Two-point Cluster Function Research Articles

Controllable image expansion of rock castings based on deep learning

Abstract Digital rock physics (DRP) offers an effective method of deriving elastic parameters from digital rock images, but its practical application is always limited to limited datasets. Recently, deep learning techniques have presented a promising avenue for generating more extensive and cost-effective samples. However, generating controllable samples according to user definition remains very difficult due to high dependence on sufficient datasets. To resolve this problem, a new network was proposed based on the UNet framework through image translation (UNet-IT) to expand rock castings by given porosity in relatively fewer datasets. Practical tests on carbonate rock images demonstrate that the proposed method can generate samples tailored to specific porosity requirements, which achieved a minimum porosity relative error of less than 1%. Compared with the unextended samples, the generated ones have completely different pore structures in terms of two-point probability, two-point cluster, and lineal path functions. Furthermore, the elastic parameters of the generated images obtained through the finite element method (FEM) and practical logging data matched well, with an average relative error of ∼9%. This indicates that the generated samples can be used as effective data to estimate fine rock physics templates and then improve inversion accuracy.

A new framework for the reconstruction of porous media based on statistical characteristics: Multiscale erosion simulated annealing method

In this paper, a multiscale erosion simulated annealing method (MESA) is proposed, which can improve the computational efficiency of microstructure reconstruction by an order of magnitude. The reconstruction is performed on two-dimensional (2D) and three-dimensional (3D) microstructure based on simulated annealing (SA) and MESA, where the two-point correlation function is selected as the morphological information descriptor. The two-point cluster function containing significant high-order statistical information is used to verify the reconstruction results. From the analysis of 2D reconstruction, it can be found that although the proposed MESA technology can improve the reconstruction efficiency, the reconstruction is still not perfect. That is because the two-point correlation function contains insufficient information. In the 3D reconstruction, the two-point correlation function of the reconstructed 3D microstructure has a good consistency with the original 2D two-point correlation function, indicating that our proposed method is effective for the 3D microstructure reconstruction. By comparing the energy and computation time of SA and MESA, it is shown that our proposed method is one order of magnitude faster than SA method. This is because only a small part of pixels in the overall scales need to be sampled.

The novel continuous reconstruction approach for reconstructing anisotropic porous rocks

In this paper, a novel continuous reconstruction method is proposed to generate a numerical porous rock model that is statistically equivalent to natural rock. First, the spatial and mechanical properties of pore microstructures in rocks are obtained by computed tomography (CT) and triaxial compression tests. Then, the reconstructed rock models are generated based on the novel continuous reconstruction method. The generated rock models preserve the interest characteristics of the microstructures in rock. Compared with the exchange of discrete and isolated elements in our previous contributions, the microstructures are directly continuously reconstructed to form the reconstruction model. With the continuous approach, the natural rock sample can be efficiently represented by the reconstruction model on a large scale. We also consider reconstructing the anisotropic microstructures by different spatial correlation functions, such as the two-point probability function (TPPF), the lineal-path function (LPF), the fractal function (FF), and the two-point cluster function (TPCF). In addition, we introduce the percolating cluster volume (PCV) to improve the characterization capabilities of the proposed reconstruction method. The comparisons among the reference model, reconstructed model and rock samples show that the proposed continuous reconstruction method is able to reconstruct porous rocks while preserving mechanical and topological properties.

Reconstruction of the microstructure based on hierarchical simulated annealing with erosion method

