Chord Length Distribution Function Research Articles

Chord-length distribution functions and Rice formulae. Application to random media

We consider a stationary and isotropic bi-phasic (pore and solid) medium, draw many lines through it, and see each line as a one-dimensional level-cut process with value 0 or 1 according to whether a regular stationary process X is less or greater than a given level. The intervals corresponding to the points at which X is in a given phase are named chords. We are interested in obtaining information on the chord-length distribution functions. Working with the Palm probability measure and using level crossings techniques, in particular Rice methods, we can obtain not only the exact analytical formula of the chord-length distribution function but also the joint distribution function of the lengths of two successive chords. Finally, we indicate some concrete applications for the computation of usual stereological parameters.

How fast can the chord length distribution decay?

The modeling of random bi-phasic, or porous, media has been, and still is, under active investigation by mathematicians, physicists, and physicians. In this paper we consider a thresholded random process X as a source of the two phases. The intervals when X is in a given phase, named chords, are the subject of interest. We focus on the study of the tails of the chord length distribution functions. In the literature concerned with real data, different types of tail behavior have been reported, among them exponential-like or power-like decay. We look for the link between the dependence structure of the underlying thresholded process X and the rate of decay of the chord length distribution. When the process X is a stationary Gaussian process, we relate the latter to the rate at which the covariance function of X decays at large lags. We show that exponential, or nearly exponential, decay of the tail of the distribution of the chord lengths is very common, perhaps surprisingly so.

How fast can the chord length distribution decay?

The modeling of random bi-phasic, or porous, media has been, and still is, under active investigation by mathematicians, physicists, and physicians. In this paper we consider a thresholded random process X as a source of the two phases. The intervals when X is in a given phase, named chords, are the subject of interest. We focus on the study of the tails of the chord length distribution functions. In the literature concerned with real data, different types of tail behavior have been reported, among them exponential-like or power-like decay. We look for the link between the dependence structure of the underlying thresholded process X and the rate of decay of the chord length distribution. When the process X is a stationary Gaussian process, we relate the latter to the rate at which the covariance function of X decays at large lags. We show that exponential, or nearly exponential, decay of the tail of the distribution of the chord lengths is very common, perhaps surprisingly so.

On the reconstruction of structural and functional properties in random heterogeneous media

Flow and transport in porous media is determined by its structure. Beside spatial correlation, especially the connectivity of heterogeneous conductivities is acknowledged to be a key factor. This has been demonstrated for well defined random fields having different topological properties. Yet, it remains an open question which morphological measures carry sufficient information to actually predict flow and transport in porous media. We analyze flow and transport in classical, two-dimensional random fields showing different topology and we determine a selection of structural characteristics including classical two-point statistics, chord-length distribution and Minkowski functions (four-point statistics) including the Euler number as a topological measure. Using the approach of simulated annealing for global optimization we generate analog random fields that are forced to reproduce one or several of theses structural characteristics. Finally we evaluate in how far the generated analogons reproduce the original flow and transport behavior as well as some more elaborate structural characteristics including percolation probabilities and the pair connectivity function. The results confirm that two-point statistics is insufficient to capture functional properties since it is not sensitive to connectivity. In contrast, the combination of Minkowski functions and chord length distributions carries sufficient information to reproduce the breakthrough curve of a conservative solute. Hence, global topology provided by the Euler number together with local clustering provided by the chord length distribution seems to be a powerful condensation of structural complexity with respect to functional properties.

Random copy of a segment within a convex domain

Under the assumption that S is a segment of the length l and D is a bounded, convex domain in the Euclidean plane ℝ2, the paper considers the randomly moving copy L of S, under the condition that it hits D. Denote by L| the length of L ∩ D. In the paper an elementary expression for the distribution function F L (x) of the random variable L| is obtained. Note that F L (x) can have a jump at the point l or can be a continuous function depending on l and the domainD. In particular, a relation between chord length distribution functions of D and F L (x) is given. Moreover, we derive explicit forms of F L (x) for the disk and regular n-gons with n = 3÷7.

