An efficient optimization based microstructure reconstruction approach with multiple loss functions
Anindya Bhaduri, Ashwini Gupta, Audrey Olivier, Lori Graham-Brady
AAn efficient optimization based microstructurereconstruction approach with multiple loss functions
Anindya Bhaduri a , Ashwini Gupta a , Audrey Olivier b , Lori Graham-Brady a Department of Civil Engineering, Johns Hopkins University, Baltimore, MD, USA b Columbia University, New York, NY, USA
Abstract
Stochastic microstructure reconstruction involves digital generation of mi-crostructures that match key statistics and characteristics of a (set of) targetmicrostructure(s). This process enables computational analyses on ensemblesof microstructures without having to perform exhaustive and costly experi-mental characterizations. Statistical functions-based and deep learning-basedmethods are among the stochastic microstructure reconstruction approachesapplicable to a wide range of material systems. In this paper, we integratestatistical descriptors as well as feature maps from a pre-trained deep neuralnetwork into an overall loss function for an optimization based reconstruc-tion procedure. This helps us to achieve significant computational efficiencyin reconstructing microstructures that retain the critically important phys-ical properties of the target microstructure. A numerical example for themicrostructure reconstruction of bi-phase random porous ceramic materialdemonstrates the efficiency of the proposed methodology. We further performa detailed finite element analysis (FEA) of the reconstructed microstructuresto calculate effective elastic modulus, effective thermal conductivity and ef-fective hydraulic conductivity, in order to analyse the algorithm’s capacity tocapture the variability of these material properties with respect to those ofthe target microstructure. This method provides an economic, efficient andeasy-to-use approach for reconstructing random multiphase materials in 2Dwhich has potential to be extended to 3D structures.
Keywords: microstructure reconstruction; transfer learning; Gram matrix;total variation; 2-point probability; stochastic simulation; two-phase Corresponding author. Email address: [email protected]
Preprint submitted to Elsevier February 5, 2021 a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b aterials
1. Introduction
The discovery and development of new materials with targeted propertieshas always been a central theme for computational material science research,which requires a fundamental understanding of materials behavior and prop-erties across different scales. Stochastic microstructure reconstruction is animportant piece of that challenging puzzle. It enables computational gener-ation of microstructures that match the statistically equivalent characteris-tic(s) of a (set of) target microstructure(s), avoiding the need for exhaustiveand costly physical microstructure characterization of every sample to beevaluated.There exists a number of different types of microstructure reconstructionapproaches in the current literature. Reconstruction via statistical functions[1, 2, 3] is a popular technique applicable to a wide range of materials suchas, particulate structures [2, 4], chalk [5], soil [6], sandstone [7, 8].The re-construction is usually done via a stochastic optimization method, knownas Yeong and Torquato (YT) method [9, 10] and tries to ensure that thechosen statistical functions of the target microstructure match closely withthat of the reconstructed microstructure. However, if the target image isanisotropic or multiphase or large in size, the convergence of the YT methodis especially slow which makes the procedure computationally expensive.Physical descriptor-based approaches [11, 12, 13], on the other hand, in-volve a relatively cheaper optimization procedure which tries to match thephysically meaningful descriptors of the target microstructure to that ofthe reconstructed one. However, their application is limited to the mi-crostructure reconstruction of crystalline and particulate structures (regulargeometries) and not to material systems with irregular geometries. Spec-tral density function (SDF)-based approaches aim to reconstruct statisti-cally equivalent microstructure by generating SDF representations of thetarget microstructure and matching it to the reconstructed one. The meth-ods [14, 15, 16, 17] are mostly analytical in nature and hence much fasterthan the optimization-based reconstruction approaches but their applicationis restricted to