This second edition has undergone substantial revision from the 1999 first edition, recognizing that a lot has changed in the multiple target tracking field. One of the most dramatic changes is in the widespread use of particle filters to implement nonlinear, non-Gaussian Bayesian trackers. This book views multiple target tracking as a Bayesian inference problem. Within this framework it develops the theory of single target tracking. In addition to providing a detailed description of a basic particle filter that implements the Bayesian single target recursion, this resource provides numerous examples that involve the use of particle filters.

Preface to Second Edition xi

Introduction xiii

Acknowledgments xix

Chapter I Tracking Problems 1

1.1 Description of Tracking Problem 1

      1.1.1 Measurement and Motion Models 2

      1.1.2 Estimation 2

      1.1.3 Filters 2

1.2 Example 1 : Tracking a Surface Ship 5

      1.2.1 Prior Distribution on Target State 6

      1.2.2 Motion Model 6

      1.2.3 Measurement Model 8

      1.2.4 Tracker Output 8

1.3 Example 2: Bearings-Only Tracking 11

      1.3.1 Description of Example 12

      1.3.2 Pnor Distribution 13

      1.3.3 Motion Model 14

      1.3.4 Measurement Model 16

      1.3.5 Particle Filter Description 18

      1.3.6 Comments 20

      1.3.7 Tracker Output 20

1.4 Example 3: Periscope Detection and Tracking 23

      1.4.1 Target Tracker 24

      1.4.2 Example 25

1.5 Example 4: Trackinε Multiple Targets 28

      1.5.1 Soft Association 29

      1.6 Summary 33

      References 38

Chapter 2 Bayesian Inference and Likelihood Functions 39

2.1 The Case for Bayesian Inference 40

      2.1.1 Frequentist Point of View 40

      2.1.2 Conditionalist Point of View 41

      2.1.3 Bayesian Point of View 42

2.2 The Likelihood Function and Bayes’ Theorem 44

      2.2.1 The Likelihood Function 45

      2.2.2 Bay Theorem 46

      2.2.3 Sequential Nature of Bayes’ Theorem 46

2.3 Examples of Likelihood Functions 47

      2.3.1 A Gaussian Contact Model 47

      2.3.2 A Gaussian Bearing Error Model 49

      2.3.3 Combining Bearin and Contact Measurements 50

      2.3.4 Negative Information 53

      2.3.5 Positive Information 56

      2.3.6 Radar and Infrared Detection 58

      2.3.7 A Signal-Plus-Noise Model 60

      2.3.8 Summary 63

References 64

Chapter 3 Single Target Tracking 65

3.1 Bayesian Filtering 66

      3.1.1 Recursive Bayesian Filtering 66

      3.1.2 Prediction and Smoothing 73

      3.1.3 Recursive Prediction 74

      3.1.4 Recursive Smoothing 74

      3.1.5 Batch Smoothing 77

      3.1.6 Land Avoidance 77

3.2 Kalman Filtering 80

      3.2.1 Discrete Kalman Filtering 81

      3.2.2 Continuous-Discrete Kalman Filtering 86

      3.2.3 Kalman Smoothing. 93

3.3 Particle Filter Implementation of Nonlinear Filtering 97

      3.3.1 Generating Particles 98

      3.3.2 Particle Filter Recursion 99

      3.3.3 Resampling 100

      3.3.4 Perturbing Target States 101

      3.3.5 Convergence 102

      3.3.6 Outliers . 103

      3.3.7 Multiple Motion Models 104

      3.3.8 High Dimensional State Spaces 105

3.4 Summary 105

References 105

Chapter 4 Classical Multiple Target Tracking 107

4.1 Multiple Target Tracking 109

      4.1.1 Multiple Target Motion Model 109

      4.1.2 Multi ple Target Likeli hood Functions 110

      4.1.3 Bayesian Recursion for Multiple Targets 112

4.2 Multiple Hypothesis Tracking 1 13

4.2.1 Contacts 113

4.2.2 Scans 115

4.2.3 Data Association Hypotheses 115

4.2.4 Scans and Scan Association Hypotheses 117

4.2.5 Multiple Hypothesis Tracking Decomposition 120

4.3 Independent Multiple Hypothesis Tracking 122

      4.3.1 Conditionally Independent Association Likelihoods 123

      4.3.2 Scan Association Likelihood Function Example 124

      4.3 3 Independence Theorem 126

      4.3.4 Independent MHT Recursion 129

4.4 Linear-Gaussian Multiple Hypothesis Tracking 130

      4.4.1 MHT Recursion for Linear-Gaussian Case 131

      4.4.2 2 Posterior Distributions and Association Probabilities 132

