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E-raamat: Bayesian Estimation and Tracking: A Practical Guide

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  • Formaat: EPUB+DRM
  • Ilmumisaeg: 29-May-2012
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781118287804
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Bayesian Estimation and Tracking: A Practical GuideSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 29-May-2012
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781118287804
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"This book presents a practical approach to estimation methods that are designed to provide a clear path to programming all algorithms. Readers are provided with a firm understanding of Bayesian estimation methods and their interrelatedness. Starting with fundamental principles of Bayesian theory, the book shows how each tracking filter is derived from a slight modification to a previous filter. Such a development gives readers a broader understanding of the hierarchy of Bayesian estimation and tracking.Following the discussions about each tracking filter, the filter is put into block diagram form for ease in future recall and reference. The book presents a completely unified approach to Bayesian estimation and tracking, and this is accomplished by showing that the current posterior density for a state vector can be linked to its previous posterior density through the use of Bayes' Law and the Chapman-Kolmogorov integral. Predictive point estimates are then shown to be density-weighted integrals of nonlinear functions. The book also presents a methodology that makes implementation of the estimation methods simple (or, rather, simpler than they have been in the past). Each algorithm is accompanied by a block diagram that illustrates how all parts of the tracking filter are linked in a never-ending chain, from initialization to the loss of track. These filter block diagrams provide a ready picture for implementing the algorithms into programmable code. In addition, four completely worked out case studies give readers examples of implementation, from simulation models that generate noisy observations to worked-out applications for all tracking algorithms. This book also presents the development and application of track performance metrics, including how togenerate error ellipses when implementing in real-world applications, how to calculate RMS errors in simulation environments, and how to calculate Cramer-Rao lower bounds for the RMS errors. These are also illustrated in the case study presentations"--

Preface xv
Acknowledgments xvii
List Of Figures
xix
List Of Tables
xxv
PART I PRELIMINARIES
1 Introduction
3(8)
1.1 Bayesian Inference
4(1)
1.2 Bayesian Hierarchy of Estimation Methods
5(1)
1.3 Scope of This Text
6(2)
1.3.1 Objective
6(1)
1.3.2
Chapter Overview and Prerequisites
6(2)
1.4 Modeling and Simulation with MATLAB®
8(3)
References
9(2)
2 Preliminary Mathematical Concepts
11(31)
2.1 A Very Brief Overview of Matrix Linear Algebra
11(5)
2.1.1 Vector and Matrix Conventions and Notation
11(1)
2.1.2 Sums and Products
12(1)
2.1.3 Matrix Inversion
13(1)
2.1.4 Block Matrix Inversion
14(1)
2.1.5 Matrix Square Root
15(1)
2.2 Vector Point Generators
16(3)
2.3 Approximating Nonlinear Multidimensional Functions with Multidimensional Arguments
19(10)
2.3.1 Approximating Scalar Nonlinear Functions
19(4)
2.3.2 Approximating Multidimensional Nonlinear Functions
23(6)
2.4 Overview of Multivariate Statistics
29(13)
2.4.1 General Definitions
29(3)
2.4.2 The Gaussian Density
32(8)
References
40(2)
3 General Concepts of Bayesian Estimation
42(14)
3.1 Bayesian Estimation
43(1)
3.2 Point Estimators
43(3)
3.3 Introduction to Recursive Bayesian Filtering of Probability Density Functions
46(3)
3.4 Introduction to Recursive Bayesian Estimation of the State Mean and Covariance
49(6)
3.4.1 State Vector Prediction
