Mimicking 3D food microstructure using limited statistical information from 2D cross-sectional image

Measuring structural nonstationarity: The use of imaging information to quantify homogeneity and inhomogeneity

Heterogeneity is the concept we encounter in numerous research areas and everyday life. While "not mixing apples and oranges" is easy to grasp, a more quantitative approach to such segregation is not always readily available. Consider the problem from a different angle: To what extent does one have to make apples more orange and oranges more "apple-shaped" to put them into the same basket (according to their appearance alone)? This question highlights the central problem of the blurred interface between heterogeneous and homogeneous, which also depends on the metrics used for its identification. This work uncovers the physics of structural stationarity quantification, based on correlation functions (CFs) and clustering based on CFs different between image subregions. By applying the methodology to a wide variety of synthetic and real images of binary porous media, we confirmed computationally that only periodically unit-celled structures and images produced by stationary processes with resolutions close to infinity are strictly stationary. Natural structures without recurring unit cells are only weakly stationary. We established a physically meaningful definition for these stationarity types and their distinction from nonstationarity. In addition, the importance of information content of the chosen metrics is highlighted and discussed. We believe the methodology as proposed in this contribution will find its way into numerous research areas dealing with materials, structures, and measurements and modeling based on structural imaging information.

Robust surface-correlation-function evaluation from experimental discrete digital images

Correlation functions (CFs) are universal structural descriptors; surface-surface F_{ss} and surface-void F_{sv} CFs are a subset containing additional information about the interface between the phases. The description of the interface between pores and solids in porous media is of particular importance and recently Ma and Torquato [Phys. Rev. E 98, 013307 (2018)2470-004510.1103/PhysRevE.98.013307] proposed an elegant way to compute these functions for a wide variety of cases. However, their "continuous" approach is not always applicable to digital experimental 2D and 3D images of porous media as obtained using x-ray tomography or scanning electron microscopy due to nonsingularities in chemical composition or local solid material's density and partial volume effects. In this paper we propose to use edge-detecting filters to compute surface CFs in the "digital" fashion directly in the images. Computed this way, surface correlation functions are the same as analytically known for Poisson disks in case the resolution of the image is adequate. Based on the multiscale image analysis we developed a C_{0.5} criterion that can predict if the imaging resolution is enough to make an accurate evaluation of the surface CFs. We also showed that in cases when the input image contains all major features, but do not pass the C_{0.5} criterion, it is possible with the help of image magnification to sample CFs almost similar to those obtained for high-resolution image of the same structure with high C_{0.5}. The computational framework as developed here is open source and available within the CorrelationFunctions.jl package developed by our group. Our "digital" approach was applied to a wide variety of real porous media images of different quality. We discuss critical aspects of surface correlation functions computations as related to different applications. The developed methodology allows applying surface CFs to describe the structure of porous materials based on their experimental images and enhance stochastic reconstructions or super-resolution procedures, or serve as an efficient metrics in machine learning applications due to computationally effective GPU implementation.

Pore-network extraction using discrete Morse theory: Preserving the topology of the pore space

Pore-scale modeling based on the 3D structural information of porous materials has enormous potential in assessing physical properties beyond the capabilities of laboratory methods. Such capabilities are pricey in terms of computational expenses, and this limits the applicability of the direct simulations to a small volume and requires high-performance computational resources, especially for multiphase flow simulations. The only pore-scale technique capable of dealing with large representative volumes of porous samples is pore-network (PNM) based modeling. The problem of the PNM approach is that 3D pore geometry first needs to be simplified into a graph of pores and throats that conserve topological and geometrical properties of the original 3D image. While significant progress has been achieved in terms of geometry representation, no methodology provides full conservation of the topological features of the pore structure. In this paper we present a pore-network extraction algorithm for binary 3D images based on discrete Morse theory and persistent homology that by design targets topology preservation. In addition to methodological developments, we also clarify the relationship between topological characteristics of constructed Morse chain complex and pore-network elements. We show that the Euler numbers calculated for PNMs based on our methodology coincide with those obtained using the direct topological analysis. The characteristics of the extracted pore network are calculated for several 3D porous binary images and compared with the results of maximum inscribed balls-based and watershed-based approaches as well as a hybrid approach to support our methodology.

