Stochastic Geometry -- Research Frontiers

Preface......Page 6
Acknowledgements......Page 8
Contents......Page 9
Contributors......Page 12
 1.1 Introduction: Geometric Probability, Integral Geometry, Stochastic Geometry......Page 13
 1.2 From Buffon\'s Needle to Integral Geometry......Page 15
  1.2.1 Starting from Buffon\'s Needle......Page 16
  1.2.3 Extension to Higher Dimension......Page 18
  1.3.1 Starting from Bertrand\'s Paradox......Page 22
  1.3.2 Random Sets of Points, Random Sets of Lines and Extensions......Page 24
  1.3.3 On Two Examples of Random Convex Tessellations......Page 25
   1.3.4.1 The Zero-Cell of a Poisson Hyperplane Tessellation......Page 29
   1.3.4.3 The Typical Cell of a Poisson-Voronoi Tessellation......Page 30
  1.4.1 Starting from Sylvester\'s Four-Point Problem......Page 32
   1.4.1.1 Calculation of Sylvester\'s Probability in the Case of the Disk......Page 33
   1.4.1.2 Calculation of Sylvester\'s Probability in the Case of the Triangle......Page 34
   1.4.1.3 Extremes of p4(K)......Page 36
  1.4.2 Random Polytopes......Page 40
   1.4.2.1 Non-asymptotic Results......Page 41
   1.4.2.2 Asymptotic Results......Page 43
  1.5.1 Starting from the Bicycle Wheel Problem and Random Covering of the Circle......Page 44
  1.5.2 A Few Basics on the Boolean Model......Page 47
   1.5.2.2 The Number of Grains in a Typical Connected Component of the Occupied Phase......Page 49
 References......Page 50
 2.1 Introduction......Page 56
  2.2.1 Definition and Theoretical Characterization of a Spatial Point Process......Page 58
  2.2.2 Moment Measures Factorial Moment Measures and Intensity Functions......Page 59
  2.2.3 Palm Distributions and Palm Intensities......Page 60
  2.2.4 Papangelou Conditional Intensities......Page 63
  2.3.1 Poisson Point Process......Page 65
  2.3.2 Gibbs Point Processes......Page 67
   2.3.3.1 Log-Gaussian Cox Processes......Page 70
   2.3.3.2 Shot Noise Cox Processes......Page 71
  2.3.4 Determinantal Point Processes......Page 73
 2.4 Estimating the Intensity Function......Page 74
   2.4.1.2 General Case......Page 75
  2.4.2 Non Parametric Estimation of the Intensity Function......Page 77
  2.4.3 Parametric Estimation of the Intensity Function......Page 78
   2.4.3.1 Poisson Likelihood Estimator......Page 79
   2.4.3.2 Numerical Implementation of the Weighted Poisson Log-Likelihood......Page 80
   2.4.3.3 Logistic Regression Likelihood......Page 81
   2.4.3.4 Quasi-Likelihood......Page 82
 2.5 Higher-Order Interaction Estimation via Conditional Intensity......Page 83
  2.5.1 Parametric Estimation with the Papangelou Conditional Intensity......Page 84
   2.5.1.1 Maximum Pseudo-Likelihood Estimator......Page 85
   2.5.1.2 Takacs–Fiksel Estimator......Page 87
   2.5.1.3 Variational Estimator......Page 89
  2.5.2 Palm Likelihood Estimation......Page 91
 2.6 Conclusion......Page 92
 References......Page 93
 3.1 Visual Perception and the Non-accidentalness Principle......Page 97
  3.1.1 Gestalt Theory of Visual Perception......Page 99
  3.1.2 The Non-accidentalness Principle......Page 103
  3.2.1 General Formulation of a Contrario Methods......Page 105
   3.2.2.1 Definitions and First Properties......Page 106
   3.2.2.2 Maximality......Page 109
   3.2.2.3 The Line Segment Detector Algorithm......Page 111