50(1)
3.4.2 State Vector Update
51(4)
3.5 Discussion of General Estimation Methods
55(1)
References
55(1)
4 Case Studies: Preliminary Discussions
56(17)
4.1 The Overall Simulation/Estimation/Evaluation Process
57(1)
4.2 A Scenario Simulator for Tracking a Constant Velocity Target Through a DIFAR Buoy Field
58(4)
4.2.1 Ship Dynamics Model
58(1)
4.2.2 Multiple Buoy Observation Model
59(1)
4.2.3 Scenario Specifics
59(3)
4.3 DIFAR Buoy Signal Processing
62(5)
4.4 The DIFAR Likelihood Function
67(6)
References
69(4)
PART II THE GAUSSIAN ASSUMPTION: A FAMILY OF KALMAN FILTER ESTIMATORS
5 The Gaussian Noise Case: Multidimensional Integration of Gaussian-Weighted Distributions
73(13)
5.1 Summary of Important Results From
Chapter 3
74(2)
5.2 Derivation of the Kalman Filter Correction (Update) Equations Revisited
76(2)
5.3 The General Bayesian Point Prediction Integrals for Gaussian Densities
78(8)
5.3.1 Refining the Process Through an Affine Transformation
80(2)
5.3.2 General Methodology for Solving Gaussian-Weighted Integrals
82(3)
References
85(1)
6 The Linear Class of Kalman Filters
86(7)
6.1 Linear Dynamic Models
86(1)
6.2 Linear Observation Models
87(1)
6.3 The Linear Kalman Filter
88(1)
6.4 Application of the LKF to DIFAR Buoy Bearing Estimation
88(5)
References
92(1)
7 The Analytical Linearization Class of Kalman Filters: The Extended Kalman Filter
93(22)
7.1 One-Dimensional Consideration
93(5)
7.1.1 One-Dimensional State Prediction
94(1)
7.1.2 One-Dimensional State Estimation Error Variance Prediction
95(1)
7.1.3 One-Dimensional Observation Prediction Equations
96(1)
7.1.4 Transformation of One-Dimensional Prediction Equations
96(2)
7.1.5 The One-Dimensional Linearized EKF Process
98(1)
7.2 Multidimensional Consideration
98(9)
7.2.1 The State Prediction Equation
99(1)
7.2.2 The State Covariance Prediction Equation
100(2)
7.2.3 Observation Prediction Equations
102(1)
7.2.4 Transformation of Multidimensional Prediction Equations
103(2)
7.2.5 The Linearized Multidimensional Extended Kalman Filter Process
105(1)
7.2.6 Second-Order Extended Kalman Filter
105(2)
7.3 An Alternate Derivation of the Multidimensional Covariance Prediction Equations
107(1)
7.4 Application of the EKF to the DIFAR Ship Tracking Case Study
108(7)
7.4.1 The Ship Motion Dynamics Model
108(1)
7.4.2 The DIFAR Buoy Field Observation Model
109(2)
7.4.3 Initialization for All Filters of the Kalman Filter Class
111(1)
7.4.4 Choosing a Value for the Acceleration Noise
112(1)
7.4.5 The EKF Tracking Filter Results
112(2)
References
114(1)
8 The Sigma Point Class: The Finite Difference Kalman Filter
115(13)
8.1 One-Dimensional Finite Difference Kalman Filter
116(4)
8.1.1 One-Dimensional Finite Difference State Prediction
116(1)
8.1.2 One-Dimensional Finite Difference State Variance Prediction
117(1)
8.1.3 One-Dimensional Finite Difference Observation Prediction Equations
118(1)
8.1.4 The One-Dimensional Finite Difference Kalman Filter Process
118(1)
8.1.5 Simplified One-Dimensional Finite Difference Prediction Equations
118(2)
8.2 Multidimensional Finite Difference Kalman Filters
120(5)
8.2.1 Multidimensional Finite Difference State Prediction
120(3)
8.2.2 Multidimensional Finite Difference State Covariance Prediction
123(1)
8.2.3 Multidimensional Finite Difference Observation Prediction Equations
124(1)
8.2.4 The Multidimensional Finite Difference Kalman Filter Process
125(1)
8.3 An Alternate Derivation of the Multidimensional Finite Difference Covariance Prediction Equations
125(3)
References
127(1)
9 The Sigma Point Class: The Unscented Kalman Filter
128(12)
9.1 Introduction to Monomial Cubature Integration Rules
128(2)
9.2 The Unscented Kalman Filter
130(7)
9.2.1 Background
130(1)
9.2.2 The UKF Developed
131(3)
9.2.3 The UKF State Vector Prediction Equation
134(1)
9.2.4 The UKF State Vector Covariance Prediction Equation
134(1)
9.2.5 The UKF Observation Prediction Equations
135(1)
