## 2 Material and Methods
### X-ray Scanning and Digital Image Processing  
Parameters such as air void content, air void number, and average void diameter have been successfully utilized in characterizing the internal structure of asphalt mixtures. These parameters were well established and verified in previous work [16; 22; 23]. However, previous researchers relied on obtaining these parameters from two-dimensional digital image slices, limiting the extent to which the development of damage by freeze-thaw can be detected. This also holds true in cases where the variations in internal structural properties were determined along the height (vertical direction) of the specimens. Considering that the internal structure evolution of asphalt mixture by freeze-damage occurs in three dimensions and without a determined direction, it is necessary to use three-dimensional parameters to characterize the internal structure evolution of asphalt mixture during freeze-thaw cycles. In this paper, the internal structure evolution of samples were characterized by their air void content, air void number, and average void diameter in 3D via image processing, as shown in Figure 3; more details can be found in my previous research [24]. For a precise characterization of the internal structure, samples were scanned before and after freeze-thaw exposure to evaluate the changes in their internal structure with a vertical interval of 0.1 mm along the height of the specimens and with a system resolution of 0.05 mm/pixel. The core sample with a diameter of 50 mm and a height of 90 mm was drilled from the surface of the specimen prepared in the gyratory compact. In addition, the algorithm of [PERSON] [25] was employed to calculate the air voids' volumetric fractals, which are an effective parameter to describe the evolution of void shape during freeze-thaw cycles. Voids' volumetric fractals are a significant parameter to quantify the complexity of void shape caused by the new micro-cracks and micro-voids during freeze-thaw cycles. When the air voids had been reconstructed by MATLAB toolbox, then 26 neighborhoods shown in Figure 4 were taken to discriminate each void and the voids were numbered by the image processing procedure shown in Figure 3, and the voxel of each void was read out. Subsequently, as each voxel represents 0.05 \(\times\) 0.05 \(\times\) 0.05 mm, the volume of each void was calculated, and all results about air void could be output from Matlab toolbox, as shown in Table 3. Finally, the total void volume was obtained by accumulating the volume of each void, and the total volume of the specimen was identified by the procedure. Meanwhile, the equivalent diameter was calculated for each irregularly shaped void volume as a regularly  
Figure 2: Normal fitting of average temperature-proportion. shaped sphere. The 3D void content and average void diameter were calculated by using Equations (1) and (2), where \(V_{Total}\) corresponds to the sum of all voids' volume. \[\text{Air voids content}(\%)=\frac{V_{Total}}{\text{Total specimens volume}}\times 100 \tag{1}\]  
\[\text{Average voids diameter}=2\times\sqrt[3]{\frac{3V_{Total}}{4\pi}} \tag{2}\]  
The voids' volumetric fractals need a specific algorithm based on Equations (3) and (4). Firstly, using a series of cubes with a length of 32 to 1 to cover the 3D void model, the relationship between length (\(\varepsilon\)) and number(N(\(\varepsilon\))) was established, as shown in Equation (3). Then, the form of Equation (3) was changed to Equation (4), and a linear function was used to fit the relationship between \(\lg\) and \(\lg\)(N(\(\varepsilon\))). The absolute value of the slope of linear function is the voids' volumetric fractals. \[N(\varepsilon)=k\varepsilon^{-D} \tag{3}\]  
\[\lg N(\varepsilon)=\lg k-D\lg\varepsilon \tag{4}\]  
Figure 4: Principle of the three-dimensional 26-neighborhood division of connected domains. Figure 3: Imaging processing technique [24].