A Markov Random Field Approach for

Microstructure Synthesis

A. Kumara, V. Sundararaghavan*, M. DeGraefb, L. Nguyenb

*Aerospace Engineering, University of Michigan, Ann Arbor, MI, USAa Oak Ridge National Lab, Oak Ridge TN, USA.

b Materials Science and Engineering, Carnegie Mellon University, Pittsburgh.

Abstract

We test the notion that many microstructures have an underlying stationary prob-ability distribution. The stationary probability distribution is ubiquitous: we knowthat different windows taken from a polycrystalline microstructure generally ‘lookalike’. To enable computation of such a probability distribution, we represented mi-crostructures in the form of undirected probabilistic graphs called Markov RandomFields (MRFs). In the model, pixels take up integer or vector states and interactwith multiple neighbors over a window. Using this lattice structure, we developedalgorithms to sample the conditional probability density for the state of each pixelgiven the known states of its neighboring pixels. The sampling is performed usingreference experimental images. We artificially synthesized 2D microstructures usingthe sampled probabilities. We found that the statistical features such as grain sizedistribution and shape moment invariants closely match with those of the experi-mental images. The mechanical properties of the synthesized microstructures werecomputed using finite element method and were also found to match the experi-mental values.

Keywords: Reconstruction; Microstructure; Finite element analysis; Simulation; Prob-ability

1 Introduction

Microstructures are stochastic in nature and a single snapshot of the mi-crostructure does not give its complete variability. However, we know thatdifferent windows taken from a polycrystalline microstructure generally ’look

1 *Corresponding author: Associate Professor of Aerospace Engineering, Tel: +734-615-7242, Email: veeras@umich.edu

alike’. In mathematical terms, this amounts to the presence of a stationaryprobability distribution from which various microstructural snapshots are sam-pled. There are various ways of modeling this probability distribution indi-rectly. Feature–based algorithms have long been used that categorize variousmicrostructural snapshots based on a common set of underlying features, andgenerate new synthetic images with similar features [1–3]. These features couldinclude marginal histograms [1], multiresolution filter outputs (Gaussian [2]and wavelet [3] filters) and point probability functions (eg. autocorrelationfunction) [4]. These methods are good at capturing the global features of theimage, however local information in the form of per–pixel data is lost. Thus,features such as grain boundaries are smeared out when reconstructing poly-crystalline structures [3].

Alternatively, one could start with sampling the conditional probability den-sity for the state of a pixel given the known states of its neighboring pixels[5–8] using reference 2D or 3D experimental images. If only the nearest neigh-bors are chosen, this amounts to sampling from a Ising–type model [9,10]. Forgeneral microstructures, the correlation lengths can span several pixels and alarger neighbor window may be needed [4]. In this work, we employ generalizedIsing models called Markov Random Fields (MRFs) to model the probabilitydistribution. While in Ising models, a lattice is constructed with pixels (withbinary states) interacting with its nearest neighbors, in MRFs, pixels take upinteger or vector states and interact with multiple neighbors over a window.The sampling of conditional probability of a pixel given the states of its knownneighbors is based on Claude Shannon’s generalized Markov chain [11]. In theone dimensional problem, a set of consecutive pixels is used as a template todetermine the probability distribution function (PDF) of the next pixel. Sun-dararaghavan (2014) [12] used this approach for synthesizing 3D microstruc-tures from 2D micrographs. Here, pixel variables in a 3D microstructure wereoptimized so as to match the conditional probability densities on orthogonal2D sections with the reference images. In this work, an alternate approach isemployed where microstructures are grown layer–by–layer from a small seedimage (3x3 pixels) taken randomly from the experimental micrograph. The al-gorithm first finds all windows in an experimental micrograph that are similarto an unknown pixel’s neighborhood window. One of these matching windowsis chosen and its center pixel is taken to be the newly synthesized pixel.

