
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
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
III Approximate Multitarget Filtering 539
Chapter 15 Multitarget Particle 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 Appropriate?
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
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 Examples of PHDs
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 Survey of 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
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 Survey of 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
A.1 Transparent Notational System
683
A.2 General Mathematics
684
A.3 Set Theory
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.1 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.1 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.1 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 Unification of DS Combinations
721
G.5 Unification of Rule-Firing
722
G.6 Generalized Likelihoods: 30 Is Finite
723
G.7 NOTA for Fuzzy DS Measurements
724
G.8 KEF Predictor
726
G.9 KEF Corrector (Fuzzy DS Measurements)
729
G.10 Likelihoods for AGA Fuzzy Measurements
732
G.11 Likelihoods for AGA Generalized Fuzzy Measurements
733
G.12 Likelihoods for AGA Fuzzy DS Measurements
734
G.13 Interval Argsup Formula
735
G.14 Consonance of the Random State Set Γz
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 Likelihoods
747
G.21 Continuity of Likelihoods 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


Ronald P.S. Mahler is a staff scientist at Lockheed Martin MS2 Tactical Systems with over 25 years of industry experience. He earned his B.E.E. degree at the University of Minnesota and his Ph.D. in Mathematics at Brandeis University. He has served on technology planning workshops for many prominent organizations, including the Electronics Division of the Army Research Office. Dr. Mahler was also a reviewer of the DARPA Dynamic Data Base (DDB) project.
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