Stochastic Geometry and Its Applications

Foreword to the first edition xiii From the preface to the first edition xvii Preface to the second edition xix Preface to the third edition xxi Notation xxiii 1 Mathematical foundations 1 1.1 Set theory 1 1.2 Topology in Euclidean spaces 3 1.3 Operations on subsets of Euclidean space 5 1.4 Mathematical morphology and image analysis 7 1.5 Euclidean isometries 9 1.6 Convex sets in Euclidean spaces 10 1.7 Functions describing convex sets 17 1.8 Polyconvex sets 24 1.9 Measure and integration theory 27 2 Point processes I: The Poisson point process 35 2.1 Introduction 35 2.2 The binomial point process 36 2.3 The homogeneous Poisson point process 41 2.4 The inhomogeneous and general Poisson point process 51 2.5 Simulation of Poisson point processes 53 2.6 Statistics for the homogeneous Poisson point process 55 3 Random closed sets I: The Boolean model 64 3.1 Introduction and basic properties 64 3.2 The Boolean model with convex grains 78 3.3 Coverage and connectivity 89 3.4 Statistics 95 3.5 Generalisations and variations 103 3.6 Hints for practical applications 106 4 Point processes II: General theory 108 4.1 Basic properties 108 4.2 Marked point processes 116 4.3 Moment measures and related quantities 120 4.4 Palm distributions 127 4.5 The second moment measure 139 4.6 Summary characteristics 143 4.7 Introduction to statistics for stationary spatial point processes 145 4.8 General point processes 156 5 Point processes III: Models 158 5.1 Operations on point processes 158 5.2 Doubly stochastic Poisson processes (Cox processes) 166 5.3 Neyman-Scott processes 171 5.4 Hard-core point processes 176 5.5 Gibbs point processes 178 5.6 Shot-noise fields 200 6 Random closed sets II: The general case 205 6.1 Basic properties 205 6.2 Random compact sets 213 6.3 Characteristics for stationary and isotropic random closed sets 216 6.4 Nonparametric statistics for stationary random closed sets 230 6.5 Germ-grain models 237 6.6 Other random closed set models 255 6.7 Stochastic reconstruction of random sets 276 7 Random measures 279 7.1 Fundamentals 279 7.2 Moment measures and related characteristics 284 7.3 Examples of random measures 286 8 Line, fibre and surface processes 297 8.1 Introduction 297 8.2 Flat processes 302 8.3 Planar fibre processes 314 8.4 Spatial fibre processes 330 8.5 Surface processes 333 8.6 Marked fibre and surface processes 339 9 Random tessellations, geometrical networks and graphs 343 9.1 Introduction and definitions 343 9.2 Mathematical models for random tessellations 346 9.3 General ideas and results for stationary planar tessellations 357 9.4 Mean-value formulae for stationary spatial tessellations 367 9.5 Poisson line and plane tessellations 370 9.6 STIT tessellations 375 9.7 Poisson-Voronoi and Delaunay tessellations 376 9.8 Laguerre tessellations 386 9.9 Johnson-Mehl tessellations 388 9.10 Statistics for stationary tessellations 390 9.11 Random geometrical networks 397 9.12 Random graphs 402 10 Stereology 411 10.1 Introduction 411 10.2 The fundamental mean-value formulae of stereology 413 10.3 Stereological mean-value formulae for germ-grain models 421 10.4 Stereological methods for spatial systems of balls 425 10.5 Stereological problems for nonspherical grains (shape-and-size problems) 436 10.6 Stereology for spatial tessellations 440 10.7 Second-order characteristics and directional distributions 444 References 453 Author index 507 Subject index 521