4.5 Nonlinear Joint Probabilistic Data Association 135

      4.5.1 Scan Association Hypotheses 136

      4.5.2 Scan Association Probability 136

      4 5.3 JPDA Posterior 139

      4.5.4 Allowing New Targets and Deleting Existing Ones 139

      4.5.5 Particle Fi lter Impl ementation 140

      4.5.6 Example 141

4.6 Probabil istic Multi ple Hypothesis Tracking (PMJ-IT) 142

      4.6.1 PMJ-IT Assumption s 143

      4.6.2 Posterior Distribution on Associations 146

      4.6.3 Expectation Maximization 147

      4.6.4 Nonlinear PMHT 149

      4.6.5 Linear-Gau ssian PMHT 152

      4.6.6 Proof (4.81) 153

4.7 Summary 155

4.8 Notes 156

References 159

Chapter 5 Multitarget I ntensity Filters 161

5.1 Point Process Model of Multitarget State 163

      5.1.1 Basic Properties of PPPs 164

      5.1.2 Probability Distribution Function for a PPP 166

      5.1.3 Superposition of Point Processes 166

      5.1.4 Target Motion Process. 167

      5.1.5 Sensor Measurement Process 167

      5.1.6 Thinning a Process 1 67

      5.1.7 Augmented Spaces 168

5.2 Intensity Filter 169

      5.2.1 Augmented State Space Modeling 169

      5.2.2 Predicted Detected and Undetected Target Processes 170

      5.2.3 Measurement Process. 171

      5.2.4 Bayes Posterior Point Process (Information Update) 172

      5.2.5 PPP Approximation 173

      5.2.6 Correlation Losses in the PPP Approximation 174

      5.2.7 The iFilter 174

      5.2.8 Transformations of PPPs are PPPs 175

5.3 Probability Hypothesis Density (PHD) Filter 178

5.4 PGF Approach to the iFilter 180

      5.4.1 Brief Review of PGFs 181

      5.4.2 η1e iFilter on Finite Grids 185

      5.4.3 Joint PGF of Gridded States and Histogram Data 185

      5.4.4 Small Cell Size Limits 194

5.5 Extended Target Filters 197

5.6 Summary 197

5.7 Notes 199

      5.7.1 Other Topics 199

      5.7.2 Background 200

References 200

Chapter 6 Multiple Target Tracking Using Tracker-Generated Measurements 203

6.1 Maximum A Posteriori Penalty Function Tracking 204

      6.1.1 MAP-PF Formulation 205

      6.1.2 fterative Optimization . 209

      6.1.3 Algorithm 112

      6.1.4 Variations 213

6.2 Particle Filter Implementation 215

6.3 Linear-Gaussian Implementation 216

6.4 Examples 117

      6.4.1 Model 217

      6.4.2 MAP-PF Implementation 221

      6.4.3 JPDA Implementation 225

      6.4.4 Summary of Examples 227

6.5 Summary 227

6.6 Notes 229

6.7 Sensor Array Observation Model and Signal Processing 230

      6.7.1 Sensor Observation Model 230

      6.7.2 Array Signal Processing 232

      6.7.3 Cramer-Rao Bound (CRB） 236

References 237

Chapter 7 Likelihood Ratio Detection and Tracking 239

7.1 Basic Definitions and Relations 240

      7.1.1 Likelihood Ratio 242

      7.1.2 Measurement Likelihood Ratio 242

7.2 Likelihood Ratio Recursions 243

7.2.1 Simplified Likelihood Ratio Recursion 245

7.2.2 Log-Likeli hood Ratios 247

7.3 Declaring a Target Present 247

7.3.1 Minimizing Bayes’ Risk 248

7.3.2 Target Declaration at a Given Confidence Level 249

7.3.3 Neyman-Pearson Criter ion for Declaration 249

7.3.4 Track Before Detect 249

7.4 Low-SNR Examples of LRDT 250

      7.4.1 Simple Example 250

      7.4.2 Periscope Detection Example 257

7.5 Thresholded Data with High Clutter Rate 262

7.5.1 Measurement and Fa lse Alarm Model 262

7.5.2 Mult istatic Sonar Example 264

7.6 Grid-Based Implementation 269

      7.6.1 Prior Likelihood Ratio 270

      7.6.2 Motion Model 270

      7.6.3 In formation Update 272

7.7 Multi ple Target Tracking Using LRDT 272

      7.7.1 Local Property for Measurement Likelihood Ratios 273

      7.7.2 LRDT as Detector for a Multiple Target Tracker 274

7.8 iLRT 275

      7.8.1 Particle Filter Implementation of Intensity Filtering 275

      7.8.2 Target Detection and Track Estimation 278

      7.8.3 Example 279

7.9 Summary 282

7.10 Notes 283

References 284

Appendix 285

About the Authors 287

Index 289

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