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Morphological characterization of porous media is of paramount interest, mainly due to the connections between their physicochemical properties and their porous microstructure geometry. Heterogeneity can be seen as a geometric characteristic of porous microstructures. In this paper, two novel topological descriptors are proposed, based on the M-tortuosity formalism. Using the concept of geometric tortuosity or morphological tortuosity, a first operator is defined, the H-tortuosity. It estimates the average variations of the morphological tortuosity as a function of the scale, based on Monte Carlo method and assessing the heterogeneity of porous networks. The second descriptor is an extension, named the H-tortuosity-by-iterativeerosions, taking into account different percolating particle sizes. These two topological operators are applied on Cox multi-scale Boolean models, to validate their behaviors and to highlight their discriminative power.

The intact grains of the dead leaves model enables us to generate random media with non overlapping grains. Using the time non homogeneous sequential model with convex grains, theoretically very dense packings can be generated, up to a full covering of space. For these models, the theoretical volume fraction, the size distribution of grains, and the pair correlation function of centers of grains are given.

Pigments made of metal particles of around 10 µm or 20 µm produce sparkling effects in paints, due to the specular reflection that occurs at this scale. Overall, the optical aspect of paints depend on the density and distribution in space of the particles. In this work, we model the dispersion of metal particles of size up to 50 µm, visible to the eyes, in a paint layer. Making use of optical and scanning electron microscopy (SEM) images, we estimate the dispersion of particles in terms of correlation functions. Particles tend to aggregate into clusters, as shown by the presence of oscillations in the correlation functions. Furthermore, the volume fraction of particles is non-uniform in space. It is highest in the middle of the layer and lowest near the surfaces of the layer. To model this microstructure, we explore two models. The first one is a deposit model where particles fall onto a surface. It is unable to reproduce the observed measurements. We then introduce a "stack" model where clusters are first modeled by a 2D Poisson point process, and a bi-directional deposit model is used to implant particles in each cluster. Good agreement is found with respect to SEM images in terms of correlation functions and density of particles along the layer height.

In the area of tessellation models, there is an intense activity to fully understand the classical models of Voronoi, Laguerre and Johnson-Mehl. Still, these models are all simulations of isotropic growth and are therefore limited to very simple and partly convex cell shapes. The here considered microstructure of martensitic steel has a much more complex and highly non convex cell shape, requiring new tessellation models. This paper presents a new approach for anisotropic tessellation models that resolve to the well-studied cases of Laguerre and Johnson-Mehl for spherical germs. Much better reconstructions can be achieved with these models and thus more realistic microstructure simulations can be produced for materials widely used in industry like martensitic and bainitic steels.

We propose a new computer aided detection framework for tumours acquired on DCE-MRI (Dynamic Contrast Enhanced Magnetic Resonance Imaging) series on small animals. To perform this approach, we consider DCE-MRI series as multivariate images. A full multivariate segmentation method based on dimensionality reduction, noise filtering, supervised classification and stochastic watershed is explained and tested on several data sets. The two main key-points introduced in this paper are noise reduction preserving contours and spatio temporal segmentation by stochastic watershed. Noise reduction is performed in a special way to select factorial axes of Factor Correspondence Analysis in order to preserves contours. Then a spatio-temporal approach based on stochastic watershed is used to segment tumours. The results obtained are in accordance with the diagnosis of the medical doctors.

X-ray microtomography from solid propellant allows studying the microstructure of fragmented grains in damaged samples. A new reconstruction algorithm of fragmented grains for 3D images is introduced. Based on a watershed transform of a morphological closing of the input image, the algorithm can be used with different sets of markers. Two of them are compared. After the grain reconstruction, a multiscale segmentation algorithm is used to extract each fragment of the damaged grains. This allows an original quantitative study of the fragmentation of each grain in 3D. Experimental results on X-ray microtomographic images of a solid propellant fragmented under compression are presented and validated.

