Cross-Correlation Function for Accurate Reconstruction of Heterogeneous Media

Pejman Tahmasebi1 and Muhammad Sahimi2,*1Department of Energy Resources Engineering, Stanford University, Stanford, California 94305-2220, USA2Mork Family Department of Chemical Engineering & Materials Science, University of Southern California,

Los Angeles, California 90089-1211, USA(Received 5 October 2012; published 14 February 2013)

Porous media, heterogeneous materials, and biological tissues are examples of ubiquitous disordered

systems, the understanding of which and any physical phenomenon in them entails having an accurate

model. We show that a new reconstruction method based on a cross-correlation function and a one-

dimensional raster path provides an accurate description of a wide variety of such materials and media.

The reconstruction uses a single 2D slice of data to reconstruct an entire 3D medium. Seventeen examples

are reconstructed accurately, as indicated by two connectivity functions that we compute for them. The

reconstruction method may be used for both unconditioned and conditioned problems, and is highly

efficient computationally.

DOI: 10.1103/PhysRevLett.110.078002 PACS numbers: 81.05.Rm, 02.70.-c, 05.20.-y

Disordered multiphase media, ranging from porousmaterials to biological tissues, are ubiquitous and theirunderstanding is of fundamental importance [1,2]. Hence,their modeling has been a long-standing problem. Inprinciple, one needs an infinite set of n-point correlationfunctions in order to obtain a complete statistical charac-terization of such disordered media. Chief among suchfunctions is [1,2] the standard n-point correlation function[1–3] Snðx1;x2; � � � ;xnÞ, the probability of locating npoints at positions x1;x2; � � � ;xn, all in one phase. But,measuring Sn for n � 5 is impractical, while it is a formi-dable challenge for n ¼ 3 and 4, although [4] S3 and S4 donot provide much more information beyond what S2 does.Describing a disordered medium based solely on S2 is notaccurate [4,5]. Thus, a long-standing problem has been thedevelopment of correlation functions that not only can bemeasured, but also provide an accurate description of dis-ordered media.

In this Letter we introduce a cross-correlation function(CCF) that is shown to provide an accurate description ofdisordered media. To demonstrate this we use an inversemethod, usually referred to as the reconstruction technique[1,2,6–10]: given some data for a target system (TS), onetries to construct a model for it that closely matches thedata. Complete reconstruction is impractical. Thus, onegenerates an ensemble of realizations to compute the per-tinent properties. We incorporate the CCF in a new recon-struction algorithm that is different from those of the past,and produces accurate realizations of the TS, even if it ishighly complex. The reconstruction does not use computa-tionally intensive methods, such as simulated annealing[11], which have been used in the past reconstructionmethods. As such, it is highly efficient computationally.

Let G represent the computational grid used in thereconstruction. The template (blocks) of G by whichthe TS is reconstructed is denoted by T, while DTðuÞ is

the data event at position u in T. We use the data eventbecause during the reconstruction DT changes. Finally, Orepresents the overlap regions between neighboring tem-plates. We describe the CCF and the reconstruction methodfor two-dimensional (2D) media; the extension to 3Dmedia is explained in the Supplemental Material [12].Let TS(x, y) represent the datum at point (x, y) of a TSof size Lx � Ly, with x 2 f0; � � � ; Lx � 1g and y 2f0; � � � ; Ly � 1g. Examining the TS, one focuses on a por-

tion DTðuÞ of size ‘x � ‘y and regenerates it based on the

data, such that it matches the corresponding portion in theTS. We introduce a CCF to quantify the similarity betweenthe TS and DT ,

Cði; jÞ ¼ X‘x�1

x¼0

X‘y�1

y¼0

TSðxþ i; yþ jÞDTðx; yÞ; (1)

with 0 � i < Lx þ ‘x � 1 and 0 � j < Ly þ ‘y � 1.

Equation (1) indicates that the desired position of (i, j)—the best match with the TS—is one that maximizes Cði; jÞ.If the size of the TS is not too large, all the computationsare carried out in the spatial domain. When the TS is verylarge, calculations in the Fourier space result in significantsavings in the computations [13].Consider, first, unconditional reconstruction—one in

which the algorithm must not honor exactly specific hard(quantitative) data (HD) in the TS. G is partitioned intoblocks of size Lx � Ly. Between every two neighboring

blocks is an O region of size ‘x � ‘y representing DT ,

where ‘xð‘yÞ ¼ LxðLyÞ and ‘yð‘xÞ � LyðLxÞ, if the O

region is between two templates that are neighbors inthe x (y) direction. Suppose, for example, that the 1Draster path is along the vertical (y) direction; see theSupplemental Material [12] for an illustration. The algo-rithm begins at the path’s origin inG—the leftmost bottomtemplate in G—and moves along it. A realization of the

