## 2 Materials and Methods
### Image Processing Method
#### 2.2.1 Region Growing Method (RG)  
Original CT images are segmented by the region growing method [46; 47; 48]. An image analysis software VGstudioMax3.1 (Volume Graphics GmbH, Kyoto, Japan) is used to conduct the region growing process. An effective voxel selection in a region growing process is essential [49; 50; 51; 52]. The process of region growing method is schematically illustrated in Figure 5. The final segmentation image can be obtained until the total number of seed voxels stops increasing, then the segmented CT image slice can be achieved.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Confining Stress (MPa)** & **Voltage (kV)** & **Current** (\(\mu\)A)** & **Voxel Size** (\(\mu\)m\({}^{3}\))** & **Projection** & **Scan Area** & **Integrated Images for** \\ \hline
0 & 150 & 65 & \(15.1^{2}\times 17.0\) & 4506 & 15.462 & 10 \\ \hline
3 & 160 & 90 & \(18.4^{2}\times 21.0\) & 2253 & 18.824 & 15 \\ \hline \hline \end{tabular}
\end{table}
Table 1: CT scan conditions.  
Figure 4: The extracted CT images (Slice A to Slice E) from the granite rock sample.  
The voxel connectivity (i.e., in this study 26-connected neighborhood) will assimilate the same voxel number in \(x\)-, \(y\)-, and \(z\)-direction into the representative value, and the voxel connectivity in 3-D can be ensured. In contrast, if the CT voxel membership probability is over the threshold, those voxel values can be distinguished. The value of different region in a CT image is listed in Table 3. It includes the mean value of each phase (i.e., \(\sigma_{1}\) and \(\sigma_{2}\)); the standard of the mean value (i.e., \(\mu_{1}\) and \(\mu_{2}\)); and a threshold value of 85 that is decided through the Gaussian distribution. The representative voxel value 118.6 is chosen to represent the rock initial value. The neighboring voxels are assimilated through the distance (i.e., the tolerance \(T=33\)) between the threshold value and the mean value of the rock region. If the adjacent voxels around the initial value are larger than the threshold value of 85, these voxels will also be assimilated into the rock-phase seed groups with this region growing. Then, the rock aperture is extracted from the rock region by this segmentation process.  
The CT value of rock and void region in the 2-D CT image is measured by counting CT voxel number of each region. The mean values of void and rock region are equal to 4.5 and 118.6, and the standard deviation of them are 8.6 and 29.5, respectively. The mean value and the standard deviation are used to determine the threshold parameter of region growing. The normal distribution of void and rock are calculated from the CT value, as the dashed line presented in Figure 6. Therefore, the threshold value is decided from this normal  
Figure 5: The principle of region growing method in separating a CT image (yellow color represents the rock phase, and gray color is the void phase). (**a**) Initial seed voxel of rock. (**b**) Neighbor voxels within the tolerance of rock. (**c**) Neighbor voxels in the rock criteria. (**d**) Neighbor voxels not in the rock criteria. (**e**) Rock seed group search. (**f**) RG segmentation result.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Mean Value** \\ **of Rock \(\sigma_{1}\)** \\ \end{tabular} } & **Mean Value** & \begin{tabular}{c} **Standard** \\ **of Void \(\sigma_{2}\)** \\ \end{tabular} & \
\begin{tabular}{c} **Standard** \\ **Distribution** \\ **of Rock \(\mu_{1}\)** \\ \end{tabular} &
\begin{tabular}{c} **Tolerance** \\ **of Void \(\mu_{2}\)** \\ \end{tabular} & **Threshold** \\ \hline
118.6 & 4.5 & 29.5 & 8.6 & 33 & 85 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Histogram values of CT images.  
distribution, which is equal to 33. From Figure 6, it is noted that the RG segmentation image value corresponds well with the original CT image value.