# Research on Damage Localization of Steel Truss-Concrete Composite Beam Based on Digital Orthoimage
## 3 Static Test of a Steel Truss-Concrete Composite Beam
### Image Data Acquisition
\right]\right) \tag{10}\]  
The following equations can be obtained. \[-k\left[ \begin{array}{c}x\\ y\end{array} \right]=u\left[ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right]+v\left[ \begin{array}{c}v_{1}\\ v_{2}\end{array} \right]+\left[ \begin{array}{c}x_{0}\\ y_{0}\end{array} \right]=\left[ \begin{array}{ccc}u_{1}&v_{1}&x_{0}\\ u_{2}&v_{2}&y_{0}\end{array} \right]\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right] \tag{11}\]  
Figure 17: Image perspective transformation model. \[kf=uu_{3}+vv_{3}+z_{0}-f=\left[ \begin{array}{cc}u_{3}&v_{3}&z_{0}-f\end{array}  \right]\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right] \tag{12}\]  
From Equation (12), we then obtain the following:  
\[-k=\left[ \begin{array}{cc}-\frac{u_{3}}{f}&-\frac{v_{3}}{f}&-\frac{z_{0}-f}{ f}\end{array} \right]\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right] \tag{13}\]  
Combining this with Equation (11) gives  
\[-k\left[ \begin{array}{c}x\\ y\\ 1\end{array} \right]=\left[ \begin{array}{ccc}u_{1}&v_{1}&x_{0}\\ u_{2}&v_{2}&y_{0}\\ -\frac{u_{3}}{f}&-\frac{v_{3}}{f}&-\frac{z_{0}-f}{f}\end{array} \right]\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right] \tag{14}\]  
Introducing the parameter matrix \(M\), we obtain  
\[M=\left[ \begin{array}{ccc}u_{1}&v_{1}&x_{0}\\ u_{2}&v_{2}&y_{0}\\ -\frac{u_{3}}{f}&-\frac{v_{3}}{f}&-\frac{z_{0}-f}{f}\end{array} \right] \tag{15}\]  
If the measuring plane does not pass through the focal point \([0,0,f]^{T}\), then the matrix M must be an invertible matrix. Under normal operation, the focal point does not lie on the measuring plane, so the matrix \(M\) is always an invertible matrix. When the camera moves to a new position to photograph the target structure, the relative spatial position of the camera and the target structure changes. The change can be equated to the condition whereby the camera imaging plane does not move, while the focal length and the actual spatial position of the target structure change accordingly. Let the coordinates of the camera focus become \([0,0,f^{\prime}]^{T}\) and the origin of the measuring plane become \([x^{\prime}_{0},y^{\prime}_{0},z^{\prime}_{0}]^{T}\). The unit vectors in the \(x\), \(y\) axis of the scenic plane become \([u^{\prime}_{1},u^{\prime}_{2},u^{\prime}_{3}]^{T}\) and \([v^{\prime}_{1},v^{\prime}_{2},v^{\prime}_{3}]^{T}\), respectively. Similarly, \(\exists k^{\prime}\in R\) makes the coordinate point \([u,v]^{T}\) on the measuring plane and the point \([x^{\prime},y^{\prime},0]^{T}\) of the imaging plane corresponding to the point \([u,v]^{T}\) satisfy the following equation. \[-k^{\prime}\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\\ 1\end{array} \right]=\left[ \begin{array}{ccc}u^{\prime}_{1}&v^{\prime}_{1}& x^{\prime}_{0}\\ u^{\prime}_{2}&v^{\prime}_{2}&y^{\prime}_{0}\\ -\frac{u^{\prime}_{3}}{f}&-\frac{v^{\prime}_{3}}{f}&-\frac{z^{\prime}_{0}-f^{ \prime}}{f}\end{array} \right]\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right] \tag{16}\]  
Then, the parameter matrix \(M^{\prime}\) is shown as  
\[M^{\prime}=\left[ \begin{array}{ccc}u^{\prime}_{1}&v^{\prime}_{1}&x^{\prime }_{0}\\ u^{\prime}_{2}&v^{\prime}_{2}&y^{\prime}_{0}\\ -\frac{u^{\prime}_{3}}{f}&-\frac{v^{\prime}_{3}}{f}&-\frac{z^{\prime}_{0}-f^{ \prime}}{f}\end{array} \right] \tag{17}\]  
Combining Equations (14) and (16) yields  
\[-k^{\prime}\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\\ 1\end{array} \right]=M^{\prime}\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right]=-kM^{\prime}M^{-1}\left[ \begin{array}{c}x\\ y\\ 1\end{array} \right] \tag{18}\]  
Assuming  
\[M^{\prime}\cdot M^{-1}=\left[ \begin{array}{ccc}m_{11}&m_{12}&m_{13}\\ m_{21}&m_{22}&m_{23}\\ m_{31}&m_{32}&m_{33}\end{array} \right] \tag{19}\]  
Then, we can obtain \[\left\{ \begin{array}{l}k^{\prime}x^{\prime}=k(m_{11}x+m_{12}y+m_{13})\\ k^{\prime}y^{\prime}=k(m_{21}x+m_{22}y+m_{23})\\ k^{\prime}=k(m_{31}x+m_{32}y+m_{33})\end{array} \right. \tag{20}\]  
Therefore, there is  
\[\left\{