Boolean Random Sets Research Articles

Mean covariogram of cylinders and applications to Boolean random sets

This work focuses on the variance properties of isotropic Boolean random sets containing randomly-oriented cylinders with circular cross-section. Emphasis is put on cylinders with large aspect ratios, of the oblate and prolate types. A link is established between the powerlaw decay of the covariance function and the variance of the estimates of the volume fraction of cylinders. The covariance and integral range of the Boolean mixtures are expressed in terms of the orientation-averaged covari-ogram of cylinders, for which exact analytical formulas and approximate expressions are provided.

Indicator Variogram Models: Do We Have Much Choice?

Many variogram (or covariance) models that are valid—or realizable—models of Gaussian random functions are not realizable indicator variogram (or covariance) models. Unfortunately there is no known necessary and sufficient condition for a function to be the indicator variogram of a random set. Necessary conditions can be easily obtained for the behavior at the origin or at large distance. The power, Gaussian, cubic or cardinal-sine models do not fulfill these conditions and are therefore not realizable. These considerations are illustrated by a Monte Carlo simulation demonstrating nonrealizability over some very simple three-point configurations in two or three dimensions. No definitive result has been obtained about the spherical model. Among the commonly used models for Gaussian variables, only the exponential appears to be a realizable indicator variogram model in all dimensions. It can be associated with a mosaic, a Boolean or a truncated Gaussian random set. In one dimension, the exponential indicator model is closely associated with continuous-time Markov chains, which can also lead to more variogram models such as the damped oscillation model. One-dimensional random sets can also be derived from renewal processes, or mosaic models associated with such processes. This provides an interesting link between the geostatistical formalism, focused mostly on two-point statistics, and the approach of quantitative sedimentologists who compute the probability distribution function of the thickness of different geological facies. The last part of the paper presents three approaches for obtaining new realizable indicator variogram models in three dimensions. One approach consists of combining existing realizable models. Other approaches are based on the formalism of Boolean random sets and truncated Gaussian functions.

Morphological modelling of three-phase microstructures of anode layers using SEM images.

A general method is proposed to model 3D microstructures representative of three-phases anode layers used in fuel cells. The models are based on SEM images of cells with varying morphologies. The materials are first characterized using three morphological measurements: (cross-)covariances, granulometry and linear erosion. They are measured on segmented SEM images, for each of the three phases. Second, a generic model for three-phases materials is proposed. The model is based on two independent underlying random sets which are otherwise arbitrary. The validity of this model is verified using the cross-covariance functions of the various phases. In a third step, several types of Boolean random sets and plurigaussian models are considered for the unknown underlying random sets. Overall, good agreement is found between the SEM images and three-phases models based on plurigaussian random sets, for all morphological measurements considered in the present work: covariances, granulometry and linear erosion. The spatial distribution and shapes of the phases produced by the plurigaussian model are visually very close to the real material. Furthermore, the proposed models require no numerical optimization and are straightforward to generate using the covariance functions measured on the SEM images.

Stokes Flow Through a Boolean Model of Spheres: Representative Volume Element

The Stokes flow is numerically computed in porous media based on 3D Boolean random sets of spheres. Two configurations are investigated in which the fluid flows inside the spheres or in the complementary set of the spheres. Full-field computations are carried out using the Fourier method of Wiegmann (2007). The latter is applied to large system sizes representative of the microstructure. The overall permeability of the two models as well as the representative volume element are estimated as a function of the pore volume fraction. We give numerical estimates for the asymptotic behavior of the permeability in the dilute limit for the solid phase, and close to the percolation threshold of the pores. FFT maps of the velocity field are presented, for increasing values of the pore volume fraction. The patterns of the local velocity field is analyzed using various morphological criteria. The tortuosity of the streamlines is found to be much higher than the geometrical tortuosity, for both models. The histograms of the velocity field are computed at increasing pore volume fraction. The covariance of orientation is used to characterize the spatial correlation of the velocity field.

