Fractals, Random Shapes and Point Fields Methods of Geometncal Statistics

Dietrich Stoyan and Helga Stoyan Freiberg University of Mining and Technology (TU Bergakademie Freiberg), Germany

JOHN WILEY & SONS Chichester • New York • Brisbane • Toronto • Singapore

Contents

Preface xi

List of Symbols . xiii

PART I FRACTALS AND METHODS FOR THE DETERMINATION OF FRACTAL DIMENSIONS 1

Chapter 1 Introduction 3

Chapter 2 Hausdorff Measure and Dimension 11 2.1 The Hausdorff Measure in R1 11 2.2 Fractal Dimension 14 2.3 Local Hausdorff Dimension and Dimension Distribution 18 2.4 Fractals 20

Chapter 3 Deterministic Fractals 21 3.1 General Properties of Fractals 21 3.2 Examples of Deterministic Fractals 21

3.2.1 Curves of fractal dimension 21 3.2.2 Self-similar sets 22 3.2.3 A program for the generation of generalized

von Koch snow flakes 26

Chapter 4 Random Fractals 29 4.1 Random Self-similar Sets 29 4.2 Mandelbrot-Zähle Cut-outs 33 4.3 Random Fractals Connected with Brownian Motion 33 4.4 Self-similar Stochastic Processes 36

Chapter 5 Methods for the Empirical Determination of Fractal Dimension 39 5.1 Introduction 39 5.2 Divider Stepping Method 40 5.3 Box-counting Method 41 5.4 Sausage Method 44

VI CONTENTS

5.5 Estimation of Local Dimension 46 5.6 Further Methods 47 5.7 Estimating the Fractal Dimension

of the Boundary of a Graphite Particle 48

PART II THE STATISTICS OF SHAPES AND FORMS 51

Chapter 6 Fundamental Concepts 53

Chapter 7 Representation of Contours 61 7.1 Introduction 61 7.2 Definition and Measurement of Contour Functions 62

7.2.1 Cross-section functions for Symmetrie figures 62 7.2.2 Radius-vector functions 63 7.2.3 Support functions 65 7.2.4 Tangent-angle functions 69 7.2.5 Comparison of the three variants of

contour functions 70 7.2.6 Smoothing of contours 70

7.3 Invariant Parameters of Contour Functions for Particles 71 7.4 Two Classes of Figures 73

7.4.1 Superellipses 73 7.4.2 Radial-rhombi 76

7.5 Determination of Approximating Contour Functions 77 7.5.1 Introduction 77 7.5.2 Approximation by ellipses and radial-rhombi 78 7.5.3 Fourier analysis 80

7.6 Stochastic Models for the Contour Function Approach 88 7.6.1 Invariant contour funetion parameters of

random figures 88 7.6.2 Random radial-rhombi 89 7.6.3 Randomly disturbed figures 90 7.6.4 Three disturbance modeis 91

7.7 Statistical Analysis for the Contour Function Approach 94 7.7.1 Values of the invariant parameters of Single figures 94 7.7.2 Statistical determination of distributional

characteristics of the invariant parameters 95 7.7.3 Statistics for randomly disturbed functions 96 7.7.4 Statistics for random radial-rhombi and

related figures 100

Chapter 8 Set Theoretic Analysis 103 8.1 Introduction 103 8.2 Simple Geometrical Shape Ratios 103 8.3 Characteristics of Random Compact Sets 107

8.3.1 Introduction 107 8.3.2 Random compact sets 107 8.3.3 Mean value formulae for compact convex sets 108 8.3.4 Means of random compact sets 108 8.3.5 Variances of random compact sets 113

CONTENTS VII

8.3.6 Medians of random compact sets 115 8.3.7 Some simple Statistical methods 115

8.4 Four Functions for Descnption of Figures 116 8.4.1 Introduction 116 8.4.2 Chord length distribution functions 117 8.4.3 Isotropized set covariance function 122 8.4.4 Erosion functions 123

8.5 Stochastic Models of Random Compact Sets 125 8.5.1 Poisson polygon 125 8.5.2 Dirichlet polygons 128 8.5.3 Rounded polygons 132 8.5.4 Convex hulls of random figures 134 8.5.5 Gaussian random sets 138 8.5.6 Inhomogeneous Boolean modeis 138 8.5.7 Vorob'ev's forest-fire model 139

Chapter 9 Point Description of Figures 141 9.1 Introduction 141 9.2 Description of Landmark Configurations and their Size 141 9.3 Distances and Transformations of Landmark Configurations 142

