Fast reconstruction of multiphase microstructures based on statistical descriptors

In this paper, we propose a hierarchical simulated annealing of erosion method (HSAE) to improve the computational efficiency of multiphase microstructure reconstruction, whose computational efficiency can be improved by an order of magnitude. Reconstruction of the two-dimensional (2D) and three-dimensional (3D) multiphase microstructures (pore, grain, and clay) based on simulated annealing (SA) and HSAE are performed. In the reconstruction of multiphase microstructure with HSAE and SA, three independent two-point correlation functions are chosen as the morphological information descriptors. The two-point cluster function which contains significant high-order statistical information is used to verify the reconstruction results. From the analysis of 2D reconstruction, it can find that the proposed HSAE technique not only improves the quality of reconstruction, but also improves the computational efficiency. The reconstructions of our proposed method are still imperfect. This is because the used two-point correlation functions contain insufficient information. For the 3D reconstruction, the two-point correlation functions of the 3D generation are in excellent agreement with those of the original 2D image, which illustrates that our proposed method is effective for the reconstruction of 3D microstructure. The comparison of the energy vs computational time between the SA and HSAE methods shows that our presented method is an order of magnitude faster than the SA method. That is because only some of the pixels in the overall hierarchy need to be considered for sampling.

Figure 1

Schematic illustration of different correlation functions.

Figure 2

The synthesis rule of reference image and the refinement rule of a reconstructed image. (a) is the synthesis rule of the reference image; (b) is the refinement rule of the reconstructed image.

Figure 3

Black pixels represent structural elements. (a) shows the 2D structural elements; (b) shows the 3D structural elements.

Figure 4

The general scheme of hierarchical simulated annealing of erosion method (HSAE) and the comparison of its results against simulated annealing (SA) reconstructions based on the two-point correction functions. In the scheme, N refers to the image width and height in pixels, $s$ is the hierarchical layer, and the gray pixels (the value is 200) are nonfrozen states. The final results for HSAE and SA are the best reconstructions for these methods among five replicas.

Figure 5

(a) is the original image at a full resolution of $256\times 256$. Its corresponding coarser scales are (b) $128\times 128$ and (c) $64\times 64$. (d) is the result of SA reconstruction. The results of the HSAE reconstruction are (e) $64\times 64$, (g) $128\times 128$, and (i) $256\times 256$. The erosion structure as the initial structure of scale 1 and scale 0 are (f) $128\times 128$ and (h) $256\times 256$.

Figure 6

(a) is the original image at a full resolution of $128\times 128$. Its corresponding coarser scales are (b) $64\times 64$ and (c) $32\times 32$. (d) is the result of SA reconstruction. The results of the HSAE reconstruction are (e) $32\times 32\times 32$, (g) $64\times 64\times 64$, and (i) $128\times 128\times 128$. The erosion structure as the initial structure of scale 1 and scale 0 are (f) $64\times 64\times 64$ and (h) $128\times 128\times 128$.

Figure 7

Comparison diagrams of the two-point correlation function (TPCF) for the original and the reconstructed images. (a) is the two-point correlation function $P_{11}$ for the clay phase; (b) is the two-point (clay-pore) correlation function $P_{12}$; (c) is the two-point correlation function $P_{22}$ for the pore phase.

Figure 8

Energy vs computational time (on a HP 348 G7 computer with 1.60 GHz Intel core i5 class machine): the comparison of 3D reconstruction for HSAE and SA methods.