Stochastic and Integral Geometry

by Schneider, Rolf; Weil, Wolfgang

• ISBN13: 9783540788584
• ISBN10: 3540788581
• Format: Hardcover
• Copyright: 2008-11-03
• Publisher: Springer Nature

Summary

Stochastic geometry has in recent years experienced considerable progress, both in its applications to other sciences and engineering, and in its theoretical foundations and mathematical expansion. This book, by two eminent specialists of the subject, provides a solid mathematical treatment of the basic models of stochastic geometry - random sets, point processes of geometric objects (particles, flats), and random mosaics. It develops, in a measure-theoretic setting, the integral geometry for the motion and the translation group, as needed for the investigation of these models under the usual invariance assumptions. A characteristic of the book is the interplay between stochastic and geometric arguments, leading to various major results. Its main theme, once the foundations have been laid, is the quantitative investigation of the basic models. This comprises the introduction of suitable parameters, in the form of functional densities, relations between them, and approaches to their estimation. Much additional information on stochastic geometry is collected in the section notes.

Author Biography

Rolf Schneider: Born 1940, Studies of Mathematics and Physics in Frankfurt/M, Diploma 1964, PhD 1967 (Frankfurt), Habilitation 1969 (Bochum), 1970 Professor TU Berlin, 1974 Professor Univ. Freiburg, 2003 Dr. h.c. Univ. Salzburg, 2005 EmeritusWolfgang Weil: Born 1945, Studies of Mathematics and Physics in Frankfurt/M, Diploma 1968, PhD 1971 (Frankfurt), Habilitation 1976 (Freiburg), 1978 Akademischer Rat Univ. Freiburg, 1980 Professor Univ. Karlsruhe

Table of Contents

Prolog	p. 1
Introduction	p. 1
General Hints to the Literature	p. 8
Notation and Conventions	p. 10
Foundations of Stochastic Geometry
Random Closed Sets	p. 17
Random Closed Sets in Locally Compact Spaces	p. 17
Characterization of Capacity Functionals	p. 22
Some Consequences of Choquet's Theorem	p. 31
Random Closed Sets in Euclidean Space	p. 37
Point Processes	p. 47
Random Measures and Point Processes	p. 48
Poisson Processes	p. 58
Palm Distributions	p. 70
Palm Distributions - General Approach	p. 79
Marked Point Processes	p. 82
Point Processes of Closed Sets	p. 95
Geometric Models	p. 99
Particle Processes	p. 100
Germ-grain Processes	p. 109
Germ-grain Models, Boolean Models	p. 117
Processes of Flats	p. 124
Surface Processes	p. 140
Associated Convex Bodies	p. 145
Integral Geometry
Averaging with Invariant Measures	p. 167
The Kinematic Formula for Additive Functionals	p. 168
Translative Integral Formulas	p. 180
The Principal Kinematic Formula for Curvature Measures	p. 190
Intersection Formulas for Submanifolds	p. 203
Extended Concepts of Integral Geometry	p. 211
Rotation Means of Minkowski Sums	p. 211
Projection Formulas	p. 220
Cylinders and Thick Sections	p. 223
Translative Integral Geometry, Continued	p. 228
Spherical Integral Geometry	p. 248
Integral Geometric Transformations	p. 265
Flag Spaces	p. 266
Blaschke-Petkantschin Formulas	p. 270
Transformation Formulas Involving Spheres	p. 287
Selected Topics from Stochastic Geometry
Some Geometric Probability Problems	p. 293
Historical Examples	p. 293
Convex Hulls of Random Points	p. 298
Random Projections of Polytopes	p. 328
Randomly Moving Bodies and Flats	p. 335
Touching Probabilities	p. 349
Extremal Problems for Probabilities and Expectations	p. 359
Mean Values for Random Sets	p. 377
Formulas for Boolean Models	p. 379
Densities of Additive Functionals	p. 393
Ergodic Densities	p. 404
Intersection Formulas and Unbiased Estimators	p. 413
Further Estimation Problems	p. 429
Random Mosaics	p. 445
Mosaics as Particle Processes	p. 446
Voronoi and Delaunay Mosaics	p. 470
Hyperplane Mosaics	p. 484
Zero Cells and Typical Cells	p. 493
Mixing Properties	p. 515
Non-stationary Models	p. 521
Particle Processes and Boolean Models	p. 522
Contact Distributions	p. 534
Processes of Flats	p. 543
Tessellations	p. 550
Appendix
Facts from General Topology	p. 559
General Topology and Borel Measures	p. 559
The Space of Closed Sets	p. 563
Euclidean Spaces and Hausdorff Metric	p. 570
Invariant Measures	p. 575
Group Operations and Invariant Measures	p. 575
Homogeneous Spaces of Euclidean Geometry	p. 581
A General Uniqueness Theorem	p. 593
Facts from Convex Geometry	p. 597
The Subspace Determinant	p. 597
Intrinsic Volumes and Curvature Measures	p. 599
Mixed Volumes and Inequalities	p. 610
Additive Functionals	p. 617
Hausdorff Measures and Rectifiable Sets	p. 633
References	p. 637
Author Index	p. 675
Subject Index	p. 681
Notation Index	p. 689
Table of Contents provided by Ingram. All Rights Reserved.