The microstructure of porous media directly affects its macroscopic properties (such as mechanical properties, electromagnetic properties, capillary properties and transport properties). As for sandstone, the distribution of pore has an important effect on the conductivity and transport properties. In this paper, a novel modeling scheme based on hierarchical simulated annealing with erosion method (HSAE), which can reconstruct more accurate microstructure than that of hierarchical simulated annealing method (HSA) and simulated annealing method (SA), is proposed and applied to the optimization of microstructure reconstruction. In the paper, the two-point correlation function (S2(r)) obtained from two-dimensional (2D) reference image is used to reconstruct its corresponding microstructure. From the result of the 2D reconstruction, it is found that the phase distribution for the reconstructed image of our proposed method is more similar to the original image than that of HSA reconstruction and SA reconstruction, which illustrates that our proposed method can generate more similar morphological information as that of the original image. The comparison of the two-point cluster function between the reconstructed microstructures and the reference image indicates that our proposed method can improve the accuracy of reconstruction. Next, HSAE, HSA and SA methods are used to reconstruct the three-dimensional (3D) microstructures from the 2D slices. The comparison of the permeability between the original 3D microstructures and those of the reconstructions shows that the permeability of the reconstructions by our proposed method is more close to that of the original 3D microstructures. The reconstruction time of HSAE method is shorter than that of SA method, and longer than that of HSA method. That is because the pixels participating in HSAE reconstruction is less than that of SA reconstruction, and more than that of HSA reconstruction.

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.

A New Statistical Descriptor for the Physical Characterization and 3D Reconstruction of Heterogeneous Materials

3D reconstruction of heterogeneous materials from 2D images is essential for a precise characterization of their physical properties (mechanical, thermal, electrical, and so on). For this, statistical descriptors such as two-point correlation function (TPCF), lineal path function (LPF), or two-point correlation cluster function (TPCCF) are frequently used. But the effective properties of the reconstructed microstructures are not always corresponding to the real ones as the statistical distribution functions may distribute the material microstructure in a different way from the original one. This is more pronounced for cellular and porous materials such as trabecular bone, fuel cell, and rocks where the connectivity between clusters is not well correlated with the one of real material and degrades the materials physical behavior predictions. This paper proposes a new statistical descriptor, called quality of connection function (QCF), able to determine the quality of connections between clusters and has detailed statistical information about the microstructure distribution. The proposed descriptor is tested on trabecular bone obtained from X-ray micro-computed tomography and used as an example of heterogeneous material having a complex microstructure. Effective properties such as Young’s modulus were calculated for these microstructures and compared with real ones. The new descriptor shows improved capacity to describe the material microstructure distribution and prediction of its physical properties.

A modified simulated annealing algorithm for hybrid statistical reconstruction of heterogeneous microstructures

Mathematical descriptors of complex microstructures easily obtained from limited microstructural information offer next-generation, inexpensive solutions for digital material reconstruction. In this work, we propose a versatile method for accurate digital reconstruction and design of material’s microstructure. The method uses a combination of the full set of two-point correlation function (TPCF) and two-point correlation cluster function (TPCCF) applicable to two-dimensional (2D) and three-dimensional (3D) reconstructions. We employ a modified Simulated annealing (SA) algorithm to accelerate the optimization procedure using a novel probability-based neighborhood generator consisting of three different neighborhood functions. We demonstrate the applications of the method in reconstruction of solid oxide fuel cells (SOFC) and similar complex-microstructure heterogeneous materials. In each case, we demonstrate the accuracy of the method using several distinctive statistical measures, namely TPCF, TPCCF, porosity, and specific surface in 2D and 3D spaces.

Numerical Simulation of GUE Two-Point Correlation and Cluster Functions

Numerical simulations of the two-point eigenvalue correlation and cluster functions of the Gaussian unitary ensemble (GUE) are carried out directly from their definitions in terms of deltas functions. The simulations are compared with analytical results which follow from three analytical formulas for the two-point GUE cluster function: (i) Wigner's exact formula in terms of Hermite polynomials, (ii) Brezin and Zee's approximate formula which is valid for points with small enough separations and (iii) French, Mello and Pandey's approximate formula which is valid on average for points with large enough separations. It is found that the oscillations present in formulas (i) and (ii) are reproduced by the numerical simulations if the width of the function used to represent the delta function is small enough and that the non-oscillating behaviour of formula (iii) is approached as the width is increased.

Super-resolution of real-world rock microcomputed tomography images using cycle-consistent generative adversarial networks.

