Factorization Calculus and Geometric Probability
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This unique book develops the classical subjects of geometric probability and integral geometry, and the more modern one of stochastic geometry, in rather a novel way to provide a unifying framework in which they can be studied. The author focuses on factorization properties of measures and probabilities implied by the assumption of their invariance with respect to a group, in order to investigate nontrivial factors. The study of these properties is the central theme of the book. Basic facts about integral geometry and random point process theory are developed in a simple geometric way, so tha...
This unique book develops the classical subjects of geometric probability and integral geometry, and the more modern one of stochastic geometry, in rather a novel way to provide a unifying framework in which they can be studied. The author focuses on factorization properties of measures and probabilities implied by the assumption of their invariance with respect to a group, in order to investigate nontrivial factors. The study of these properties is the central theme of the book. Basic facts about integral geometry and random point process theory are developed in a simple geometric way, so that the whole approach is suitable for a non-specialist audience. Even in the later chapters, where the factorization principles are applied to geometrical processes, the prerequisites are only standard courses on probability and analysis. The main ideas presented here have application to such areas as stereology and tomography, geometrical statistics, pattern and texture analysis. This book will be well-suited as a starting point for individuals working in those areas to learn about the mathematical framework. It will also prove valuable as an introduction to geometric probability theory and integral geometry based on modern ideas.
Table of contents:
Preface; 1. Cavalieri principle and other prerequisites; 2. Measures invariant with respect to translations; 3. Measures invariant with respect to Euclidean motions; 4. Haar measures on groups of affine transformations; 5. Combinatorial integral geometry; 6. Basic integrals; 7. Stochastic point processes; 8. Palm distributions of point processes; 9. Poisson-generated geometrical processes; 10. Section through planar geometrical processes.
Table of contents:
Preface; 1. Cavalieri principle and other prerequisites; 2. Measures invariant with respect to translations; 3. Measures invariant with respect to Euclidean motions; 4. Haar measures on groups of affine transformations; 5. Combinatorial integral geometry; 6. Basic integrals; 7. Stochastic point processes; 8. Palm distributions of point processes; 9. Poisson-generated geometrical processes; 10. Section through planar geometrical processes.
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