isotropic binary materials. Another class of microstructurereconstruction methods involves using deep learning [18, 19], a machine learn-ing approach that can be used amongst other tasks for surrogate modeling[20, 21, 22, 23, 24, 25, 26] and thus has been implemented successfully for a2ide range of classification and regression tasks. Deep learning approaches,in particular convolutional neural networks, are particularly well-suited tohandle image data, and have thus received attention lately from the materi-als research community to process microstructures image data for a varietyof tasks [27, 28]. Deep learning approaches for reconstructions are of two dif-ferent types: material-system-dependent and material-system-independentapproaches. Material-system dependent approaches include work by Canget al. [29] which uses a convolutional deep belief network [30] and work byLi et al. [31] where a Generative Adversarial Network (GAN) [32] model isemployed. These approaches train the weights of the network with imagesspecific to a material system and thus need to be retrained for a new mate-rial system. Alternatively, transfer learning approaches are material-system-independent and does not require training weights with a set of materialsdata. Instead, deep learning models, pre-trained using benchmark datasetsin the field of computer vision, are used to generate microstructure recon-structions. However, it is noted that, given the fixed pretrained weights,there still may be a need for hyperparameter tuning to get the most optimalresults for a given material data set. The work by Li et al. [33] is one suchapproach where a deep convolutional network VGG-19 [34] pretrained onImageNet [35] dataset has been used for the reconstruction based on a sin-gle given target microstructure. A feature-matching optimization has beenperformed using a Gram-matrix loss function to generate statistically equiv-alent microstructures. The reconstruction approach presented in this paperis derived from the work by Li et al [33].In this paper, we propose a modified version of the existing transfer learn-ing approach [33]. The modifications include: a) different combinations ofGram matrix layers, b) a weighted combination of the gram matrix (GM)loss components, c) elimination of the potentially costly simulated-annealingbased volume fraction matching process, d) inclusion of microstructural de-scriptor metrics in the overall loss function in the main optimization step.The modifications aim to enhance the efficiency, accuracy as well as inter-pretability of the proposed modified approach compared to that of the exist-ing approach. It is noted here that the performance of the proposed approachis assessed via a statistical analysis of both the physics-based descriptors ofthe microstructure as well as the Finite Element Analysis (FEA) simulatedaveraged material properties. The analysis is demonstrated using a true mi-crostructure dataset consisting of 185 images of size 560 pixels ×
902 pixels,corresponding to 185 successive slices of a porous ceramic microstructure3easured by x-ray tomography similar to those found in [36, 37]. The targetmicrostructures are described in section 2. Section 3 discusses the denoisingof the original target microstructures and compares the statistical propertiesbetween the original and the denoised versions. Section 4 explains the sixmicrostructural descriptors considered in this study to assess the reconstruc-tion quality. The microstructure reconstruction procedure with different lossfunction combination and different slices (original and denoised) as the tar-get image is discussed in details in section 5. In section 6, FEA simulatedmaterial properties are compared between the original and the reconstructedimages. Section 7 provides a discussion of the additional advantages of theproposed approach. Section 8 provides the conclusions.
2. Target microstructures
The target microstructure is that of a two-phase porous ceramic materialand there are images of 185 successive slices of size 560 pixels ×
902 pixels.Thus a binary indicator function I (1) :( X, Y ) ∈ Ω → { , } represents themicrostructure and is defined as I (1) ( X, Y ) = (cid:40) , if ( X, Y ) ∈ Ω (1) , if ( X, Y ) ∈ Ω (0) where Ω is the material domain, Ω (0) ⊂ Ω is the black solid phase domainand Ω (1) ⊂ Ω is the white porous phase domain such that Ω (0) + Ω (1) = Ω andΩ (0) ∩ Ω (1) = ∅ . This is discretized in a computational setting by an array of(0,1) values. The slice 1 microstructure is shown in figure 1.
