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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
The purpose of this book is the development of probabilistic models giving a simplified representation of heterogeneous structures (Part I and Part II) and enabling us to predict some change of scale effects (micro–macro scaling) for the physical behavior of heterogeneous media (Part III). Our approach is directed towards probabilistic models. Other books were published on these subjects over years. To mention some monographs, and excluding proceedings, books [113, 591] review basic models of random sets, as introduced in Stochastic Geometry. Books [497–499] are concentrated on a basic type of model (namely the Boolean model), mainly from a theoretical point of view. Books [532, 515] concern mainly image analysis, and are mostly algorithmic. The present book is a large extension of earlier works [257, 275], of a preliminary unpublished version, and of lecture notes on courses on random media [287], [297], [308]. We will make use of the probabilistic modeling paradigm, developing a synthetic approach, in contrast with data analysis. The main advantages of this approach are to make possible a representation and simulation of microstructures (based on the notion of « virtual materials» in materials science), after identification of models by image analysis. Once a proper model is available, it can be implemented for the prediction of probabilistic properties of heterogeneous media, and in the next step for the optimization of microstructures with respect to their physical behavior.
Dominique Jeulin

### Chapter 2. Introduction to Random Closed Sets and to Semi-Continuous Random Functions

Abstract
This chapter recalls the construction and the main properties of random closed sets: i) The Choquet topology, for the closed sets of a space E, enables us to build a probability space of closed random sets, characterized by their Choquet capacity. ii) From the same approach extended to upper or lower semi continuous numerical functions, it is possible to build random functions for which the supremum or the infimum of its values inside a compact set K is a random variable.
Dominique Jeulin

### Chapter 3. Quantitative Analysis of Random Structures

Abstract
The measurements used to analyze the morphology of random media, experimentally available from image analysis, are introduced: measurements are obtained in a two-steps approach: transformation and basic measurement; their prototypes are the basic operations of mathematical morphology (erosion and dilation) by appropriate structuring elements, coupled with the Minkowski functionals. They are used to define various criteria for characterizing the sizes, the spatial distribution and the connectivity of objects like sets, functions or graphs. Tools introduced in this chapter can be used for pattern and texture recognition by means of machine learning algorithms, as illustrated by some examples.
Dominique Jeulin

### Chapter 4. Excursion Sets of Gaussian RF

Abstract
Random sets models derived from truncated RF, with an application to Gaussian random functions, are introduced. Covariance and order m central correlation functions are given. The model is illustrated by examples of application.
Dominique Jeulin

### Chapter 5. Stochastic Point Processes and Random Trees

Abstract
Models of stochastic point processes are basic models of random sets with a rather simple geometry. The archetype model is the Poisson point process, which simulates a population of particles without any spatial interaction. It can be used as seeds for other models developed in other chapters, called grain models. Variants of the Poisson point process use constraints like repulsion, as for the hard-core process. Multi scale models are obtained by means of Cox processes. Iterated Poisson varieties provide models with clusters like alignments on lines or in planes. A short introduction to Gibbs and to Determinantal point processes is given. Finally models of random trees are obtained by branching processes.
Dominique Jeulin

### Chapter 6. Boolean Random Sets

Abstract
The models of random sets introduced in this chapter are based on the Poisson point process, used as seeds for the Boolean RACS, which gives good simulations of two phase connected media, for which a percolation threshold can be estimated. Properties of this model and examples of application with proper identification of its parameters are given. Generalizations include Poisson varieties (which are Poisson point processes in appropriate spaces), to build fibre or strata Boolean models, a multi-scale version based on the Cox point process, and a multi component version.
Dominique Jeulin

### Chapter 7. Random Tessellations

Abstract
The probabilistic characterization of random tessellations is introduced. The construction and the main properties of some random tessellations (Voronoi, Johnson-Mehl, Laguerre, Poisson, Cauwe, and iterated tessellation models) and of their extensions are given.
Dominique Jeulin

### Chapter 8. The Mosaic Model

Abstract
This chapter introduces the mosaic model, where subdomains of space have a constant value (univariate or multivariate). It is built on a random tessellation of space. Its Choquet capacity is given and illustrated by order two and order three probability distributions. Special cases are the Poisson and the STIT mosaic models.
Dominique Jeulin