PRL 110, 078002 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

15 FEBRUARY 2013

0031-9007=13=110(7)=078002(5) 078002-1 � 2013 American Physical Society

disorder in the TS, equal in size to T, is generated andinserted into the first block, part of which represents the Oregion with the next block. The purpose of the O region isto preserve the continuity near the common boundarybetween two blocks. Thus, one generates the best realiza-tion of the disorder for the next neighboring block alongthe path whose bottom O section makes a seamlesstransition between the two. If several realizations of thedisorder have the same degree of similarity with O, one ofthem is selected randomly. The procedure continues untilreconstruction of the first vertical column of the raster pathis finished. The algorithm then moves to the bottom blockof the next vertical column of G and uses the same proce-dure, taking into account itsO region with the neighboringblock on the left. The next block is more complex as it hastwo O regions, at the bottom and on the left. Thus, thereconstruction proceeds as follows. (i) Since we do notknow a priori the location of maximum of the CCF, we use1=C and set a threshold 0 � � � 1. If � ¼ 0, the similaritybetween the O regions and the corresponding portions inthe TS is perfect, whereas � > 0 generates an ensemble ofrealizations that do not match the TS exactly. After somepreliminary simulations we used, � ¼ 0:2. (ii) Generate forthe first block ofG along the raster path a realization of thedisorder in the TS, based on and constrained by the statis-tics of the TS (volume fractions of the phases), or bysampling TS. (iii) Compute the CCF between the O andthe entire TS. (iv) If 1=C< �, the realization is accepted. Ifnot, other realizations are made, or � is increased some-what, until 1=C< �. Typically, one generates 30–50 real-izations in order to satisfy the similarity criterion. (v) Fillup the next block with a realization of the disorder, identifythe new O with the next block to be reconstructed, andrepeat (ii)–(iv). The entire procedure continues until G iscompletely reconstructed.

In conditional reconstruction G contains some HD thatmust be honored exactly, due to which the data event DT isthe entire block, not just the O regions. Thus, conditionalreconstruction is a two-step process. First, one identifiesthe realizations that honor the HD, and then determines thematching DT . The algorithm first computes C and checkswhether 1=C< �. Those that honor the HD are identified,one of which is selected randomly and inserted in the T.

A problem may arise if T is very large, or if a realizationdoes not honor the HD, leading to discontinuities andfailure of the reconstruction. In such a rare case one mayincrease the threshold � to obtain new realizations that,although they may have significant differences with the Oregions, they are still acceptable. � is increased until aproper data eventDT is identified. But, doing so also insertsincorrect structures in the ensemble. To overcome theproblem, we use an adaptive algorithm described below.The algorithm’s parameters are the templates’ size and

the threshold �. Our study indicates that the templates’ sizeis the most important factor, which depends on the hetero-geneity of the TS; for a relatively homogeneous TS acoarse grid suffices, whereas highly disordered mediarequire grids with small blocks. We have developed amethod for determining the optimal block size based onthe Shannon entropy S [14]. The TS is partitioned intolarge coarse blocks, for each of which S is computed,where S ¼ P

ni¼1 pi lnpi, with pi being the probability of

having states i in the TS, and n the number of pixels inthe block. pi is given by pi ¼ histogram of samplei=length of the sample. The entropies are normalized by

their maximum value, and denoted by S. A quadtree-

based partitioning based on S is then used. If S � 1,the coarse block is homogeneous enough that it need not

be partitioned any further. But, if S is close to 1, thecoarse block is divided into 4 (8 in 3D) blocks, theprocedure is repeated for the new smaller blocks, and itis continued until the optimal block size is determined.The optimal block size for the binary images of the TSanalyzed here was 20� 20 with an O region of size20� 4, while the corresponding sizes for the continuousTS were 25� 25 and 25� 5.To quantify the accuracy of the reconstruction, we

compute two connectivity functions for the TS and itsreconstruction. One [4,15] is the two-point cluster (TPC)function C2ðrÞ, the probability of finding two points sepa-rated by a distance r in the same cluster of the phase ofinterest. The second measure is the multiplepoint connec-tivity (MPC) function [16], the probability pðr;mÞ ofhaving a sequence of m points in a phase in a direction r.