3. Denoising of target microstructure
The need for denoising of the original target microstructure arises fromthe fact that spurious inclusions of only one or a few pixels appear in the im-age. These spurious pixels significantly complicate finite element (FE) anal-yses used to calculate effective properties. While denoising enables coarserdiscretization of the material domain and more efficient FE analysis, toomuch denoising can lead to significant changes in the local microstructure.Keeping this in mind, all cluster sizes for the black solid phase less than 10pixels are converted to the white porous phase. Figure 2 shows a comparisonof the original and the corresponding denoised microstructures for slices igure 1: Target microstructure (slice
4. Microstructural descriptors
Microstructural descriptors are considered key in determining the effec-tive physical properties of random heterogeneous materials [38]. Thus, itis important to match the microstructural descriptors of the reconstructedmicrostructure with that of the original microstructure. In this study, sixsuch descriptors are considered: porosity, correlation length, specific bound-ary length, mean pore size, 90 th -percentile pore size and 97 th -percentile poresize. Each of these descriptors are described below and some are shown infigure 3.Porosity is the fraction of the porous phase in the microstructure givenby: φ = (cid:80) N x i =1 (cid:80) N y j =1 I (1) ( x i , y j ) N x N y (1)where ( x i , y j ) is any pixel location in the microstructural domain Ω, while N x and N y are the number of pixels along the horizontal and the vertical di-rections respectively. Figure 4(a) shows the porosity of each of the 185 slicesof the material. Denoising slightly increases porosity since small regions ofthe black solid phase are replaced by the white porous phase.While the volume fraction provides information about the microstruc-tural phase at individual points, it does not provide information about thespatial patterns of the microstructure that can be explained to some ex-tent by the two-point probability function. The two-point probability func-5ion S (1)2 (∆ l , ∆ m ) calculates the probability that any two points ( x i , y j ) and( x i + l , y j + m ) with a separation vector (∆ l , ∆ m ) fall in the white porous phase (a) slice igure 3: A schematic showing different microstructural descriptors in a binary-phase(black solid phase and white porous phase) microstructure. The two-point correlationfunction measure for the white phase is represented by a blue solid line. The red solidlines denote the phase boundaries, the total length of which gives the boundary length.The pore size distribution function is denoted by P ( δ ) where δ is the pore radius variable. and is given by: S (1)2 (∆ l , ∆ m ) = P (cid:2) I (1) ( x i , y j ) = 1 , I (1) ( x i + l , y j + m ) = 1 (cid:3) = (cid:80) N x − li =1 (cid:80) N y − mj =1 I (1) ( x i , y j ) I (1) ( x i + l , y j + m )( N x − l )( N y − m ) (2)It thus provides a measure of how the two end points of a vector (denoted bya blue solid line in figure 3) in a phase (in this case, the white porous phase)are correlated. It is noted that the above description of the two-point prob-ability function similarly applies for the solid phase. In the above definition,it is assumed that the material is spatially stationary (or statistically homo-geneous), which may not be valid for materials in which the microstructuralstatistics vary spatially (e.g., in a functionally graded material). Correlationlength is one scalar measure associated with the two-point probability func-7 a) Porosity (b) Correlation length (c) Specific boundary length(d) Mean pore size (e) 90 th -percentile pore size (f) 97 th -percentile pore sizeFigure 4: Variation of statistical properties across original and denoised slices tion, which is calculated by integrating the normalized two-point probabilityfunction over the entire domain [39] and then taking its square root: α = (cid:115) S (1)2 (∆ l , ∆ m ) − φ φ − φ (3)Figure 4(b) shows the correlation length of each of the 185 slices of the ma-terial. It is noted that denoising slightly increases correlation length becauseit removes small correlation length inclusions.Specific boundary length is the cumulative length of the boundary ofseparation between the black solid phase and the white porous phase, nor-malized by the total microstructural area. The cumulative boundary lengthdenoted by red solid lines in figure 3 is obtained by generating the correspond-ing morphological gradient of the microstructure by taking the difference ofits dilation and erosion. The morphological gradient of the microstructure is8 matrix of the same size as the microstructural image with non-zero valuedpixels along the boundaries of the two phases and zero-valued pixels else-where. The cumulative boundary length is thus obtained by counting allthe non-zero pixels in the morphological gradient image. Figure 4(c) showsthe specific boundary length of each of the 185 slices of the material for theoriginal and denoised cases. It is seen that denoising slightly decreases thespecific boundary length because the replacement of small clusters of blacksolid phase by white porous pixels leads to a decrease in the cumulativeboundary length.The pore-size probability density function P ( δ ) is used to characterize thevoid or “pore” space in porous media [40]. The function P ( δ ) for isotropicmedia is defined such that P ( δ ) dδ is the probability that a randomly chosenpoint in the porous phase domain Ω (1) lies at a distance between δ and δ + dδ from the nearest point on the pore-solid interface. Thus, (cid:82) ∞ P ( δ ) dδ = 1.The mean pore size is defined as: (cid:104) δ (cid:105) = (cid:90) ∞ δP ( δ ) dδ (4)The associated complementary cumulative distribution function F ( δ ) isthe fraction of pore space that has a pore radius larger than δ and is givenby: F ( δ ) = (cid:90) ∞ δ P ( δ ) dδ (5)90 th -percentile pore size is defined as the pore size such that 90% of thepore space has a pore radius less than or equal to that pore size, i.e., F ( δ ) =0 .
