## 2 Image Preprocessing and Shape Measurement  
Prior to the virtual particles' reconstitution, several steps were performed that could provide the proposed algorithm with the fundamental data. Firstly, enough aggregate particles were selected and were scanned by the AIMS to obtain their shape image and the angularity index. The particles were put in the turntable of the AIMS and were rotated slowly to go through the scanning area. A digital camera above the particles was utilized to capture the image of the maximum section from the top side. Then, the original images of the particles were converted to the binary ones within the MATLAB routine as shown in Figure 1. As shown, two forms of binary images were acquired via the image processing technology including the shape (Figure 1a) and its contour (Figure 1b).  
Due to the size differences between the binary images and realistic particles, the processed images should be scaled back based on the actual ones. Because the binary images consisted of pixels only, the sizes were represented by the pixel number at different directions in the images rather than the actual length. In this proposed method, the relationship between the actual length and \"pixel length\" were quantified by uniform standards as one pixel equal to 0.005 mm. After this standard was developed, based on the actual particle size, the required reduction scale of binary images could be calculated to match the actual size, as shown in Table 1. Furthermore, after the images' adjustment completed, the actual area of various particles was measured by the ImagePro software. The black area of the shape image in Figure 1a could be distinguished within the software, and the corresponding area was calculated and stored as shown in Table 1, as well.  
Although there was an original coordinate system after image processing, another two coordinate systems were introduced in this proposed method to help the virtual particles' reconstitution further, including the local coordinate system and global coordinate system. The local one will be introduced in the algorithm part, while the global one is developed as the following. It is necessary to develop a uniform global coordinate system for all particles because the original coordinate system in each image was not the same at all. The central position of each particle was defined firstly as shown in Equation (1):  
\[cen\_x=\frac{\sum_{1}^{n}x_{i}}{n}\ cen\_y=\frac{\sum_{1}^{n}y_{i}}{n} \tag{1}\]  
where \(cen\_x\), \(cen\_y\) are the X- and Y-coordinate values of the particle central position, respectively, in the original coordinate system; \(n\) is the total number of pixels in Figure 1b; \(x_{i}\), \(y_{i}\) are the X- and Y-coordinate values of the \(i\)-th pixel in Figure 1b in the original coordinate system.  
The original coordinate system of each image was converted to the uniform global coordinate system by setting the central positions as the new origin of the coordinates, and the adjustments of the original coordinate are summarized as shown in Table 2. It is noted that the data in Tables 1 and 2 are only a part of the total virtual particles limited by space. The data shown were utilized to illustrate the preparation procedure.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample** & **Angularity** & **Reduction Scale** & **Particle Area (m\({}^{2}\))** \\ \hline
1 & 2864.41 & 4109.07 & 0.00033 \\
2 & 3358.80 & 2882.90 & 0.00040 \\
3 & 1803.25 & 1692.92 & 0.00045 \\
4 & 2354.54 & 2015.89 & 0.00053 \\
5 & 4329.94 & 2174.40 & 0.00038 \\
6 & 3641.75 & 1925.21 & 0.00056 \\
7 & 2192.07 & 3589.41 & 0.00036 \\
8 & 3716.14 & 3527.46 & 0.00043 \\
9 & 2040.78 & 4591.80 & 0.00028 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summaries of the two-dimensional images.  
Figure 1: Binary images of aggregate particles: (**a**) shape; (**b**) shape contour.