Introduction | p. 1 |
Chapter Parade | p. 2 |
Part I: Fundamentals | p. 2 |
Part II: Applications | p. 3 |
Part III: Beyond the Poisson Point Process | p. 5 |
Appendices | p. 5 |
The Real Line Is Not Enough | p. 6 |
General Point Processes | p. 6 |
An Alternative Tradition | p. 8 |
Fundamentals | |
The Poisson Point Process | p. 11 |
The Event Space | p. 12 |
Intensity | p. 12 |
Realizations | p. 13 |
Likelihood Function | p. 17 |
Expectations | p. 18 |
Definition | p. 19 |
Random Sums | p. 21 |
Campbell's Theorem2 | p. 23 |
Characterization of PPPs | p. 25 |
Probability Generating Functional | p. 27 |
Superposition | p. 28 |
Independent (Bernoulli) Thinning | p. 30 |
Declarations of Independence | p. 33 |
Independent Scattering | p. 33 |
Poisson's Gambit | p. 36 |
Inevitability of the Poisson Distribution | p. 38 |
Connection to Stochastic Processes | p. 41 |
Nonlinear Transformations | p. 42 |
Stochastic Transformations | p. 46 |
Transition Processes | p. 46 |
Measurement Processes | p. 47 |
PPPs on Other Spaces | p. 50 |
Discrete Spaces | p. 50 |
Discrete-Continuous Spaces | p. 53 |
Intensity Estimation | p. 57 |
Maximum Likelihood Algorithms | p. 58 |
Necessary Conditions | p. 59 |
Gaussian Crosshairs and Edge Effects | p. 60 |
Superposed Intensities with Sample Data | p. 63 |
EM Method with Sample Data | p. 64 |
Interpreting the Weights | p. 67 |
Simple Examples | p. 67 |
Affine Gaussian Sums | p. 69 |
Superposed Intensities with Histogram Data | p. 73 |
EM Method with Histogram Data | p. 73 |
Affine Gaussian Sums | p. 76 |
Regularization | p. 78 |
Parametric Tying | p. 78 |
Bayesian Methods | p. 80 |
Cramér-Rao Bound (CRB) for Intensity Estimates | p. 81 |
Background | p. 82 |
Unbiased Estimators | p. 83 |
Fisher Information Matrix and the Score Vector | p. 83 |
CRB and the Cauchy-Schwarz Inequality | p. 84 |
Spinoffs | p. 86 |
CRB for PPP Intensity with Sample Data | p. 88 |
CRB for PPP Intensity with Histogram Data | p. 90 |
CRB for PPP Intensity on Discrete Spaces | p. 93 |
Gating: Gauss on a Pedestal | p. 95 |
Joint CRB for Gaussian Sums | p. 97 |
Mean Vectors in a Gaussian Sum | p. 98 |
Means and Coefficients in a Gaussian Sum | p. 99 |
Observed Information Matrices | p. 100 |
General Sums | p. 101 |
Affine Gaussian Sums | p. 103 |
Applications to Imaging, Tracking, and Distributed Sensing | |
Tomographic Imaging | p. 109 |
Positron Emission Tomography | p. 110 |
PET: Time-of-Flight Data | p. 112 |
Image Reconstruction | p. 113 |
Small Cell Limit | p. 117 |
Intuitive Interpretation | p. 118 |
PET: Histogram Data | p. 118 |
Detectors as a Discrete Space | p. 119 |
Shepp-Vardi Algorithm | p. 119 |
Single-Photon Computed Emission Tomography (SPECT) | p. 124 |
Gamma Cameras | p. 124 |
Image Reconstruction | p. 126 |
Transmission Tomography | p. 134 |
Background | p. 134 |
Lange-Carson Algorithm | p. 135 |
CRBs for Emission and Transmission Tomography | p. 142 |
Regularization | p. 143 |
Grenander's Method of Sieves | p. 143 |
Multiple Target Tracking | p. 147 |
Intensity Filters | p. 149 |
PPP Model Interpretation | p. 149 |
Predicted Target and Measurement Processes | p. 150 |
Information Updates | p. 153 |
The Final Filter | p. 156 |
Relationship to Other Filters | p. 159 |
Probability Hypothesis Density (PHD) Filter | p. 159 |
Marked Multisensor Intensity Filter (MMIF) | p. 160 |
Implementation | p. 161 |
Particle Methods | p. 161 |
Mean Shift Algorithm | p. 163 |
Multimode Algorithms | p. 165 |
Covariance Matrices | p. 166 |
Gaussian Sum Methods | p. 167 |
Regularization | p. 168 |
Estimated Target Count | p. 171 |
Sources of Error | p. 171 |
Variance Reduction | p. 171 |
Multiple Sensor Intensity Filters | p. 172 |
Identical Coverage Sensors | p. 173 |
Heterogeneous Sensor Coverages | p. 176 |
Historical Note | p. 178 |
Distributed Sensing | p. 179 |
Distance Distributions | p. 180 |
From Sensors To Target | p. 181 |
Between Sensors | p. 185 |
Communication Diversity | p. 189 |
Detection Coverage | p. 190 |
Stationary Sensor Fields | p. 191 |
Drifting Fields and Anisotropy | p. 195 |
Stereology | p. 198 |
Beyond the Poisson Point Process | |
A Profusion of Point Processes | p. 203 |
Marked Processes | p. 204 |
Product Space and Marking Theorem | p. 205 |
Filtered Processes | p. 207 |
FIM for Unbiased Estimators | p. 207 |
Hard Core Processes | p. 208 |
Cluster Processes | p. 210 |
Poisson Cluster Processes | p. 210 |
Neyman-Scott Processes | p. 211 |
Cox (Doubly Stochastic) Processes | p. 213 |
Equivalent Neyman-Scott Process | p. 214 |
Intensity Function as Solution of an SDE | p. 215 |
Markov Modulated Poisson Processes | p. 216 |
Gibbs Point Processes | p. 216 |
The Cutting Room Floor | p. 219 |
Further Topics | p. 219 |
Possible Trends | p. 221 |
Expectation-Maximization (EM) Method | p. 223 |
Formulation | p. 223 |
E-step | p. 224 |
M-step | p. 225 |
Convergence | p. 225 |
Iterative Majorization | p. 227 |
Observed Information | p. 228 |
Solving Conditional Mean Equations | p. 229 |
Bayesian Filtering | p. 233 |
General Recursion | p. 233 |
Special Case: Kalman Filtering | p. 235 |
Multitarget Tracking | p. 237 |
Bayesian Derivation of Intensity Filters | p. 239 |
Posterior Point Process | p. 239 |
PPP Approximation | p. 241 |
Altogether Now | p. 243 |
First Moment Intensity and Janossy Densities | p. 243 |
MMIF: Marked Multitarget Intensity Filter | p. 245 |
Target Modeling | p. 245 |
Joint Measurement-Target Intensity Function | p. 246 |
Likelihood Function | p. 248 |
MMIF Recursion | p. 250 |
Linear Filter Model | p. 253 |
PPP Signal Model | p. 253 |
Poisson Limit | p. 254 |
Utility | p. 256 |
Glossary | p. 257 |
List of Acronyms | p. 263 |
References | p. 265 |
Index | p. 271 |
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