## 2 Methods
### Evaluation Metrics Calculation  
Due to the huge amount of point-cloud data, it is difficult to evaluate the denoising effect of the denoised point cloud manually. Some researchers evaluate the denoising effect of the point cloud by calculating the distribution difference of the point-cloud data before and after denoising. [PERSON], [PERSON] [34] and other researchers quantitatively evaluated the denoising effect of the point cloud by calculating the maximum relative deviation (MRD), average deviation (AD), and mean square error (MSE). [PERSON], [PERSON] [35] evaluated the denoising effect of the point cloud by calculating the maximum positive displacement and maximum negative displacement of the point-cloud projection before and after denoising. To quantitatively evaluate the effectiveness of the Huber loss function-based point-cloud denoising method, this study, based on the aforementioned literature and mathematical statistical principles, employs three evaluation metrics--mean squared error (MSE), root-mean-squared error (RMSE), and mean absolute error (MAE)--to assess whether the fitted ellipse accurately matches the distribution of the original point cloud, and thereby evaluates whether the denoised point cloud retains the geometric features of the tunnel lining. However, due to the presence of non-structural facility point clouds (such as noise points from auxiliary structures) in the original point cloud, these non-structural point clouds introduce calculation errors when evaluating the entire section. This ultimately affects the judgment of the denoising effect for the section. Therefore, directly calculating the three metrics for the entire section point cloud cannot accurately reflect the fitting and denoising performance. In addition to calculating the evaluation metrics for the entire section, this study also selects the local point cloud at the tunnel crown (from 60\({}^{\circ}\) to 120\({}^{\circ}\) of the tunnel cross-section, with the positive x-axis as the 0\({}^{\circ}\) direction) to compute the three evaluation metrics. This part of the tunnel point cloud, from the 60\({}^{\circ}\) to 120\({}^{\circ}\) range at the top of the tunnel, effectively reflects the denoising results of the tunnel lining structure since there are no non-structuralfacilities (noise points). As shown in Figure 10, the red area represents the selected partial point cloud. Due to the presence of noise points (outliers), calculating the error using only the x and y coordinates in the Cartesian coordinate system for each point cloud may result in sudden value changes that affect the evaluation of the point-cloud denoising effect. To mitigate the influence of outliers and comprehensively evaluate the denoising effect of the method in both the \(x\) and \(y\) coordinate directions, this study uses the distance \(r_{0}\) from the points to the center of the fitted circle before and after projection, instead of calculating the evaluation metrics based on the \(x\) and \(y\) coordinates, as shown in Equation (23). \[r_{0}=\sqrt{(x_{i}-h)^{2}+(y_{i}-k)^{2}} \tag{23}\]  
This equation evaluates the difference between the fitted denoised point cloud and the tunnel-lining-structure point cloud by calculating the difference in the radius from each point to the center of the fitted circle before and after projection. Therefore, the calculation of evaluation metrics MSE, RMSE, and MAE based on the radial distances \(r_{0}\) of the point cloud and the projected point-cloud radial distances \(r_{0}\)\({}^{\prime}\) is as follows:  
The MSE is the average of the squared differences between the fitted values and the original values, used to quantify the overall magnitude of the error, as shown in Equation (24). The RMSE is the square root of the MSE, and it has the same units as the original data, making it more intuitive when interpreting the error magnitude. As shown in Equation (25), the lower the value of RMSE, the smaller the error.