10. 97 th -percentile pore size, similarly, is defined as the pore size suchthat 97% of the pore space has a pore radius less than or equal to thatpore size, i.e., F ( δ ) = 0 .
03. Figures 4(d), 4(e) and 4(f) show the variation ofthe mean pore size, the 90 th -percentile pore size and the 97 th -percentile poresize, respectively, across all 185 slices for the original and their correspondingdenoised microstructures. It is seen that denoising has very little effect onthese three statistical properties of the microstructure.
5. Microstructure reconstruction
In this section, the reconstruction procedure and the corresponding re-sults are discussed. The algorithm is an optimization based procedure and9ifferent loss function combinations are implemented with an optimal set ofhyperparameters for the reconstruction of original slice / , ,
0] and the 3D white pixelcoordinate [255 , , In this section, a transfer learning based pattern-matching optimizationis performed using a pre-trained VGG-19 network shown in figure 5. VGG-19 is a deep convolutional neural network and in total has 16 convolutionallayers and 5 max pooling layers, divided into 5 blocks. The original network,in addition, has 3 fully connected layers at the end which are excluded here.It is noted that 3-channels representation is a requirement for the input im-
Figure 5: VGG-19 network architecture [34] F ipr and ˜ F ipr denote the feature maps of the p th filter at position r in layer i for the originaland the reconstructed microstructures, respectively. The Gram matrix G ipq of the original microstructure and ˜ G ipq of the reconstructed microstructure,in layer i , is thus defined by the following inner product: G ipq = (cid:88) r F ipr F iqr (6)˜ G ipq = (cid:88) r ˜ F ipr ˜ F iqr (7)The Gram matrix loss for the layer i is given by: E i = 14 N i M i (cid:88) j,k ( G ijk − ˜ G ijk ) (8)where N i is the number of filters and M i is the size of the vectorized featuremaps in layer i . The total Gram matrix loss across the selected layers is thusgiven by: L G = (cid:88) i w i E i (9)11 igure 6: Optimization framework with gram matrix loss where the weight of each layer i is given by w i = min i (cid:80) j,k ( G ijk ) (cid:80) j,k ( G ijk ) . The firstconvolutional layers of each of the 5 blocks of the VGG-19 network as shownin figure 5 is found to be most optimal. The overall optimization frameworkis shown in figure 6. The reconstruction algorithm was run for 50 differentinitializations of the reconstructed microstructure. Figure 7 compares the (a) Two-point probability functionfor the porous phase (b) Pore size complementary cdfFigure 7: Comparison of statistical properties between the original slice two-point probability function and the pore size cumulative density function(CDF) for the original slice igure 8: Reconstructed microstructure (checkerboard patterns) using gram matrix lossfunction based optimization To address the checkerboard patterns, total variation (TV) loss betweenthe target image and the reconstructed image is included in the optimizationframework as shown in figure 9. The total variation for an image is definedas the sum of the absolute differences in the pixel values between neighboringpixels in that image. It is a measure of noise in an image that is sensitive tofeatures like checkerboard patterns. The total variation loss is the anisotropicversion of the total variation norm [48] and is given by: L V = (cid:88) k (cid:88) i,j | y i +1 ,j,k − y i,j,k | + | y i,j +1 ,k − y i,j,k | (10) Figure 9: Optimization framework with a combination of gram matrix loss and totalvariation loss igure 10: Reconstructed microstructure (no checkerboard patterns) using a combinationof gram matrix loss and total variation loss for optimization where y i,j,k is the ( i, j, k )-th pixel value of the 3D representation of the recon-structed image. Figure 10 shows the reconstructed image corresponding toone of the initializations, and it is evident that the inclusion of the total vari-ation loss eliminates the checkerboard patterns found in the reconstructedimages in section 5.1. It is also noted that similar observations are found inall the other initialization cases. The total loss L T in this case is given by: L T = w G L G + w V L V (11)where w G = 1 and w V = 1e − In this section, we try to improve the reconstruction quality further byincluding another loss function in the optimization, the two-point probabil-ity (2PP) loss L P . This loss is defined as the mean squared error betweenthe 2-point probability functions [refer to eq . (2)] of the original and thereconstructed microstructures. The two-point probability is sensitive to thespatial distribution of a given phase. The overall optimization framework is14 able 1: Error in reconstruction with GM loss and “GM+TV” loss GM loss GM+TV lossMean error in porosity 0.0027 0.0048Mean error in correlation length 0.2923 0.0580Mean error in specific boundary length 0.0144 0.0050Mean error in mean pore size 0.1225 0.0228Mean error in 90-th percentile pore size 0.3846 0.0947Mean error in 97-th percentile pore size 0.8641 0.1953shown in figure 11. The total loss L T in this case is given by: L T = w G L G + w V L V + w P L P (12)where w G = 1, w V = 1e − w P = 1e8 are the weight combinationsthat seem to produce the best results using ‘trial and error’ approach. Itis noted that a loss function corresponding to any other physical propertyof interest can be added to the optimization framework in a similar fashion.Figure 12 shows the box plot of the errors in the statistical properties ofinterest corresponding to each of the loss function cases: “GM”, “GM+TV”and “GM+TV+2PP” for the reconstruction of original slice Figure 11: Optimization framework with a combination of gram matrix loss, total variationloss and two-point correlation function loss a) Porosity (b) Correlation length (c) Specific boundary length(d) Mean pore size (e) 90 th -percentile pore size (f) 97 th -percentile pore sizeFigure 12: Comparison of error in statistical functions for the three different loss functioncases are calculated over 50 different initializations of the reconstructed image.It is clear from the box plots that, with the exception of the very smallporosity errors, all the other statistical properties match more accurately forthe “GM+TV” and “GM+TV+2PP” loss cases compared to the “GM” losscase. However, the performance of the “GM+TV” and “GM+TV+2PP”loss cases are very similar to each other and the addition of the two-pointprobability loss function to the optimization framework does not improvethe reconstruction results significantly, which is evident from the mean andcoefficient of variation (CV) values of the absolute error in the propertiesshown in table 2. In the previous sections, reconstructions were performed with originalslice able 2: Error in reconstruction with “GM+TV” loss and “GM+TV+2PP” loss
GM+TV loss GM+TV+2PP lossMean CV Mean CVError in porosity 0.0048 0.6402 0.0046 0.6169Error in correlation length 0.0580 0.6698 0.0615 0.6437Error in specific boundary length 0.0050 0.1587 0.0050 0.1867Error in mean pore size 0.0228 0.7957 0.0247 0.7321Error in 90-th percentile pore size 0.0947 0.6825 0.0992 0.6295Error in 97-th percentile pore size 0.1953 0.6432 0.1873 0.6288as for slice (a) Porosity (b) Correlation length (c) Specific boundary length(d) Mean pore size (e) 90 th -percentile pore size (f) 97 th -percentile pore sizeFigure 13: Errors in the statistical properties across three different slices (1, 90, 185) forGM+TV and GM+TV+2PP loss In this section, reconstruction is performed by considering all 185 originalmicrostructure slices and their corresponding denoised versions individuallyas the target images. Three different initializations of the reconstructedimage (leading to three distinct statistically equivalent images) are chosenfor each of target image. Therefore, overall there are 555 reconstructionscases and their results are shown in figure 14. Figure 14 shows the compar-ison of the error in the statistical properties after reconstructions with the“GM+TV” and “GM+TV+2PP” loss cases, once with the original slices asthe target microstructures and in the other instance, with the denoised slicesas the target images. It is found that the comparison of almost all statisticalproperties are very similar to each other for all four reconstruction scenar-ios. The only exception is for the specific boundary length values where theoverall error for the reconstruction cases with the denoised target slices arelower than that with the original target slices.
