## 1 Introduction  
The mechanical properties and structural safety of rocks are relevant to many fields, including mining engineering, civil engineering and petroleum engineering. Many accidents occur due to the strength of rocks exceeding their ultimate bearing capacity, and rocks' instantaneous failure often leads to catastrophic consequences. Due to the heterogeneity and opacity of brittle materials, it is difficult to accurately visualize their internal structure and characterize the fracture process [1, 2, 3]. Traditional fracture mechanism research mainly focuses on macro-mechanical tests, such as the uniaxial/triaxial compression test [4, 5, 6, 7, 8] and indoor fracturing model experiments [9, 10, 11, 12]. However, in these studies, it is impossible to predict or determine which components of the sample will interact with the cracks before testing. Numerical methods have been widely used [13, 14] to solve the above problem. However, in most cases, to simplify the numerical calculation, the inhomogeneity of the material is ignored, which will cause a deviation between the simulated crack morphology and the actual crack morphology [15]. Therefore, to consider the influence of heterogeneity in the model, many scholars have adopted different statistical methods to incorporate heterogeneity into the numerical model, and to assign different mechanical properties to the units in the numerical model [16; 17; 18; 19]. For example, heterogeneity was introduced into material through the Weibull statistical model [14]. The above methods are used to characterize the heterogeneity of rock materials; they are random-level models that cannot entirely reflect the actual geometric changes taking place in the microstructures of the materials [20]. Additionally, few documents describe how the virtual structure generated by this method reflects the natural structure of the rocks [21]. The internal microstructure and composition determine the macroscopic mechanical properties of rocks, and the contours of different components determine their macroscopic fracture mechanisms [22]. Therefore, the actual internal structure of rocks should be considered as much as possible in mechanical analysis and fracture research [23]. Reconstructing digital models based on digital images has been applied in fields such as petroleum engineering [15; 24]. Previous studies have shown that digital image data, such as mineral and fracture data, can be used to characterize the heterogeneity of materials quantitatively [25]. Digital image processing (DIP) technology can accurately introduce the heterogeneity of materials into numerical models, and depict the physical qualities and microstructures of actual materials more accurately [26; 27; 28]. These investigations have shown that models that take into account real microscopic heterogeneity may better predict materials' mechanical behavior and fracture reactions. However, these studies are limited to the establishment and research of two-dimensional (2D) numerical models. The accuracy and efficiency of numerical simulation research depend on the computing performance of numerical codes. This method has reduced advantages in three-dimensional (3D) numerical research due to the limitations of computational performance. The coarse mesh of the 3D dimensional model fails to characterize the actual structure of rocks, so this method is not widely used in mechanical and fracture behaviors research. For instance, [PERSON] et al. [28] reconstructed a 3D finite element (FE) numerical model of a jointed rock mass based on X-ray microcomputed tomography (micro-CT), and studied its mechanical behavior. The pixel size of the micro-CT image is 30 \(\upmu\)m, and the element size of the reconstructed finite element numerical model is \(2/3\,\mathrm{mm}\times 2/3\,\mathrm{mm}\times 2/3\,\mathrm{mm}\). The element size of the reconstructed finite element model is about 22 times larger than the pixel size of the original image, which affects the accuracy of the reconstructed finite element model. Ref.