We use this method to generate polycrystalline microstructures. Voronoi con-struction has been typically used for synthesis of such microstructures [13–16].However, Voronoi constructions are largely an idealization and do not accountfor the complexity of real microstructures (eg. non-convex grain shapes). MRFalgorithm can be an attractive replacement for these models as it capturesfine scale structure such as grain boundaries realistically. In this paper, wetest the synthetic microstructures generated by this method by comparing itsstatistical measures including lower order statistics (eg. grain size distribu-

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tion, orientation distribution function) and finer features (eg. shape momentinvariants) with those of experimental images. In addition, physical propertiessuch as elastic moduli and stress distribution are also calculated using image–based finite element models and compared against experiments. The resultsare discussed in section 3.

2 Mathematical modeling of microstructures as Markov Randomfields

Some of early attempts at microstructure modeling were based on Ising models[9]. In the Ising model, a N × N lattice (L) is constructed with values Xi

assigned for each particle i on the lattice, i ∈ [1, .., N2]. In an Ising model, Xi

is a binary variable equal to either +1 or −1 (eg. magnetic moment [9]). Inthis work, the values Xi may contain any one of G color levels in the range{0, 1, .., G − 1} (following the integer range extension of the Ising model byBesag [17]). A coloring of L denoted by X maps each particle in the latticeL to a particular value in the set {0, 1, .., G− 1}. Ising models fall under theumbrella of undirected graph models in probability theory. In order to rewritethe Ising model as a graph, we assign neighbors to particles and link pairs ofneighbors using a bond as shown in Fig. 1(a). The rule to assign neighborsis based on a pairwise Markov property. A particle j is said to be a neighborof particle i only if the conditional probability of the value Xi given all otherparticles (except (i, j), i.e., p(Xi|X1, X2, .., Xi−1, Xi+1, .., Xj−1, Xj+1, .., XN2))depends on the value Xj.

(a) (b)

Fig. 1. Markov random field as an undirected graph model, circles are pixels in theimage and bonds are used to connect neighbors: (a) Ising model with nearest neighborinteractions (b) Microstructure modeled by including higher order interactions in theIsing model.

Note that the above definition does not warrant the neighbor particles to beclose in distance, although this is widely employed for physical reasons. For ex-

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ample, in the classical Ising model, each particle is bonded to the next nearestneighbor as shown in Fig. 1(a). In this work, we assume that a microstructureis a higher order Ising model (Fig. 1(b)). The particles of the microstructurecorrespond to pixels of the 2D image. The neighborhood of a pixel is mod-eled using a square window around that pixel and bonding the center pixelto every other pixel within the window. Using this graph structure, a Markovrandom field can be defined as the joint probability density P (X) on theset of all possible colorings X, subject to a local Markov property. The localMarkov property states that the probability of value Xi, given its neighbors, isconditionally independent of the values at all other particles. In other words,P (Xi|all particles except i) = p(Xi|neighbors of particle i). Next, we describea method based on Ref. [8] to sample from the conditional probability densityp(Xi|neighbors of voxel i).

2.1 Sampling algorithm

In the following discussion, the color (Xi) of a pixel i is represented usingG color levels in the range {0, 1, .., G − 1} each of which maps to an RGBtriplet. The number of color levels is chosen based on the microstructure tobe reconstructed, eg. for binary images G = 2. Let E and S denote theexperimental and synthesized microstructure, respectively. Let v be a pixel inS whose color needs to inferred using the sampling procedure. Let Sv denotethe colors in a neighborhood window around pixel v. Let Ew denote the colorsof pixels in a window of the same size in the input 2D micrograph.

Input microstructure (E)

Initial synthesized image (S) Progress during first iteration

search

v Sv

Ew

Fig. 2. The Markov random field approach [8,18]: The image is grown from a 3x3seed image (center). As the algorithm progresses along the path shown (right), theunknown output pixel (shown in blue) is computed by searching for a pixel with asimilar neighborhood in the input image (left).