Contact and Chord Length Distribution Functions of the Poisson-Voronoi Tessellation in High Dimensions

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

Contact and Chord Length Distribution Functions of the Poisson-Voronoi Tessellation in High Dimensions

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

The chord length distribution function for regular polygons

In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

The chord length distribution function for regular polygons

In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

Characterisation of granule structure and strength made in a high shear granulator

Results of a study of the influence of impeller speed on the strength, structure and morphology of granules produced in a type of high shear mixer granulators are reported. Calcium carbonate particles (Durcal 65) have been granulated in a Cyclomix with a capacity of 5 L. An aqueous solution of polyethylene glycol was used as the binder. The granules produced have been dried and their structure visualized using X-ray micro-tomography equipment, Nanotom, with a resolution of less than 1 μm. It is shown that the operation of the granulator at high impeller tip speeds produces granules with a higher strength and lower porosity than those produced at medium and low impeller speeds. Two different granule micro-structures and morphologies are produced at high and low impeller speeds. Structure descriptors such as phase volume fraction (as representative of porosity), chord length distribution and auto-correlation function (as indices of homogeneity of structure) are used to quantify the internal structure of granules in 3D, which in turn affects the granule strength.

The chord length transform and the segmentation of crossing fibres

Segmentation of crossing fibres is a complex problem of image processing. In the present paper, various solutions are presented basing on tools of morphological image processing. Two new image transforms are introduced--the lineal distance transform and the chord length transform. Both transforms are applied to two-dimensional images and their results are three-dimensional images. Thus, the segmentation problem originally formulated for crossing fibres observed in a two-dimensional image can be reformulated as a segmentation problem in a three-dimensional image. This can be solved by a segmentation in the three-dimensional image. Algorithms for the lineal distance transform and the chord length transform are given and their use in image analysis is demonstrated. Furthermore, the chord length distribution function of the foreground of a binary image can efficiently be estimated via the chord length transform.

Monte-Carlo simulation of the chord length distribution function across convex bodies, non-convex bodies and random media

The study of chord length distributions across various kinds of geometrical shapes, including stochastic mixtures, is a topic of great interest in many research fields ranging from ecology to neutronics. We have tried here to draw links between theoretical results and actual simulations for simple objects like disks, circular rings, spheres, hollow spheres, as well as for random media consisting of stochastic mono- or polydisperse spheres packing (three different packing algorithms were tested). The Monte Carlo simulations which were performed for simple objects fit perfectly theoretical formulas. For stochastic binary mixtures the simulations were still in rather good agreement with known analytical results.

Diffusion and Flow in Porous Domains Constructed Using Process-Based and Stochastic Techniques

Two model mesoporous materials, namely Vycor glass and alumina membrane made by compaction of spherical monosize alumina particles, are reconstructed and the transport properties (Knudsen diffusivity, molecular diffusivity and permeability) of the resulting 3-dimensional binary domains are investigated through computer simulations. For the alumina membrane, two reconstruction alternatives are used, a ballistic deposition of spherical particles (process-based approach) and a stochastic procedure based on the porosity and the two-point correlation function of the porous matrix. For Vycor glass, only the stochastic reconstruction technique is employed. The obtained 3-D samples are structurally characterized in terms of their two-point correlation and chord length distribution functions. This type of information along with the comparisons between computed and reported transport coefficients indicate that the random sphere pack obtained from the ballistic-deposition procedure represents quite well the porous structure of the alumina membrane while the stochastically reconstructed domain fails to do so. In contrast, for the Vycor glass it is shown that the stochastic reconstruction technique is sufficient for a faithful representation of the porous matrix. It is furthermore argued that in the case of Vycor the stochastic reconstruction technique constitutes a kind of process-based approach.

Lineal measures of clustering in overlapping particle systems.

The lineal-path function L(z) gives the probability of finding a line segment of length z entirely in one of the phases of a disordered multiphase medium. We develop an exact methodology to determine L(z) for the particle phase of systems of overlapping particles, thus providing a measure of particle clustering in this prototypical model of continuum percolation. We describe this procedure for systems of overlapping disks and spheres with a polydispersivity of sizes and for randomly aligned equal-sized overlapping squares. We also study the effect of polydispersivity on the range of the lineal-path function. We note that the lineal-path function L(z) is a rigorous lower bound on the two-point cluster function ${\mathit{C}}_{2}$(z), which is not available analytically for overlapping particle models for spatial dimension d\ensuremath{\geqslant}2. By evaluating the second derivative of L(z), we then evaluate the chord-length distribution function for the particle phase. Computer simulations that we perform are in excellent agreement with our theoretical results. \textcopyright{} 1996 The American Physical Society.

Structure and transport properties of a porous magnetic gel via x-ray microtomography.

