## 2 Orthogonal Projection and Global Deformation Acquisition Method of Structures  
### Perspective Transformation of Digital Image  
On the basis of unchanged image content, the image pixel position is transformed, which is called image geometric transformation [19]. It mainly includes translation,rotation, zooming, reflection, and slicing. Usually, compound transformations, such as the perspective transformation, can be divided into a series of basic transformations. According to the perspective principle [20], when photographed under the condition of non-orthogonal projection, the image of the measured structure will deform. As a result, the true shape of the structure can be obtained only when the camera is in the orthogonal projection position of the measured surface. The process of mathematical transformation from oblique projection center to orthogonal projection center is called the perspective transformation. Figure 1 illustrates the basic model of perspective transformation. A 3D Cartesian coordinate system can be established, in which the projection center of the camera is selected as the origin, called the camera coordinate. Meanwhile, the image plane is set as the x-y plane, and the focus of the plane is located at \([0,0,f](f>0)\). A 2D Cartesian coordinate system can be established on the plane where the object is measured, called the measuring coordinate. The origin of the measuring coordinate system is \([x_{0},y_{0},z_{0}]^{T}\) in the camera coordinate. The unit vectors in the \(x\)-axis direction are \([u_{1},u_{2},u_{3}]^{T}\), the unit vectors in the \(y\)-axis direction are \([v_{1},v_{2},v_{3}]^{T}\), and the vector relation can be written as follows:  
\[\left\{ \begin{array}{c}u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}=0\\ u_{1}^{2}+u_{2}^{2}+u_{3}^{2}=v_{1}^{2}+v_{2}^{2}+v_{3}^{2}=1\end{array} \right.. \tag{1}\]  
The points of coordinates \([u,v]^{T}\) in the measuring plane can be expressed in the camera coordinate system as the vector below,  
\[u\left[ \begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array} \right]+v\left[ \begin{array}{c}v_{1}\\ v_{2}\\ v_{3}\end{array} \right]+\left[ \begin{array}{c}x_{0}\\ y_{0}\\ z_{0}\end{array} \right]. \tag{2}\]  
Assuming that the coordinate of the point in the original imaging plane is \([x,y,0]^{T}\), \(\exists k\in R\) the following expression can be derived,  
\[u\left[ \begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array} \right]+v\left[ \begin{array}{c}v_{1}\\ v_{2}\\ v_{3}\end{array} \right]+\left[ \begin{array}{c}x_{0}\\ y_{0}\\ z_{0}\end{array} \right]-\left[ \begin{array}{c}0\\ 0\\ f\end{array} \right]=k\left(\left[ \begin{array}{c}0\\ 0\\ f\end{array} \right]-\left[ \begin{array}{c}x\\ y\\ 0\end{array} \right]\right). \tag{3}\]  
Comparing the preceding formula, it yields  
\[-k\left[ \begin{array}{c}x\\ y\end{array} \right]=u\left[ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right]+v\left[ \begin{array}{c}v_{1}\\ v_{2}\end{array} \right]+\left[ \begin{array}{c}x_{0}\\ y_{0}\end{array} \right]=\left[ \begin{array}{ccc}u_{1}&v_{1}&x_{0}\\ u_{2}&v_{2}&y_{0}\end{array} \right]\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right] \tag{4}\]  
Figure 1: Perspective transformation. \[kf=uu_{3}+vv_{3}+z_{0}-f=\left[ \begin{array}{cc}u_{3}&v_{3}&z_{0}-f\end{array}  \right]\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right]. \tag{5}\]  
Equation (5) can be rewritten as the following,  
\[-k=\left[ \begin{array}{cc}-\frac{u_{3}}{f}&-\frac{v_{3}}{f}&-\frac{z_{0}-f}{f }\end{array} \right]\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right]. \tag{6}\]  
Combining Equations (6) and (4), it leads to  
\[-k\left[ \begin{array}{c}x\\ y\\ 1\end{array} \right]=\left[ \begin{array}{ccc}u_{1}&v_{1}&x_{0}\\ u_{2}&v_{2}&y_{0}\\ -\frac{u_{3}}{f}&-\frac{v_{3}}{f}&-\frac{z_{0}-f}{f}\end{array} \right]\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right]. \tag{7}\]  
For convenience, a parameter matrix M is introduced, as shown below,  
\[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].