6. FE analysis based material properties
In this section, quality of the reconstructions are assessed in terms of a fewaveraged material properties of the porous ceramic material. The materialproperties of interest are the effective Young’s modulus ( ¯ E ), effective thermalconductivity (¯ λ ) and effective hydraulic conductivity ( ¯ K ). After an image isreconstructed based on a target image, the properties of the reconstructedmicrostructure are simulated using the commercial FE software ABAQUS[49]. 560 ×
902 elements (each pixel assigned an element) are used to modeleach microstructure in 2D. 18 static linear elastic analysis is performed to calculate the effectiveYoung’s modulus by subjecting the finite element model of the microstruc-ture to a tensile load as shown in figure 15. The macroscopic stress ¯ σ yy inthe vertical direction (direction of applied displacement) is calculated [50]by summing all the nodal reaction forces on the top face along the verticaldirection and dividing it by the cross sectional area A of the face:¯ σ yy = (cid:80) Ni F Ri A (13)where F Ri is the reaction force on node i along the vertical direction, N is thenumber of nodes on the surface where displacement is applied. The effective (a) Porosity (b) Correlation length (c) Specific boundary length(d) Mean pore size (e) 90 th -percentile pore size (f) 97 th -percentile pore sizeFigure 14: Errors in the statistical properties across all 185 slices a) Finite element model (b) Von Mises stress mapFigure 15: Linear elastic analysis. (a) Finite element model: displacement is applied atthe top face ( δ y = 1) with the bottom face held fixed ( δ y = 0), both along the verticaldirection; the left face is also fixed along the horizontal direction ( δ x = 0); the in-planeYoung’s modulus for the solid phase is given by E solid = 100 MPa, and that of the porousphase is given by E pore = 1 MPa. (b) Contour map of Von Mises stress obtained from thelinear elastic analysis is shown here. Young’s modulus is then obtained using:¯ E = ¯ σ yy ¯ (cid:15) yy (14)where macroscopic strain ¯ (cid:15) yy = δ y L and L is the edge length along the verticaldirection.The effective thermal conductivity is obtained by applying a constanttemperature difference between the top and bottom faces of the finite elementmodel of the microstructure [51] as shown in figure 16. The macroscopic heatflux density ¯ q y is calculated by summing the nodal heat fluxes on the top face.The effective thermal conductivity is then calculated using Fourier’s law ofheat conduction [52]: ¯ λ = ¯ q y L ∆ T (15)where ∆ T = T − T is the temperature difference between the top and bot-tom face, and L is the length of the edge along the vertical direction.The effective hydraulic conductivity for a laminar fluid flow throughthe porous medium of the material is simulated by specifying the hydraulichead between the top and bottom faces of the finite element model of themicrostructure and then calculated from Darcy’s equation as shown in fig-ure 17. The macroscopic specific discharge ¯ q y is calculated by summing up20 a) Finite element model (b) Heat flux density mapFigure 16: Heat conduction analysis. (a) Finite element model: Temperature difference isapplied between the top face ( T = 300 K ) and the bottom face ( T = 0 K ); the thermalconductivities for the solid phase and the porous phase are given by λ solid = 400 W/m/Kand λ pore = 1 W/m/K respectively. (b) Contour map of the heat flux density obtainedfrom the heat conduction analysis is shown here.(a) Finite element model (b) Specific discharge mapFigure 17: Fluid flow analysis. (a) Finite element model: Hydraulic head is appliedbetween the top face ( h = 300 m ) and the bottom face ( h = 0 m ); the hydraulic con-ductivities for the porous phase and the solid phase are given by K solid = 400 m/s and K pore = 1 m/s respectively. (b) Contour map of the specific discharge obtained from thefluid flow analysis is shown here. the nodal specific discharge values on the top face. The effective hydraulicconductivity is then calculated using Darcy’s law of fluid flow [53]:¯ K = ¯ q y L ∆ h (16)where ∆ h = h − h is the hydraulic head between the top and bottom face,and L is the length of the edge along the vertical direction.21 a) Effective modulus (b) Effective thermalconductivity (c) Effective hydraulicconductivityFigure 18: Errors in the material properties across all 185 slices Figure 18 shows the boxplot of the relative absolute errors in the effec-tive modulus, conductivity and hydraulic conductivity of the reconstructedmicrostructures with respect to the corresponding target microstructures. Itis seen in figure 18(a) that the effective Young’s modulus error values for allthe cases are in an acceptable range. The reconstruction quality is similarbetween the two loss function cases when the original images are used astarget slices. However, when the denoised images are used, the “GM+TV”loss case performs better than the “GM+TV+2PP” case. The error valuescorresponding to the effective thermal conductivity in figure 18(b) and theeffective hydraulic conductivity in figure 18(c) are significantly higher thanthat of the elastic modulus. This suggests that the reconstruction algorithmis not able to capture these properties efficiently. This can be attributed tothe fact that the effective modulus is a function of the overall pattern of themicrostructure, i.e., the relative arrangement of the black solid and the whiteporous phase, which is well matched in the original and the reconstructedmicrostructure. However, the effective thermal conductivity and hydraulicconductivity depend significantly on localizations in the microstructures thatenable heat flow and fluid flow respectively. The goal of the reconstructionalgorithm is to achieve statistical equivalence by matching the pattern andstatistics of the original and reconstructed microstructure across the entireimage domain. Thus, even with a statistically equivalent reconstructed im-age, differences in small localized regions can lead to significant differences in22he heat and fluid flow channels, resulting in very different effective thermaland hydraulic conductivity. Overall, if the error values are compared amongall the four reconstruction cases for each material property, it is seen thatthe “GM+TV” loss case with the denoised slices as the target images yieldsthe most accurate reconstructions.