In order to find the coloring of pixel v, one needs to compute the conditionalprobability density p(Xv|Sv). Explicit construction of such a probability den-sity is often computationally intractable. Instead, the most likely value of v is

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identified by first finding a window Ew∗ in the input 2D micrograph that ismost similar to Sv (see Fig. 2). This is done by solving the following problem(where Sv,u denotes the color of pixel u in Sv and Ew

u denotes the color ofpixel u in Ew):

Ew∗ = arg minEw

D (1)

where, D=∑u

ωu‖Sv,u −Ewu ‖2

In the above equation, D is a distance measure defined as the normalized sumof weighted squared differences of pixel colors. The weights for nearby pixelsare taken to be greater than for pixels farther away based on a Gaussianweighting function (ω). If the pixel u is located at position (x, y) (in latticeunits) with respect to the center pixel (located at (0, 0)), ωu is given as:

ωu =exp(− (x2+y2)

2σ2 )∑i

∑j exp(− (i2+j2)

2σ2 )(2)

Here, the summation in the denominator is taken over all the known pixelsin Sv. The weights ωu for the unknown pixels in Sv are taken to be zero.This ensures that the distance measure is computed only using the knownvalues and is normalized by the total number of known pixels. The standarddeviation (σ) is taken to be 0.16w.

The problem in Eq. 1 is solved using an exhaustive search by comparing allthe windows in the input 2D micrograph to the corresponding neighborhoodof pixel v. In our approach, a measure of stochasticity is introduced by storingall matches with a distance measure that is within 1.3 times that of the bestmatching window [8]. The center pixel colors of all these matches give a his-togram for the color of the unknown pixel (Xv), which is then sampled usinga uniform random number.

The microstructure is grown layer–by–layer starting from a small seed image(3x3 pixels) taken randomly from the experimental micrograph (Fig. 2). Inthis way, for any pixel the values of only some of its neighborhood pixels willbe known. The fundamental approximation in this numerical implementationis that the probability distribution function (PDF) of an unfilled pixel is as-sumed to be independent of the PDF of its unfilled neighbors. Each iterationin the algorithm involves coloring the unfilled pixels along the boundary offilled pixels in the synthesized image as shown in Fig. 2. An upper limit of0.1 is enforced for the distance measure initially. If the matching window fora unfilled pixel has a larger distance measure, then the pixel is temporarily

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skipped while the other pixels on the boundary are filled. If none of the pix-els on the boundary can be filled during an iteration, then the threshold isincreased by 10% for the next iteration.

Sample Window Size 3 Window Size 5

Window Size 7 Window Size 9 Window Size 11

Fig. 3. Effect of Window size: The sample image of a W-Ag composite is shownand its reconstruction using our algorithm is shown for various window sizes. Thereconstructed images and the input image intuitively ‘look alike’ at higher windowsizes. In the later examples, we quantify this notion by comparing the features.

Although there are several free parameters in this model, we choose the win-dow size to be the only adjustable parameter for different microstructures forsimplicity. Window size plays an important role in the MRF model. At win-dow sizes much smaller than the correlation lengths, false matches lead to highnoise in the reconstructions. At very high window sizes, not enough matchingwindows can be identified. Hence, there is an ideal window size that needs tobe found through numerical trial. In Fig 3, we have demonstrated the effectof window size on the quality of the synthesized image. Note that only oddvalues are allowed for the window size such that the window is symmetricaround the center pixel. This example is of a two phase W-Ag composite fromRef. [19].

3 Examples

In order to test the physical features and properties of the synthesized mi-crostructure, we are interested in the following criteria:

• The synthetic microstructure must ‘look like’ the seed image. The similaritymeasures in this work come from the field of metallography/crystallography

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including lower order statistics (eg. grain size distribution, orientation dis-tribution function) and finer statistical features (eg. grain boundary con-nectivity descriptors, higher order orientation correlations).• Physical properties of the synthetic microstructure such as elastic moduli

must be within reasonable bounds to the properties of the original mi-crostructure. The properties were tested using finite element models andcompared to experiments and bounding theories.

3.1 Example 1. Polycrystal

A polycrystalline structure with grayscale data from [20] was chosen for recon-struction. The free parameter in the reconstruction is the sampling windowsize which is taken to be w = 13 × 13 pixels. To compare the sample imagewith reconstructed image, two global feature vectors were extracted from theinput microstructure.

(1) Heyn’s intercept histogram [21,22] is employed for assessing the grainsizes. Histograms of the intercept length distribution (mean interceptlength versus number of test lines possessing the mean intercept length)is used as the feature vector.

