Bayesian Estimation and Tracking – A Practical Guide

ANTON J. HAUG, PhD, is member of the technical staff at the Applied Physics Laboratory at The Johns Hopkins University, where he develops advanced target tracking methods in support of the Air and Missile Defense Department. Throughout his career, Dr. Haug has worked across diverse areas such as target tracking; signal and array processing and processor design; active and passive radar and sonar design; digital communications and coding theory; and time- frequency analysis.

Preface Acknowledgments List of Figures xi List of Tables xxi Part I. Prelininaries 1. Introduction 3 1.1 Bayesian Inference 5 1.2 Bayesian Hierarchy of Estimation Methods 7 1.3 Scope of this Text 8 1.4 Modeling and Simulation with Matlab(r) 13 2. Preliminary Mathematical Concepts 19 2.1 A Very Brief Overview of Matrix Linear Algebra 20 2.2 Vector Point Generators 27 2.3 Approximating Nonlinear Multidimensional Functions with Multidimensional Arguments 32 2.4 Overview of Multivariate Statistics 47 3. General Concepts of Bayesian Estimation 69 3.1 Bayesian Estimation 70 3.2 Point Estimators 72 3.3 Introduction to Recursive Bayesian Filtering of Probability Density Functions 76 3.4 Introduction to Recursive Bayesian Estimation of the State Mean and Covariance 81 3.5 Discussion of General Estimation Methods 88 4. Case Studies: Preliminary Discussions 93 4.1 The Overall Simulation/Estimation/Evaluation Process 94 4.2 A Scenario Simulator for Tracking a Constant-Velocity Target Through a DIFAR Buoy Field 97 4.3 DIFAR Buoy Signal Processing 102 4.4 The DIFAR Likelihood Function 111 Part II. The Gaussian Assumption: A Family of Kalman Filter Estimators 5. The Gaussian Noise Case: Multidimensional Integration of Gaussian-Weighted Distributions 119 5.1 Summary of Important Results From Chapter 3 122 5.2 Derivation of the Kalman Filter Correction (Update) Equations Revisted 124 5.3 The General Bayesian Point Prediction Integrals for Gaussian Densities 128 6. The Linear Class of Kalman Filters 141 6.1 Linear Dynamic Models 142 6.2 Linear Observation Models 143 6.3 The Linear Kalman Filter 144 6.4 Application of the LKF to DIFAR Buoy Bearing Estimation 146 7. The Analytical Linearization Class of Kalman Filters: The Extended Kalman Filter 153 7.1 One-Dimensional Consideration 154 7.2 Multidimensional Consideration 159 7.3 An Alternate Derivation of the Multidimensional Covariance Prediction Equations 172 7.4 Application of the EKF to the DIFAR Ship Tracking Case Study 174 8. The Sigma Point Class: The Finite Difference Kalman Filter 187 8.1 One-Dimensional Finite Difference Kalman Filter 189 8.2 Multidimensional Finite Difference Kalman Filters 195 8.3 An Alternate Derivation of the Multidimensional Finite Difference Covariance Prediction Equations 201 9. The Sigma Point Class: The Unscented Kalman Filter 207 9.1 Introduction to Monomial Cubature Integration Rules 207 9.2 The Unscented Kalman Filter 211 9.3 Applications of the UKF to the DIFAR Ship Tracking Case Study 221 10. The Sigma Point Class: The Spherical Simplex Kalman Filter 227 10.1 One-Dimensional Spherical Simplex Sigma Points 228 10.2 Two-Dimensional Spherical Simplex Sigma Points 229 10.3 Higher-Dimensional Spherical Simplex Sigma Points 233 10.4 The Spherical Simplex Kalman Filter 233 10.5 The Spherical Simplex Kalman Filter Process 236 10.6 Application of the SSKF to the DIFAR Ship Tracking Case Study 236 11. The Sigma Point Class: The Gauss-Hermite Kalman Filter 241 11.1 One-Dimensional Gauss-Hermite Quadrature 242 11.2 One-Dimensional Gauss-Hermite Kalman Filter 248 11.3 Multidimensional Gauss-Hermite Kalman Filter 251 11.4 Sparse Grid Approximation for High Dimension/High Polynomial Order 257 11.5 Application of the GHKF to the DIFAR Ship Tracking Case Study 261 12. The Monte Carlo Kalman Filter 265 12.1 The Monte Carlo Kalman Filter 268 13. Summary of Gaussian Kalman Filters 273 13.1 Analytical Kalman Filters 274 13.2 Sigma-Point Kalman Filters 276 13.3 A More Practical Approach to Utilizing the Family of Kalman Filters 284 14. Performance Measures for the Family of Kalman Filters 289 14.1 Error Ellipses 290 14.2 Root Mean Squared Errors 299 14.3 Divergent Tracks 301 14.4 Cramer-Rao Lower Bound 302 14.5 Performance of Kalman Class DIFAR Track Estimators 315 Part III. Monte Carlo Methods 15. Introduction to Monte Carlo Methods 323 15.1 Approximating a Density From a Set of Monte Carlo Samples 325 15.2 General Concepts Importance Sampling 340 15.3 Summary 347 16. Sequential Importance Sampling Particle Filters 351 16.1 General Concept of Sequential Importance Sampling 351 16.2 Resampling and Regularization (Move) for SIS Particle Filters 357 16.3 The Bootstrap Particle Filter 372 16.4 The Optimal SIS Particle Filter 378 16.5 The SIS Auxiliary Particle Filter 385 16.6 Approximations to the SIS Auxiliary Particle Filter 393 16.7 Reducing the Computational Load Through Rao-Blackwellization 396 17. The Generalized Sequential Monte Carlo Particle Filter 403 17.1 The Gaussian Particle Filter 404 17.2 The Combination Particle Filter 406 17.3 Performance Comparison of all DIFAR Tracking Filters 411 Part IV Additional Case Studies 18. A Spherical Constant Velocity Model for Target Tracking in Three Dimensions 421 18.1 Tracking a Target in Cartesian Coordinates 426 18.2 Tracking a Target in Spherical Coordinates 433 18.3 Implementation of Cartesian and Spherical Tracking Filters 443 18.4 Performance Comparison for Various Estimation Methods 453 18.5 Some Observations and Future Considerations 469 19. Tracking a Falling Rigid Body Using Photogrammetry 497 19.1 Introduction 497 19.2 The Process (Dynamic) Model for Rigid Body Motion 502 19.3 Components of the Observation Model 513 19.4 Estimation Methods 517 19.5 The Generation of Synthetic Data 529 19.6 Performance Comparison Analysis 538 20. Sensor Fusion using Photogrammetric and Inertial Measurements 559 20.1 Introduction 559 20.2 The Process (Dynamic) Model for Rigid Body Motion 562 20.3 The Sensor Fusion Observational Model563 20.4 The Generation of Synthetic Data 569 20.5 Estimation Methods 572 20.6 Performance Comparison Analysis 577 20.7 Conclusions 585 20.8 Future Work 586 References 589