## 4 Study of the Microscopic Void Characteristics of PAC
### Void Characteristics Analysis  
The CT images were processed to identify the void structures in the samples [27; 28; 29; 30]. In this paper, the analysis of void characteristics was conducted using CT images that are vertical to the X-axis and the Y-axis. The characteristic parameters of the void were identified by writing programs in MATLAB, including \(N\) as the average number of voids  
Figure 4: CT image segmentation: (**a**) Original CT scan image; (**b**) Identified void structure. Figure 3: This is a figure. Schemes follow the same formatting. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & **Void Content (\%)** & **Peak \(\alpha\)** & **Peak Corresponding Frequency (Hz)** & **Mean \(\alpha\)** \\ \hline A1 & 16.29 & 0.45 & 725 & 0.39 \\ A2 & 18.51 & 0.59 & 910 & 0.42 \\ A3 & 20.78 & 0.88 & 1050 & 0.55 \\ A4 & 23.59 & 0.93 & 1140 & 0.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistical results of the sound absorption coefficient \(\alpha\). within each image, \(a\) as the average area of individual voids and \(d\) as the equivalent diameter of voids. The calculations are as follows:  
\[N=\frac{\sum_{i=1}^{n}N_{i}}{n} \tag{1}\]  
\[A=\frac{\sum_{i=1}^{n}A_{i}}{n} \tag{2}\]  
\[a=\frac{A}{N} \tag{3}\]  
\[d=\frac{a}{I} \tag{4}\]  
where \(N_{i}\) is the total number of voids in the sample's \(i\)-th layer cross-sectional image; \(n\) is the total number of layers in the CT scan; \(A_{i}\) is the total area of voids in the sample's ith layer cross-sectional image; \(A\) is the average area of voids in all cross-sectional pictures of each sample and \(l\) is the average length of each void. To compute the void content of CT scan identification, the total area of voids within each cross-sectional picture was transformed into the void content of each layer, and the average value was determined. The aforementioned formulas only describe the size and number of voids, not their morphological properties. Because the internal voids of PAC have obvious fractal properties [31], in this paper, the fractal dimension was used to evaluate the internal complexity of voids, with the box-counting method in fractal theory as the quantification criterion. The CT images were divided by a square grid with a side length \(\delta\); the number of boxes \(N(\delta)\) occupied by the image boundary with the corresponding side length \(\delta\) was counted separately. The side lengths of the square grid were then changed, which generates different side lengths with corresponding grid numbers. The side lengths \(\delta\) and the grid numbers \(N(\delta)\) were linearly regressed in logarithmic coordinates, with the slope \(D\) of the regression line representing the fractal dimension measured using the box-counting method. The formulations are as follows:  
\[N(\delta)=k\delta^{-D} \tag{5}\]  
taking the logarithm of both sides of Equation (14) simultaneously, that is, Equation (15). \[lgN(\delta)=lgk-Dlg\delta \tag{6}\]  
\[D_{B}=\frac{\sum_{i=1}^{n}D_{i}}{n} \tag{7}\]  
where \(D_{i}\) is the fractal dimension within the i-th layer of the section images of the sample and \(D_{B}\) is the average fractal dimension within all section images of the sample. Table 6 shows the microscopic void characteristics of samples A1-A4. The void content obtained by image recognition technology is very similar to the void content measured by the volume method. The recognition void content is slightly smaller than that obtained by actual measurement, which is due to the limitations of CT scanning accuracy and algorithm recognition accuracy. There are micro-voids that are difficult to identify, which results in a smaller identification void content.