A three-dimensional digitized image of a porous magnetic gel is determined by x-ray microtomographic techniques. The complex connected pore-space topology is quantitatively characterized by measuring a variety of statistical correlation functions, including the chord-length distribution function, the pore-size distribution function, and the lineal-path function. This structural information is then employed to estimate transport properties, such as the fluid permeability and trapping rate, of the gel. \textcopyright{} 1996 The American Physical Society.

Chord-length and free-path distribution functions for many-body systems

We study fundamental morphological descriptors of disordered media (e.g., heterogeneous materials, liquids, and amorphous solids): the chord-length distribution function p(z) and the free-path distribution function p(z,a). For concreteness, we will speak in the language of heterogeneous materials composed of two different materials or ‘‘phases.’’ The probability density function p(z) describes the distribution of chord lengths in the sample and is of great interest in stereology. For example, the first moment of p(z) is the ‘‘mean intercept length’’ or ‘‘mean chord length.’’ The chord-length distribution function is of importance in transport phenomena and problems involving ‘‘discrete free paths’’ of point particles (e.g., Knudsen diffusion and radiative transport). The free-path distribution function p(z,a) takes into account the finite size of a simple particle of radius a undergoing discrete free-path motion in the heterogeneous material and we show that it is actually the chord-length distribution function for the system in which the ‘‘pore space’’ is the space available to a finite-sized particle of radius a. Thus it is shown that p(z)=p(z,0). We demonstrate that the functions p(z) and p(z,a) are related to another fundamentally important morphological descriptor of disordered media, namely, the so-called lineal-path function L(z) studied by us in previous work [Phys. Rev. A 45, 922 (1992)]. The lineal path function gives the probability of finding a line segment of length z wholly in one of the ‘‘phases’’ when randomly thrown into the sample. We derive exact series representations of the chord-length and free-path distribution functions for systems of spheres with a polydispersivity in size in arbitrary dimension D. For the special case of spatially uncorrelated spheres (i.e., fully penetrable spheres) we evaluate exactly the aforementioned functions, the mean chord length, and the mean free path. We also obtain corresponding analytical formulas for the case of mutually impenetrable (i.e., spatially correlated) polydispersed spheres.

Chord-length distribution function for two-phase random media.

A statistical correlation function of basic importance in the study of two-phase random media (such as suspensions, porous media, and composites) is the chord-length distribution function p(z). We show that p(z) is related to another fundamentally important morphological descriptor studied by us previously, namely, the lineal-path function L(z), which gives the probability of finding a line segment of length z wholly in one of the phases when randomly thrown into the sample. We derive exact series representations of the chord-length distribution function for media comprised of spheres with a polydispersivity in size for arbitrary space dimension D. For the special case of spatially uncorrelated spheres (i.e., fully penetrable spheres), we determine exactly p(z) and the mean chord length ${\mathit{l}}_{\mathit{C}}$, the first moment of p(z). We also obtain corresponding formulas for the case of impenetrable (i.e., spatially correlated) polydispersed spheres.

A distribution function for double-bit upsets

It is suggested that the chord length distribution method could be useful for predicting double-bit upset rates in certain circumstances. A chord length distribution function for simultaneous path lengths in two parallelepipeds, applicable to a unidirectional flux, is derived. A proof of the system is outlined for the case under consideration.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Pore size distributions of foams from chord distributions of random lines: Mathematical inversion and computer simulation

A method is developed to determine pore size distributions of solid foams from optical or electron micrographs of fracture surfaces. The pores are approximated by spheres, and the chord lengths of random lines drawn through the micrographs are measured. Two methods are developed for smoothing the experimental chord length distribution: one by Fourier analysis and the other by curve-fitting to a multimodal modified Gaussian distribution function. Since the chord length distribution function (CDF) is an integral transform of the sphere diameter distribution function (SDF), a mathematical procedure is derived for inverting the CDF to obtain the SDF. Computer simulation using delta-function, unimodal, and bimodal SDFs are used to verify that the method gives very good approximations of the actual SDF, even when the data base includes fewer than 300 chord lengths. The method is applicable also to determining SDFs for spherical and sphere-like particles, and lends itself readily to automation.

The Calculation of the Chord Distribution Function

A direct method for the calculation of the chord length distribution function f(l), for special as well as more general geometries, is investigated here. The method is tested against well-known results for sphere and infinite slab. Then, a derivation of P(x), the escape probability function, for an infinite square prism is presented. This method differs from Carlvik’s in that, instead of currents into the surface, f(l) is derived here based on geometrical considerations alone. For P(x) we only assume a homogeneous source in fuel.