We consider the structural and physical properties of complex model morphologies and microstructures obtained by Xray-CT imaging. The Minkowski functionals, a family of statistical measures based on the Euler-Poincar&eacute characteristic of n-dimensional space, are shown to be sensitive measures of the morphology of disordered structures. Analytic results for the Boolean model are given and used to devise a reconstruction scheme, which allows one to accurately reconstruct a complex Boolean structure given at any phase fraction for all other phase fractions. The percolation thresholds of either phase are obtained with good accuracy. From the reconstructions one can subsequently predict property curves for the material across all phase fractions from a single 3D image. We illustrate this for transport and mechanical properties of complex Boolean systems and for experimental sandstone samples. By extending the Minkowski functionals to parallel surfaces using operations from mathematical morphology, a powerful discrimination of structure is obtained. Further the sensitivity of the Minkowski functionals under experimental conditions is analysed. Accurate calculations of conductive and elastic properties directly from tomographic images are achieved by estimating and minimising several sources of numerical error. Simulations of electrical conductivity and linear elastic properties on microtomographic images of Fontainebleau sandstone are in excellent agreement with experimental measurements over a wide range of porosity. The results show the feasibility of combining digitised images with transport and elasticity calculations to accurately predict physical properties of individual material morphologies. We show that measurements of properties based on microtomographic images are more accurate than those based on conventional theories for disordered materials. We study the elastic behaviour of model clean and cemented sandstones. Results are in excellent agreement with available experimental data, and are compared to conventional theoretical and empirical laws. A new predictive empirical method is given for predicting the elastic moduli of sandstone morphologies. The method gives an excellent match to numerical and experimental data.

Les tessellations, aussi appelées mosaïques, permettent de modéliser de nombreuses structures, comme des assemblages de cellules en biologie ou de grains en science des matériaux. La tessellation aléatoire la plus connue est le diagramme de Voronoï qui à partir d'un ensemble de points, appelés germes, partitionne le plan. L'approche innovante de cette thèse est d'utiliser des champs aléatoires gaussiens pour générer des germes et des distances aléatoires, qui vont permettre de simuler une grande variété de tessellations en termes de formes et de tailles des cellules.Pour connaître les propriétés des tessellations simulées à partir de champs aléatoires gaussiens, celles-ci vont être caractérisées et comparées à d'autres tessellations. Tout d'abord par une approche ponctuelle en étudiant les germes, dont leur distribution spatiale. Puis par une approche par région, en étudiant la géométrie et la morphométrie des cellules.L'endothélium cornéen humain est une monocouche de cellules formant un pavage hexagonal régulier à la naissance, et perdant de sa régularité ensuite. La qualité du greffon cornéen est donnée par certaines observations, comme la densité, l'homogénéité de la forme et des tailles des cellules endothéliales.L'évolution avec l'âge de cette mosaïque cornéenne va être caractérisée à partir d’une base d’images de l’endothélium. L'originalité est ensuite d'effectuer une estimation de l'âge d’un endothélium à partir des différentes mesures permettant de caractériser les tessellations, et enfin de mettre en place une méthode prometteuse afin de savoir si une cornée a une évolution normale. / Tessellations, also called mosaics, are used to model many structures, for example cellular arrangements in biology or grains in material science. The most known tessellation is the Voronoï diagram which partitions the space from a set of points, called germs. The innovative approach of this thesis is to use Gaussian random fields to generate germs and random distances. The use of random fields allows to simulate a great variety of tessellations in terms of cells forms and sizes.To study the properties of each type of tessellation, they are characterized: first, by studying the germs, including their spatial distribution, and then by analyzing the cells geometry and morphometry. These tessellations are also compared to other known tessellations.The human corneal endothelium is a mono-layer of cells forming a regular hexagonal mosaic at birth, and losing his regularity later. The corneal graft quality is given by some observations made on the endothelial mosaic (cells density, the homogeneity of cells sizes and shapes).A database of endothelium images allows to characterize the evolution with age of the corneal mosaic. The originality is to estimate the age of an endothelium based on the measures computed to characterize the tessellations, and finally to set up a promising method to evaluate if a corneal evolution is normal.