Bayesian multiple target tracking

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.

Author Notes

Lawrence D. Stone received his Ph.D. and MS in mathematics from Purdue University,

Stone is Chief Operating Officer at Metron, Inc.

050

Preface to Second Edition	p. xi
Introduction	p. xiii
Acknowledgments	p. xix
Chapter 1 Tracking Problems	p. 1
1.1 Description of Tracking Problem	p. 1
1.1.1 Measurement and Motion Models	p. 2
1.1.2 Estimation	p. 2
1.1.3 Filters	p. 2
1.2 Example 1: Tracking a Surface Ship	p. 5
1.2.1 Prior Distribution on Target State	p. 6
1.2.2 Motion Model	p. 6
1.2.3 Measurement Model	p. 8
1.2.4 Tracker Output	p. 8
1.3 Example 2: Bearings-Only Tracking	p. 11
1.3.1 Description of Example	p. 12
1.3.2 Prior Distribution	p. 13
1.3.3 Motion Model	p. 14
1.3.4 Measurement Model	p. 16
1.3.5 Particle Filter Description	p. 18
1.3.6 Comments	p. 20
1.3.7 Tracker Output	p. 20
1.4 Example 3: Periscope Detection and Tracking	p. 23
1.4.1 Target Tracker	p. 24
1.4.2 Example	p. 25
1.5 Example 4: Tracking Multiple Targets	p. 28
1.5.1 Soft Association	p. 29
1.6 Summary	p. 33
References	p. 38
Chapter 2 Bayesian Inference and Likelihood Functions	p. 39
2.1 The Case for Bayesian Inference	p. 40
2.1.1 Frequentist Point of View	p. 40
2.1.2 Conditionalist Point of View	p. 41
2.1.3 Bayesian Point of View	p. 42
2.2 The Likelihood Function and Bayes' Theorem	p. 44
2.2.1 The Likelihood Function	p. 45
2.2.2 Bayes' Theorem	p. 46
2.2.3 Sequential Nature of Bayes' Theorem	p. 46
2.3 Examples of Likelihood Functions	p. 47
2.3.1 A Gaussian Contact Model	p. 47
2.3.2 A Gaussian Bearing Error Model	p. 49
2.3.3 Combining Bearing and Contact Measurements	p. 50
2.3.4 Negative Information	p. 53
2.3.5 Positive Information	p. 56
2.3.6 Radar and Infrared Detection	p. 58
2.3.7 A Signal-Plus-Noise Model	p. 60
2.3.8 Summary	p. 63
References	p. 64
Chapter 3 Single Target Tracking	p. 65
3.1 Bayesian Filtering	p. 66
3.1.1 Recursive Bayesian Filtering	p. 66
3.1.2 Prediction and Smoothing	p. 73
3.1.3 Recursive Prediction	p. 74
3.1.4 Recursive Smoothing	p. 74
3.1.5 Batch Smoothing	p. 77
3.1.6 Land Avoidance	p. 77
3.2 Kalman Filtering	p. 80
3.2.1 Discrete Kalman Filtering	p. 81
3.2.2 Continuous-Discrete Kalman Filtering	p. 86
3.2.3 Kalman Smoothing	p. 93
3.3 Particle Filter Implementation of Nonlinear Filtering	p. 97
3.3.1 Generating Particles	p. 98
3.3.2 Particle Filter Recursion	p. 99
3.3.3 Resampling	p. 100
3.3.4 Perturbing Target States	p. 101
3.3.5 Convergence	p. 102
3.3.6 Outliers	p. 103
3.3.7 Multiple Motion Models	p. 104
3.3.8 High Dimensional State Spaces	p. 105
3.4 Summary	p. 105
References	p. 105
Chapter 4 Classical Multiple Target Tracking	p. 107
4.1 Multiple Target Tracking	p. 109
4.1.1 Multiple Target Motion Model	p. 109
4.1.2 Multiple Target Likelihood Functions	p. 110
4.1.3 Bayesian Recursion for Multiple Targets	p. 112
4.2 Multiple Hypothesis Tracking	p. 113
4.2.1 Contacts	p. 113
4.2.2 Scans	p. 115
4.2.3 Data Association Hypotheses	p. 115
4.2.4 Scans and Scan Association Hypotheses	p. 117
4.2.5 Multiple Hypothesis Tracking Decomposition	p. 120
4.3 Independent Multiple Hypothesis Tracking	p. 122
4.3.1 Conditionally Independent Association Likelihoods	p. 123
4.3.2 Scan Association Likelihood Function Example	p. 124
4.3.3 Independence Theorem	p. 126
4.3.4 Independent MHT Recursion	p. 129
4.4 Linear-Gaussian Multiple Hypothesis Tracking	p. 130
4.4.1 MHT Recursion for Linear-Gaussian Case	p. 131
4.4.2 Posterior Distributions and Association Probabilities	p. 132
4.5 Nonlinear Joint Probabilistic Data Association	p. 135
4.5.1 Scan Association Hypotheses	p. 136
4.5.2 Scan Association Probability	p. 136
4.5.3 JPDA Posterior	p. 139
