# Line Structure Extraction from LiDAR Point Cloud Based on the Persistence of Tensor Feature
## 4 Experiments and Discussions
### Comparisons with Other Line Structure Extraction Methods  
With the \(LCS^{\prime}\) computed in Section 4.3, the Morse-Smale complex is constructed based on the discrete Morse theory, and the persistence of complex structures is computed. The important geometric structure will keep a high persistence value during the filtration of the Morse-Smale complex structure. Moreover, the persistence of 1-stable manifold is the curve structure for the \(LCS^{\prime}\). Hence, the higher the persistence, the more important the line segment. Finally, line segment \(\left\{p_{critical},l_{connection}\right\}\) is extracted from the \(LCS^{\prime}\) based on the persistence threshold \(\delta\). Then, a graph is built and line structure \(\left\{\iota\right\}\) is computed using the line segment and connection relations, as depicted in Figure 7. \begin{table}
\begin{tabular}{c c} \hline \hline
**Different Geometric Feature** & **The Number of Points** \\ \hline \(LCS\) by \(s_{surface}\) & 86 \\ \(LCS\) by \(s_{line}\) & 3326 \\ \(LCS\) by \(s_{point}\) & 982 \\ \(LCS\) without duplication & 4155 \\ ground truth \(LCS\) & 4006 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The statistic of \(LCS\) by different geometric features. As shown in Figure 7, the ground truth line structure is shown in Figure 7a, and it's manually drawn from the original dataset. Figure 7b shows the result constructed based on the point cloud segmentation method [23] (with parameter [Max angle: 20\({}^{\circ}\), Max relative  
Figure 6: The \(LCS\) and \(LCS^{\prime}\) with different t grid sizes \(\kappa\) of the point cloud. Figure 7: The comparison of different line structure extraction results. distance: 1.0 m, Max distance: 0.2 m, Min points per facet: 10, Max edge length 0.03 m)), and the line structure is extracted using the rim of segmented geometric objects, while the result does not seem reasonable when compared with Figure 7a, since there are many mistakes in the extracted line structure caused by the unstable quality of point cloud dataset. Besides, using different persistence threshold \(\delta\), the related line structure is extracted from the _LCS_' based on the discrete Morse theory. Moreover, the result is compared with the ground truth line structure and the 3D point cloud segmentation-based line extraction method. In Figure 7c-e, persistence thresholds of 0.01, 6, and 15 are applied to each result, respectively. Although there are more details in Figure 7c, the result seems not good as there are many over-connections in the line structure. However, it does not mean that \"the lower the threshold \(\delta\), the better the result.\" In Figure 7e, there can be observed many disconnected line segments with a high threshold \(\delta\). Fortunately, the threshold \(\delta\) can be interactively set through a trivial operation, and it performs best in the result shown in Figure 7d with threshold \(\delta\) = 6. Among the results, the line structure extracted based on the proposed method performs superior to others. The statistic of extracted line structure is conducted, as depicted in Table 2. Line extraction results in Figure 7 are quantitatively compared with the ground truth line structure, and only the line segment within the buffer area (0.05 m) of ground truth line structure is considered to be the effective line structure. The total length of the extracted and effective line structure is calculated. The length of line structure extracted by each method, ranging from Figure 7b to Figure 7e, are 186.94 m, 30.47 m, 24.49 m, and 24.46 m, respectively, while effective lengths is 33.50 m, 23.79 m, 21.39 m, and 21.36 m. Although it has a longer effective length in Figure 7b than the others, there are too many mistakes in the result. In addition, compared with the result in Figure 7d, there are some over-connected line segments in Figure 7c and some disconnected line segments in Figure 7e. Hence, the result of proposed method holds the best performance, and 87.33% of line segments have been effectively extracted from the result.