## 3 Performance Evaluation and Discussion
### Topology Evaluation and LoD Representation  
In the field of computational geometry, topology is a crucial index to determine whether the tunnel can be reconstructed via a series of cross sections. Fortunately, our methodology has the capability to maintain the topology of cross sections in two aspects: (1) For each cross section, the tunnel points are generated by sequentially intersecting a straight line with its local fitted quadric surface. Because of this, the generated points within each cross-sectional plane have topological relationships, i.e., clockwise or counter-clockwise order. Using this type of relation, the geometric shape of a specific cross section can be created. (2) For extraction of multiple cross sections (see Section 2.3.4), we also sequentially generate them along the tunnel axis once given an explicit split interval, thereby forming a series of continuous cross sections. That is, the relationship like the former and later cross sections of the currently processing one is considered, and thus the points from any two arbitrary cross sections have one-to-one correspondence. Using the relationship among multiple cross sections, the entire geometric tunnel shape is expected to be generated. As shown in Figure 12a, the purple triangles shown in the enlarged view are the wire meshes of the tunnel that are waved according to the two topologies above. Because we can simultaneously maintain the point order within each cross section and the order of multiple cross sections, which lays a solid foundation for LoD representation of the metro tunnel. That is, we can use any arbitrary number of cross sections to achieve the abstraction/representation of metro tunnel. Low levels of LoDs with fewer cross sections are lightweight and suitable for storage, web transmission, acceleration of rendering and other tunnel-related applications. In contrast, although the high level of LoDs are not as compact as the shapes with low level LoDs, the geometric accuracy is relatively high. In practice, selection of the optimal LoD is often determined by factors including data acquisition cost, time, accuracy and labor investment and specific tunnel-related applications. For example, when the tunnel models are used in equipment installation on the tunnel linings, the relatively simple tunnel models generated by the sparse cross sections can meet the requirements. When considering tunnel profile deformation, crack and defect detection, and/or water leakage detection, the most detailed tunnel models with high geometric accuracy are expected to meet particular project needs. Furthermore, in the area of tunnel interior visualization, it often needs tunnel models at different LoDs rather than a single LoD. In this case, a multiple LoDs should be generated on the fly to enhance the users' experience. As shown in Figure 12a-g, the tunnel model is rendered  
Figure 11: The evaluation of cross-sectional points through the measure of minimum Euclidean distance. The cross-sectional points are rendered by the different colors according to the deviations from their raw point clouds. To clearly show where the deviation arises from, the different categories are gradually superimposed onto their raw point clouds, denoted by purple. by different LoDs corresponding to profiles of 0.2 m, 0.5 m, 2.0 m, 4.0 m, 6.0 m, 8.0 m and 10.0 m, respectively. Figure 12: The geometric tunnel representation at different LoDs (Level of Details). (**a–g**) are the created tunnel models using the extracted cross sections at the split intervals of 0.2 m, 0.5 m, 2.0 m, 4.0 m, 6.0 m, 8.0 m and 10.0 m. Note that the created tunnel models are superimposed onto their ‘reference model’ denoted by the gray color. The reference model is produced by a screened Poisson surface reconstruction algorithm through using the raw tunnel point clouds. To get an optimal split interval \(\Delta d\), we first define the measures on two tunnel meshes \(\mathcal{X}\) and \(\mathcal{Y}\) to evaluate the similarity of two geometric shapes.