# Line Structure Extraction from LiDAR Point Cloud Based on the Persistence of Tensor Feature
## 4 Experiments and Discussions
### Line Structure Extraction in the Complex Area  
To make a further assessment of the capability for the proposed method, a complex area of the indoor stair scene is collected by the iPhone-based LiDAR sensor, and experiments are conducted. The stair dataset is in an area of 10.61 m\({}^{2}\). To deal with the irregularly distributed sampling density, the dataset is randomly resampled with minimum distance of 0.01 m, and the number of point in the dataset is 141,837, with the average number of 2400 points per m\({}^{2}\). As seen from Figure 8a, there are at least 29 facets in different directions. In addition, some structures are incomplete, and the point density is unevenly distributed, which makes it difficult to extract line structures from the quality-unstable dataset. With neighborhood searching distance \(r\) = 0.1 m, the initial tensor voting process is conducted, followed by the tensor refinement. Then, the geometric feature of each point is computed based on the refined tensor. In Figure 8b-d, the saliency of surface, line, and point of each point is labeled with the color ranging from light to dark. In Figure 8b, the darker color with the low saliency of \"surface\" is considered to be the line structure, while the darker color with high saliency of \"line\" and \"point\" in Figure 8c,d shows a high probability of being the line structure. Hence, the _LCS_ is computed using threshold values \(\left\{\sigma_{surface}=18,\sigma_{line}=14,\sigma_{point}=27\right\}\), as depicted in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Different Results** & **Total Length (m)** & **Effective Length (m)** \\ \hline Figure 7a & 17.16 & 17.16 \\ Figure 7b & 186.94 & 33.50 \\ Figure 7c & 30.47 & 23.79 \\ Figure 7d & 24.49 & 21.39 \\ Figure 7e & 24.46 & 21.36 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The statistic of extracted line structure. As seen in Table 3, the point with surface saliency \(s_{surface}\) lower than \(\sigma_{surface}\), or line saliency \(s_{line}\) higher than \(\sigma_{line}\), or point saliency \(s_{point}\) higher than \(\sigma_{point}\), is selected and taken into the \(LCS\). Besides, the number of selected points by \(\sigma_{surface}\), \(\sigma_{line}\), and \(\sigma_{point}\) and is 1225, 20,137, and 220, respectively, and the total point number without duplication is 21,106. On the other hand, the ground truth \(LCS\) is computed using the spatial relation (within the distance of 0.05 m) between the ground truth line structure and the point cloud, and the total number of points is 48,307. To deal with the quality-unstable point cloud, the \(LCS\) is resampled, and the related \(LCS\)' is constructed with different grid sizes \(\kappa\), as depicted in Figure 9. \begin{table}
\begin{tabular}{c c} \hline \hline
**Different Geometric Feature** & **The Number of Points** \\ \hline \(LCS\) by \(s_{surface}\) & 1225 \\ \(LCS\) by \(s_{line}\) & 20,137 \\ \(LCS\) by \(s_{point}\) & 220 \\ \(LCS\) without duplication & 21,106 \\ ground truth \(LCS\) & 48,307 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The statistic of \(LCS\) by different geometric features in the complex area. Figure 8: The point cloud in the complex area and the geometric saliency in different dimensions. In Figure 9, the \(LCS\)' is computed with different grid sizes \(\kappa\), ranging from 50 to 300. In Figure 9a, it's the constructed \(LCS\) based on Equation (10). In Figure 9b, with \(\kappa=64\), the details are not properly preserved. While in Figure 9d, the connectivity of line structure can be broken with the high grid size \(\kappa=256\), since the space can be over-divided. To make a balance between the details and the connectivity of line structure, the grid size \(\kappa\) is set as 128.