Digital rock imaging plays an important role in studying the microstructure and macroscopic properties of rocks, where microcomputed tomography (MCT) is widely used. Due to the inherent limitations of MCT, a balance should be made between the field of view (FOV) and resolution of rock MCT images-a large FOV at low resolution (LR) or a small FOV at high resolution (HR). However, large FOV and HR are both expected for reliable analysis results in practice. Super-resolution (SR) is an effective solution to break through the mutual restriction between the FOV and resolution of rock MCT images, for it can reconstruct an HR image from a LR observation. Most of the existing SR methods cannot produce satisfactory HR results on real-world rock MCT images. One of the main reasons for this is that paired images are usually needed to learn the relationship between LR and HR rock images. However, it is challenging to collect such a dataset in a real scenario. Meanwhile, the simulated datasets may be unable to accurately reflect the model in actual applications. To address these problems, we propose a cycle-consistent generative adversarial network (CycleGAN)-based SR approach for real-world rock MCT images, namely, SRCycleGAN. In the off-line training phase, a set of unpaired rock MCT images is used to train the proposed SRCycleGAN, which can model the mapping between rock MCT images at different resolutions. In the on-line testing phase, the resolution of the LR input is enhanced via the learned mapping by SRCycleGAN. Experimental results show that the proposed SRCycleGAN can greatly improve the quality of simulated and real-world rock MCT images. The HR images reconstructed by SRCycleGAN show good agreement with the targets in terms of both the visual quality and the statistical parameters, including the porosity, the local porosity distribution, the two-point correlation function, the lineal-path function, the two-point cluster function, the chord-length distribution function, and the pore size distribution. Large FOV and HR rock MCT images can be obtained with the help of SRCycleGAN. Hence, this work makes it possible to generate HR rock MCT images that exceed the limitations of imaging systems on FOV and resolution.

Accelerating multi-point statistics reconstruction method for porous media via deep learning

Reconstruction of three-dimensional (3D) porous media from a single two-dimensional (2D) image or limited morphological information has been an outstanding problem. Specifically, multi-point statistics (MPS)-based methods have been proved to be effective and promising to address this problem. Besides, recently sampling-based MPS methods that reconstruct 3D microstructure layer by layer have attracted increasing attention, for its relatively lower computational cost and memory demand. However, due to the nature that one has to search and match multi-point information in a data structure like binary tree or list for each simulation, it is very time-consuming, especially for large template used to reconstruct geometrically complex media. In this paper, to overcome this challenge, for the first time to the best of our knowledge, an accelerating reconstruction method based on deep learning is proposed. Specially, conditional generative adversarial networks (CGAN) is leveraged to model and characterize the relation between the sampling image which contains conditioning data and the target void-solid image. Unlike conventional MPS-based method generally performing simulation point by point, our method is able to directly translate the sampling image to the target one instantaneously, thus leading to a significant speedup factor (∼760 for 2D reconstruction and ∼25 for 3D reconstruction on CPU). Meanwhile, to assess the performance of our method, we test it on sandstone samples, and two-point correlation function, lineal-path function and two-point cluster function, as well as local porosity distribution are quantitatively compared with those of the target microstructure. The comparison indicates that our method is of high efficiency while preserving good agreement of statistical functions with that of the target system. We remark that our approach opens the door to a new methodology that incorporating machine learning/deep learning techniques into conventional methods to address CPU- or memory-demanding issues in a wide range of related fields.

