## 2 Materials and Methods
### Tunnel Cross-Section Point Cloud Data Preprocessing
#### 2.2.1 Cross-Section Lining Filtering  
Cross-section lining denoising involves various advanced techniques to improve image quality by reducing noise while preserving essential structural details [22; 23]. This process starts with tunnel cross-section lining extraction after data acquisition, which is essential for point cloud displacement analysis [24]. A circular tunnel cross-section is a standard circle, with non-aligned points appearing as deviations from the tunnel lining surface [25; 26]. For tunnel lining extraction, each tunnel cross-section point cloud is selected separately. The lower section of the cross-section point cloud, which contains more tunnel facilities, is removed, while the upper section, with no or fewer non-lining points, is retained. The Segment Tool in CloudCompare 2.7.0 was utilized to isolate specific parts of the 3D point cloud for focused analysis. This tool allows for the selection and segmentation of targeted areas within the dataset, ensuring that only the relevant portions of the point cloud, such as the tunnel cross-section lining, are extracted for further processing. By separating the segmented regions into new entities, this approach optimizes computational resources and streamlines subsequent analysis.  
Figure 2: Point cloud data collection system. (**a**) Mobile Laser Scanning. (**b**) MLS point cloud data.  
For denoising, the statistical outlier removal (SOR) filter was applied to identify and eliminate outliers based on local point density metrics [27]. The SOR filter analyzes the distribution of distances between each point and its neighbors, removing points with distances that significantly deviate from the local statistical distribution. This denoising process improved the dataset's quality and reliability, ensuring accurate subsequent analysis. For each point, \(p\), in the point cloud, the local density was calculated based on the average distance to its \(k\)-nearest neighbors (p\({}_{1}\), p\({}_{2}\), \(\ldots\), pk) [28; 29]. This metric, denoted as \(\mu_{i}\), is expressed as Equation (1).  
\[\mu_{i}=\frac{1}{k}\sum_{j=1}^{k}\left\|p-p_{j}\right\| \tag{1}\]  
where \(\left\|p-p_{j}\right\|\) represents the Euclidean distance between \(p\) and its \(j-th\) nearest neighbor and \(k\) is the number of nearest neighbors considered. Points with lower local density values typically indicate noise or outliers in the dataset. To distinguish outliers from valid points, the global mean local density, \(\mu\)[30], and standard deviation, \(\sigma\), of the entire point cloud were computed using Equation (2).  
\[\mu=\frac{1}{N}\sum_{i=1}^{N}\mu_{i} \tag{2}\]  
where \(N\) is the total number of points in the point cloud. A threshold, \(T\), was then defined to identify outliers, calculated using Equation (3).  
\[T=\mu-\alpha\cdot\sigma \tag{3}\]  
where \(\alpha\) is a scaling factor (set to 3). Points with \(\mu_{i}<T\) were classified as non-lining and removed from the dataset. This approach ensures that only points belonging to meaningful structures within the tunnel cross-sections are retained, improving the reliability of subsequent analyses, including point cloud alignment and displacement measurements. The denoising process significantly enhances the accuracy of deformation monitoring, particularly in detecting cross-section-level displacements.