## 2 Deformation Monitoring of Inverted Arch Uplifts of Soft Rock Tunnels  
### Three-Dimensional Point Cloud Splicing of Inverted Arches of Soft Rock Tunnels  
The single point monitoring method only relies on a small number of monitoring points, and cannot obtain sufficient deformation information about the inverted arches of soft rock tunnels. At the same time, in three-dimensional space, because three points can determine the coordinate transformation relationship, when positioning marker points, the number of public marker points in the overlapping area is required to be at least three. According to the pose invariance of the landmark, the relative position between the landmark and the center of the three-dimensional space composed of the landmark set remains unchanged; a reference point is obtained based on this. Set the set of public landmarks of the soft rock tunnel invert as \(P\). The number of \(P\) in public landmarks is \(N_{P}\) and the set of reference public landmarks is \(Q\). The number of \(Q\) in public landmarks is \(N_{Q}\). First, obtain the center of the three-dimensional space \(P\) formed by the mark points in \(\psi_{P}\), the coordinates of which are \(\left(\psi_{Px},\psi_{Py},\psi_{Pz}\right)\). The center of the three-dimensional space \(Q\) is formed by the mark points of \(\psi_{Q}\), the coordinates of which are \(\left(\psi_{Qx},\psi_{Qy},\psi_{Qz}\right)\). Then, use k-nearest neighbors to obtain the three closest points in \(\psi_{P}\) to form a set of landmarks \(PA=\left\{PA_{i}|PA_{i}\in P,i=1,2,3\right\}\), with the other points being \(PA_{j}\). Similarly, \(\psi_{Q}\) obtains the set of three points \(QB=\left\{QB_{j}|QB_{j}\in P,j=1,2,3\right\}\), and the other points are \(QB_{i}\). Additionally, match the \(PA\) and \(QB\) symbols according to Formula (1), where \(\delta\) is the given threshold and \(\left\|\cdot\right\|_{2}\) is the Euclidean distance. The Euclidean distance relationship between the marked point and the reference point is as follows:  
\[\left\|PA_{i}-PA_{j}\right\|_{2}-\left\|QB_{i}-QB_{j}\right\|_{2}<\delta,\ i\
eq j \tag{1}\]Finally, set \(PA\) as the datum point and calculate the point in \(P\) which is at the Euclidean distance from other landmarks. In \(QB\), the mark point is also used as the reference point \(Q\); the Euclidean distance from other marker points to the reference point is used to match other marker point pairs by comparing the distances. If the number of public landmarks is small, it can be matched directly according to Formula (1). Otherwise, automatic registration of public landmarks can be achieved by the above methods [13; 14; 15; 16]. After the automatic matching of landmarks is completed, the geometric transformation matrix needs to be solved for point cloud stitching. The corresponding relationship between the camera and the projector does not change when measuring the three-dimensional point cloud of the soft rock tunnel invert under different viewpoint conditions; thus, only translation transformations and rotation transformations are required when splicing the three-dimensional point cloud of the soft rock tunnel invert, and no scaling transformation is required. To solve the rotation matrix \(R\) and flat matrix \(T\), the improved Rodrigues parameter method is adopted. If the Rodrigues parameter is assumed, \(S\), then  
\[S= \begin{bmatrix}S_{x}&S_{y}&S_{z}\end{bmatrix}  \tag{2}\]  
According to Formula (2), the vector form of the Rodriguez parameter can be obtained [17]. Then, its antisymmetric matrix \(H\) can be expressed as:  
\[H= \begin{bmatrix}0&-S_{z}&S_{y}\\ S_{z}&0&-S_{x}\\ -S_{y}&S_{x}&0\end{bmatrix}  \tag{3}\]  
According to Formula (3), the antisymmetric matrix \(H\) is calculated through the vector form of the [PERSON] parameter, and then the rotation matrix \(R\) is calculated through the antisymmetric matrix \(H\). The calculation formula of the rotation matrix is as follows [18]:  
\[R=