### Chapter 9. Boolean Random Functions

Abstract
The Boolean RF are a generalization of the Boolean RACS. Their construction based on the combination of a sequence of primary RF by the operation ∨ (supremum) or ∧ (infimum), and their main properties (among which the supremum or infimum infinite divisibility) are given in the following cases: scalar RF built on a Poisson point process and on Poisson varieties; multivariate case to simulate multispectral data.
Dominique Jeulin

### Chapter 10. Random Tessellations and Boolean Random Functions

Abstract
Generalizations of usual random tessellation models in $${\mathbb{R}}^{n}$$ generated by Poisson point processes are introduced, and their functional probability $$P(K)$$ is given. They are obtained from Boolean random function, providing a generic way of simulation of a wide range of random tessellations.
Dominique Jeulin

### Chapter 11. Dead Leaves Models: From Space Tessellations to Random Functions

Abstract
Sequential models with support in $${\mathbb{R}}^{n}$$ are developed. For each point x in $${\mathbb{R}}^{n}$$, the models combine families of independent random sets or random functions, indexed by a parameter t. By a masking process, the Dead Leaves models (and its generalized version, the Markovian jumps sequential RF) simulate random images with objects in the foreground partially masking objects in the background, as seen in perspective views. The main probabilistic properties of the models are presented for the following cases: Dead Leaves tessellation, Color Dead Leaves, Dead Leaves RF, multivariate Dead Leaves RF and varieties, Markov Jumps RF. It is illustrated by applications to the morphological characterization of powders.
Dominique Jeulin

### Chapter 12. Sequential Cox Boolean and Conditional Dead Leaves Models

Abstract
New versions of the Boolean RACS and RF, as well as the Dead Leaves random tessellations and Color Dead Leaves are obtained by imposing conditions on the choice of primary grains and primary functions, depending on the location of sequential Poisson points. These models are iteratively generated by a time sequence of Cox Processes. Their probabilistic properties and simplified closed-form approximations are derived. Some simulations illustrate the models.
Dominique Jeulin

### Chapter 13. Sequential Alternate Random Functions

Abstract
Sequential random function models, intermediary between the Boolean RF and the Dead Leaves RF, are built from primary RF by alternating the operations ˅ and ˄. These types of models might simulate rough surfaces evolving by a sequence of depositions and abrasions. After the construction of this model, the univariate and bivariate distributions are worked out (with an illustration to cylinder primary RF). For certain types of primary RF, the laws of apparent maxima and minima are derived. Generalizations to the multivariate case and to the random varieties are given and an example of application to rough surfaces is presented.
Dominique Jeulin

### Chapter 14. Primary Grains and Primary Functions

Abstract
Models of random structures with a bounded support, that can be used as primary grains in the previous chapters, are proposed:several families of random sets are studied (populations of spheres and of ellipsoids in $${\mathbb{R}}^{3}$$), Poisson polyhedra, random aggregates; similarly,models of random functions are built according to different processes:cylinder RF, restriction of stationary RF to a random compact, spherical primary RF, Boolean and Dead Leaves RF with a compact support.
Dominique Jeulin

### Chapter 15. Dilution Random Functions

Abstract
The Dilution RF (DRF) combines primary RF located on Poisson points by addition. The Dilution RF is infinite divisible according to the addition operation. The scalar RF built on a Poisson point process and on Poisson varieties, as well as the multivariate case to simulate multispectral data are presented.
Dominique Jeulin

