Bayesian Estimation and Tracking: A Practical Guide

Table of Contents:
Preface xv Acknowledgments xvii List of Figures Xix List of Tables xxv PART I PRELIMINARIES 1 Introduction 3 1.1 Bayesian Inference 4 1.2 Bayesian Hierarchy of Estimation Methods 5 1.3 Scope of This Text 6 1.3.1 Objective 6 1.3.2 Chapter Overview and Prerequisites 6 1.4 Modeling and Simulation with MATLAB® 8 References 9 2 Preliminary Mathematical Concepts 11 2.1 A Very Brief Overview of Matrix Linear Algebra 11 2.1.1 Vector and Matrix Conventions and Notation 11 2.1.2 Sums and Products 12 2.1.3 Matrix Inversion 13 2.1.4 Block Matrix Inversion 14 2.1.5 Matrix Square Root 15 2.2 Vector Point Generators 16 2.3 Approximating Nonlinear Multidimensional Functions with Multidimensional Arguments 19 2.3.1 Approximating Scalar Nonlinear Functions 19 2.3.2 Approximating Multidimensional Nonlinear Functions 23 2.4 Overview of Multivariate Statistics 29 2.4.1 General Definitions 29 2.4.2 The Gaussian Density 32 References 40 3 General Concepts of Bayesian Estimation 42 3.1 Bayesian Estimation 43 3.2 Point Estimators 43 3.3 Introduction to Recursive Bayesian Filtering of Probability Density Functions 46 3.4 Introduction to Recursive Bayesian Estimation of the State Mean and Covariance 49 3.4.1 State Vector Prediction 50 3.4.2 State Vector Update 51 3.5 Discussion of General Estimation Methods 55 References 55 4 Case Studies: Preliminary Discussions 56 4.1 The Overall Simulation/Estimation/Evaluation Process 57 4.2 A Scenario Simulator for Tracking a Constant Velocity Target Through a DIFAR Buoy Field 58 4.2.1 Ship Dynamics Model 58 4.2.2 Multiple Buoy Observation Model 59 4.2.3 Scenario Specifics 59 4.3 DIFAR Buoy Signal Processing 62 4.4 The DIFAR Likelihood Function 67 References 69 PART II THE GAUSSIAN ASSUMPTION: A FAMILY OF KALMAN FILTER ESTIMATORS 5 The Gaussian Noise Case: Multidimensional Integration of Gaussian-Weighted Distributions 73 5.1 Summary of Important Results From Chapter 3 74 5.2 Derivation of the Kalman Filter Correction (Update) Equations Revisited 76 5.3 The General Bayesian Point Prediction Integrals for Gaussian Densities 78 5.3.1 Refining the Process Through an Affine Transformation 80 5.3.2 General Methodology for Solving Gaussian-Weighted Integrals 82 References 85 6 The Linear Class of Kalman Filters 86 6.1 Linear Dynamic Models 86 6.2 Linear Observation Models 87 6.3 The Linear Kalman Filter 88 6.4 Application of the LKF to DIFAR Buoy Bearing Estimation 88 References 92 7 The Analytical Linearization Class of Kalman Filters: The Extended Kalman Filter 93 7.1 One-Dimensional Consideration 93 7.1.1 One-Dimensional State Prediction 94 7.1.2 One-Dimensional State Estimation Error Variance Prediction 95 7.1.3 One-Dimensional Observation Prediction Equations 96 7.1.4 Transformation of One-Dimensional Prediction Equations 96 7.1.5 The One-Dimensional Linearized EKF Process 98 7.2 Multidimensional Consideration 98 7.2.1 The State Prediction Equation 99 7.2.2 The State Covariance Prediction Equation 100 7.2.3 Observation Prediction Equations 102 7.2.4 Transformation of Multidimensional Prediction Equations 103 7.2.5 The Linearized Multidimensional Extended Kalman Filter Process 105 7.2.6 Second-Order Extended Kalman Filter 105 7.3 An Alternate Derivation of the Multidimensional Covariance Prediction Equations 107 7.4 Application of the EKF to the DIFAR Ship Tracking Case Study 108 7.4.1 The Ship Motion Dynamics Model 108 7.4.2 The DIFAR Buoy Field Observation Model 109 7.4.3 Initialization for All Filters of the Kalman Filter Class 111 7.4.4 Choosing a Value for the Acceleration Noise 112 7.4.5 The EKF Tracking Filter Results 112 References 114 8 The Sigma Point Class: The Finite Difference Kalman Filter 115 8.1 One-Dimensional Finite Difference Kalman Filter 116 8.1.1 One-Dimensional Finite Difference State Prediction 116 8.1.2 One-Dimensional Finite Difference State Variance Prediction 117 8.1.3 One-Dimensional Finite Difference Observation Prediction Equations 118 8.1.4 The One-Dimensional Finite Difference Kalman Filter Process 118 8.1.5 Simplified