The morphological tortuosity of a geodesic path in a medium can be defined as the ratio between its geodesic length and the Euclidean distance between its two extremities. Thus, the minimum tortuosity of all the geodesic paths into a medium in 2D or in 3D can be estimated by image processing methodsusing mathematical morphology. Considering a medium, the morphological tortuosities of its internal paths are estimated according to one direction, which is perpendicular to both starting and ending opposite extremities of the geodesicpaths. The used algorithm estimates the morphological tortuosities from geodesic distance maps, which are obtained from geodesic propagations. The shape of the propagated structuring element used to estimate the geodesic distance maps on a discrete grid has a direct influence on the morphological tortuosity and has to be chosen very carefully. The results of our algorithm is an image with pixels p having a value equal to the length of the shortest path containing p and connected to two considered opposite boundaries A and B of the image. The analysis of the histogram of the morphological tortuosities gives access to their statistical distribution. Moreover, for each tortuosity the paths can be extracted from the original image, which highlights the location of them into the sample. However, these geodesic paths have to be reconstructed for further processing. The extraction, because applying a threshold on the tortuosities, results in disconnected components, especially for highly tortuous paths. This reconstruction consists in reconnecting these components to the geodesic path linking the two opposite faces, by means of a backtracking algorithm.

Computing an array of all pairs of geodesic distances between the pixels of an image is time consuming. In the sequel, we introduce new methods exploiting the redundancy of geodesic propagations and compare them to an existing one. We show that our method in which the source point of geodesic propagations is chosen according to its minimum number of distances to the other points, improves the previous method up to 32 % and the naive method up to 50 % in terms of reduction of the number of operations.

The normal human adult kidney contains between 300,000 and 1 million nephrons (the functional units of the kidney). Nephrons develop at the tips of the branching ureteric duct, and therefore ureteric duct branching morphogenesis is critical for normal kidney development. Current methods for analysing ureteric branching are mostly qualitative and those quantitative methods that do exist do not account for the 3- dimensional (3D) shape of the ureteric "tree". We have developed a method for measuring the total length of the ureteric tree in 3D. This method is described and preliminary data are presented. The algorithm allows for performing a semi-automatic segmentation of a set of grey level confocal images and an automatic skeletonisation of the resulting binary object. Measurements of length are automatically obtained, and numbers of branch points are manually counted. The final representation can be reconstructed by means of 3D volume rendering software, providing a fully rotating 3D perspective of the skeletonised tree, making it possible to identify and accurately measure branch lengths. Preliminary data shows the total length estimates obtained with the technique to be highly reproducible. Repeat estimates of total tree length vary by just 1-2%. We will now use this technique to further define the growth of the ureteric tree in vitro, under both normal culture conditions, and in the presence of various levels of specific molecules suspected of regulating ureteric growth. The data obtained will provide fundamental information on the development of renal architecture, as well as the regulation of nephron number.

Image analysis of rock fragmentation is used in mines and quarries to control the quality of blasting. Obtained information is the particle-size-distribution curve relating volume-proportions to the sizes of fragments. Calculation by image analysis of this particle-size-distribution is carried out in several steps, and each step has its inherent limitations. We will focus in this paper on one of themost crucial steps: reconstructing the volumes (3D). For the 3D-step, we have noticed that, due to the current acquisition method, there is no correlation between the average grey level of surfaces of the fragments and their third dimension. Consequently volumes (3D) as well as the sizes (1D) has to be calculated indirectly from the extracted projected areas of the visible fragments of images. For this purpose, we have built in laboratory a set of images of fragmented rocks resulting from blasting. Moreover, several tests based on comparisons between image analysis and screening measurements were carried out. A new stereological method, based on the comparison of the densities of probability (histograms) of the samemeasurements (with very weak covering and overlapping)was elaborated. It allows us to estimate correctly, for a given type of rock, two intrinsic laws weighing the projected areas distribution in order to predict the volumic distribution.