4.5.4 Allowing New Targets and Deleting Existing Ones	p. 139
4.5.5 Particle Filter Implementation	p. 140
4.5.6 Example	p. 141
4.6 Probabilistic Multiple Hypothesis Tracking (PMHT)	p. 142
4.6.1 PMHT Assumptions	p. 143
4.6.2 Posterior Distribution on Associations	p. 146
4.6.3 Expectation Maximization	p. 147
4.6.4 Nonlinear PMHT	p. 149
4.6.5 Linear-Gaussian PMHT	p. 152
4.6.6 Proof (4.81)	p. 153
4.7 Summary	p. 155
4.8 Notes	p. 156
References	p. 159
Chapter 5 Multitarget Intensity Filters	p. 161
5.1 Point Process Model of Multitarget State	p. 163
5.1.1 Basic Properties of PPPs	p. 164
5.1.2 Probability Distribution Function for a PPP	p. 166
5.1.3 Superposition of Point Processes	p. 166
5.1.4 Target Motion Process	p. 167
5.1.5 Sensor Measurement Process	p. 167
5.1.6 Thinning a Process	p. 167
5.1.7 Augmented Spaces	p. 168
5.2 Intensity Filter	p. 169
5.2.1 Augmented State Space Modeling	p. 169
5.2.2 Predicted Detected and Undetected Target Processes	p. 170
5.2.3 Measurement Process	p. 171
5.2.4 Bayes Posterior Point Process (Information Update)	p. 172
5.2.5 PPP Approximation	p. 173
5.2.6 Correlation Losses in the PPP Approximation	p. 174
5.2.7 The iFilter	p. 174
5.2.8 Transformations of PPPs are PPPs	p. 175
5.3 Probability Hypothesis Density (PHD) Filter	p. 178
5.4 PQF Approach to the iFilter	p. 180
5.4.1 Brief Review of PGFs	p. 181
5.4.2 The iFilter on Finite Grids	p. 185
5.4.3 Joint PGF of Gridded States and Histogram Data	p. 185
5.4.4 Small Cell Size Limits	p. 194
5.5 Extended Target Filters	p. 197
5.6 Summary	p. 197
5.7 Notes	p. 199
5.7.1 Other Topics	p. 199
5.7.2 Background	p. 200
References	p. 200
Chapter 6 Multiple Target Tracking Using Tracker-Generated Measurements	p. 203
6.1 Maximum A Posteriori Penalty Function Tracking	p. 204
6.1.1 MAP-PF Formulation	p. 205
6.1.2 Iterative Optimization	p. 209
6.1.3 Algorithm	p. 212
6.1.4 Variations	p. 213
6.2 Particle Filter Implementation	p. 215
6.3 Linear-Gaussian Implementation	p. 216
6.4 Examples	p. 217
6.4.1 Model	p. 217
6.4.2 MAP-PF Implementation	p. 221
6.4.3 JPDA Implementation	p. 225
6.4.4 Summary of Examples	p. 227
6.5 Summary	p. 227
6.6 Notes	p. 229
6.7 Sensor Array Observation Model and Signal Processing	p. 230
6.7.1 Sensor Observation Model	p. 230
6.7.2 Array Signal Processing	p. 232
6.7.3 Cramér-Rao Bound (CRB)	p. 236
References	p. 237
Chapter 7 Likelihood Ratio Detection and Tracking	p. 239
7.1 Basic Definitions and Relations	p. 240
7.1.1 Likelihood Ratio	p. 242
7.1.2 Measurement Likelihood Ratio	p. 242
7.2 Likelihood Ratio Recursions	p. 243
7.2.1 Simplified Likelihood Ratio Recursion	p. 245
7.2.2 Log-Likelihood Ratios	p. 247
7.3 Declaring a Target Present	p. 247
7.3.1 Minimizing Bayes' Risk	p. 248
7.3.2 Target Declaration at a Given Confidence Level	p. 249
7.3.3 Neyman-Pearson Criterion for Declaration	p. 249
7.3.4 Track Before Detect	p. 249
7.4 Low-SNR Examples of LRDT	p. 250
7.4.1 Simple Example	p. 250
7.4.2 Periscope Detection Example	p. 257
7.5 Thresholded Data with High Clutter Rate	p. 262
7.5.1 Measurement and False Alarm Model	p. 262
7.5.2 Multistage Sonar Example	p. 264
7.6 Grid-Based Implementation	p. 269
7.6.1 Prior Likelihood Ratio	p. 270
7.6.2 Motion Model	p. 270
7.6.3 Information Update	p. 272
7.7 Multiple Target Tracking Using LRDT	p. 272
7.7.1 Local Property for Measurement Likelihood Ratios	p. 273
7.7.2 LRDT as Detector for a Multiple Target Tracker	p. 274
7.8 iLRT	p. 275
7.8.1 Particle Filter Implementation of Intensity Filtering	p. 275
7.8.2 Target Detection and Track Estimation	p. 278
7.8.3 Example	p. 279
7.9 Summary	p. 282
7.10 Notes	p. 283
References	p. 284
Appendix	p. 285
About the Authors	p. 287
Index	p. 289

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