One-Dimensional Finite Difference Prediction Equations 118 8.2 Multidimensional Finite Difference Kalman Filters 120 8.2.1 Multidimensional Finite Difference State Prediction 120 8.2.2 Multidimensional Finite Difference State Covariance Prediction 123 8.2.3 Multidimensional Finite Difference Observation Prediction Equations 124 8.2.4 The Multidimensional Finite Difference Kalman Filter Process 125 8.3 An Alternate Derivation of the Multidimensional Finite Difference Covariance Prediction Equations 125 References 127 9 The Sigma Point Class: The Unscented Kalman Filter 128 9.1 Introduction to Monomial Cubature Integration Rules 128 9.2 The Unscented Kalman Filter 130 9.2.1 Background 130 9.2.2 The UKF Developed 131 9.2.3 The UKF State Vector Prediction Equation 134 9.2.4 The UKF State Vector Covariance Prediction Equation 134 9.2.5 The UKF Observation Prediction Equations 135 9.2.6 The Unscented Kalman Filter Process 135 9.2.7 An Alternate Version of the Unscented Kalman Filter 135 9.3 Application of the UKF to the DIFAR Ship Tracking Case Study 137 References 138 10 The Sigma Point Class: The Spherical Simplex Kalman Filter 140 10.1 One-Dimensional Spherical Simplex Sigma Points 141 10.2 Two-Dimensional Spherical Simplex Sigma Points 142 10.3 Higher Dimensional Spherical Simplex Sigma Points 144 10.4 The Spherical Simplex Kalman Filter 144 10.5 The Spherical Simplex Kalman Filter Process 145 10.6 Application of the SSKF to the DIFAR Ship Tracking Case Study 146 Reference 147 11 The Sigma Point Class: The Gauss–Hermite Kalman Filter 148 11.1 One-Dimensional Gauss–Hermite Quadrature 149 11.2 One-Dimensional Gauss–Hermite Kalman Filter 153 11.3 Multidimensional Gauss–Hermite Kalman Filter 155 11.4 Sparse Grid Approximation for High Dimension/High Polynomial Order 160 11.5 Application of the GHKF to the DIFAR Ship Tracking Case Study 163 References 163 12 The Monte Carlo Kalman Filter 164 12.1 The Monte Carlo Kalman Filter 167 Reference 167 13 Summary of Gaussian Kalman Filters 168 13.1 Analytical Kalman Filters 168 13.2 Sigma Point Kalman Filters 170 13.3 A More Practical Approach to Utilizing the Family of Kalman Filters 174 References 175 14 Performance Measures for the Family of Kalman Filters 176 14.1 Error Ellipses 176 14.1.1 The Canonical Ellipse 177 14.1.2 Determining the Eigenvalues of P 178 14.1.3 Determining the Error Ellipse Rotation Angle 179 14.1.4 Determination of the Containment Area 180 14.1.5 Parametric Plotting of Error Ellipse 181 14.1.6 Error Ellipse Example 182 14.2 Root Mean Squared Errors 182 14.3 Divergent Tracks 183 14.4 Cramer–Rao Lower Bound 184 14.4.1 The One-Dimensional Case 184 14.4.2 The Multidimensional Case 186 14.4.3 A Recursive Approach to the CRLB 186 14.4.4 The Cramer–Rao Lower Bound for Gaussian Additive Noise 190 14.4.5 The Gaussian Cramer–Rao Lower Bound with Zero Process Noise 191 14.4.6 The Gaussian Cramer–Rao Lower Bound with Linear Models 191 14.5 Performance of Kalman Class DIFAR Track Estimators 192 References 198 PART III MONTE CARLO METHODS 15 Introduction to Monte Carlo Methods 201 15.1 Approximating a Density From a Set of Monte Carlo Samples 202 15.1.1 Generating Samples from a Two-Dimensional Gaussian Mixture Density 202 15.1.2 Approximating a Density by Its Multidimensional Histogram 202 15.1.3 Kernel Density Approximation 204 15.2 General Concepts Importance Sampling 210 15.3 Summary 215 References 216 16 Sequential Importance Sampling Particle Filters 218 16.1 General Concept of Sequential Importance Sampling 218 16.2 Resampling and Regularization (Move) for SIS Particle Filters 222 16.2.1 The Inverse Transform Method 222 16.2.2 SIS Particle Filter with Resampling 226 16.2.3 Regularization 227 16.3 The Bootstrap Particle Filter 230 16.3.1 Application of the BPF to DIFAR Buoy Tracking 231 16.4 The Optimal SIS Particle Filter 233 16.4.1 Gaussian Optimal SIS Particle Filter 235 16.4.2 Locally Linearized Gaussian Optimal SIS Particle Filter 236 16.5 The SIS Auxiliary Particle Filter 238 16.5.1 Application of the APF to DIFAR Buoy Tracking 242 16.6 Approximations to the SIS Auxiliary Particle