## 2 Orthogonal Projection and Global Deformation Acquisition Method of Structures
### Perspective Transformation of Digital Image
\tag{8}\]  
If the focus \([0,0,f]^{T}\) is not on the measuring plane, the matrix M is a nonsingular matrix. Under normal working conditions, the focus will not be on the measuring plane, so the matrix M can usually be treated as a nonsingular matrix. The focal length and spatial position of the camera will change when the camera moves to a new position to capture the target structure. It can also be considered that the camera imaging plane is fixed, the focal length and the actual spatial position of the structure are changed. Make the coordinates of the camera focus change to \([0,0,f^{\prime}]^{T}\), and the original coordinates of the measuring plane change to \(\left[x^{\prime}_{0^{\prime}},y^{\prime}_{0^{\prime}},z^{\prime}_{0}\right]^{T}\). The unit vectors of the \(x,y\) axes in the measuring plane become \(\left[u^{\prime}_{1},u^{\prime}_{2},u^{\prime}_{3}\right]^{T}\), \(\left[v^{\prime}_{1},v^{\prime}_{2},v^{\prime}_{3}\right]^{T}\). Similarly, there is \(\exists k\in R\), making the coordinate point \(\left[u,v\right]^{T}\) on the measuring plane, and its corresponding imaging point \(\left[x^{\prime},y^{\prime},0\right]^{T}\) should satisfy:  
\[-k^{\prime}\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\\ 1\end{array} \right]=\left[ \begin{array}{ccc}u^{\prime}_{1}&v^{\prime}_{1}&x ^{\prime}_{0}\\ u^{\prime}_{2}&v^{\prime}_{2}&y^{\prime}_{0}\\ -\frac{u^{\prime}_{3}}{f^{\prime}}&-\frac{z^{\prime}_{3}}{f^{\prime}}&-\frac{ z^{\prime}_{0}-f^{\prime}}{f^{\prime}}\end{array} \right]\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right]. \tag{9}\]  
The parameter matrix M\({}^{\prime}\) is denoted as:  
\[M^{\prime}=\left[ \begin{array}{ccc}u^{\prime}_{1}&v^{\prime}_{1}&x^{\prime }_{0}\\ u^{\prime}_{2}&v^{\prime}_{2}&y^{\prime}_{0}\\ -\frac{u^{\prime}_{3}}{f^{\prime}}&-\frac{z^{\prime}_{3}}{f^{\prime}}&-\frac{z^ {\prime}_{0}-f^{\prime}}{f^{\prime}}\end{array} \right]. \tag{10}\]  
Comparison of Equations (7) and (9) leads to  
\[-k^{\prime}\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\\ 1\end{array} \right]=M^{\prime}\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right]=-kM^{\prime}M^{-1}\left[ \begin{array}{c}x\\ y\\ 1\end{array} \right] \tag{11}\]assuming:  
\[M^{{}^{\prime}}\cdot M^{-1}=\left[ \begin{array}{ccc}m_{11}&m_{12}&m_{13}\\ m_{21}&m_{22}&m_{23}\\ m_{31}&m_{32}&m_{33}\end{array} \right]. \tag{12}\]  
Accordingly, the following expansions are introduced:  
\[\left\{ \begin{array}{l}k^{\prime}x^{{}^{\prime}}=k(m_{11}x+m_{12}x+m_{13})\\ k^{\prime}x^{{}^{\prime}}=k(m_{21}x+m_{22}x+m_{23})\\ k^{\prime}=k(m_{31}x+m_{32}y+m_{33})\end{array} \right.. \tag{13}\]  
Therefore, there is:  
\[\left\{ \begin{array}{l}x^{{}^{\prime}}=\frac{m_{11}x+m_{12}x+m_{12}}{m_{31} x+m_{32}x+m_{33}}\\ y^{{}^{\prime}}=\frac{m_{21}x+m_{22}x+m_{33}}{m_{31}x+m_{32}y+m_{33}}\end{array}  \right.. \tag{14}\]  
The coordinates \((x,y)\) are the imaging point of the original image, which is transformed into a new imaging point \((x^{\prime},y^{\prime})\) after the perspective transformation. Based on the above analysis, the proposed process can convert the original oblique structural image into orthophoto-projection image, which provides technological foundation for monitoring the overall deformation of the structure.