7. Discussion
Results in the previous sections have shown the performance of the pro-posed reconstruction approach to be reasonably accurate and efficient. Onefurther advantage of the proposed approach is that the reconstructed imageis not constrained to be of the same size as the original image. Obtaining ahigh resolution image of a microstructure can be expensive, which limits thesize and/or resolution of the obtained image. It is often of interest to analyzea microstructural image of dimensions larger than that of the original im-age. One of the useful features of this approach is the ability to reconstructmicrostructure of spatial dimensions different from that of the original mi-crostructural image. Here, as a demonstration, reconstruction of an image ofsize 700 pixels × (a) Original slice ×
902 pixels ) (b) Reconstructed image (700 × )Figure 19: Comparison of the original and a larger reconstructed microstructure a) Two-point probability functionfor the porous phase (b) Pore size complementary cdfFigure 20: Comparison of statistical properties between the original slice pore size complementary cdf plot for the two images are compared and shownin figure 20.Another advantage of the proposed approach is that it is more efficientand accurate than other existing approaches based on statistical features ofthe microstructure. The approach is thus compared here with a modified ver-sion of the YT method [9, 10, 3] in terms of efficiency and accuracy. The YTmethod initiates by generating an image having the same porosity measure asthe target microstructure. It is an iterative method where in each iteration,a randomly chosen black solid phase pixel and a white porous phase pixel areinterchanged in the image. If the pixel swapping leads to an improvement inthe statistical measure (equivalent to a reduction in loss function measure) ofthe new image compared to the previous one, the new image is updated to bethe current reconstructed image. Otherwise, the previous image remains thecurrent reconstructed image at the end of that iteration. The pixel swappingis also allowed if a random number generator, generating random numbersuniformly between 0 and 1, returns a value greater than or equal to p I . Here p I is defined as the pixel interchange probability. The iterative convergenceof the reconstructed image to the target image is measured by the root ofsum of squares error (RSSE) between the two-point correlation function ofthe reconstructed image and the target image. With increase in the num-ber of iterations, RSSE has a decreasing trend and the reconstructed image24 a) Original slice converges closer to the target microstructure with respect to the two-pointcorrelation function measure. Figure 21 shows a visual comparison of thereconstructed images formed by using the proposed reconstruction approachand the YT method. Slice . Conclusions In this paper, a modified version of an existing transfer learning recon-struction approach [33] is presented by introducing additional loss functionsin the optimization framework. It is then used to reconstruct a binary porousceramic material. A thorough statistical study was done to check the ro-bustness of the algorithm over different initializations and over different mi-crostructure slices (original and denoised) of the same material. Further-more, the reconstruction quality is assessed not only based on the statisticalproperties of the microstructure but also the FEA simulated effective mate-rial properties. This algorithm has the advantage of being easily extendedto 3D microstructure reconstruction [55, 56, 57] if a suitable pretrained 3Dconvolutional network is available. It also has the flexibility of generatingmicrostructures of size different from that of the target microstructure. Apotential future work can be incorporating the simulated effective materialproperty information within the reconstruction algorithm framework in anefficient manner, in order to generate microstructures with effective materialproperties closer to that of the target microstructure.
Acknowledgements
Research was sponsored by the Army Research Laboratory and was ac-complished under Cooperative Agreement Number W911NF-12-2-0023 andW911NF-12-2-0022. The views and conclusions contained in this documentare those of the authors and should not be interpreted as representing the offi-cial policies, either expressed or implied, of the Army Research Laboratory orthe U.S. Government. The U.S. Government is authorized to reproduce anddistribute reprints for Government purposes notwithstanding any copyrightnotation herein.