9.2.6 The Unscented Kalman Filter Process
135(1)
9.2.7 An Alternate Version of the Unscented Kalman Filter
135(2)
9.3 Application of the UKF to the DIFAR Ship Tracking Case Study
137(3)
References
138(2)
10 The Sigma Point Class: The Spherical Simplex Kalman Filter
140(8)
10.1 One-Dimensional Spherical Simplex Sigma Points
141(1)
10.2 Two-Dimensional Spherical Simplex Sigma Points
142(2)
10.3 Higher Dimensional Spherical Simplex Sigma Points
144(1)
10.4 The Spherical Simplex Kalman Filter
144(1)
10.5 The Spherical Simplex Kalman Filter Process
145(1)
10.6 Application of the SSKF to the DIFAR Ship Tracking Case Study
146(2)
Reference
147(1)
11 The Sigma Point Class: The Gauss-Hermite Kalman Filter
148(16)
11.1 One-Dimensional Gauss-Hermite Quadrature
149(4)
11.2 One-Dimensional Gauss-Hermite Kalman Filter
153(2)
11.3 Multidimensional Gauss-Hermite Kalman Filter
155(5)
11.4 Sparse Grid Approximation for High Dimension/High Polynomial Order
160(3)
11.5 Application of the GHKF to the DIFAR Ship Tracking Case Study
163(1)
References
163(1)
12 The Monte Carlo Kalman Filter
164(4)
12.1 The Monte Carlo Kalman Filter
167(1)
Reference
167(1)
13 Summary of Gaussian Kalman Filters
168(8)
13.1 Analytical Kalman Filters
168(2)
13.2 Sigma Point Kalman Filters
170(4)
13.3 A More Practical Approach to Utilizing the Family of Kalman Filters
174(2)
References
175(1)
14 Performance Measures for the Family of Kalman Filters
176(25)
14.1 Error Ellipses
176(6)
14.1.1 The Canonical Ellipse
177(1)
14.1.2 Determining the Eigenvalues of P
178(1)
14.1.3 Determining the Error Ellipse Rotation Angle
179(1)
14.1.4 Determination of the Containment Area
180(1)
14.1.5 Parametric Plotting of Error Ellipse
181(1)
14.1.6 Error Ellipse Example
182(1)
14.2 Root Mean Squared Errors
182(1)
14.3 Divergent Tracks
183(1)
14.4 Cramer-Rao Lower Bound
184(8)
14.4.1 The One-Dimensional Case
184(2)
14.4.2 The Multidimensional Case
186(1)
14.4.3 A Recursive Approach to the CRLB
186(4)
14.4.4 The Cramer-Rao Lower Bound for Gaussian Additive Noise
190(1)
14.4.5 The Gaussian Cramer-Rao Lower Bound with Zero Process Noise
191(1)
14.4.6 The Gaussian Cramer-Rao Lower Bound with Linear Models
191(1)
14.5 Performance of Kalman Class DIFAR Track Estimators
192(9)
References
198(3)
PART III MONTE CARLO METHODS
201(58)
15 Introduction to Monte Carlo Methods
201(17)
15.1 Approximating a Density From a Set of Monte Carlo Samples
202(8)
15.1.1 Generating Samples from a Two-Dimensional Gaussian Mixture Density
202(1)
15.1.2 Approximating a Density by Its Multidimensional Histogram
202(2)
15.1.3 Kernel Density Approximation
204(6)
15.2 General Concepts Importance Sampling
210(5)
15.3 Summary
215(3)
References
216(2)
16 Sequential Importance Sampling Particle Filters
218(29)
16.1 General Concept of Sequential Importance Sampling
218(4)
16.2 Resampling and Regularization (Move) for SIS Particle Filters
222(8)
16.2.1 The Inverse Transform Method
222(4)
16.2.2 SIS Particle Filter with Resampling
226(1)
16.2.3 Regularization
227(3)
16.3 The Bootstrap Particle Filter
230(3)
16.3.1 Application of the BPF to DIFAR Buoy Tracking
231(2)
16.4 The Optimal SIS Particle Filter
233(5)
16.4.1 Gaussian Optimal SIS Particle Filter
235(1)
16.4.2 Locally Linearized Gaussian Optimal SIS Particle Filter
236(2)
16.5 The SIS Auxiliary Particle Filter
238(5)
16.5.1 Application of the APF to DIFAR Buoy Tracking
242(1)
16.6 Approximations to the SIS Auxiliary Particle Filter
243(2)
16.6.1 The Extended Kalman Particle Filter
243(1)
16.6.2 The Unscented Particle Filter
243(2)
16.7 Reducing the Computational Load Through Rao-Blackwellization
245(2)
References
245(2)
17 The Generalized Monte Carlo Particle Filter
247(12)
17.1 The Gaussian Particle Filter
248(2)
17.2 The Combination Particle Filter
250(3)
17.2.1 Application of the CPF-UKF to DIFAR Buoy Tracking
252(1)
17.3 Performance Comparison of All DIFAR Tracking Filters
253(6)
References
255(4)