Reconstruction of three-dimensional heterogeneous media from a single two-dimensional section via co-occurrence correlation function

Accurately modeling and reconstructing three-dimensional (3D) microstructure of heterogeneous media from limited morphological information obtained from a single two-dimensional (2D) section remains an outstanding problem. Specially, based on simulated annealing (SA) procedure, two-point correlation function S2 is widely used to reconstruct various 3D microstructures in many different cases. However, two-point correlation function is a lower-order statistical descriptor and is generally not sufficient to accurately characterize a microstructure. From the perspective of texture description, a co-occurrence correlation function (CCF) based on gray-level co-occurrence matrix is presented, and four feature functions are derived from it and used as objective function in SA procedure. Moreover, the CCF can be both manageably measured from a single 2D image and embody non-trivial higher-order morphological information. Furthermore, to avoid completely re-computing the CCF in the reconstruction, a fast updating method for CCF is proposed, and thus accelerating the reconstruction process. The four feature functions are utilized to reconstruct different types of media, including square array of nonoverlapping disks, anode microstructure of solid oxide fuel cell and sandstone core. Meanwhile, the two-point correlation function reconstructions are used for comparison. Unconstrained functions including lineal-path function and two-point cluster function are used to quantitatively ascertain the accuracy of the reconstructions, and the comparison of run time is also demonstrated. The results indicate that the proposed method can accurately reconstruct the microstructures with correlation functions being in good agreement with those of the target.

Measuring the food microstructure by two-point cluster function

The clustering of pores of three types of bread was studied by measuring the two point cluster function, C2(r). The ability of this function in describing the connectivity among the structure elements of bread was proved by comparing the C2(r) of a “reference” and a “reconstructed” bread image having the same porosity fraction and lineal path distribution function, L(r). The two point cluster function of the void phase of two commercial, “Pan Brioscè” and “Pancarrè”, and a non commercial, “White”, bread enabled to highlight significant differences of their topological properties. “White” bread was characterized from more homogeneous pores with a lower degree of connectivity while the “Pancarrè” samples exhibited voids highly connected producing pores bigger in size and with a shape extremely complex. Furthermore, for the first time it was proved as the microstructure information of the two point cluster function allows discriminating the 90% of the total variace of bread samples.

Evaluation of effect of glass beads on thermal conductivity of insulating concrete using micro CT images and probability functions

Insulating concrete is a type of concrete that is designed to reduce thermal conductivity. Insulating concrete contains numerous voids that play an important role in reducing heat conduction. Therefore, appropriate nondestructive methods are required to examine the spatial distribution of voids and constituents in a concrete specimen. In this study, an insulating concrete specimen containing hollow glass beads to increase the insulating effect is adopted. Then, micro computed tomography (CT) is used to investigate the spatial distribution of the voids in this specimen. By using a micro CT device, a series of cross-sectional images of the specimen at micrometer-order pixel size are generated by X-rays. To quantitatively describe the spatial distribution of voids in the specimen, probability functions such as two-point correlation, lineal-path, and two-point cluster functions are adopted. In addition, the thermal conductivity of the specimen is evaluated using finite element simulation. The results clarify the insulating effect of glass beads on the concrete specimen and reveal a strong relationship between the probabilistic characteristics of the void distribution and the material responses of insulating concrete.

Universal spatial correlation functions for describing and reconstructing soil microstructure.

Structural features of porous materials such as soil define the majority of its physical properties, including water infiltration and redistribution, multi-phase flow (e.g. simultaneous water/air flow, or gas exchange between biologically active soil root zone and atmosphere) and solute transport. To characterize soil microstructure, conventional soil science uses such metrics as pore size and pore-size distributions and thin section-derived morphological indicators. However, these descriptors provide only limited amount of information about the complex arrangement of soil structure and have limited capability to reconstruct structural features or predict physical properties. We introduce three different spatial correlation functions as a comprehensive tool to characterize soil microstructure: 1) two-point probability functions, 2) linear functions, and 3) two-point cluster functions. This novel approach was tested on thin-sections (2.21×2.21 cm2) representing eight soils with different pore space configurations. The two-point probability and linear correlation functions were subsequently used as a part of simulated annealing optimization procedures to reconstruct soil structure. Comparison of original and reconstructed images was based on morphological characteristics, cluster correlation functions, total number of pores and pore-size distribution. Results showed excellent agreement for soils with isolated pores, but relatively poor correspondence for soils exhibiting dual-porosity features (i.e. superposition of pores and micro-cracks). Insufficient information content in the correlation function sets used for reconstruction may have contributed to the observed discrepancies. Improved reconstructions may be obtained by adding cluster and other correlation functions into reconstruction sets. Correlation functions and the associated stochastic reconstruction algorithms introduced here are universally applicable in soil science, such as for soil classification, pore-scale modelling of soil properties, soil degradation monitoring, and description of spatial dynamics of soil microbial activity.