Transition aluminas, and especially the gamma type, are largely used as catalyst supports in refining and petrochemicals. Most studies focus on properties resulting from material texture and casting (specific surface, porous volume, pore shape and diameter). However, surface properties of alumina should be considered as well, as the catalytic activity is tightly related to the structure of exposed crystalline faces. As γ alumina results from controlled thermal treatment of boehmite γ-AlOOH by a topotactic transformation, the nature of exposed crystalline planes is related to the starting material. Therefore, the synthesis of the oxihydroxide γ-AlOOH, and especially size and shape of these particles, is critical in determining the relevant surface properties. Unlike often aggregated alumina, boehmite nanoparticles can be observed by TEM. Analysis of these TEM images can be performed to estimate the size of the boehmite nanoparticles. Information about morphology of the nanoparticles is obtained by the analysis of the covariance, modeling micrographs by a dilution model.

In the framework of the Silent Wall ANR project, the CMM and the US2B are associated in order to characterize and to model fibrous media studying 3D images acquired with an X-Ray tomograph used by the US2B. The device can make 3D images of maximal 23043 voxels with resolutions in the range of 2 μm to 15 μm. Using mathematical morphology, measurements on the 3D X-Ray CT images are used to characterize materials. For example measuring the covariance on these images of an acoustic insulating material made of wooden fibres highlights the isotropy of the fibres orientations in the longitudinal planes which are perpendicular to the compression Oz axis. Moreover, it is possible to extract other morphological properties from these image processing methods such as the size distribution either of the fibres or of the pores by estimating the morphological opening granulometry of the considered medium. Using the theory of random sets introduced by Georges Matheron in the early 1970's, the aim of this work is to model such a fibrous material by parametric random media in 3D according to the prior knowledge of its morphological properties (covariance, porosity, size distributions, etc.). A Boolean model of random cylinders in 3D stacked in planes parallel to each other and perpendicular to the Oz compression axis is first considered. The granulometry results provide gamma distributions for the radii of the fibres. In addition, a uniform distribution of the orientations is chosen, according to the experimental isotropy measurements in the longitudinal planes. Finally the third statistical factor is the length distribution of the fibres which can be fitted by an exponential distribution. Thus it is possible to estimate the validity of this model first by trying to fit the experimental transverse and longitudinal covariances of the pores with the theoretical ones taking into account the statistical distributions of the dimensions of the random cylinders. The second method to validate the model consists in comparing morphological measurements (density profiles, covariance, opening granulometry, tortuosity, specific surface area) processed on real and on simulated media.

In this paper we present algorithms for measuring local characteristics of random fiber systems. The calculation of the local directions and radii is based on directional distance transforms and evaluation of the inertia moments and axes of the resulting extremities of the centralized, directed chords. The method provides continuous results while minimizing the runtime by using few sampled directions. Furthermore several steps of improvement for the computation of orientation and radius information are presented. The algorithms are evaluated using synthetic data and applied to images of realmicrostructures obtained by computer tomography.

Starting from scanning electron microscope images of some food products, we generate binary images of composite materials. After measuring the covariance and the probability for segments and for squares to be included in the dominant component, we develop a modelling of the microstructure from random sets obtained by thresholding Gaussian random functions. The covariance function of the underlying Gaussian random function is estimated from the experimental covariance of the food products. The validity of the model is checked by comparison of the probability curves for segments and for squares, measured on simulated and on initial images. The approach enables us to generate 3D realisations of the microstructure.

This paper presents a method to simulate high density non overlapping sphere structures with anisotropy and heterogeneity.