PART IV ADDITIONAL CASE STUDIES
18 A Spherical Constant Velocity Model for Target Tracking in Three Dimensions
259(49)
18.1 Tracking a Target in Cartesian Coordinates
261(4)
18.1.1 Object Dynamic Motion Model
262(1)
18.1.2 Sensor Data Model
263(1)
18.1.3 Gaussian Tracking Algorithms for a Cartesian State Vector
264(1)
18.2 Tracking a Target in Spherical Coordinates
265(8)
18.2.1 State Vector Position and Velocity Components in Spherical Coordinates
266(1)
18.2.2 Spherical State Vector Dynamic Equation
267(3)
18.2.3 Observation Equations with a Spherical State Vector
270(1)
18.2.4 Gaussian Tracking Algorithms for a Spherical State Vector
270(3)
18.3 Implementation of Cartesian and Spherical Tracking Filters
273(5)
18.3.1 Setting Values for q
273(1)
18.3.2 Simulating Radar Observation Data
274(2)
18.3.3 Filter Initialization
276(2)
18.4 Performance Comparison for Various Estimation Methods
278(15)
18.4.1 Characteristics of the Trajectories Used for Performance Analysis
278(4)
18.4.2 Filter Performance Comparisons
282(11)
18.5 Some Observations and Future Considerations
293(15)
Appendix 18.A Three-Dimensional Constant Turn Rate Kinematics
294(1)
18.A.1 General Velocity Components for Constant Turn Rate Motion
294(3)
18.A.2 General Position Components for Constant Turn Rate Motion
297(2)
18.A.3 Combined Trajectory Transition Equation
299(1)
18.A.4 Turn Rate Setting Based on a Desired Turn Acceleration
299(2)
Appendix 18.B Three-Dimensional Coordinate Transformations
301(1)
18.B.1 Cartesian-to-Spherical Transformation
302(3)
18.B.2 Spherical-to-Cartesian Transformation
305(1)
References
306(2)
19 Tracking a Falling Rigid Body Using Photogrammetry
308(38)
19.1 Introduction
308(3)
19.2 The Process (Dynamic) Model for Rigid Body Motion
311(7)
19.2.1 Dynamic Transition of the Translational Motion of a Rigid Body
311(2)
19.2.2 Dynamic Transition of the Rotational Motion of a Rigid Body
313(3)
19.2.3 Combined Dynamic Process Model
316(1)
19.2.4 The Dynamic Process Noise Models
317(1)
19.3 Components of the Observation Model
318(3)
19.4 Estimation Methods
321(7)
19.4.1 A Nonlinear Least Squares Estimation Method
321(2)
19.4.2 An Unscented Kalman Filter Method
323(2)
19.4.3 Estimation Using the Unscented Combination Particle Filter
325(1)
19.4.4 Initializing the Estimator
326(2)
19.5 The Generation of Synthetic Data
328(6)
19.5.1 Synthetic Rigid Body Feature Points
328(1)
19.5.2 Synthetic Trajectory
328(5)
19.5.3 Synthetic Cameras
333(1)
19.5.4 Synthetic Measurements
333(1)
19.6 Performance Comparison Analysis
334(8)
19.6.1 Filter Performance Comparison Methodology
335(3)
19.6.2 Filter Comparison Results
338(3)
19.6.3 Conclusions and Future Considerations
341(1)
Appendix 19.A Quaternions, Axis-Angle Vectors, and Rotations
342(1)
19.A.1 Conversions Between Rotation Representations
342(1)
19.A.2 Representation of Orientation and Rotation
343(1)
19.A.3 Point Rotations and Frame Rotations
344(2)
References
345(1)
20 Sensor Fusion Using Photogrammetric and Inertial Measurements
346(21)
20.1 Introduction
346(1)
20.2 The Process (Dynamic) Model for Rigid Body Motion
347(1)
20.3 The Sensor Fusion Observational Model
348(4)
20.3.1 The Inertial Measurement Unit Component of the Observation Model
348(2)
20.3.2 The Photogrammetric Component of the Observation Model
350(1)
20.3.3 The Combined Sensor Fusion Observation Model
351(1)
20.4 The Generation of Synthetic Data
352(2)
20.4.1 Synthetic Trajectory
352(1)
20.4.2 Synthetic Cameras
352(1)
20.4.3 Synthetic Measurements
352(2)
20.5 Estimation Methods
354(3)
20.5.1 Initial Value Problem Solver for IMU Data
354(3)
20.6 Performance Comparison Analysis
357(4)
20.6.1 Filter Performance Comparison Methodology
359(1)
20.6.2 Filter Comparison Results
360(1)
20.7 Conclusions
361(1)
20.8 Future Work
362(5)
References
364(3)
Index 367
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.
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