The Joshi-Quiblier-Adler (JQA) Approach

The basic idea of statistical reconstruction is to generate a three dimensional (3D) model of a random composite using only statistical information measured from a two dimensional (2D) image. Since 2D cross-sections are readily obtained by common experimental methods this provides a very attractive method of modeling porous and composite media. Quiblier [30] developed a method capable of producing a three-dimensional model with a specified volume fraction and two-point correlation function. The method was first studied in two dimensions by Joshi [24] and has been extended by Adler [1]. Thus it has been called the JQA model. We first give a summary of the procedure to demonstrate its equivalence to the single-cut GRF model discussed in the last section. First p and p ⁽²⁾_expt, are measured from an experimental image. The level-cut parameter $\beta$ is fixed by the volume fraction [Eqn. (4) with = - $\alpha=-\infty$]. Second g_expt(r_i) is obtained at the set of discrete points where p⁽²⁾_expt is measured by inverting Eqn. (5) with the left hand side set to p⁽²⁾_expt (r_i) and $\alpha=-\infty$ = - . To carry out the inversion the right hand side of Eqn. (5) is expanded as a series in powers of g;

$$p^{(2)}_{\rm expt}(r_i)=p^2+p(1-p)\sum_{m=0}^\infty C_m^2 [g_{\rm expt}(r_i)]^m \end{displaymath} img56.gif$$ (7)

where the coefficients C_m depend on certain integrals of Hermite polynomials [30]. The series converges very slowly for g near one, and Alder [1] has discussed the inversion procedure in this case, and possible situations where the equation has no solution. We note that the inversion can also be simply carried out by numerical integration of Eqn. (5) and a standard non-linear equation solver.

Next it is necessary to generate a GRF with field-field correlation function g_expt(r). Quibler's original formulation involved the solution of a very large system of non-linear equations for the terms of a convolution operator used in his definition of a GRF. Adler [1] simplified the procedure by reformulating the problem in terms of Fourier transforms, which is equivalent to a three-dimensional version [37] of Rice's [32] method for Gaussian processes in one dimension. In terms of the definition given in Eqn. (3) the inversion is equivalent to a numerical integration of P(k) = 2/ ₀ g(r)rk sin krdr, where g is known only at a discrete set of points. The inversion methods described above for g(r) do not guarantee that P(k) (or its equivalent in other formulations) is greater than zero. However Adler [1] found that if P(k) is negative at some points it is also small, and can be replaced with zero. Finally the GRF is thresholded in the usual way to obtain the reconstructed microstructure. Thus the JQA method produces a single-cut Gaussian random field, which we have termed model N(c=0).

In this paper we employ a different implementation of the JQA method [34]. At this stage we restrict attention to model N(c=0). First, the volume fraction of the model is set to that of an image: p_mod= p_expt. Second, the experimental two point correlation function is fitted by varying the morphological parameters of a given g(r) [Eqn. (2)] to minimize the non-linear least squares error

$$[Ep^{(2)}]^2 = \frac{\sum_{i=1}^{N_f} [p^{(2)}_{\rm {mod}}(r... ...m_{i=1}^{N_f} [p^{(2)}_{\rm {expt}}(r_i)-p_{\rm {expt}}^2]^2}.$$ (8)

where N_f is the number of experimental points to be fitted. Numerical integration is used to find p⁽²⁾_mod (r_i) [Eqn. (5)]. The minimization is very fast, but several starting points should be used as a check against local minima. Once the parameters of g(r) are known an analytic form of P(k) is used to generate the coefficients of the GRF (3). The reconstructed model is obtained by thresholding the random field as described above. Note that p⁽²⁾_mod (r_i) will not match p⁽²⁾_expt (r_i) exactly at each point as would be the case with Quiblier's procedure. However with this choice of g(r), P(k) is guaranteed to be positive. More general functional forms of g(r) can also be employed.

For isotropic materials p and p⁽²⁾(r) can be exactly measured from a two (or one) dimensional image. Hence the application of the JQA method results in a model which shares p, s_v and p⁽²⁾(r) with a real composite. The question is whether or not the model provides an accurate and useful representation of the original microstructure. In certain cases it appears to, in other it does not. First, predictions of transport properties (conductivity and permeability) obtained from reconstructed porous models under-estimate experimental and numerical data. Second, the percolation threshold (the volume fraction at which the pore space or inclusion phase is no longer macroscopically connected) of model N(c=0) is around 10% [37] for the model, but many materials exhibit lower thresholds [36]. Both points indicate that the pores (or inclusions) of the model are not sufficiently well connected to mimic many physical materials. Third, we can test the model by trying to reconstruct several of the distinct models defined in the previous section. The results are shown in Fig 1. Even though model N(c=0) is able to reproduce the correlation functions reasonably well, the reconstructions (i) do not in all cases appear to reproduce the original microstructure and (ii) look quite similar to one another. This indicates that irrespective of g(r), and the original image, model N(c=0) can only generate microstructures that are similar to those shown in the final row of Fig. 1.

Figure 1: The statistical reconstructions (bottom row) of four different two-dimensional images by adjusting the parameters (r_c, $\xi$, d and $\beta$ ) of model N(c=0) to fit the auto-correlation functions (middle row) of the original models (shown in the top row). The procedure used is very similar to that of Quiblier. The results suggest that model N(c=0) cannot mimic all types of microstructure, even though it can reproduce the two-point correlation functions of all the cases shown. The models in the top row are (from left to right), overlapping spheres, models N(c=1), I(c=1) and U(c=1) (see Sec. 2). The images are 128 x 128 pixels, and the length scale of the correlation functions shown in the 2nd row extends to 32 pixels.

Quiblier [30] has suggested that the ability of the method to generate a reasonable model depends on the validity of the hypothesis that: all the necessary information about the morphology is contained in the auto-correlation function. Suppose this were true. Then since the JQA method is sufficiently general to re-produce all reasonable two-point correlation functions (see Fig. 1), it must also be able to generate all types of morphology. The discussion above (and Fig. 1) indicates that model N(c=0) can only produce a limited class of microstructure and therefore that the hypothesis is false. This does not mean that the the method cannot produce useful models, but we argue it will do so only when the original material is approximately contained in the same limited class. If it is not, then a model from a different class needs to be considered. We discuss this issue in the following section.