Spatial Distribution of Voids in Insulating Concrete Analyzed by Micro-CT Images and Probability Functions

Insulating concrete is a multiphase material designed for reduced thermal conductivity, and the void distribution in concrete strongly affects its physical properties such as mechanical response and heat conduction. Therefore, it is essential to develop a method for identifying the spatial distribution of voids. To examine the voids of insulating concrete specimens, micro-CT (computed tomography) images can be effectively used. The micro-CT images are binarized to visualize the void distribution and stacked to generate 3D specimen images. From the obtained images, the spatial distribution of the voids and the microscopic constituents inside the insulating concrete specimens can be identified. The void distribution in the material can be characterized using low-order probability functions such as two-point correlation, lineal-path, and two-point cluster functions. It is confirmed that micro-CT images and low-order probability functions are effective in describing the relative degree of void clustering and void connectivity in insulating concrete.

Reconstruction of multiphase microstructure based on statistical descriptors

The microscopic structural features of porous media directly affect their macroscopic properties (e.g., mechanical, electromagnetic, capillary, and transport properties). As for sandstone, the distributions of the clay and the pore in three-dimensional (3D) space have important effects on its electrical conductivity and transport property. In this paper, the experiment of the two-dimensional (2D) and 3D reconstructions of multiphase microstructure (pore, grain, and clay) based on 2D images was carried out. In the reconstruction procedure of simulated annealing for multiphase microstructure, three independent two-point correlation functions and two lineal-path functions are chosen as the morphological information descriptors. Another important morphological descriptor of microstructure is the two-point cluster function, which is used to validate the reconstruction results. From the result of the 2D reconstruction, it is found that the 2D generation is similar with the original 2D image in morphological information and their correlation functions match very well, which illustrate that the presented method has the ability to generate the 2D microstructure from the original 2D image for multiphase microstructure. In the case of 3D reconstruction, the two-point correlation functions and lineal-path functions of the 3D generation are in excellent agreement with that of the original 2D image, and comparison of the two-point cluster functions from the slices of the 3D generation with the original image reflects the similarity and the variability of the slices in real sandstone, which illustrate that the presented methodology is effective and proper for the reconstruction of 3D microstructure based on the 2D image.

Correlation between low-order probability distribution functions and percolation of porous concrete

Concrete is a multi-phase material whose material properties are strongly affected by phase distribution. It is important to identify the spatial distribution of voids in concrete because voids have a significant effect on mechanical behaviour and percolation. Therefore, a proper method to describe the void distribution of the material is needed. This paper describes how computed tomography (CT) was used to examine and quantify the void distribution of porous concrete specimens. Virtual three-dimensional porous concrete specimens were created using CT. Then, low-order probability distribution functions such as two-point correlation, lineal-path and two-point cluster functions were used to characterise the void distribution. Based on CT image analysis, the correlation of hydraulic conductivity between experiment results and probability distribution functions of porous concrete is examined. It is confirmed that low-order probability functions and percolation of porous concrete are correlated and that the low-order probability functions are efficient and effective in describing percolation within porous concrete.

