## 5 Analysis of Roughness Detection Method by Kernel
### Discussion  
In this paper, 10 subsamples each having 100,000, 50,000, 30,000, 10,000, 7500, 5000, 2500, and 1000 were randomly extracted from the original point clouds of three samples to identify the density effect in more detail. The distance between the randomly extracted 10 subsampled point clouds was calculated by the Cloud to Cloud (C2C) distance estimation method. The C2C distance is sensitive to the roughness of the point cloud because it directly compares the distance between a point and a point. When flat point clouds with low roughness are compared with each other, the density of the point cloud is an influencing factor to affect the distance estimation ([PERSON], 2021 [14]). However, when calculating the distance between two rough point clouds, the C2C distance is not only affected by the density but also the roughness. A sparse point cloud is obtained on a rough surface under the same resolution conditions. Therefore, the distance between subsamples of a rough surface is estimated to be larger than the distance between subsamples of a flat surface (see Figure 15a). In order to identify the effect of the roughness of grouting, sand, and gravel samples on the C2C distance, the C2C distance between the subsamples was calculated in conjunction with the density of the point cloud. As shown in Figure 15b, the C2C distance is decreased with the decrease of point density in all three cases. The subsampled point clouds having 100,000 points had a C2C distance of about 0.3-0.4 mm, and all three samples obtained almost the same results. However, the C2C distance increased rapidly with the decrease of the number of points, resulting in a C2C distance exceeding 2 mm in the 1000 point result (see Figure 15b). Table 2 shows that the difference of the average C2C distance among the three samples is increased with the decrease of the density. As the density of the point cloud decreases, the distance between the points of the gravel subsamples becomes farther than that of the other samples. Although more points are required to show the proper roughness in the gravel sample than the other samples, the roughness is inevitably underestimated under low density conditions. Therefore, the point  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**No. of Points**} & \multicolumn{3}{c}{**Roughness, mm (95\% of Points)**} & \multicolumn{3}{c}{**Error (\%)**} \\ \cline{2-7}  & **Grouting** & **Sand** & **Gravel** & **Grouting** & **Sand** & **Gravel** \\ \hline
135,000 & 5.73 & 5.94 & 12.09 & - & - & - \\ \hline
100,000 & 5.75 & 5.94 & 12.02 & 0.4 & 0.0 & 0.6 \\ \hline
10,000 & 5.76 & 6.06 & 11.88 & 0.5 & 2.0 & 1.8 \\ \hline
5000 & 5.62 & 5.83 & 11.98 & 1.9 & 1.8 & 0.9 \\ \hline
1000 cloud density of each sample required for proper roughness estimation analysis can be calculated as following the C2C distance presented in Table 2.  
Figure 14: Color scale of roughness with different density of point cloud.  
In this paper, P2 PH and kernel methods were proposed to determine the three-dimensional roughness of the failure surface of pile samples mixed with the sand and the gravel. The failure surfaces of grouting, sand, and gravel samples were distinguished by the proposed methods, and it is expected to contribute to this study to determine not only one-dimensional or two-dimensional roughness but also three-dimensional roughness of failure surface.