Morphology and effective properties of multi-scale random sets: A review

Complex microstructures in materials often involve multi-scale heterogeneous textures, modeled by random sets derived from Mathematical Morphology. Our approach starts from 2D or 3D images; a complete morphological characterization by image analysis is performed, and used for the identification of a model of random structure. Morphological models enter into the prediction of effective properties by estimation, bounds, or from numerical simulations. Simulations of realistic microstructures are introduced in a numerical solver to compute appropriate fields (electric, elastic, …) and to estimate the effective properties by numerical homogenization, accounting for scale dependent statistical fluctuations of the fields. Our approach is illustrated by various examples of multi-scale models: Boolean random sets based on Cox point processes and various random grains (spheres, cylinders), showing a very low percolation threshold, and therefore a high conductivity or high elastic moduli for a low volume fraction of a second phase. Multi-scale iterations of random media provide another source of morphologies with interesting overall properties.

Multi-scale random sets: from morphology to effective properties and to fracture statistics

Complex microstructures in materials often involve multi-scale heterogeneous textures, modelled by random sets derived from Mathematical Morphology. Starting from 2D or 3D images, a complete morphological characterization by image analysis is performed, and used for the identification of a model of random structure. From morphological models, simulations of realistic microstructures are introduced in a numerical solver to compute appropriate fields (electric, elastic stress or strain, …) and to estimate the effective properties by numerical homogenization, accounting for scale dependent statistical fluctuations of the fields. Our approach is illustrated by various examples of multi-scale models: Boolean random sets based on Cox point processes and various random grains (spheres, cylinders), showing a very low percolation threshold, and therefore a high conductivity or high elastic moduli for a low volume fraction of a second phase. Multiscale Cox point processes are also a source of instructive models of fracture statistics, such as multiscale weakest link models.

On the Existence of Mosaic and Indicator Random Fields with Spherical, Circular, and Triangular Variograms

This paper focuses on two specific families of stationary random fields (namely, mosaic and indicator random fields) and examines the question of whether the variogram of a member of these two families could be spherical, circular, or triangular. It is shown that there are such examples in one-dimensional spaces, while there are counterexamples in higher-dimensional spaces for mosaic random fields, indicators of Boolean random sets and indicators of excursion sets of Gaussian random fields. Further results concerning the spherical plus nugget and circular plus nugget variograms are provided.

Multi Scale Random Models of Complex Microstructures

Many nanocomposite materials are obtained by dispersing a charge in a matrix. Due to the conditions of mixing, the arrangement of the charge usually presents some heterogeneity at different scales. In order to predict the effective properties of such composites (like the dielectric permittivity or the elastic moduli), it is necessary to know the properties of the two components (charge and matrix), and their spatial distribution. To fulfil this project, we developed a general methodology in several steps: the morphology is summarized by multi-scale random models accounting for the heterogeneous distribution of aggregates. The identification of models is made from image analysis. It is then used for the prediction of effective properties by estimation, or by numerical simulations. Our approach is illustrated by various examples of multi-scale models: Boolean random sets based on Cox point processes and various random grains (spheres, cylinders), showing a very low percolation threshold and therefore a high conductivity or elastic moduli for a low charge content; multi-scale iterations of random media.

Modelling multi-scale microstructures with combined Boolean random sets: A practical contribution

Boolean random sets are versatile tools to match morphological and topological properties of real structures of materials and particulate systems. Moreover, they can be combined in any number of ways to produce an even wider range of structures that cover a range of scales of microstructures through intersection and union. Based on well-established theory of Boolean random sets, this work provides scientists and engineers with simple and readily applicable results for matching combinations of Boolean random sets to observed microstructures. Once calibrated, such models yield straightforward three-dimensional simulation of materials, a powerful aid for investigating microstructure property relationships. Application of the proposed results to a real case situation yield convincing realisations of the observed microstructure in two and three dimensions.

Elastic behavior of composites containing Boolean random sets of inhomogeneities

The overall mechanical response as well as strain and stress field statistics of an heterogeneous material made of two randomly distributed, linear elastic phases, are investigated numerically. The Boolean model of spheres is used to generate microstructures consisting of either porous or rigid inclusions, at any volume fraction of the phases. Stress and strain field integral ranges, or equivalently the representative volume element, are computed and linked to features of the field statistics, and to the microstructure geometry.

Regression models for Boolean random sets

In this paper we consider the regression problem for random sets of the Boolean-model type. Regression modeling of the Boolean random sets using some explanatory variables are classified according to the type of these variables as propagation, growth or propagation-growth models. The maximum likelihood estimation of the parameters for the propagation model is explained in detail for some specific link functions using three methods. These three methods of estimation are also compared in a simulation study.