3-D microstructure reconstruction of polymer nano-composite using FIB–SEM and statistical correlation function

Abstract 3-D reconstruction of Halloysite nanotube (HNT) polypropylene composite has been performed using two different methods. In the first method, several slices of the composite material were obtained using focused ion beam (FIB), and scanning electron microscopy (SEM). A representative volume element (RVE) of the real material’s micro/nanostructures was then constructed by stacking these morphological images using VCAT® software. In the second method, SEM images of the nano-composite were used to extract statistical two-point correlation function (TPCF), for reconstruction of an RVE of the nano-composite. The resulting RVEs obtained from both methods were meshed for finite element (FE) simulation of deformation under tension and shear loadings. The FE results were then used to compute the stiffness tensor of the nano-composite. In the statistical approach, the TPCF was obtained from a none-Eigen microstructure which can partially reflect statistical information of the microstructure. The mechanical constants obtained from statistical RVEs using FEM approach shows a 5.7% error compared with those obtained from real RVE, which could be attributed to the approximation using TPCF [1] . It is concluded that the statistical method using TPCF alone can produce an approximate microstructure that should be modified using other statistical descriptor such as two-point cluster function and lineal path function to have better reconstruction of heterogeneous nano-composites [2] .

투수 콘크리트의 투수성과 확률 분포 함수의 상관관계 분석

다상(multi-phase) 재료인 콘크리트는, 각 구성 성분의 공간적 분포에 의해 재료의 특성이 큰 영향을 받는다. 공극(pore)은 콘크리트의 강도 및 투수성 등에 큰 영향을 주는 요인으로서, 콘크리트 재료의 물성 파악을 위해서 콘크리트에 존재하는 공극의 공간적 분포를 살펴보는 것은 매우 중요하다. 본 연구에서는 투수 콘크리트 시편에 존재하는 공극 분포를 분석하기 위해서 CT(computed tomography)를 사용하여 얻은 단면 이미지를 적층하여 3차원 디지털 시편을 생성하고, 공극의 분포를 관찰하였다. 공극의 공간적 분포 상태를 분석하기 위해서 확률 분포 함수(two-point correlation function, lineal-path function, two-point cluster function)을 사용하여 투수 콘크리트의 공극 분포를 확률적으로 묘사하였다. 서로 다른 공극비(pore ratio)와 공극 분포를 가진 투수 콘크리트 시편들에 대해 CT 이미지를 활용해서 확률적인 공극 분포를 묘사하고, 실제 투수 실험 결과와의 비교를 통해서 투수 콘크리트의 투수성(percolation)과 확률 분포 함수의 상관관계를 분석하였다. 그 결과, CT 이미지 분석과 확률 분포 함수를 이용하여 투수 콘크리트의 투수성 및 공극의 연속적인 분포를 효과적으로 파악할 수 있음을 확인하였다. Concrete is a multi-phase material of which material properties are strongly affected by the phase distribution. Spatial distribution of pore in concrete affects mechanical behavior and percolation significantly. Therefore, a proper method to describe the pore distribution of a material is needed. CT(computed tomography) image is used to examine and to quantify the pore distribution of porous concrete specimens. 3D concrete digital specimens base on the real concrete with different pore ratio are created by subsequent stacking of 2D cross-sectional images from CT. Probability distribution functions such as two-point correlation, lineal-path and two-point cluster functions are used for pore distribution characterization. Correlation between hydraulic conductivity from experiment and probability distribution functions of porous concrete is examined using CT image processing. It is confirmed that probability distribution functions based on CT images are effective in characterizing pore distributions including pore clustering and percolation.

Three-dimensional reconstruction and homogenization of heterogeneous materials using statistical correlation functions and FEM

In this study, a previously developed reconstruction methodology is extended to three-dimensional reconstruction of a three-phase microstructure, based on two-point correlation functions and two-point cluster functions. The reconstruction process has been implemented based on hybrid stochastic methodology for simulating the virtual microstructure. While different phases of the heterogeneous medium are represented by different cells, growth of these cells is controlled by optimizing parameters such as rotation, shrinkage, translation, distribution and growth rates of the cells. Based on the reconstructed microstructure, finite element method (FEM) was used to compute the effective elastic modulus and effective thermal conductivity. A statistical approach, based on two-point correlation functions, was also used to directly estimate the effective properties of the developed microstructures. Good agreement between the predicted results from FEM analysis and statistical methods was found confirming the efficiency of the statistical methods for prediction of thermo-mechanical properties of three-phase composites.