Porous media are ubiquitous in both nature and engineering applications. Therefore, their modeling and understanding is of vital importance. In contrast to direct acquisition of three-dimensional (3D) images of this type of medium, obtaining its subregion (s) such as 2D images or several small areas can be feasible. Therefore, reconstructing whole images from limited information is a primary technique in these types of cases. Given data in practice cannot generally be determined by users and may be incomplete or only partially informed, thus making existing reconstruction methods inaccurate or even ineffective. To overcome this shortcoming, in this study we propose a deep-learning-based framework for reconstructing full images from their much smaller subareas. In particular, conditional generative adversarial network is utilized to learn the mapping between the input (a partial image) and output (a full image). To ensure the reconstruction accuracy, two simple but effective objective functions are proposed and then coupled with the other two functions to jointly constrain the training procedure. Because of the inherent essence of this ill-posed problem, a Gaussian noise is introduced for producing reconstruction diversity, thus enabling the network to provide multiple candidate outputs. Our method is extensively tested on a variety of porous materials and validated by both visual inspection and quantitative comparison. It is shown to be accurate, stable, and even fast ($\sim 0.08$ s for a $128\times 128$ image reconstruction). The proposed approach can be readily extended by, for example, incorporating user-defined conditional data and an arbitrary number of object functions into reconstruction, while being coupled with other reconstruction methods.

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.

A random three-dimensional (3D) porous medium can be reconstructed from a two-dimensional (2D) image by reconstructing an image from the original 2D image, and then repeatedly using the result to reconstruct the next 2D image. The reconstructed images are then stacked together to generate the entire reconstructed 3D porous medium. To perform this successfully, a very important issue must be addressed, i.e., controlling the continuity and variability among adjacent layers. Continuity and variability, which are consistent with the statistics characteristic of the training image (TI), ensure that the reconstructed result matches the TI. By selecting the number and location of the sampling points in the sampling process, the continuity and variability can be controlled directly, and thus the characteristics of the reconstructed image can be controlled indirectly. In this paper, we propose and develop an original sampling method called three-step sampling. In our sampling method, sampling points are extracted successively from the center of $5\times 5$ and $3\times 3$ sampling templates and the edge area based on a two-point correlation function. The continuity and variability of adjacent layers were considered during the three steps of the sampling process. Our method was tested on a Berea sandstone sample, and the reconstructed result was compared with the original sample, using tests involving porosity distribution, the lineal path function, the autocorrelation function, the pore and throat size distributions, and two-phase flow relative permeabilities. The comparison indicates that many statistical characteristics of the reconstructed result match with the TI and the reference 3D medium perfectly.