   3.2.3.1 Meaningful Boundaries......Page 112
   3.2.3.2 Meaningful Good Continuations......Page 115
  3.2.4 Detection of Vanishing Points......Page 118
  3.2.5 Detection of the Similarity of a Scalar Attribute......Page 121
  3.2.6 Discussion......Page 123
 3.3 Stochastic Models of Images: The Problem of Modeling and Synthesizing Texture Images......Page 124
  3.3.2 Texture Synthesis......Page 125
  3.3.3 Discrete Fourier Transform and the RPN Algorithm......Page 128
  3.3.4 Gaussian Models for Texture Images......Page 131
  3.3.5 Shot Noise Model and Dead Leaves Model......Page 134
 References......Page 136
 4.1 Random Fields and Scale Invariance......Page 138
   4.1.1.1 Definitions and Distribution......Page 139
   4.1.1.2 Gaussian Processes......Page 140
   4.1.1.3 Gaussian Fields Defined from Processes......Page 142
  4.1.2 Stationarity and Invariances......Page 144
   4.1.2.1 Stationarity and Isotropy......Page 145
   4.1.2.2 Self-Similarity or Scale Invariance......Page 148
   4.1.2.3 Stationary Increments......Page 149
   4.1.2.4 Operator Scaling Property......Page 155
 4.2 Sample Paths Properties......Page 157
   4.2.1.1 Hölder Regularity......Page 158
   4.2.1.2 Critical Hölder Exponent......Page 161
   4.2.1.3 Directional Hölder Regularity......Page 162
   4.2.2.1 Hausdorff Measures and Dimensions......Page 163
   4.2.2.2 Upper Bound of Graphs Hausdorff Dimension......Page 164
   4.2.2.3 Lower Bound of Graphs Hausdorff Dimension......Page 165
  4.3.1 Simulation......Page 166
   4.3.1.1 Fast and Exact Synthesis of Fractional Brownian Motion......Page 167
   4.3.1.2 Turning Band Method for 2-Dimensional Anisotropic Self-Similar Fields......Page 168
   4.3.1.3 Stein Method for 2-Dimensional Operator Scaling fields......Page 171
   4.3.2.1 1D Estimation Based on Variograms......Page 172
   4.3.2.2 Application to Random Fields by Line Processes......Page 176
  4.3.3 Application in Medical Imaging Analysis......Page 177
   4.3.3.1 Osteoporosis and Bone Radiographs......Page 178
   4.3.3.2 Mammograms and Density Analysis......Page 179
  4.4.1 Random Measures......Page 180
  4.4.2 Chentsov\'s Representation: Lévy and Takenaka Constructions......Page 182
  4.4.3 Fractional Poisson Fields......Page 184
 References......Page 187
 5.1 Introduction......Page 190
 5.2 Finite Volume Gibbs Point Processes......Page 192
  5.2.1 Poisson Point Process......Page 193
  5.2.2 Energy Functions......Page 194
  5.2.3 Finite Volume GPP......Page 197
  5.2.4 DLR Equations......Page 199
  5.2.5 GNZ Equations......Page 200
  5.2.6 Ruelle Estimates......Page 202
 5.3 Infinite Volume Gibbs Point Processes......Page 203
  5.3.1 The Local Convergence Setting......Page 204
  5.3.2 An Accumulation Point Pz,β......Page 205
  5.3.3 The Finite Range Property......Page 207
  5.3.4 DLR Equations......Page 208
  5.3.5 GNZ Equations......Page 210
  5.3.6 Variational Principle......Page 212
  5.3.7 A Uniqueness Result......Page 215
  5.3.8 A Non-uniqueness Result......Page 218
 5.4 Estimation of Parameters......Page 223
  5.4.1 Maximum Likelihood Estimator......Page 224
  5.4.2 Takacs-Fiksel Estimator......Page 226
  5.4.3 Maximum Pseudo-Likelihood Estimator......Page 229
  5.4.4 Solving an Unobservable Issue......Page 231
  5.4.5 A Variational Estimator......Page 232
 References......Page 236