## 4 Discussion
To further discuss the effectiveness of the ellipse-fitting and point-cloud denoising method proposed in this study, we compare it with two widely used fitting and denoising methods: the least-squares method and random sample consensus (RANSAC) method. Figure 16: CloudCompare user interface diagram. The least-squares method is a fitting approach based on the sum of squared errors. It minimizes the squared differences between the actual data values and the fitted function to obtain the optimal fitting ellipse. By projecting the point cloud onto the fitted ellipse, denoising is achieved. This method is widely used in various data fitting tasks due to its simple algorithmic structure and other advantages. As described in the previous section, this study extracted five sections from a 40 m long tunnel for denoising. In order to more intuitively and scientifically compare the denoising effects of the least-squares method and the method proposed in this study, this study performed denoising on the same section (Section 1 in 3. Results). The denoising result is shown in Figure 18, where the blue points represent the original point cloud and the red points represent the denoised point cloud after fitting. Random sample consensus (RANSAC) is an iterative fitting method based on random sampling. It randomly selects a subset of data, treating points that are close to the fitted ellipse as inliers and points that are far from the fitted ellipse as outliers (noise points). Through iterative sampling and estimation of model parameters, RANSAC ultimately  
Figure 17: CloudCompare software SOR filter denoising effect diagram (unit: meters). Figure 18: Denoising effect of point-cloud fitting using the least-squares method (unit: meters). obtains the optimal fitted ellipse. This method is widely used in fitting and denoising tasks due to its robustness to noise, ability to handle various fitting models, and other advantages. In order to compare the denoising effect of the method proposed in this study, the RANSAC method is used to denoise Section 1 extracted in the previous stages (or steps) of the work. The denoising results are shown in Figure 19, where the blue points represent the original point cloud and the red points represent the denoised point cloud after fitting. As shown in the above figures, both the least-squares method and RANSAC method can perform fitting and denoising on a single subway tunnel section. To quantitatively assess the denoising effectiveness of these two methods in comparison with the method proposed in this study, the evaluation metrics are calculated for the point cloud of all cross-sections of the entire tunnel as well as the selected top tunnel point cloud (60\({}^{\circ}\)-120\({}^{\circ}\)) of each section. The average values of the evaluation metrics for all cross-sections are calculated, and the results are compared with those of the Huber loss-based method proposed in this study. The calculation results are shown in Tables 6 and 7. In addition, this study also recorded the time required for these three methods to process the same data (the entire 40 m long tunnel point cloud) under the same experimental conditions, including computer performance and system environment. The results are shown in Table 8. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fitting Denoising Method** & **MSE(m\({}^{2}\))** & **RMSE(m)** & **MAE(m)** \\ \hline Huber loss-based method & 0.04920 & 0.21655 & 0.13257 \\ Least-squares method & 0.01958 & 0.13964 & 0.10187 \\ RANSAC method & 0.01961 & 0.13969 & 0.10066 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistical table of evaluation metrics for denoising of the whole tunnel section point cloud. Figure 19: Denoising effect of point-cloud fitting using the RANSAC method (unit: meters).