Stochastic Geometry: Selected Topics

✍ Scribed by Viktor Benes, Jan Rataj

• Publisher: Springer
• Year: 2004
• Tongue: English
• Leaves: 234
• 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

Cover
Title
Contents
Preface
Acknowledgments
1. PRELIMINARIES
1.1 Geometry and measure in the Euclidean space
1.1.1 Measures
1.1.2 Convex bodies
1.1.3 Hausdorff measures and rectifiable sets
1.1.4 Integral geometry
1.2 Probability and statistics
1.2.1 Markov chains
1.2.2 Markov chain Monte Carlo
1.2.3 Point estimation
2. RANDOM MEASURES AND POINT PROCESSES
2.1 Basic definitions
2.2 Palm distributions
2.3 Poisson process
2.4 Finite point processes
2.5 Stationary random measures on R[sup(d)]
2.6 Application of point processes in epidemiology
2.7 Weighted random measures, marked point processes
2.8 Stationary processes of particles
2.9 Flat processes
3. RANDOM FIBRE AND SURFACE SYSTEMS
3.1 Geometric models
3.1.1 Projection integral-geometric measures
3.1.2 The Campbell measure and first order properties
3.1.3 Second-order properties
3.1.4 H[sup(k)]-processes and Palm distributions
3.1.5 Poisson process
3.1.6 Flat processes
3.2 Intensity estimators
3.2.1 Direct probes
3.2.2 Indirect probes
3.2.3 Application - fibre systems in soil
3.3 Projection measure estimation
3.3.1 Convergence in quadratic mean
3.3.2 Examples
3.4 Best unbiased estimators of intensity
3.4.1 Poisson line processes
3.4.2 Poisson particle processes
3.4.3 Comparison of estimators of length intensity of Poisson segment processes
3.4.4 Asymptotic normality
4. VERTICAL SAMPLING SCHEMES
4.1 Randomized sampling
4.1.1 IUR sampling
4.1.2 Application - effect of steel radiation
4.1.3 VUR sampling
4.1.4 Variances of estimation of length
4.1.5 Variances of estimation of surface area
4.1.6 Cycloidal probes
4.2 Design-based approach
4.2.1 VUR sampling design
4.2.2 Further properties of intensity estimators
4.2.3 Estimation of average particle size
4.2.4 Estimation of integral mixed surface curvature
4.2.5 Gradient structures
4.2.6 Microstructure of enamel coatings
5. FIBRE AND SURFACE ANISOTROPY
5.1 Introduction
5.2 Analytical approach
5.2.1 Intersection with x[sub(1)]-axis in R[sup(2)]
5.2.2 Relating roses of directions and intersections
5.2.3 Estimation of the rose of directions
5.3 Convex geometry approach
5.3.1 Steiner compact in R[sup(2)]
5.3.2 Poisson line process
5.3.3 Curved test systems
5.3.4 Steiner compact in R[sup(d)], d ≥ 3
5.3.5 Anisotropy estimation using MCMC
5.4 Orientation-dependent direction distribution
6. PARTICLE SYSTEMS
6.1 Stereological unfolding
6.1.1 Planar sections of a single particle
6.1.2 Planar sections of stationary particle processes
6.1.3 Unfolding of particle parameters
6.2 Bivariate unfolding
6.2.1 Platelike particles
6.2.2 Numerical solution
6.2.3 Analysis of microcracks in materials
6.3 Trivariate unfolding
6.3.1 Oblate spheroids
6.3.2 Prolate spheroids
6.3.3 Trivariate unfolding, EM algorithm
6.3.4 Damage initiation in aluminium alloys
6.4 Stereology of extremes
6.4.1 Sample extremes – domain of attraction
6.4.2 Normalizing constants
6.4.3 Extremal size in the corpuscule problem
6.4.4 Shape factor of spheroidal particles
6.4.5 Prediction of extremal shape factor
6.4.6 Farlie-Gumbel-Morgenstern distribution
6.4.7 Simulation study of shape factor extremes
References
Index
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