# Stochastic Geometry and Its Applications, 3rd Edition

# Stochastic Geometry and Its Applications, 3rd Edition

ISBN: 978-1-118-65822-2 September 2013 584 Pages

## Description

**An**** extensive update to a classic text**

Stochastic geometry and spatial statistics play a fundamental role in many modern branches of physics, materials sciences, engineering, biology and environmental sciences. They offer successful models for the description of random two- and three-dimensional micro and macro structures and statistical methods for their analysis.

The previous edition of this book has served as the key reference in its field for over 18 years and is regarded as the best treatment of the subject of stochastic geometry, both as a subject with vital applications to spatial statistics and as a very interesting field of mathematics in its own right.

*This editi**on*:

- Presents a wealth of models for spatial patterns and related statistical methods.
- Provides a great survey of the modern theory of random tessellations, including many new models that became tractable only in the last few years.
- Includes new sections on random networks and random graphs to review the recent ever growing interest in these areas.
- Provides an excellent introduction to theory and modelling of point processes, which covers some very latest developments.
- Illustrate the forefront theory of random sets, with many applications.
- Adds new results to the discussion of fibre and surface processes.
- Offers an updated collection of useful stereological methods.
- Includes 700 new references.
- Is written in an accessible style enabling non-mathematicians to benefit from this book.
- Provides a companion website hosting information on recent developments in the field www.wiley.com/go/cskm

*Stochastic Geometry and its Applications* is ideally suited for researchers in physics, materials science, biology and ecological sciences as well as mathematicians and statisticians. It should also serve as a valuable introduction to the subject for students of mathematics and statistics.

## Related Resources

### Student

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