Bayesian Multiple Target Tracking

by Lawrence D. Stone, Thomas L. Corwin, Carl A. Barlow

Overview

Get the solutions to your most challenging tracking problems with this up-to-date resource. Using the Bayesian inference framework, the book helps you design and develop mathematically sound algorithms for dealing with tracking problems involving multiple targets, multiple sensors, and multiple platforms. The book shows you how non-linear Multiple Hypothesis Tracking and the Theory of Unified Tracking are successful methods when multiple target tracking must be performed without contacts or association.

With detailed examples illustrating the developed concepts, algorithms, and approaches — the book helps you:

>Track when observations are non-linear functions of target site, when the target state distributions or measurement error distributions are not Gaussian, in low data rate and low signal to noise ratio situations, and when notions of contact and association are merged or unresolved among more than one target

>Detect and track when a single sensor response is not strong enough to call a contact

>Determine bounds on tracker performance from the characteristics of the targets and sensors

>Set optimal threshold levels for calling contacts in likelihood ratio detection and tracking, and compute association probabilities of joint observations and non-geometric information

Product Details

ISBN-13:	9781580530248
Publisher:	Artech House, Incorporated
Publication date:	07/31/1999
Series:	Radar Library
Pages:	324
Product dimensions:	6.00(w) x 9.00(h) x 0.88(d)

About the Author

Lawrence D. Stone is Chief Operating Officer at Metron, Inc. He received his Ph.D. and MS in mathematics from Purdue University. Carl A. Barlow is an independent scientific consultant. He holds S.B. and S.M. degrees in theoretical physics from MIT. Thomas L. Corwin is Chief Executive Officer of Metron, Inc. He received his Ph.D and MS in statistics from Princeton University.