If an indicator function IðiÞðuÞ is defined by IðiÞðuÞ ¼ 1, if

u 2 phase i, and IðiÞ ¼ 0 otherwise, then

pðr;mÞ ¼ ProbfIðiÞðuÞ ¼ 1; IðiÞðuþ rÞ ¼ 1; . . . ; IðiÞðuþmrÞ ¼ 1g: (2)

Reproducing pðr;mÞ represents a most stringent test of themethod’s accuracy. We used m ¼ 100 and, for conve-nience, delete m and denote the MPC function by pðrÞ.We note that the previous methods that used optimizationtechniques based on, for example, simulated annealing,become computationally intractable if the number of datapoints is very large, whereas the computational cost of the

present algorithm is very low, even for a very large 3Dsystem.A most difficult problem in reconstruction is using a

single 2D section of an entire 3D medium in order toreconstruct it. Such 2D sections are obtained by, for ex-ample, x-ray computed tomography, whereas 3D imagesare difficult and costly to produce. Thus, an algorithm that

PRL 110, 078002 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

15 FEBRUARY 2013

078002-2

utilizes 2D data to reconstruct a 3D medium is a powerfultool for studying disordered materials and media. We showthat our reconstruction method is such an algorithm, hencedemonstrating that the amount of the data used in thealgorithm is not the crucial factor for its accuracy.

As the first example, we reconstruct a sample of Bereasandstone of porosity 0.22, whose complete 3D digitizedimage is available [17]. One 2D slice is used for recon-structing the next layer and adding it to the medium underreconstruction. Since the first layer represents actual harddata (HD) taken from the 3D image, the reconstruction isconditional. The newly reconstructed layer acts as a sourceof the HD for the next layer to be reconstructed. Theimportant questions are the following. (i) How shouldone select the HD in each layer? (ii) How many HD pointsfrom the 2D section should one select for reconstructingthe next 2D section? If too much information is used, thenext layer will be unrealistically similar to the current one,whereas if the selected HD is too little, the connectivitywill be lost. (iii) Where in the 2D slice should one pick theHD? To address these we proceed as follows.

We first use the quadtree partitioning and the Shannonentropy to superimpose a nonuniform grid on the 2D slice.The smallest cell in the 2D grid is of size 3� 3, fromwhich a single data point at the center is used, while fromall other larger cells 2%–5% of the data are selected atrandom. Because the HD must be honored exactly, theconnectivity is preserved automatically. Every time a 2Dslice is reconstructed and added to the system, it is used inthe same way for reconstructing the next slice.

The single 2D section of the 3D image of Berea sand-stone that we used had a size of 200� 200 pixels, repre-senting 2% of the total data for the 3D sample. A total of199 2D sections were reconstructed and added to theinitially selected 2D section. Figure 1 presents the con-nectivity functions C2ðrÞ and pðrÞ, computed for both theBerea sandstone and its reconstruction. The agreement isexcellent. The total computation time for the reconstruc-tion was about 500 CPU seconds.Next, we generated a synthetic 3D heterogeneous porous

medium on a 2003 grid using geostatistical techniques [18];see Fig. 2. The local permeabilities vary by 3 orders ofmagnitude.We used one 200� 200 slice at the grid’s centerand reconstructed the 3D porous medium. The results areshown in Fig. 2. The agreement is again excellent.The next 2D-to-3D reconstruction example that we

present is a granular open-cell copper foam that contains

FIG. 2 (color online). The TS is a 3D heterogeneous porousmedium. The color bar indicates the local porosities.

FIG. 3. The TS is a 3D sample of copper foam.

FIG. 1 (color online). The target system (TS) is a 3D sample ofBerea sandstone. Shown is the comparison between the con-nectivity functions C2ðrÞ and pðrÞ for both TS and the recon-structed medium.

PRL 110, 078002 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

15 FEBRUARY 2013

078002-3

about 800 cells with a porosity of 0.33. The size of itsbinary image is 1283 pixels, and we used a 2D slice ofsize 128� 128 to reconstruct the entire image. Figure 3presents the results. Once again, the agreement is excellent.

It is useful to make a comparison between the perform-ance of the present reconstruction and that of an accuratemethod developed previously. Thus, we reconstructed a2D piece of concrete [19] with nanometer-sized pores andcentimeter-sized aggregates, using a binarized image of across section of the material with a linear size of 170 pixels.The results are shown in Fig. 4. We also computed the TPCand MPC functions for another reconstruction of the samematerial due to Jiao et al. [4]. Whereas the CCF reproducespðrÞ very accurately, Jiao et al.’s does not, because theirs isbased on C2ðrÞ, a radially averaged TPC function thatcontains much less microstructural information than theCCF. (A recently proposed [20] dilation or erosion proce-dure using a generalizedC2ðrÞ yields a better reconstructionof the same sample.) Depending on the template size, the2D CCF-based reconstruction might use more data than themethod used by Jiao et al. and, thus, one might argue that itmust produce more accurate results. Even if true, as wedemonstrate for the 2D-to-3D reconstructions, it is not theprimary factor for the method’s accuracy. The amount ofthe data used is based on the templates’ size. If the templatesize is much smaller than the TS (as is the case here), thenthe amount of data used is also small.

