Reconstruction of three-dimensional anisotropic microstructures from two-dimensional micrographs imaged on orthogonal planes
- Veera Sundararaghavan^{1}Email author
DOI: 10.1186/s40192-014-0019-3
© Sundararaghavan.; licensee springer. 2014
Received: 23 December 2013
Accepted: 23 May 2014
Published: 29 June 2014
Abstract
A pervasive method for reconstructing microstructures from two-dimensional microstructures imaged on orthogonal planes is presented. The algorithm reconstructs 3D images through matching of 3D slices at different voxels to the representative 2D micrographs and an optimization procedure that ensures patches from the 2D micrographs meshed together seamlessly in the 3D image. We show that the method effectively models the three-dimensional features in the microstructure using three cases (i) disperse spheres, (ii) anisotropic lamellar microstructure, and (iii) a polycrystalline microstructure. The method is validated by comparing the point probability functions of the reconstructed images to the original 2D image, as well as by comparing the elastic properties of reconstructed image to the experimental data.
Keywords
Microstructure Markov random field Ising model Sampling Reconstruction Statistical descriptorsBackground
Three-dimensional microstructural information is essential for understanding the relationships between the material structure and its properties. Three-dimensional microstructures experimentally characterized by serial sectioning or X-ray computed tomography are expensive for routine applications due to the time and effort involved. The direct problem of measuring 2D surface images using optical or micro-diffraction methods is relatively easier. Using these 2D images, inverse models could be developed that would allow the generation of full 3D microstructural maps and speeding-up the development of microstructure databases for the purposes of microstructure selection and design.
An inverse problem of specific interest in this paper is the reconstruction of 3D microstructures from three orthogonal 2D sectional images taken along the x-, y-, and z-planes. The information contained in these three 2D micrographs is in the form of pixels containing colors corresponding to different constituent phases. The outcome of the inverse problem is a 3D microstructure containing voxels colored consistently such that any arbitrary x-, y-, or z-slice ‘looks’ similar to the corresponding input micrographs. This reconstruction problem leads to anisotropic microstructures, which is in contrast to other such works in literature that use a single reference (2D) image and make assumptions of microstructural isotropy, i.e., slices in every direction look similar to a single input image [1]. The most popular among these methods involves matching statistical features like two-point correlation functions of a single planar image to a random 3D image using optimization procedures like simulated annealing [2],[3]. Extension of these methods to achieve anisotropic microstructures has been proposed in the past using directionally dependent statistical features [4]. However, these methods are restricted to simple two-phase microstructures and are not applicable to more complex microstructures such as metallic polycrystals.
The approach proposed here involves maximizing the similarity between the solid microstructure and the 2D sectional microstructures by minimizing a neighborhood cost function. This cost function ensures that the local neighborhood on 2D slices taken along the x-, y-, or z-directions through the 3D microstructure is similar to some neighborhood in the 2D micrograph imaged along that plane. The approach is similar to those proposed in the computer graphics community [5] based on Markov random field assumption. This assumption simply states that microstructures have a stationary probability distribution or, in other words, different windows taken from a large microstructure ‘look alike’. To synthesize a voxel in the 3D image, the window in the 2D micrograph that best matches the unknown voxel’s neighborhood is chosen. The color of the voxel is decided based on the color indicated by the matching window in the 2D input image. The result is a simple method for generating 3D microstructures from 2D micrographs that generates visually striking 3D reconstructions of anisotropic microstructures, is computationally efficient, and is applicable to diverse microstructures.
Methods
Mathematical modeling of microstructures as Markov random fields
Note that the above definition does not warrant the neighbor particles to be close in distance, although this is widely employed for physical reasons. For example, in the classical Ising model, each particle is bonded to the next nearest neighbor as shown in Figure 1a. In this work, we assume that a microstructure is a higher-order Ising model (Figure 1b). The particles of the microstructure correspond to pixels of the 2D image (or voxels in 3D). The neighborhood of a pixel is modeled using a square window around that pixel and bonding the center pixel to every other pixel within the window. The window size is a parameter that is chosen based on the scale of the biggest regular feature (e.g., grain size). Using this graph structure, a Markov random field can be defined as the joint probability density P(X) on the set of all possible colorings X, subject to a local Markov property. The local Markov property states that the probability of value X_{ i }, given its neighbors, is conditionally independent of the values of all other particles. In other words, P(X_{ i }|all particles except i)=p(X_{ i }|neighbors of particle i). The microstructures are obtained by sampling the Markov random field P(X). In this paper, we present an algorithm to sample 3D microstructures by selecting the color of pixel X_{ i } by sampling the conditional probability density p(X_{ i }|neighbors of voxel i) from available 2D experimental data.
Algorithm
In the following discussion, let S^{ x }, S^{ y }, and S^{ z } denote the set of orthogonal (x, y, and z, respectively) slices of the microstructure. Let V denote the solid (3D) microstructure. The color of voxel v in the 3D microstructure is denoted by V_{ v }. In addition to the color (e.g., RGB triplet), the vector V_{ v } may also contain other values including grain orientation and phase index. In this work, the color is represented using G color levels in the range {0,1,..,G−1}, each of which maps to an RGB triplet. The number of color levels is chosen based on the microstructure to be reconstructed, e.g., for binary images G=2.
Here, ${\omega}_{v,u}^{i}$ denotes a per pixel weight. In order to preserve the short-range correlations of the microstructure as much as possible, the weight for the nearby pixel is taken to be greater than those of the pixels farther away (Gaussian weighting is used).