Filter 243 16.6.1 The Extended Kalman Particle Filter 243 16.6.2 The Unscented Particle Filter 243 16.7 Reducing the Computational Load Through Rao-Blackwellization 245 References 245 17 The Generalized Monte Carlo Particle Filter 247 17.1 The Gaussian Particle Filter 248 17.2 The Combination Particle Filter 250 17.2.1 Application of the CPF–UKF to DIFAR Buoy Tracking 252 17.3 Performance Comparison of All DIFAR Tracking Filters 253 References 255 PART IV ADDITIONAL CASE STUDIES 18 A Spherical Constant Velocity Model for Target Tracking in Three Dimensions 259 18.1 Tracking a Target in Cartesian Coordinates 261 18.1.1 Object Dynamic Motion Model 262 18.1.2 Sensor Data Model 263 18.1.3 GaussianTracking Algorithms for a Cartesian StateVector 264 18.2 Tracking a Target in Spherical Coordinates 265 18.2.1 State Vector Position and Velocity Components in Spherical Coordinates 266 18.2.2 Spherical State Vector Dynamic Equation 267 18.2.3 Observation Equations with a Spherical State Vector 270 18.2.4 GaussianTracking Algorithms for a Spherical StateVector 270 18.3 Implementation of Cartesian and Spherical Tracking Filters 273 18.3.1 Setting Values for q 273 18.3.2 Simulating Radar Observation Data 274 18.3.3 Filter Initialization 276 18.4 Performance Comparison for Various Estimation Methods 278 18.4.1 Characteristics of the Trajectories Used for Performance Analysis 278 18.4.2 Filter Performance Comparisons 282 18.5 Some Observations and Future Considerations 293 APPENDIX 18.A Three-Dimensional Constant Turn Rate Kinematics 294 18.A.1 General Velocity Components for Constant Turn Rate Motion 294 18.A.2 General Position Components for Constant Turn Rate Motion 297 18.A.3 Combined Trajectory Transition Equation 299 18.A.4 Turn Rate Setting Based on a Desired Turn Acceleration 299 APPENDIX 18.B Three-Dimensional Coordinate Transformations 301 18.B.1 Cartesian-to-Spherical Transformation 302 18.B.2 Spherical-to-Cartesian Transformation 305 References 306 19 Tracking a Falling Rigid Body Using Photogrammetry 308 19.1 Introduction 308 19.2 The Process (Dynamic) Model for Rigid Body Motion 311 19.2.1 Dynamic Transition of the Translational Motion of a Rigid Body 311 19.2.2 Dynamic Transition of the Rotational Motion of a Rigid Body 313 19.2.3 Combined Dynamic Process Model 316 19.2.4 The Dynamic Process Noise Models 317 19.3 Components of the Observation Model 318 19.4 Estimation Methods 321 19.4.1 A Nonlinear Least Squares Estimation Method 321 19.4.2 An Unscented Kalman Filter Method 323 19.4.3 Estimation Using the Unscented Combination Particle Filter 325 19.4.4 Initializing the Estimator 326 19.5 The Generation of Synthetic Data 328 19.5.1 Synthetic Rigid Body Feature Points 328 19.5.2 Synthetic Trajectory 328 19.5.3 Synthetic Cameras 333 19.5.4 Synthetic Measurements 333 19.6 Performance Comparison Analysis 334 19.6.1 Filter Performance Comparison Methodology 335 19.6.2 Filter Comparison Results 338 19.6.3 Conclusions and Future Considerations 341 APPENDIX 19.A Quaternions Axis-Angle Vectors and Rotations 342 19.A.1 Conversions Between Rotation Representations 342 19.A.2 Representation of Orientation and Rotation 343 19.A.3 Point Rotations and Frame Rotations 344 References 345 20 Sensor Fusion Using Photogrammetric and Inertial Measurements 346 20.1 Introduction 346 20.2 The Process (Dynamic) Model for Rigid Body Motion 347 20.3 The Sensor Fusion Observational Model 348 20.3.1 The Inertial Measurement Unit Component of the Observation Model 348 20.3.2 The Photogrammetric Component of the Observation Model 350 20.3.3 The Combined Sensor Fusion Observation Model 351 20.4 The Generation of Synthetic Data 352 20.4.1 Synthetic Trajectory 352 20.4.2 Synthetic Cameras 352 20.4.3 Synthetic Measurements 352 20.5 Estimation Methods 354 20.5.1 Initial Value Problem Solver for IMU Data 354 20.6 Performance Comparison Analysis 357 20.6.1 Filter Performance Comparison Methodology 359 20.6.2 Filter Comparison Results 360 20.7 Conclusions 361 20.8 Future Work 362 References 364 Index 367

About the Author :
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.