(2) Rose of intersections [23] is used as the feature vector for assessing grainshapes. To obtain the rose of intersections, a network of parallel equidis-tant lines is placed over the microstructure image at several angles andthe number of grain boundary intersections with each test line is mea-sured. The histogram of intersections with the angle of orientation of thelines is called the rose of intersection.

Fig. 4. Initial microstructure (left) and the synthesized microstructure (right) fromour Markov random field code. Note that local features such as grain boundaries areeffectively captured.

These features were compared with similar features from a few snapshots(with the same size as the input image) extracted from the reconstructedmicrostructure. Our results for case (1) (Fig. 5) look quite impressive. The

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MRF model is not only able to reconstruct the local features such as grainboundaries and connectivities, the global feature vectors (intercept histogramand rose of intersections) compare favorably with the input image.

0 5 10 15 200

100

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300

400

500

Mean intercept length

Nu

mbe

r o

f lin

es

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image 1

image 2

image 3

image 4

image 5

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Image 4 Image 5

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Image 2 Image 3

Fig. 5. Statistics of synthesized images are compared with the seed image. The meanintercept length and rose of intersections are shown. Note that none of the synthe-sized images are identical to the seed image, yet global statistics of the seed imageare well captured.

3.1.1 Comparison of moment invariants

In this section, we compare descriptors that describe the shape distributionof individual grains in the experimental and synthesized grains in Example 1,rather than the global (average) feature represented in the histograms of Fig. 5.Moment Invariants(MIs) are non-linear combinations of moments of an objectshape that are invariant with respect to a class of coordinate transformations[27]. For 1D distribution of data, the second order moment is analogous tostandard deviation while the 4th order moment invariant is similar to kurtosis.For 2D shapes, DeGraef and coworkers [24–26] have identified 2 MI’s (ω1, ω2)of second order, details about their values for various possible 2D shapes aregiven in [24]. Graphical representation for these two second order momentinvariants in a x–y plot is known as Second Order Moment Invariant map(SOMIM). The SOMIM for the image in Fig. 4 and its reconstruction usingMRFs is shown in Fig.6. To compare the SOMIMs, we need to introduce anappropriate similarity metric. The modified Bhattacharya coefficient H(p, q),

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also known as Hellinger distance, provides a metric to distinguish between twodifferent density maps p and q. The regular Bhattacharya coefficient β(p, q) isa measure of the similarity between two normalized distributions and can bewritten in discrete form as [28,29]

β(p, q) =N∑i=1

√p(i)q(i),

(with

N∑i=1

p(i) =N∑i=1

q(i) = 1

)(3)

where the summation runs over the N bins of the SOMIM. The larger the valueof β, more similar the two distributions are. The Hellinger distance H(p, q) isdefined by [30] as

H(p, q) =√

1− β(p, q) (4)

Value of H for SOMIM is 0.47 for the polycrystalline case in Example 1. Thisvalue indicates good reproduction of polycrystalline shapes using the MRFsynthesis approach.

Alternately, a shape descriptor called projected moment invariant maps(PMIM)can also be plotted for the individual grains [24]. In this plot, averaged MIsof second through fourth order are plotted graphically in 2D or 3D plots. Tomake a comparison of synthesized with original image, we used combinationof second and fourth order moments in a 2D PMIM. The density maps forPMIM for the original and reconstructed image are compared in Fig.6. TheHellinger distance of PMIM is 0.37, again indicating a good reconstruction.

ω1 ω1

ω2 ω2

Fig. 6. SOMIM density map original(left) and synthetic(right) microstructure

3.2 Example 2: Aluminum alloy AA3002 representing the rolling plane

In the second example, microstructure of the Aluminum alloy AA3002 mea-sured using polarised light microscopy (Ref. [31]) was used. The microstructurerepresents the rolling plane and reveals a fully recrystallised grain structurewith randomly distributed intermetallic phases (dark spots in the image). Themicrostructure is colored based on the occurrence of near-cube and non-cubeorientations. This analysis is based on observed contrast effects when the ob-ject is rotated relative to the polarised light directions. Purple regions are

