This book covers methods of Mathematical Morphology to model and simulate random sets and functions (scalar and multivariate). The introduced models concern many physical situations in heterogeneous media, where a probabilistic approach is required, like fracture statistics of materials, scaling up of permeability in porous media, electron microscopy images (including multispectral images), rough surfaces, multi-component composites, biological tissues, textures for image coding and synthesis. The common feature of these random structures is their domain of definition in n dimensions, requiring more general models than standard Stochastic Processes.The main topics of the book cover an introduction to the theory of random sets, random space tessellations, Boolean random sets and functions, space-time random sets and functions (Dead Leaves, Sequential Alternate models, Reaction-Diffusion), prediction of effective properties of random media, and probabilistic fracture theories.
1. Introduction.- Part I Tools for Random Structures.- 2 Introduction to
Random Closed Sets and to Semi-continuous Random Functions.- 3 Quantitative
Analysis of Random Structures.- Part II Models of Random Structures.- 4
Excursion sets of Gaussian RF.- 5 Stochastic Point Processes and Random
Trees.- 6 Boolean Random Sets.- 7 Random Tessellations.- 8 The Mosaic
Model.- 9 Boolean Random Functions.- 10 Random Tessellations and Boolean
Random Functions.- 11 Dead Leaves Models: from Space Tessellations to Random
Functions.- 12 Sequential Cox Boolean and Conditional Dead Leaves Models.- 13
Sequential Alternate Random Functions.- 14 Primary Grains and Primary
Functions.- 15 Dilution Random Functions.- 16 Reaction-Diffusion and Lattice
Gas Models.- 17 Texture Segmentation by Morphological Probabilistic
Hierarchies.- Part III Random Structures and Change of Scale.- 18 Change of
Scale in Physics of Random Media.- 19 Digital Materials.- 20 Probabilistic
Models for Fracture Statistics.- 21 Crack Paths in Random Media.
Dominique Jeulin is Director of Research and Professor at Mines-ParisTech (Ecole des Mines de Paris). Currently Scientific Adviser at Centre de Morphologie Mathématique, he was teaching courses on models of random structures, and on physics and mechanics of random media. His main current interests cover the theoretical prediction of physical properties of random media, models and simulations of random media, 3D image analysis, probabilistic segmentation of images, applications to materials science, biology and vision.
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