Stochastic geometry: selected topics

✍ Scribed by Viktor Beneš, Jan Rataj

• Publisher: Springer
• Year: 2004
• Tongue: English
• Leaves: 234
• Edition: 1
• Category: Library

✦ Synopsis

Stochastic geometry, based on current developments in geometry, probability and measure theory, makes possible modeling of two- and three-dimensional random objects with interactions as they appear in the microstructure of materials, biological tissues, macroscopically in soil, geological sediments etc. In combination with spatial statistics it is used for the solution of practical problems such as the description of spatial arrangements and the estimation of object characteristics. A related field is stereology, which makes possible inference on the structures, based on lower-dimensional observations. Unfolding problems for particle systems and extremes of particle characteristics are studied. The reader can learn about current developments in stochastic geometry with mathematical rigor on one hand and find applications to real microstructure analysis in natural and material sciences on the other hand.

✦ Table of Contents

Contents......Page 6
Preface......Page 10
Acknowledgments......Page 12
1.1 Geometry and measure in the Euclidean space......Page 14
1.1.1 Measures......Page 15
1.1.2 Convex bodies......Page 16
1.1.3 Hausdorff measures and rectifiable sets......Page 18
1.1.4 Integral geometry......Page 21
1.2 Probability and statistics......Page 25
1.2.1 Markov chains......Page 27
1.2.2 Markov chain Monte Carlo......Page 29
1.2.3 Point estimation......Page 30
2. RANDOM MEASURES AND POINT PROCESSES......Page 34
2.1 Basic definitions......Page 35
2.2 Palm distributions......Page 38
2.3 Poisson process......Page 41
2.4 Finite point processes......Page 43
2.5 Stationary random measures on R[sup(d)]......Page 45
2.6 Application of point processes in epidemiology......Page 48
2.7 Weighted random measures, marked point processes......Page 51
2.8 Stationary processes of particles......Page 53
2.9 Flat processes......Page 56
3. RANDOM FIBRE AND SURFACE SYSTEMS......Page 58
3.1 Geometric models......Page 60
3.1.1 Projection integral-geometric measures......Page 61
3.1.2 The Campbell measure and first order properties......Page 63
3.1.3 Second-order properties......Page 65
3.1.4 H[sup(k)]-processes and Palm distributions......Page 68
3.1.5 Poisson process......Page 71
3.1.6 Flat processes......Page 73
3.2 Intensity estimators......Page 74
3.2.1 Direct probes......Page 76
3.2.2 Indirect probes......Page 80
3.2.3 Application - fibre systems in soil......Page 85
3.3 Projection measure estimation......Page 88
3.3.1 Convergence in quadratic mean......Page 89
3.3.2 Examples......Page 93
3.4 Best unbiased estimators of intensity......Page 94
3.4.1 Poisson line processes......Page 95
3.4.2 Poisson particle processes......Page 98
3.4.3 Comparison of estimators of length intensity of Poisson segment processes......Page 99
3.4.4 Asymptotic normality......Page 101
4. VERTICAL SAMPLING SCHEMES......Page 106
4.1.1 IUR sampling......Page 108
4.1.2 Application - effect of steel radiation......Page 110
4.1.3 VUR sampling......Page 112
4.1.4 Variances of estimation of length......Page 115
4.1.5 Variances of estimation of surface area......Page 117
4.1.6 Cycloidal probes......Page 124
4.2.1 VUR sampling design......Page 127
4.2.2 Further properties of intensity estimators......Page 130
4.2.3 Estimation of average particle size......Page 133
4.2.4 Estimation of integral mixed surface curvature......Page 137
4.2.5 Gradient structures......Page 143
4.2.6 Microstructure of enamel coatings......Page 145
5.1 Introduction......Page 148
5.2.1 Intersection with x[sub(1)]-axis in R[sup(2)]......Page 149
5.2.2 Relating roses of directions and intersections......Page 151
5.2.3 Estimation of the rose of directions......Page 153
5.3 Convex geometry approach......Page 156
5.3.1 Steiner compact in R[sup(2)]......Page 158
5.3.2 Poisson line process......Page 163
5.3.3 Curved test systems......Page 165
5.3.4 Steiner compact in R[sup(d)], d ≥ 3......Page 168
5.3.5 Anisotropy estimation using MCMC......Page 172
5.4 Orientation-dependent direction distribution......Page 174
6.1 Stereological unfolding......Page 182
6.1.1 Planar sections of a single particle......Page 183
6.1.2 Planar sections of stationary particle processes......Page 184
6.1.3 Unfolding of particle parameters......Page 186
6.2.1 Platelike particles......Page 189
6.2.2 Numerical solution......Page 192
6.2.3 Analysis of microcracks in materials......Page 194
6.3 Trivariate unfolding......Page 195
6.3.1 Oblate spheroids......Page 197
6.3.2 Prolate spheroids......Page 201
6.3.3 Trivariate unfolding, EM algorithm......Page 204
6.3.4 Damage initiation in aluminium alloys......Page 206
6.4 Stereology of extremes......Page 209
6.4.1 Sample extremes – domain of attraction......Page 210
6.4.2 Normalizing constants......Page 211
6.4.3 Extremal size in the corpuscule problem......Page 212
6.4.4 Shape factor of spheroidal particles......Page 213
6.4.5 Prediction of extremal shape factor......Page 216
6.4.6 Farlie-Gumbel-Morgenstern distribution......Page 218
6.4.7 Simulation study of shape factor extremes......Page 220
References......Page 224
F......Page 232
S......Page 233
Z......Page 234