## 3 Analysis of Rock Damage Characteristics
### Mesoscopic Analysis of Rock Damage Characteristics  
Rock masses contain both macroscopic and mesoscopic fractures [17]. Taking the CT image at \(h=20\) mm under 0 MPa in situ stress as an example, the CT scan images after segmentation only contain two colors, gray and white, with considerable noise, making it difficult to distinguish whether the white pixels around the cracks are noise or meso-cracks (Figure 8b). During the denoising process, there is a risk of misclassifying noise as meso-cracks or vice versa, which could affect the accuracy of the fractal calculations (Figure 8c). In contrast, using pseudo-color images enhances the visualization of the cracks and allows for a clearer distinction between meso-cracks and noise based on the color gradient (Figure 8d). This facilitates a more accurate denoising process, where noise can be easily separated from the cracks. Additionally, the variation in color intensity in the pseudo-color images reflects the differences in the material density around the cracks, providing further insight into the crack characteristics and improving the accuracy of the quantitative analyses. The calculation of the conventional fractal dimension typically uses segmented images after denoising. However, as mentioned earlier, the denoising process of segmented images may introduce certain errors, which could result in the calculated data not accurately reflecting the actual situation. In this study, the method used is based on the multifractal dimension of grayscale images [18]. Pseudo-color images, after manual denoising, can be converted back into grayscale images while retaining the details of meso-cracks, which reduces some of the errors. Additionally, the multifractal calculation method extends the representation of complex geometric images from a single fractal dimension to multiple fractal dimensions [19]. This approach facilitates a comprehensive analysis of the macroscopic and mesoscopic damage in the rock masses, and characterizes the probability weights of the different parts of the image [20]. In conclusion, this study adopts the multifractal calculation method to investigate the mesoscopic damage characteristics of rocks. There has been substantial research on methods for calculating multifractal dimensions. This study adopts the multifractal calculation methods proposed by [PERSON] [21], [PERSON] [22], and [PERSON] [23] for the calculation and analysis. First, to demonstrate that the blasting damage images of the sandstone specimens conform to multifractal theory, Figure 9 presents the multifractal spectrum of a randomly selected CT image from the stemming segment of the sandstone after blasting under 0 MPa in situ stress. If the \(\delta\) in the partition function \(\chi_{q}(\delta)\) is scale-independent, the mass function \(D(q)\) has a nonlinear relationship with \(q\), and the multifractal spectrum function \(f(\alpha)\) forms a unimodal curve, then the image conforms to multifractal theory. Figure 8: Comparison of meso-cracks between pseudo-color images (bule) and denoised segmented images (white). (**a**) CT scan original image; (**b**) Segmentation image before denoising; (**c**) Segmentation image after denoising; (**d**) Pseudo-colored image. From Figure (a)a, it can be observed that \(\ln\chi_{q}(\delta)\) and \(\ln\delta\) exhibit a linear relationship, indicating that \(\delta\) is scale-independent. From Figure (b)b, the mass function is nonlinear. From Figure (c)c, the multifractal spectrum function \(f(\alpha)\) forms a unimodal curve. Thus, the image conforms to multifractal theory, and the blasting damage of the sandstone can be analyzed using multifractal theory. The multifractal spectrum function \(f(\alpha)\) characterizes the complexity of the fractal structure. When \(f(\alpha)\)'s slope is maximal on the left end, the corresponding horizontal coordinate is defined as the maximum fluctuation singularity index \(\alpha_{\max}\).