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Aver

age

4th O

rder

Mom

ent

Aver

age

4th O

rder

Mom

ent

Average 2nd Order Moment Average 2nd Order Moment

Fig. 7. PMIM density map original(left) and synthetic(right)

cube or near-cube orientations, whereas the yellow/red regions are non-cube.The Markov random field reconstruction and the original microstructure areindicated in Fig. 8. Only a small part (in 8(a)) of the larger experimentalimage (in 8(c)) was used for the reconstruction and the reconstructed imageof larger size (in 8(b)) was compared with the larger experimental image. Themicrostructure was reconstructed using a 150× 170 pixel input image shownin Fig 8(left) using a window size of w = 7 × 7 pixels. The fraction of cubeversus non-cube orientations and distribution of intermetallic phases was stud-ied using color histograms and color clouds. The color cloud used here is anattempt at showing the pixels in ‘color space’ rather than Euclidean space(‘microstructure’). Color densities are converted into scattered random dotsaround the spatial position assigned to the color, with the extent of the spatialposition determined by the frequency with which that RGB triplet appears inthe image.

Input image

Original experimental imageReconstruction with MRF

(a) (b) (c)

Fig. 8. Reconstruction of an experimentally measured AA3002 Aluminum alloy mi-crostructure [31] using Markov random Field algorithm.(a) Input micrograph (b)MRF reconstruction Purple regions are cube/near cube grain orientations, yel-low/red regions are non-cube orientations. The fine dark spots are the intermetallicphases. (c) The larger microstructure from which the input image is taken is alsoshown for comparison.

The color blot and color histogram results shown in Fig 9 show good visual

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Input microstructure Reconstructed microstructure

cube/near-cube

components

non-cube

components

Intermetallic phase

cube/near-cube

components

Fig. 9. Color histograms of the input microstructure and reconstructed microstruc-ture are compared. A novel color blot method is used to compare the distributionof cube/near cube regions, and intermetallic phases in the input and synthesizedimages.

correlation of the reconstructed image with the experimental input image. Thetexture components (cube versus non–cube orientations) are well reproducedin the larger synthesized image.

3.2.1 Comparison of microstructure sensitive properties

We performed finite element calculations on synthesized and sample image inFig. 8 and compared the Young’s Modulus with angle of rotation for sampleand synthesized image. The calculations were based on a crystal plasticitymodel for Aluminum (from [32]) and was performed at a constant strain rateof 6.667e-4 s−1 and a temperature of 300 K. To measure the statistically sim-ilarity between sample and synthesized image, we calculated the orientationdistribution function(ODF) by assigning a unique orientation to each pixelbased on its color. In Fig 11 we have shown ODF for sample and two synthe-sized images and we observed that ODF for synthesized images is very close tothat of sample image. In Fig 10(b), we have plotted the variation of Young’sModulus (E) with sample rotation angle for the original and reconstructedimages. In Fig 10(a) we have plotted the equivalent stress-strain response forsample and synthesized image for a shear test. We also compared the stresshistograms for the original and reconstructed image. The histogram plots the

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0 0.5 1 1.5 2 2.5

x 10-4

0

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Equ

ival

ent S

tress

(Mpa

)

Equivalent Strain

SampleSynthesis 1Synthesis 2

0 10 20 30 40 50 60 70 80 9069.5

70

70.5

71

71.5

72

72.5

73

73.5

Rotation Angle(Degree)

You

ng M

odul

us (G

Pa)

SampleSynthesized 1Synthesized 2

(a)

(b)

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x 10-4

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0 10 20 30 40 50 60 70 80 9069.5

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You

ng M

odul

us (G

Pa)

SampleSynthesized 1Synthesized 2

(a)

(b)

Fig. 10. (left) Comparison of the equivalent stress-strain curve predicted throughhomogenization for sample and two synthesized image (right) Variation of Young’sModulus with angle of rotation for sample and synthesized images

number of pixels in the microstructure within a given stress range. The colorhistogram of stresses shown in Fig.12 reveal that the global response of thesynthesized image is very similar to that of sample microstructure.