The Influence of Spatial Distributions of Inhomogeneities On Effective Dielectric Properties of Composite Materials (Effective Field Approach) - Abstract

The version of the effective field method developed in the previous work of the authors is applied to investigation of the influence of spatial distributions of inhomogeneities on the effective electrodynamic properties of the composites with sets of isolated spherical inclusions. Propagation of monochromatic electromagnetic waves through composites with generalized Boolean random sets of inclusions and with periodic distributions of inclusions is studied. The comparison of the one and two scale Boolean models discovers the influence of increasing the order in spatial positions of inclusions on the properties of the mean wave field in the composite. On the one hand there appear new types of waves in the medium, on the other hand the attenuation of these waves decrease as the order in the system increases. For composites with regular lattices of spherical inclusions the method gives narrow bands of attenuation in the vicinities of Bragg's frequencies. Outside these bands electromagnetic waves propagate through the medium without attenuation. It is shown that the effective field method allows us to describe the main features of the wave propagation phenomena in the matrix-inclusion composites.

Optimal nonlinear filter for signal-union-noise and runlength analysis in the directional one-dimensional discrete Boolean random set model

A one-dimensional discrete Boolean model is a random process on the discrete line where random-length line segments are positioned according to the outcomes of a Bernoulli process. Points on the discrete line are either covered or left uncovered by a realization of the process. An observation of the process consists of runs of covered and not-covered points, called black and white runlengths, respectively. The black and white runlengths form an alternating sequence of independent random variables. We show how a Boolean model is completely determined by probability distributions of these random variables by giving explicit formulas linking the marking probability of the Bernoulli process and the segment length distribution with the runlength distributions. The black runlength density is expressed recursively in terms of the marking probability and segment length distribution and white runlengths are shown to have a geometric probability law. This equivalence allows us to do runlength analysis of the one-dimensional Boolean model. Filtering for the Boolean model can also be done via the runlengths. The windowed optimal minimum-mean-absolute-error filter for union noise is computed as the binary conditional expectation. It is expressible as a function of the observed black runlengths.

Recursive maximum-likelihood estimation in the one-dimensional discrete Boolean random set model

The exact probability density for a windowed observation of a discrete one-dimensional Boolean process having convex grains is found via recursive probability expressions. This observation density is used as the likelihood function for the process and numerically yields the maximum-likelihood estimator for the process intensity and the parameters governing the distribution of the grain lengths. The only restriction on the derivation is that the length distribution not be too heavy tailed. Maximum-likelihood estimation is applied in the cases of uniformly and Poisson distributed lengths. The entire approach applies to unions of independent Boolean processes.

A Geometric Model of Marbling in Beef Longissimus Dorsi

The Boolean random set model with isotropic, convex primary grain was investigated as a means of describing the spatial distribution of fat in beef longissimus dorsi. Estimates of contact distribution functions (CDF) for 10 images were compared to theoretical values. Results indicated that CDF estimates were not statistically different from what was expected given the assumption of the Boolean model was correct. Estimates of model parameters assuming circular primary grains with random radius tended to be different from those estimated without constraining their shape. The assumption of circular primary grains was concluded to be incorrect.

VISUAL CHARACTERIZATION OF MARBLING IN BEEF RIBEYES AND ITS RELATIONSHIP TO TASTE R\RAMETERS

ABSTRACT The prediction of human assigned marbling scores and taste panel evaluations of meat palatability using information about the visual appearance of fat in beef carcass ribeyes was investigated. The distribution of fat within the ribeye was modeled as a realization of a Boolean random set. Model parameters estimated from seventy carcasses, representing six USDA marbling classes, were used as pattems in a hierarchical clustering scheme. Results of the clustering analysis showed that two distinct marbling classes could be distinguished with an acceptable error rate. These results were contrasted with a classification scheme based solely on the area of marbling within the ribeye. The comparison showed that only slight improvements in classification rate were realized by using the more complex model of fat distribution. In addition, very little correspondence was found between the visual appearance of the marbling, as encapsulated in the Boolean model, and taste parameters. It was concluded that the human grader used in the experiments could resolve two levels of marbling, and the differentiation was not significantly influenced by fat distribution.