This is an exhaustive search that compares all the windows in the input 2D micrograph to the corresponding x-slice neighborhood of voxel v and identifies a window that leads to a minimum weighted squared distance. In this process, for 2D images of size 64×64 with a 16×16 neighborhood window, a matrix of size 16^{2}×(64−16)^{2} is built containing all possible neighborhoods of pixels that have a complete 16^{2} window around it. The column in this matrix that has a minimum distance to the 3D slice ${\mathit{V}}_{v}^{x}$ is then found through a k-nearest neighbor algorithm [9]. Note that we are only given a limited (in this work, a single) 2D experimental sample along each cross-section, which means that the best match may not be an exact match for ${\mathit{V}}_{v}^{x}$.
Note that the subscripts u and v are switched in the above expression as compared to Equation 1. This implies that the optimal color of the voxel v is the weighted average of the colors at locations corresponding to voxel v in the best matching windows (${\mathit{S}}_{u}^{i}$) of voxels (u) in the solid microstructure. Since V_{ v } changes after this step, the set of closest input neighborhoods ${\mathit{S}}_{v}^{i}$ will also change. Hence, these two steps were repeated until convergence, i.e., until the set ${\mathit{S}}_{v}^{i}$ stops changing. As a starting condition, a random color from the input 2D images is assigned to each voxel v. The process is carried out in a multiresolution (or multigrid) fashion [10]: starting with a coarse voxel mesh and interpolating the results to a finer mesh once the coarser 3D image has converged to a local minimum. Three resolution levels (16^{3}, 32^{3}, and 64^{3}) were used. Synthesizing a 64^{3} solid microstructure took between 10 and 15 min on a 3-GHz desktop computer, with about two-thirds of the time spend in step 1 (search) algorithm.
Results and discussion
- 1.
Case 1. An isotropic distribution of solid circles;
- 2.
Case 2. An anisotropic case with solid circles in the z-slice (similar to case 1) but an interconnected lamellar structure in the x- and y-slices;
- 3.
Case 3. A polycrystalline microstructure.
Validation tests
For testing the validity of the 3D reconstructions, quantitative comparisons were made between the original 2D image and the reconstructed image, through comparison of the statistical correlation functions as described in [11]. Rotationally invariant probability functions are employed as the microstructural features. Rotationally invariant N-point correlation measure (${S}_{\left(N\right)}^{i}$) can be interpreted as the probability of finding the N vertices of a polyhedron separated by relative distances x_{1},x_{2},...,x_{ N } in phase i when tossed, without regard to orientation, in the microstructure. The simplest of these probability functions is the one-point function, S_{(1)}, which is just the volume fraction (V) of phase i. The two-point correlation measure, ${S}_{\left(2\right)}^{i}\left(r\right)$, can be obtained by randomly placing line segments of length r within the microstructure and counting the fraction of times the end points fall in phase i. These statistical descriptors occur in rigorous expressions for the effective electromagnetic, mechanical, and transport properties like effective conductivity, magnetic permeability, effective elastic modulus, Poisson’s ratio, and fluid permeability of such microstructures ([4],[12]). All the required correlation measures needed for comparison and property bound calculation are obtained using a Monte Carlo sampling procedure [1]. The procedure involves initially selecting a large number of initial points in the microstructure. For every initial point, several end points at various distances are randomly sampled, and the number of successes (of all points falling in the i^{ t h } phase) are counted to obtain the required correlation measures. Statistical measures were extracted from the microstructures by sampling 15,000 initial points.
Two- and three-point correlations of the isotropic distribution of solid circles
Elastic properties of two-phase composite
Elastic properties of silver and tungsten phases as a function of temperature (from [[14]])
T(°C) | E_{silver}(GPa) | ν _{silver} | E_{tungsten}(GPa) | ν _{tungsten} |
---|---|---|---|---|
25 | 71 | 0.36 | 400 | 0.28 |
200 | 69 | 0.36 | 392 | 0.28 |
400 | 63 | 0.36 | 383 | 0.28 |
600 | 54 | 0.36 | 373 | 0.28 |
800 | 45 | 0.37 | 363 | 0.28 |
860 | 42 | 0.37 | 361 | 0.28 |
910 | 39 | 0.37 | 359 | 0.28 |
950 | 37 | 0.37 | 357 | 0.28 |
Conclusions
A method for reconstructing diverse microstructure from two-dimensional microstructures imaged on orthogonal planes is presented. The algorithm reconstructs 3D images through matching of 3D slices at different voxels to the representative 2D micrographs. This is posed as an iterative optimization problem where the first step involves searching of patches in the 2D micrographs that look alike to the 3D voxel neighborhood, followed by a second step involving the optimization of an energy function that ensures various patches from the 2D micrographs meshed together seamlessly in the 3D image. The method is particularly promising for anisotropic cases where the x-, y-, and z-slices look different. The results demonstrate that the method can effectively model three-dimensional features in the microstructure including complex interconnectivity of the features and complex shapes that are not intuitive at first sight. The approach can be useful to rapidly build a library of 3D microstructures for modeling purposes from 2D micrographs. Although, this preliminary study shows significant promise as to the feasibility of the approach, future work will focus on increasing the resolution of the reconstruction and code optimization. In addition, future work will focus on more rigorous testing of the stereological features (e.g., grain-size histograms) and other engineering properties (yield strength) of reconstructed anisotropic microstructures.
Availability of supporting data
The executables and data files for the methodology described here are available upon request.
Author’s information
VS is an Associate Professor at the University of Michigan and is a lifetime member of The Minerals, Metals and Materials Society (TMS). VS developed the methods and examples shown in this work.
Declarations
Acknowledgements
The author would like to acknowledge the Air Force Office of Scientific Research, MURI contract FA9550-12-1-0458, for the financial support.
Authors’ Affiliations
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