UID:
edocfu_9959240278802883
Format:
1 online resource (xi, 286 pages) :
,
digital, PDF file(s).
ISBN:
1-139-88179-5
,
1-107-10259-6
,
1-107-08789-9
,
1-107-10007-0
,
1-107-09412-7
,
1-139-08656-1
Series Statement:
Encyclopedia of mathematics and its applications ;
Content:
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 factorisation properties of measures and probabilities implied by the assumption of their invariance with respect to a group, in order to investigate non-trivial 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 factorisation 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.
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
,
Cover; Half Title; Title; Copyright; CONTENTS; PREFACE; 1 Cavalieri principle and other prerequisites; 1.1 The Cavalieri principle; 1.2 Lebesgue factorization; 1.3 Haar factorization; 1.4 Further remarks on measures; 1.5 Some topological remarks; 1.6 Parametrization maps; 1.7 Metrics and convexity; 1.8 Versions of Crofton's theorem; 2 Measures invariant with respect to translations; 2.1 The space G of directed lines on R2; 2.2 The space G of (non-directed) lines in R2; 2.3 The space E of oriented planes in R3; 2.4 The space E of planes in R3; 2.5 The space Γ of directed lines in R3
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2.6 The space Γ of (non-directed) lines in R32.7 Measure-representing product models; 2.8 Factorization of measures on spaces with slits; 2.9 Dispensing with slits; 2.10 Roses of directions and roses of hits; 2.11 Density and curvature; 2.12 The roses of T3-invariant measures on E; 2.13 Spaces of segments and flats; 2.14 Product spaces with slits; 2.15 Almost sure T-invariance of random measures; 2.16 Random measures on G; 2.17 Random measures on E; 2.18 Random measures on Γ; 3 Measures invariant with respect to Euclidean motions; 3.1 The group W2 of rotations of R2; 3.2 Rotations of R3
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3.3 The Haar measure on W33.4 Geodesic lines on a sphere; 3.5 Bi-invariance of Haar measures on Euclidean groups; 3.6 The invariant measure on G and G; 3.7 The form of dg in two other parametrizations of lines; 3.8 Other parametrizations of geodesic lines on a sphere; 3.9 The invariant measure on Γ and Γ; 3.10 Other parametrizations of lines in R3; 3.11 The invariant measure in the spaces E and E; 3.12 Other parametrizations of planes in R3; 3.13 The kinematic measure; 3.14 Position-size factorizations; 3.15 Position-shape factorizations; 3.16 Position-size-shape factorizations
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3.17 On measures in shape spaces3.18 The spherical topology of Σ; 4 Haar measures on groups of affine transformations; 4.1 The group Ag and its subgroups; 4.2 Affine deformations of R2; 4.3 The Haar measure on Ag; 4.4 The Haar measure on A2; 4.5 Triads of points in R2; 4.6 Another representation of d(r)V; 4.7 Quadruples of points in R2; 4.8 The modified Sylvester problem: four points in R2; 4.9 The group Ag and its subgroups; 4.10 The group of affine deformations of R3; 4.11 Haar measures on Ag and A3; 4.12 V 3-invariant measure in the space of tetrahedral shapes
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4.13 Quintuples of points in R34.14 Affine shapes of quintuples in R3; 4.15 A general theorem; 4.16 The elliptical plane as a space of affine shapes; 5 Combinatorial integral geometry; 5.1 Radon rings in G and G; 5.2 Extension of Crofton's theorem; 5.3 Model approach and the Gauss-Bonnet theorem; 5.4 Two examples; 5.5 Rings in E; 5.6 Planes cutting a convex polyhedron; 5.7 Reconstruction of the measure from a wedge function; 5.8 The wedge function in the shift-invariant case; 5.9 Flag representations of convex bodies; 5.10 Flag representations and zonoids
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5.11 Planes hitting a smooth convex body in R3
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English
Additional Edition:
ISBN 0-521-08978-6
Additional Edition:
ISBN 0-521-34535-9
Language:
English