The studied material is a 5xxx aluminium alloys containing 2 types of intermetallic particles : Alx(Fe;Mn) and Mg2Si. It is usually used in car industry as reinforcement pieces or in packaging industry, such as bottle liquid box lid. Scanning electronic microscope coupled with EDX analysis shows complex shapes of intermetallic particles. The particle shape is obtained during the solidification of alloys. Particles fill vacant spaces between aluminium grains. Therefore final sheet properties depend on intermetallic particles shapes and notably on the matrix-particle interface properties. The goal of the present study is to classify intermetallic particles versus their shapes using local curvature information. The aluminium alloys sample is observed by X ray micro tomography performed at the ESRF. Three dimensional images are segmented, and intermetallic particles are identified in a data base. Each particle is stored as a set of voxels. The surface of each particle is meshed by a marching cubes triangular meshing with the software Amira©. A simplification of the surface is performed by an algorithm contracting the edges. Finally, principal curvatures: kmin and kmax are estimated by Amira© on each facet centre of the mesh. From the full intermetallic population, the bivariate distribution of kmin and kmax is estimated. The obtained graph kmin ¡kmax shows geometrical properties of interface portions of the surface of particles. A factorial correspondence analysis is performed to summarize the information on all intermetallic particles. In the obtained subspace, particles are classified into five shape families, in relation with their interface geometrical properties.

Composite materials containing aggregates of spherical inclusions are studied from 3D images obtained by X-ray microtomography. Using two point statistics in different directions, and the empirical distribution of orientations of pairs of inclusions, interesting details concerning the anisotropy of the distribution of inclusions are obtained and are related to the method of construction for these materials. Some 3D morphological properties, available on the 3D images, give new information on the shape and the distribution of aggregates: tortuosity of shortest paths in the matrix, local volume fraction, geodesic distance function, local histograms of numbers of objects.

The percolation threshold ρc of Boolean models of cylinders with their axis parallel to a given direction is studied by means of simulations. An efficient method of construction of percolating connected components was developed, and is applied to one or two scales Boolean model, in order to simulate the presence of aggregates. The invariance of the percolation threshold with respect to affine transformations in the common direction of the axis of cylinders is approximately satisfied on simulations. The prediction of the model (ρc close to 0.16) is consistent with experimental measurements on plasma spray coatings, which motivated this study.

The present paper develops a general methodology for the morphological segmentation of hyperspectral images, i.e., with an important number of channels. This approach, based on watershed, is composed of a spectral classification to obtain the markers and a vectorial gradient which gives the spatial information. Several alternative gradients are adapted to the different hyperspectral functions. Data reduction is performed either by Factor Analysis or by model fitting. Image segmentation is done on different spaces: factor space, parameters space, etc. On all these spaces the spatial/spectral segmentation approach is applied, leading to relevant results on the image.

On a microscopic scale, a pyrotechnic material is made of a polymer matrix containing grains with different sizes and shapes. Its physical behaviour can be predicted by homogenization. Information about the morphology of the grains can be obtained by different ways. One of these ways is 3D image processing. This has been made easier by the use of a new imaging technique, the microtomography, allowing fast threedimensional reconstruction and processing. In this paper images of two granular materials are segmented by means of 3D mathematicalmorphology algorithms, based on a multi-scale extraction of markers for watershed segmentation.

We use a method to estimate local orientations in the n-dimensional space from the covariance matrix of the gradient, which can be implemented either in the image space or in the Fourier space. In a second step, two methods allow us to detect sudden changes of orientation in images. The first one uses an index of confidence of the estimated orientation, and the second one the detection of minima of scalar products in a neighbourhood. This is illustrated on 2D Transmission Electrons Microscope images of cellulose cryofracture (to display the organisation of cellulose whiskers and the points of germination), and to 3D images of a TA6V alloy (lamellar microstructure) obtained by microtomography.

Basic random structure models (random sets and random function models) are introduced for the simulation of images and of microstructures. Their implementation requires the use of image analysis tools defined in mathematical morphology.They can be used for solving problems of physics of random media.

Corrigendum of A MIXED BOOLEAN AND DEPOSIT MODEL FOR THE MODELING OF METAL PIGMENTS IN PAINT LAYERS (2015 IMAGE ANAL STEREOL 34:125–34).