Adaptive phase-retrieval stochastic reconstruction with correlation functions: Three-dimensional images from two-dimensional cuts

Precise characterization of three-dimensional (3D) heterogeneous media is indispensable in finding the relationships between structure and macroscopic physical properties (permeability, conductivity, and others). The most widely used experimental methods (electronic and optical microscopy) provide high-resolution bidimensional images of the samples of interest. However, 3D material inner microstructure registration is needed to apply numerous modeling tools. Numerous research areas search for cheap and robust methods to obtain the full 3D information about the structure of the studied sample from its 2D cuts. In this work, we develop an adaptive phase-retrieval stochastic reconstruction algorithm that can create 3D replicas from 2D original images, APR. The APR is free of artifacts characteristic of previously proposed phase-retrieval techniques. While based on a two-point S_{2} correlation function, any correlation function or other morphological metrics can be accounted for during the reconstruction, thus, paving the way to the hybridization of different reconstruction techniques. In this work, we use two-point probability and surface-surface functions for optimization. To test APR, we performed reconstructions for three binary porous media samples of different genesis: sandstone, carbonate, and ceramic. Based on computed permeability and connectivity (C_{2} and L_{2} correlation functions), we have shown that the proposed technique in terms of accuracy is comparable to the classic simulated annealing-based reconstruction method but is computationally very effective. Our findings open the possibility of utilizing APR to produce fast or crude replicas further polished by other reconstruction techniques such as simulated annealing or process-based methods. Improving the quality of reconstructions based on phase retrieval by adding additional metrics into the reconstruction procedure is possible for future work.

Mesoscopic Seepage Simulation and Analysis of Unclassified Tailings Pores Based on 3D Reconstruction Technology

Taking the unclassified tailings as the research object, the three-dimensional (3D) pore model was established using computed tomography (CT) scanning technology, image processing, and the 3D reconstruction method. The model was imported into Flac3D software for mesoscopic seepage simulation and analysis. Combined with the laboratory seepage experiment, the influence of tailings' mesoscopic parameters on permeability was explored. The results show that there is a high correlation between the fractal dimension and fragmentation index of tailings pores and the mesoscopic seepage coefficient, with correlation coefficients of 0.987 and 0.973, respectively. When the porosity difference of the pore model is small, the permeability is mainly affected by pore connectivity. The mathematical model between the permeability coefficient and the fragmentation index of tailings is established. The average error between the permeability coefficient calculated by the model and the measured value is reduced to 4.98%, which proves that the mathematical model has guaranteed reliability.

Calculation of tensorial flow properties on pore level: Exploring the influence of boundary conditions on the permeability of three-dimensional stochastic reconstructions

While it is well known that permeability is a tensorial property, it is usually reported as a scalar property or only diagonal values are reported. However, experimental evaluation of tensorial flow properties is problematic. Pore-scale modeling using three-dimensional (3D) images of porous media with subsequent upscaling to a continuum scale (homogenization) is a valuable alternative. In this study, we explore the influence of different types of boundary conditions on the external walls of the representative modeling domain along the applied pressure gradient on the magnitude and orientation of the computed permeability tensor. To implement periodic flow boundary conditions, we utilized stochastic reconstruction methodology to create statistically similar (to real porous media structures) geometrically periodic 3D structures. Stochastic reconstructions are similar to encapsulation of the porous media into statistically similar geometrically periodic one with the same permeability tensor. Seven main boundary conditions (BC) were implemented: closed walls, periodic flow, slip on the walls, linear pressure, translation, symmetry, and immersion. The different combinations of BCs amounted to a total number of 15 BC variations. All these BCs significantly influenced the resulting tensorial permeabilities, including both magnitude and orientation. Periodic boundary conditions produced the most physical flow patterns, while other classical BCs either suppressed crucial transversal flows or resulted in unphysical currents. Our results are crucial to performing flow properties upscaling and will be relevant to computing not only single-phase but also multiphase flow properties. Moreover, other calculation of physical properties such as some mechanical, transport, or heat conduction properties may benefit from the technique described in this study.

Hierarchical Optimization: Fast and Robust Multiscale Stochastic Reconstructions with Rescaled Correlation Functions

Stochastic reconstructions based on universal correlation functions allow obtaining spatial structures based on limited input data or to fuse multiscale images from different sources. Current application of such techniques is severely hampered by the computational cost of the annealing optimization procedure. In this study we propose a novel hierarchical annealing method based on rescaled correlation functions, which improves both accuracy and computational efficiency of reconstructions while not suffering from disadvantages of existing speeding-up techniques. A significant order of magnitude gains in computational efficiency now allows us to add more correlation functions into consideration and, thus, to further improve the accuracy of the method. In addition, the method provides a robust multiscale framework to solve the universal upscaling or downscaling problem. The novel algorithm is extensively tested on binary (two-phase) microstructures of different genesis. In spite of significant improvements already in place, the current implementation of the hierarchical annealing method leaves significant room for additional accuracy and computational performance tweaks. As described here, (multiscale) stochastic reconstructions will find numerous applications in material and Earth sciences. Moreover, the proposed hierarchical approach can be readily applied to a wide spectrum of constrained optimization problems.