# Real-Time Tunnel Deformation Monitoring Technology Based on Laser and Machine Vision
## 2 Methodology
\right.. \tag{1}\]  
In the three-dimensional world coordinate system \(o\)-_xyz_, the laser beam equation is shown as Equation (2), and the equation of the intersecting surface near the measured point is shown as Equation (3), where \(A\), \(B\), \(C\), and \(d\) are constant parameters and are related to the laser position:  
\[A\times x=B\times y=C\times z, \tag{2}\]  
\[z=d. \tag{3}\]  
Figure 3: Mathematical model. Figure 2: Image acquisition module. According to Equations (1)-(3), the coordinates of the laser spot in the image coordinate system \(o^{\prime}\)-\(x^{\prime}y^{\prime}\) can be obtained as shown in Equation (4):  
\[\left\{ \begin{array}{l}x^{\prime}=\frac{C\times d}{A}+c_{0}\\ y^{\prime}=\frac{C\times d}{B}+c_{1}\end{array} \right.. \tag{4}\]  
According to Equation (4), when the positions of laser and image acquisition module are invariant, that is, when the parameters \(A\), \(B\), \(C\), and \(d\) are invariant, the coordinates of the laser spot in the image coordinate system \(\left(\frac{C\times d}{A}+c_{0},\frac{C\times d}{B}+c_{1}\right)\) remain fixed. Therefore, the monitoring reference can be transmitted from the fixed point of the earth to the measured points. When the image acquisition module sinks during tunnel deformation, the image coordinate system moves relative to the world coordinate system \(o\)-\(xyz\). Supposing that the vertical and horizontal deformation amounts of the image acquisition module are \(\Delta x\) and \(\Delta y\), respectively, the laser spot position in the image coordinate system also shifts \(\Delta x\) and \(\Delta y\) in the vertical and horizontal directions, respectively. Therefore, when the image acquisition module shifts in the vertical and horizontal directions along with the measured point, the change of the laser spot position in the image coordinate system reflects the deformation. In an actual situation, the laser beam is not absolutely perpendicular to the laser receiving screen. As a result, when the image acquisition module moves along the \(z\)-axis with the measured point, the laser spot will produce the displacement component in the vertical and horizontal directions. As shown in Figure 3, \(\alpha\) is the angle between the laser beam and the \(ox\) axis, \(\beta\) is the angle between the laser beam and the \(oy\) axis, and \(\theta\) is the angle between the laser beam and the \(\alpha z\) axis. Assuming that the displacement change of image acquisition module along the \(oz\) axis is \(\Delta z\), the errors of the horizontal and vertical deformation measurements are as shown in Equation (5):  
\[\left\{ \begin{array}{l}\Delta x=\Delta z\times\frac{cos\alpha}{cos\beta}\\ \Delta x=\Delta z\times\frac{cos\beta}{cos\beta}\end{array} \right.. \tag{5}\]  
Assuming that the laser beam points obliquely to the receiving screen, the angles \(\alpha\) = 73.3\({}^{\circ}\), \(\beta\) = 81.5\({}^{\circ}\), and \(\theta\) = 18.9\({}^{\circ}\) (in fact, the tilt of the laser beam is not that particularly obvious). When the image acquisition module shifts 1 mm along the \(oz\) axis, \(\Delta x\) = 0.3037 mm and \(\Delta y\) = 0.1562 mm. Therefore, when the shifting displacement of the image acquisition module along the \(z\)-axis is not large (and in fact it is not large) [29], this error can be ignored. The laser transmitter module is mounted on the fixed point of the earth distant from the construction site, and it should remain stable. However, in cases of emergency during construction, such as sudden vibrating, when the laser beam reference is shifted and rotated, the position of the laser spot in the image coordinate will change.