The last example is a digitized realization of an equilib-rium distribution of equal-sized hard spheres [4], generatedby the Metropolis Monte Carlo method, with particlevolume fraction of 0.446 and a linear size of 100 pixels.Figure 5 presents the results and compares them with theTPC and MPC functions that we also computed for another

reconstruction of the sample, due to Jiao et al. [4]. Whileboth reconstructions produce the C2ðrÞ function so accu-rately that it is nearly impossible to distinguish them, theCCF-based reconstructed sample also reproduces pðrÞ veryaccurately.We emphasize a few important points. (i) The recon-

struction of a continuous TS based onC2ðrÞ [4] is not only afunction of the separation distance r, but also of the valuesof the pixels, whereas in the CCF-based method the com-putations for a continuous TS and binary images remain thesame. (ii) Themethod’s aim is not reproducing explicitly allthe statistical properties of a TS, but reproducingmultiscalestructures stochastically, such that they reconstruct themultiple-point statistics. It is this feature of the methodthat generates accurate reconstruction. (iii) The method isparticularly accurate for highly heterogeneous media.In addition to the results presented here, twelve

other heterogeneous materials and media were studied.The results have been deposited as the SupplementalMaterial [12].Summarizing, the advantages of the approach, appli-

cable to any type of disordered material and media,are (i) low computational cost, but high accuracy,(ii) the ability to include hard data in the computations,(iii) the ability to do the computations in parallel mode,(iv) reconstructing a 3D medium based on a single 2Dsection, and (v) the applicability of the method to nonsta-tionary media, which are most prevalent in practice, buthave been very difficult to reconstruct. This aspect will bereported in the near future.Work at the USC was supported in part by the

Department of Energy. We thank M. Saadatfar for thedata of Fig. 3, and Yang Jiao and Sal Torquato forthe data of Fig. 5.

FIG. 4. The TS is the cross section of a binarized image ofconcrete microstructure. Comparison is made between the com-puted connectivity functions for the CCF-based reconstructionand that developed by Jiao et al. [4].

FIG. 5. The TS is a digitized realization of a hard-spherepacking with porosity of 0.446.

PRL 110, 078002 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

15 FEBRUARY 2013

078002-4

*moe@usc.edu[1] S. Torquato, Random Heterogeneous Materials (Springer,

New York, 2002).[2] M. Sahimi, Heterogeneous Materials (Springer, New York,

2003), Vols. I and II.[3] S. Torquato and G. Stell, J. Chem. Phys. 78, 3262 (1983).[4] Y. Jiao, F. H. Stillinger, and S. Torquato, Proc. Natl. Acad.

Sci. U.S.A. 106, 17634 (2009).[5] P. Debye and A.M. Bueche, J. Appl. Phys. 20, 518 (1949).[6] N. F. Berk, Phys. Rev. Lett. 58, 2718 (1987); P. Spanne,

J.-F. Thovert, C. J. Jacquin, W.B. Lindquist, K.W. Jones,and P.M. Adler, ibid. 73, 2001 (1994); A. P. Roberts andM. Teubner, Phys. Rev. E 51, 4141 (1995); B. Biswal,P.-E. Øren, R. J. Held, S. Bakke, and R. Hilfer, ibid. 75,061303 (2007).

[7] C. L. Y. Yeong and S. Torquato, Phys. Rev. E 57, 495(1998); 58, 224 (1998).

[8] H. Kumar, C. L. Briant, and W.A. Curtin, Mech. Mater.38, 818 (2006).

[9] Y. Jiao, F. H. Stillinger, and S. Torquato, Phys. Rev. E 76,031110 (2007); 77, 031135 (2008).

[10] H. Hamzehpour and M. Sahimi, Phys. Rev. E 74, 026308(2006); H. Hamzehpour, M. R. Rasaei, and M. Sahimi,ibid. 75, 056311 (2007).

[11] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Science220, 671 (1983).

[12] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.110.078002 for re-construction of 12 systems, including biological, ecologi-cal, and geological systems, and examples from materialsscience and cosmology.

[13] P. Tahmasebi and M. Sahimi, Comput. Geosci. (to bepublished).

[14] C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948).[15] S. Torquato, J. D. Beasley, and Y. C. Chiew, J. Chem. Phys.

88, 6540 (1988).[16] S. Krishna and A.G. Journel, Math. Geol. 35, 915

(2003).[17] http://www3.imperial.ac.uk/earthscienceandengineering/

research/perm/porescalemodeling.[18] M. Sahimi, Flow and Transport in Porous Media and

Fractured Rock (Wiley-VCH, Weinheim, 2011), 2nd ed.,Chap. 5.

[19] E. J. Garboczi and D. P. Bentz, Adv. Cem. Based Mater. 8,77 (1998).

[20] C. E. Zachary and S. Torquato, Phys. Rev. E 84, 056102(2011).

PRL 110, 078002 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

15 FEBRUARY 2013

078002-5