Sample Synthesized 1 Synthesized 2

Fig. 11. ODFs of sample micrograph is compared with the two synthesized images

Stress Value (GPa)

p(σ

)

Stress Value (GPa)

p(σ

)

Fig. 12. Comparison of the distribution of the equivalent stress in a finite elementsimulation using a color histogram (left) Experimental image (right) Synthesizedimage 1

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4 Conclusions

It is human intuition that different windows taken from a polycrystalline mi-crostructure generally ‘look alike’. This can be quantified through an underly-ing stationary probability distribution that generates all possible microstruc-tural windows. While quantifying this high dimensional joint probability distri-bution of all pixel colors is computationally intractable, we looked at samplingmethods to model this distribution. For many microstructures, the probabilityof a pixel color depends only on the pixel state of its close neighbors. Thus, onecould represent microstructures in the form of undirected probabilistic graphscalled Markov Random Fields (MRFs) with pixels interacting with neighborsover a sampling window.

With this lattice structure, we employed a sampling algorithm to computethe conditional probability density for the state of each pixel given the knownstates of its neighboring pixels. In the implementation, the microstructure isgrown layer–by–layer from a small seed image (3x3 pixels) taken randomlyfrom the sample. To synthesize a pixel, the algorithm first finds all windows inthe sample image that are similar to the unknown pixels neighborhood win-dow. One of these matching windows is chosen and its center pixel is takento be the newly synthesized pixel. We have artificially synthesized 2D poly-crystalline microstructures using the sampled probabilities. Previous methodsfor reconstructing polycrystals using algorithms based on a common set ofunderlying features such as marginal histograms and point probability func-tions have often failed to capture the local information such as sharp grainboundaries. We find that the MRF approach is an attractive solution in thisregard.

We show that not only are the global features such as grain size/shape dis-tribution captured but the localized grain features such as shape momentsalso closely match with those of the experimental images. In addition, thetexture distribution and mechanical (elastic) properties of the synthesized mi-crostructures as computed using finite element method were also found toclosely reproduce the experimental values. Finally, we plotted the stress his-tograms from a finite element test and these were also well reproduced. Ourresults evidence to the notion of a stationary probability distribution under-lying microstructure formation. Future efforts will aim to analytically modelsuch a distribution by using high dimensional regression to fit the results of thesampling algorithm. In addition, it will be of interest to use this tool to studythe temporal evolution of such a distribution during microstructure evolution.

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References

[1] D. J. Heeger and J. R. Bergen. Pyramid-based texture analysis/synthesis. InSIGGRAPH 95, pages 229–238, 1995.

[2] J. S. D. Bonet. Multiresolution sampling procedure for analysis and synthesisof texture images. In SIGGRAPH 97, pages 361–368, 1997.

[3] E. P. Simoncelli and J. Portilla. Texture characterization via joint statistics ofwavelet coefficient magnitudes. In Proc. 5th Intl Conf. on Image ProcessingChicago, IL, 1998.

[4] S. Torquato, Random Heterogeneous Materials: Microstructure andMacroscopic Properties, Springer-Verlag, New York, 2002.

[5] K. Popat and R. Picard. Novel cluster-based probability model for texturesynthesis, classification, and compression. In Visual Communications andImage Processing, pages 756–768, 1993.

[6] S. Zhu, Y. Wu, and D. Mumford. Filters, random fields and maximun entropy(FRAME) - towards a unified theory for texture modeling. InternationalJournal of Computer Vision, 27(2):107–126, 1998.

[7] R. Paget and I. Longstaff. Texture synthesis via a noncausal nonparametricmultiscale Markov random field. IEEE Transactions on Image Processing,7(6):925–931, June 1998.

[8] A. Efros and T. Leung. Texture synthesis by non-parametric sampling. InInternational Conference on Computer Vision, Vol 2, pages 1033–1038, Sep1999.

[9] E. Ising, Beitrag zur Theorie des Ferromagnetismus, Zeitschrift Physik, 31(1925) 253–258.

[10] M. Grigoriu, Nearest Neighbor Probabilistic Model for Aluminum Polycrystals,Journal of Engineering Mechanics, Vol. 136, No. 7, July 2010, pp. 821-829.

[11] C. E. Shannon. A mathematical theory of communication. Bell Sys. Tech.Journal, 27, 1948.