9.3.1 The general problem 142 9.3.2 Formulae for some classes of transformations 144

9.4 Means of Planar Configurations of Points 148 9.4.1 A metric for point configurations 148 9.4.2 Calculation of mean configurations 149

9.5 Procrustean Analysis 150 9.5.1 The problem 150 9.5.2 Using the results of Procrustean analysis 151

9.6 Shape Analysis for Triangles and Point Triplets 153 9.6.1 Introduction 153 9.6.2 Triangles 153

9.7 Point Triplets 158 9.8 Statistics for the Bookstein Model 161

9.8.1 The Bookstein model 161 9.8.2 Estimation of distances and variances 162 9.8.3 Shape statistics for the Bookstein model 164

Chapter 10 Examples 167 10.1 The Form of Sand Grains 167

10.2 The Form of Hands 178

PART III POINT FIELD STATISTICS 187

Chapter 11 Fundamental 189 Chapter 12 Finite Point Fields 197

12.1 Introduction 197 12.2 Point Fields of a Fixed Number of Points 197

12.2.1 Two stochastic modeis 197 12.2.2 Two geological examples 200

Vlll CONTENTS

12.3 Point Fields with a Random Number of Points 203 12.3.1 Introduction 203 12.3.2 Some distributions of random numbers 204

Chapter 13 Poisson Point Fields 211 13.1 Introduction 211 13.2 The Homogeneous Poisson Field 212

13.2.1 Fundamental properties 212 13.2.2 Some important formulae 213 13.2.3 Second-order characteristics 216 13.2.4 Simulation of a homogeneous Poisson field 217 13.2.5 Statistics for the homogeneous Poisson field 218

13.3 Inhomogeneous Poisson Fields 228 13.3.1 Fundamental properties 228 13.3.2 Important formulae 229 13.3.3 Simulating an inhomogeneous Poisson field 230 13.3.4 Statistics for an inhomogeneous Poisson field 231

Chapter 14 Fundamentals of the Theory of Point Fields 241 14.1 Introduction 241 14.2 The Intensity Measure 241 14.3 Emptiness Probabilities 243 14.4 Second-Order Characteristics 244

14.4.1 Definitions and formulae 244 14.4.2 Interpretation of pair correlation functions 250

14.5 Characteristics of Third and Higher Order 259 14.6 Second-order Characteristics of Marked Point Fields 262 14.7 Nearest-neighbour Correlation 266 14.8 Distances to Neighbours 266 14.9 Palm Characteristics 268 14.10 Anisotropy Characteristics for Marked and

Non-marked Point Fields 269

Chapter 15 Statistics for Homogeneous Point Fields 275 15.1 Introduction 275 15.2 Estimating the Intensity and the Intensity Measure 276 15.3 Estimating Mark Distributions and Related Quantities 277 15.4 Estimating Second-order Characteristics 279

15.4.1 Estimating Ä(5) , K(r) and L{r) 279 15.4.2 Estimating the pair correlation function 284 15.4.3 Individual L- and g-functions 290 15.4.4 Estimating mark correlation functions 291 15.4.5 Orientation analysis 293

15.5 Estimating Third-order Characteristics 294 15.6 Estimating Nearest-neighbour Distance Distributions 296 15.7 Estimating Palm Means 299 15.8 Goodness-of-Fit" Tests for Point Field Models 300 15.9 Methods for Estimating Model Parameters 304

CONTENTS IX

Chapter 16 Point Field Models 307 16.1 Introduction 307 16.2 Cluster Fields: Neyman-Scott Fields 309

16.2.1 Model description 309 16.2.2 Formulae for Neyman-Scott fields 310 16.2.3 Statistical methods for Matern Cluster fields 314

16.3 Gibbs Fields 317 16.3.1 Describing the modeis 317 16.3.2 Simulating Gibbs fields . 323 16.3.3 Statistical methods for Gibbs fields 326

Appendices 335

A Measure and Content 337

B sup and inf, lim sup and lim inf 339

C Basic Ideas in Topology 341

D Set Operations 343

E The Euclidean and Hausdorff Metrics 347

F Boolean Models 349

G The Convex Hüll 353

H Random Lines and Line Fields 355

I The Dirichlet Mosaic and the Delaunay Triangulation 361

J Germ-Grain Models 363

K The Area of Intersection of Two Discs 365

L Kernel Estimators for Density Functions 367

References 369

Index 387