### Chapter 16. Reaction-Diffusion and Lattice Gas Models

Abstract
Spatial and temporal multivariate RF models, where interactions between components act through chemical reactions, are introduced.These models give rise to morphogenesis. A first part gives a general description of Reaction-Diffusion models at two different scales:on a macroscopic scale, chemical concentration variables are solution of a system of non linear parabolic partial differential equations. On a microscopic scale, discrete versions of these models are obtained as Markov jump processes (for the chemical reaction) and random walks (for the diffusion).A second part studies the linear Reaction-Diffusion model, well suited to a particular kind of chemical reaction ( $${{X_{i}} \rightleftharpoons {X_{j}}}$$) or to a damping effect. Models obtained from a stationary RF as initial conditions and from a spatial-temporal random source are considered, with in particular a specific Dilution RF. Then examples of simulations of non-linear Reaction-Diffusion RF, generating time oscillating or chaotic behaviors,are illustrated. In a third part is introduced a discrete implementation based on lattice gas simulations, to simulate complex flows in random media, random aggregates, and multi species chemical reactions.
Dominique Jeulin

### Chapter 17. Texture Segmentation by Morphological Probabilistic Hierarchies

Abstract
A general methodology is introduced for probabilistic texture segmentation in binary, scalar, or multispectral images. Textural information is obtained from morphological operations on images. Starting from a fine partition of the image in regions, hierarchical segmentations are set up in a probabilistic framework by means of probabilistic distances conveying the textural or morphological information, and of random markers accounting for the morphological content of the regions and of their spatial arrangement. The probabilistic hierarchies are built from binary or multiple fusion of regions.
Dominique Jeulin

### Chapter 18. Change of Scale in Physics of Random Media

Abstract
Problems of change of scale in physics of random media handled in this chapter concern the prediction of the macroscopic behavior of a physical system from its microscopic behavior. Most physical properties of random media are concerned by this approach, like for instance the dielectric permittivity ϵ in electrostatics, the permeability K or flows in porous media, or elastic moduli of composites. This topic concerns the definition an calculation of effective (or macroscopic) properties of an equivalent homogeneous medium from lower scale information by homogenization tools. In a first step, models of RS or of RF are used to represent maps of properties on a microscopic level. Then, the homogenization problem is addressed by a perturbation expansion of the physical fields of interest and by the calculation of bounds of effective properties for models of random structures by means of correlation functions of local properties. This illustrated in the case of linear constitutive behavior by third order bounds and their application to models of RF and of random sets studied in previous chapters. Finally local fluctuations of fields on a point level are characterized by their second order moments derived from the effective properties.
Dominique Jeulin

### Chapter 19. Digital Materials

Abstract
A prediction of apparent properties of random structures is obtained from the numerical solution of the PDE concerning the physical fields on real images or on simulations of random media in bounded domains. For this purpose, numerical calculations are made by Finite Elements or by FFT. This numerical approach involves a statistical definition of a representative volume element RVE to provide intervals of confidence of the estimated apparent properties. Furthermore, from full-field simulations a morphological analysis of the fields can be performed to promote connections between the random microstructure and its local response to physical solicitations. This is illustrated by examples of application to various microstructures of industrial materials, as well as to models of random sets.
Dominique Jeulin

### Chapter 20. Probabilistic Models for Fracture Statistics

Abstract
As a consequence of microstructural heterogeneities, fluctuations in the mechanical properties of materials are observed in experiments. This requires a probabilistic approach to relate the microstructure to the overall materials properties, and to predict scale effects in the fluctuations of properties. In this chapter, the problem of the strength of materials is addressed, and various models of random structures developed for fracture statistics are introduced. As any fracture criterion is sensitive to microstructural heterogeneities, such as flaws with low strength, or defects inducing a local stress concentration, large effects of small scale heterogeneities are observed for fracture phenomena. The approach combines the selection of appropriate fracture criteria and random structure models. It enables us to predict the probability of fracture of heterogeneous media under various loading conditions.
Dominique Jeulin

### Chapter 21. Crack Paths in Random Media

Abstract
Simple probabilistic models of fracture of 3D polycrystals involving transgranular and intergranular cracks are based on undamaged paths and appropriate percolation thresholds. In a second part, theoretical extensions of phase field models for crack initiation and propagation to the case of locally heterogeneous and anisotropic fracture energy are presented. They are implemented by means of iterations of Fourier transforms, replacing the standard Finite Elements approach, to generate full field solutions. When the phase field problem is numerically solved on simulations of a random medium, its effective fracture energy can be estimated and its fluctuations characterized with the corresponding RVE.
Dominique Jeulin

### Backmatter

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