## 3 Results
### Denoising Effect of Individual Cross-Sectional Point Clouds  
To evaluate the denoising effect of individual cross-sections, five cross-sections were randomly extracted from the tunnel point-cloud data for assessment. At the same time, for the convenience of management and experimental needs, this study named these five cross-sections Sections 1-5, as shown in Figure 13. Each section in the figure is a tunnel point-cloud dataset with a thickness of 0.1 m. In order to ensure the rigor of the experiment, the distance between sections in the figure is a random value. The number of point clouds contained in each cross section is shown in Table 1.  
Figure 12: 3D illustration of original point-cloud data.  
Figure 13: Schematic diagram of cross-section extraction. (unit: meters).  
To visualize the denoising effect of individual cross-section point clouds, Section 1 is taken as an example. The ellipse-fitting and point-cloud denoising effects are shown in Figure 14, where the blue points represent the original point cloud and the red points represent the denoised point cloud after fitting.  
To quantitatively assess the effectiveness of the point-cloud denoising method in this study, three evaluation metrics--MSE, RMSE, and MAE--are introduced. These metrics reflect the similarity between the denoised point cloud and the entire-tunnel cross-section's lining structure. The evaluation results for the entire cross-section are shown in Table 2.  
MSE(m2)RMSE(m)MAE(m) The analysis of the evaluation metric results for the entire cross-section indicates that the point cloud processed by the Huber loss-based ellipse-fitting denoising method is in general agreement with the original tunnel point-cloud lining  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Section Number** & **Section 1** & **Section 2** & **Section 3** & **Section 4** & **Section 5** \\ \hline Point-cloud quantity & 65,583 & 75,196 & 67,447 & 66,256 & 67,689 \\ \hline \end{tabular}
\end{table}
Table 1: Statistics of point-cloud quantity for individual cross-sections.  
Figure 14: Fitting effect of Section 1 (XY plane, unit: meters).  
structure. However, due to the presence of noise points, these evaluation metrics do not accurately reflect the fitting denoising effect on the tunnel lining structure without noise points. Therefore, this study also separately selects the tunnel-top (60\({}^{\circ}\)-120\({}^{\circ}\)) point cloud for evaluation metric calculation, and the results are shown in Table 3.  
From the evaluation metrics in Table 3, it can be seen that the geometric parameters of the tunnel cross-sectional top point cloud, after denoising using the Huber loss-based point-cloud denoising method, are highly consistent with those of the actual tunnel lining structure's top point cloud, with errors reduced to the millimeter level.  
At the same time, the evaluation results in Table 3 are significantly lower than those in Table 2. This is because, in the evaluation metrics calculated for Table 2, the original point cloud used as the benchmark contains noise points such as tunnel pipelines, and these noise points introduce errors in the evaluation metrics, erroneously reflecting the congruence between the denoised point cloud and the actual tunnel lining structure's point cloud, thus leading to inflated values. In contrast, the tunnel-top point cloud contains fewer noise points, so the evaluation metrics computed for this portion more accurately reflect the denoising effect on the tunnel lining structure.  
From the above results, it can be concluded that the ellipse-fitting denoising method based on the Huber loss function achieves good fitting and denoising results for the lining structure point cloud of a single tunnel section.