# Three Dimensional Digital Sieving of Asphal Mixture Based on X-ray Computed Tomography
## 5 Calibration of Digital Sieving Technique  
As the shape of aggregate particles is quite complex and diverse, further investigation of the difference between mechanical sieve analysis and the digital sieving technique developed in this research is essential. In order to accomplish this, four groups of epoxy-filled aggregate specimens were prepared using aggregates retained on four different sieve sizes (4.75 mm, 9.5 mm, 13.2 mm and 16 mm, respectively). The number and type of aggregates for different specimens are shown in Table 1. The specimen dimension was 104 \(\pm\) 1 mm in diameter, and the gaps between aggregates were filled using epoxy resin (see Figure 5) which would significantly enhance the contrast between the aggregates and background in the X-ray CT images. After the voxel data of each aggregate was extracted from X-ray CT images of epoxy-filled aggregate specimens, the properties such as volume, equivalent diameter, and DSS were computed. The following equations were used to calculate volume and equivalent diameter:  
\[V\ =\ N_{\text{voxels}}dxdydz, \tag{3}\]  
\[ED\ =\ 2(\frac{3\text{V}}{4\pi})^{\frac{1}{3}}, \tag{4}\]  
where \(V\) is the volume, \(ED\) is the equivalent diameter, \(N_{\text{voxels}}\) is the number of voxels that the given aggregate is made of, and \(dx\), \(dy\) and \(dz\) are the dimensions of one voxel (mm). Figure 6 shows the results of the DSS for four different groups. It can be seen that, in each specimen, there was a certain percent of aggregates whose DSS was larger than the upper limit of their mechanical size. That percentage (counted by number) ranged from 5.69% to 31.43% in seven specimens, with the maximum ratio achieved by Group 1, Specimen 1. Additionally, those aggregates having the largest DSS in Group 2, Specimen 2 and Group 3, both specimens, should be noted because their DSS were over the mid-value of their superior sieve size range. For instance, the largest DSS in Group 3, Specimen 1 was 14.8985 mm, and its superior sieve size range was 13.2-16 mm. As the smallest DSS in Group 3, Specimen 1 was quite close to the lower limit (9.5 mm), the above phenomenon was not caused by the selection of the threshold value in the segmentation process that may increase the dimension of aggregate data. It was also observed that there were overlaps between the DSS distribution ranges of three different groups of granite. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Grouping** & **Specimen 1** & **Specimen 2** & **Type** \\ \hline Group 1 (16\(\sim\)19 mm) & 35 & 36 & Limestone \\ Group 2 (13.2\(\sim\)16 mm) & 73 & 80 & Granite \\ Group 3 (9.5\(\sim\)13.2 mm) & 104 & 138 & Granite \\ Group 4 (4.75\(\sim\)9.5 mm) & 211 & - & Granite \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The number and type of aggregates in specimens. Figure 5: Epoxy-filled aggregate specimens. In order to investigate the simplification of voxel data of each aggregate to its convex hull, 70 aggregates (10 from each specimen) were chosen to calculate the DSS with or without the simplification. It was found that the relative error brought by the simplification was less than one billionth, indicating that treating the aggregate as convex hulls was entirely feasible for the DSS computation. Considering the definition of DSS involved orthographic projection and minimum bounding square, both did the convex basis, the above approaching-zero relative error was not unexpected. It can be concluded that the concave surface of aggregate was the possible reason for the disparity between DSS and mechanical sieve size, as shown in Figure 7.  
Due to difference between the digital and mechanical sieve size, the digital sieving technique was required to be calibrated before it can be employed to evaluate the gradation of the asphalt mixture. For granite, the maximum and minimum DSS in each specimen and their corresponding upper and lower bounds of mechanical sieve size were used for calibration.