## 2 Deformation Monitoring of Inverted Arch Uplifts of Soft Rock Tunnels
### 3D Point Cloud Reconstruction of Inverted Arches of Soft Rock Tunnels  
The 3D point cloud data of soft rock tunnel inverts collected through multi-station field work [19] need reconstruction to reduce the gross error and improve the plane fitting accuracy of soft rock tunnel inverts. The Mallat algorithm is used to reconstruct the 3D point clouds of inverted arches of soft rock tunnels after splicing [20]. The Mallat algorithm can quickly decompose the 3D point cloud data of soft rock tunnel inverts with wavelets, remove high-frequency information, quickly reconstruct middle- and low-frequency information, and complete 3D point cloud reconstruction of soft rock tunnel inverts. The Mallat algorithm is a fast wavelet decomposition and reconstruction algorithm. The two-scale equation formed by a scale function \(\phi(t)\) and wavelet function \(\varpi(t)\) is given by the multi-scale feature of the wavelet transform, which describes two adjacent scale spaces \(O_{f-1}\) and \(O_{f}\), or adjacent scale space \(O_{f-1}\) and wavelet space \(W_{f}\). The basic functions are \(\phi_{f-1}(t)\), \(\phi_{f}(t)\), and \(\varpi_{f}(t)\) and the internal connection in \(j^{\prime}\) represents the scale. Specifically, the two-scale equation describes the relationship between the low-pass and high-pass filters in the wavelet transform. It represents how the original signal is decomposed and frequency components of different scales are obtained through the low-pass and high-pass filters. This decomposition process can be realized by the multilayer wavelet transform, where each layer of decomposition is performed based on two-scale equations. Additionally, the Mallat algorithm uses the two-scale equations as the basis for signal decomposition. In this algorithm, the signal can be progressively decomposed into frequency components at different scales by repeatedly applying the process of two-scale equations. This allows for a multi-scale analysis and thus a more comprehensive understanding of the time and frequency domain characteristics of the signal. According to the characteristics of multi-scale analysis, \(\phi(t)\) is the wavelet space and \(W_{0}\) is the scale space for the orthogonal basis of \(O_{0}\). Additionally, because \(W_{0}\subset O_{-1}\) and \(O_{0}\subset O_{-1}\), then \(\phi(t)\) and \(\varpi(t)\) must belong to \(O_{-1}\). In other words, \(\phi(t)\), the orthogonal basis of \(\varpi(t)\) and the available space \(\phi_{-1,n}(t)\) can be linearly expanded:  
\[\phi(t)=\sum_{n=1}^{t}h(n)\phi_{-1,n}(t) \tag{7}\]  
\[\varpi(t)=\sum_{n=1}^{t}g(n)T\phi(t) \tag{8}\]  
where \(n\) is the total number of scales and \(h(n)\) and \(g(n)\) are the filter coefficients. Formulas (7) and (8) describe the relationship between adjacent two-scale spatial basis functions [21], and these two equations are called two-scale equations. The two-scale Formula (7) is used to stretch and shift the time:  
\[\phi\left(2^{-j^{\prime}}t-k\right)=\sum_{n=2}^{j^{\prime}}h(n)\sqrt{2}\phi \left(2^{-j^{\prime}+1}t-2k-n\right) \tag{9}\]  
Here, \(k\) is the telescoping and translation time. Set \(m=2k+n\), then:  
\[\phi^{\prime}\left(2^{-j^{\prime}}t-k\right)=\sum_{k=2}^{j^{\prime}}h(m-2k) \sqrt{2}\phi\left(2^{-j^{\prime}+1}t-m\right) \tag{10}\]  
\[O_{j^{\prime}-1}=\overline{span\left\{2^{-\frac{j^{\prime}+1}{2}}\phi^{\prime }\left(2^{-j^{\prime}+1}t-k\right)\right\}} \tag{11}\]  
On the basis of a multi resolution analysis, the 3D point cloud data of any soft rock tunnel's \(\rho(t)\in O_{j^{\prime}-1}\) inverted arch are obtained. The expansion formula of \(O_{j^{\prime}-1}\) is:  
\[\rho(t)=\sum_{k=2}^{j^{\prime}}c_{j^{\prime}-1}2^{-\frac{j^{\prime}+1}{2}} \varpi(t)\phi\left(2^{-j^{\prime}+1}t-k\right) \tag{12}\]  
Here, \(c_{j^{\prime}-1}\) is the expansion coefficient.