[12] Sundararaghavan V, Reconstruction of three dimensional anisotropicmicrostructures from two-dimensional micrographs imaged on orthogonalplanes. Integrating Materials and Manufacturing Innovation, 2014, 3:19 (pages1-11).

[13] K. S. Zhang, M. S. Wu, R. Feng, Simulation of microplasticity-induceddeformation in uniaxially strained ceramics by 3-D Voronoi polycrystalmodeling, International Journal of Plasticity 21(4), 801–834, 2005.

[14] R.A. Lebensohn, O. Castelnau, R. Brenner, P. Gilormini, Study of theantiplane deformation of linear 2-D polycrystals with different microstructures,International Journal of Solids and Structures 42(20), 5441–5459, 2005.

14

[15] O. Watanabe, H. M. Zbib, E. Takenouchi, Crystal plasticity: micro-shearbanding in polycrystals using voronoi tessellation, International Journal ofPlasticity 14(8), 771–788, 1998.

[16] A. Brahme, M.H. Alvi, D. Saylor, J. Fridy, A.D. Rollett, 3D reconstructionof microstructure in a commercial purity aluminum, Scripta Materialia 55(1),75–80, 2006.

[17] J. Besag, Spatial Interaction and the Statistical Analysis of Lattice Systems,Journal of the Royal Statistical Society. Series B (Methodological), 36(2)(1974) 192–236.

[18] L-Y Wei, S Lefebvre, V Kwatra, G Turk, State of the Art in Example-basedTexture Synthesis: in EUROGRAPHICS 2009, State of the Art Report, EG-STAR - 2009.

[19] S. Umekawa, R. Kotfila, O.D. Sherby, Elastic properties of a tungsten-silvercomposite above and below the melting point of silver, J. Mech. Phys. Solids13 (4) (1965) 229230.

[20] Lefebvre S., Hoppe H.: Appearance-space texture synthesis. In SIGGRAPH06, pp. 541–548, 2006.

[21] Vander Voort GF. Examination of some grain size measurement problems, inMetallography: Past, Present and Future (75th Anniversary Volume) ASTMSTP 1165. Philadelphia : American Society for testing and materials, 1993.p.266.

[22] Vander Voort GF. Metallography: principles and practice. New York: McGraw-Hill Book Company;1984.

[23] Saltykov SA. Stereometrische metallographie. Leipzig: Deutscher Verlag furGrundstoffindustrie;1974.

[24] P G Callahan, J.P. Simmons and M De Graef A quantitative description ofthe morphological aspects of materials structures suitable for quantitativecomparisons of 3D microstructures, Modelling Simul. Mater. Sci. Eng. 20(2012)

[25] MacSleyne J P, Simmons J P and De Graef M 2008 On the use of momentinvariants for the automated analysis of 3-D particle shapes Modelling Simul.Mater. Sci. Eng 16 045008

[26] MacSleyne J P, Simmons J P and De Graef M 2008 On the use of 2-D momentinvariants for the automated classification of particle shapes Acta Mater 5642737

[27] Hu M K 1962 Visual pattern recognition by moment invariantsIRE Trans.Inform. Theory 8 17987

[28] Bhattacharyya A On a measure of divergence between two statisticalpopulations defined by their probability distribution Bull. Calcutta Math. Soc.35 99110

15

[29] Aherne F, Thacker N and Rockett P 1997 The Bhattacharyya metric as anabsolute similarity measure for frequency coded data Kybernetika 32 17

[30] Comaniciu D, Ramesh V and Meer P 2003 Kernel-based object tracking IEEETrans. Pattern Anal. Mach. Intell.25 56477

[31] N.J. Wittridge, R.D. Knutsen, A microtexture based analysis of the surfaceroughening behaviour of an aluminium alloy during tensile deformation, Mater.Sci. Eng. A 269 (1–2),205–216, 1999.

[32] V. Sundararaghavan and N. Zabaras, Design of microstructure-sensitiveproperties in elasto-viscoplastic polycrystals using multi-scale homogenization,Int. J. Plasticity, 22, 1799–1824, 2006.

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