Table of Contents

	Preface	ix
	Introduction	xiii
	Acknowledgments	xix
Chapter 1	Tracking Problems	1
1.1	Tracking a Single Target	1
1.1.1	Tracking a Surface Ship	1
1.1.2	Submarine Versus Submarine Tracking	7
1.1.3	Periscope Detection and Tracking	13
1.2	Tracking Multiple Targets	17
1.2.1	Tracking Aircraft	17
1.2.2	Underwater Surveillance	19
1.3	Classification of Tracking Systems	23
1.3.1	Target Assumptions	24
1.3.2	Information Assumptions	25
1.4	Emphasis	26
	References	27
Chapter 2	Bayesian Inference and Likelihood Functions	29
2.1	The Case for Bayesian Inference	29
2.2	The Likelihood Function and Bayes' Theorem	33
2.2.1	The Likelihood Function	33
2.2.2	Bayes' Theorem	33
2.3	Examples of Likelihood Functions	35
2.3.1	A Gaussian Contact Model	35
2.3.2	A Gaussian Bearing-Error Model	36
2.3.3	Combining Bearing and Contact Data	39
2.3.4	A Signal-Plus-Noise Model	42
2.3.5	Negative Information	46
2.3.6	Positive Information	49
2.3.7	Radar and Infrared Detection	51
	References	53
Chapter 3	Single Target Tracking	55
3.1	Bayesian Filtering	56
3.1.1	Recursive Bayesian Filtering	56
3.1.2	Recursive Bayesian Prediction and Smoothing	62
3.2	Kalman Filtering	67
3.2.1	Discrete Kalman Filtering	68
3.2.2	Continuous-Discrete Kalman Filtering	72
3.3	Discrete Bayesian Filtering	78
3.3.1	Nodestar Implementation	78
3.3.2	Correlated-Bearing Likelihood Function	86
3.3.3	Three-Dimensional Bearing Likelihood Function	93
3.3.4	Detection-No Detection Likelihood Function	95
3.3.5	Land Avoidance Likelihood Function	100
3.3.6	Elliptical Contact Likelihood Function	101
3.3.7	ELINT Likelihood Function	101
	References	102
Chapter 4	Classical Multiple Target Tracking: Multiple Hypothesis Tracking	103
4.1	Multiple Target Tracking Problem	105
4.1.1	Multiple Target Motion Model	105
4.1.2	Multiple Target Likelihood Functions	106
4.1.3	Contacts, Scans, and Association Hypotheses	107
4.1.4	Scan and Data Association Likelihood Functions	111
4.2	General Multiple Hypothesis Tracking	114
4.2.1	Conditional Target Distributions	115
4.2.2	Association Probabilities	116
4.2.3	General MHT Recursion	117
4.2.4	Association Probabilities for Gaussian Distributions	119
4.2.5	Association Probabilities for Non-Gaussian Distributions	122
4.2.6	Joint Association of Multiple Attribute Observations	124
4.2.7	Summary of Assumptions for General MHT Recursion	125
4.3	Independent Multiple Hypothesis Tracking	126
4.3.1	Conditionally Independent Scan Association Likelihood Functions	126
4.3.2	Independent MHT Recursion	130
4.3.3	Linear Gaussian Multiple Hypothesis Tracking	131
4.4	Example of Nonlinear MHT	136
4.4.1	Description of Tracking Problem	136
4.4.2	Operation of Tracker	138
4.4.3	Tracker Output	141
4.5	Notes	157
	References	158
Chapter 5	Multiple Target Tracking Without Contacts or Association	161
5.1	Unified Tracking Model	162
5.1.1	Multiple Target Motion and Likelihood Function Assumptions	162
5.1.2	Posterior Distribution	162
5.1.3	Unified Tracking Recursion	163
5.1.4	Summary of Assumptions for Unified Tracking Recursion	164
5.2	Relationship of Unified Tracking to Multiple Hypothesis Tracking	165
5.2.1	MHT is a Special Case of Unified Tracking	165
5.2.2	Extensions of MHT	168
5.3	Applications of Unified Tracking	169
5.3.1	Examples for Which Association is Meaningful	172
5.3.2	Examples for Which Association is Not Meaningful	178
5.3.3	An Example With an Unknown Number of Targets	180
5.4	Relationship of Unified Tracking to Other Tracking Algorithms	204
	References	206
Chapter 6	Likelihood Ratio Detection and Tracking: Theoretical Foundations	209
6.1	Basic Definitions and Relations	209
6.1.1	Likelihood Ratio	212
6.1.2	Measurement Likelihood Ratio	212
6.1.3	Likelihood Ratio Recursion	213
6.1.4	Log-Likelihood Ratios	215
6.1.5	Declaring a Target Present	216
6.2	Example of Likelihood Ratio Detection and Tracking	219
6.2.1	Simulated Detection and Tracking Results	221
6.2.2	Comparison to Matched Filter Detection	224
6.3	Measurement Likelihood Ratios	229
6.3.1	Additive Target Effects in Gaussian Noise	230
6.3.2	Modification for Random Target Strength	231
6.3.3	Maximum Likelihood for Unknown Target Strength	233
6.3.4	Designing for a Marginally Detectable Target	234
6.3.5	Additive Target Effects in Multivariate Gaussian Noise	236
6.3.6	Targets With Additive Small Signals	236
6.3.7	Additive Target Effects in Complex Gaussian Noise	241
6.3.8	Variance Modifying Targets in Gaussian Noise	242
6.3.9	Targets Modifying the Mean and Covariance of Gaussian Data	243
6.3.10	Exponential Distributions	244
6.3.11	Thresholded Data	245
6.3.12	Binomial Distributions: M of N Test Statistics	247
6.3.13	Dealing with Nuisance Parameters	248
6.4	Models of Likelihood Ratio Propagation	248
6.4.1	Transitions to and from the Null State	249
6.4.2	Continuous Transition Models Within the State Space	254
6.4.3	Discrete Transition Models Within the State Space	255
6.4.4	Deterministic Evolutions	257
6.5	State Entropy and Information Measures	258
6.5.1	Some Theorems Regarding Measurement Log-Likelihood Ratios	258
6.5.2	Information and Entropy in State Propagation	262
	References	267
Chapter 7	Likelihood Ratio Detection and Tracking: Implementation Issues	269
7.1	Framework for Limiting False Alarms	269
7.1.1	Measurement Likelihood Ratios in the Presence of Noise Only	269
7.1.2	False Alarm Rate and Target Detection Rate Relations	271
7.1.3	Likelihood Ratio Density in the Presence of Noise Only	273
7.2	Performance Prediction Methodology	274
7.2.1	Approximate Determination of Detection Performance	275
7.2.2	The Role of Motion Updates	276
7.2.3	The Role of the Information Updates	277
7.2.4	The Role of Averaging or Cell Formulations	277
7.3	Numerical Implementation of Likelihood Ratio Trackers	278
7.3.1	Sampled Field Approach	279
7.3.2	Cell-Based Approach	283
7.3.3	Kalman-Like Approach	284
	Appendix	291
	About the Authors	293
	Index	295

Editorial Reviews

Two mathematicians and an engineer with a consulting firm trace their experience with tracking to the CASP search and rescue planning program they developed for the Coast Guard in the 1970s, applying a stochastic model to predict the target's position and motion over time. While most books on tracking present algorithms for problems in which contacts are received at a high data rate with good localization information about the targets, the focus here is on low data rate, low signal-to-noise ratio situations where sensor responses provide ambiguous information about the target's state. They explain Bayesian inference as a statistical decision theory framework from which to view and design tracking algorithms. Includes an appendix on Gaussian density lemma. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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