## 2 Basic Theory
### Initial Displacement Excitation Method
\begin{array}{l}\left(u_{1}m_{31}^{1}-m_{11}^{1}\right)x_{w}+\left(u_ {1}m_{32}^{1}-m_{12}^{1}\right)y_{w}+\left(u_{1}m_{33}^{1}-m_{13}^{1}\right)z_ {w}=m_{14}^{1}-u_{1}m_{34}^{1}\\ \left(v_{1}m_{31}^{1}-m_{21}^{1}\right)x_{w}+\left(v_{1}m_{32}^{1}-m_{12}^{1} \right)y_{w}+\left(v_{1}m_{33}^{1}-m_{13}^{1}\right)z_{w}=m_{14}^{1}-v_{1}m_{34 }^{1}\\ \left(u_{2}m_{31}^{2}-m_{11}^{2}\right)x_{w}+\left(u_{2}m_{32}^{2}-m_{12}^{2} \right)y_{w}+\left(u_{2}m_{33}^{2}-m_{13}^{2}\right)z_{w}=m_{14}^{2}-u_{2}m_{34 }^{2}\\ \left(v_{2}m_{31}^{2}-m_{21}^{2}\right)x_{w}+\left(v_{2}m_{32}^{2}-m_{22}^{2} \right)y_{w}+\left(v_{2}m_{33}^{2}-m_{23}^{2}\right)z_{w}=m_{14}^{2}-v_{2}m_{34 }^{2}\end{array} \right. \tag{15}\]  
As shown in Figure 2, select the square image sub-area to be calculated in the digital image taken by the left camera before deformation. And find the corresponding position in the digital image taken by the right camera before deformation according to the correlation matching between the left and right cameras. According to the pre-calibrated internal and external parameters of the camera, the three-dimensional coordinates (\(x_{0}\), \(y_{0}\), \(z_{0}\)) of the center point of the image subregion. In the same way, taking the picture taken by the left camera before the deformation as the reference picture, the digital images collected by the left and right cameras after the deformation are also accurately tracked to the corresponding position of the calculation area, and the space three-dimensional coordinates of the deformed point can also be obtained (\(x_{1}\), \(y_{1}\), \(z_{1}\)), the difference between the space coordinates before and after the deformation is the three-dimensional displacement (\(x\), \(y\), \(z\)) of this point. In engineering applications, the measured object may undergo major deformation or rotation, and a square reference subregion may no longer be square after deformation. Therefore, the shape function is introduced to correspond the pixels in the reference subregion and the target subregion. The first order shape function allows translation and rotation of the target subregion, as well as uniform shearing and stretching deformation, which is suitable for most situations. Its expression is:  
\[w(\xi,p)=\left[ \begin{array}{ccc}1+u_{x}&u_{y}&u\\ v_{x}&1+v_{y}&v\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}\Delta x\\ \Delta y\\ 1\end{array} \right] \tag{16}\]  
The incremental function \(w=(\xi,\Delta p)\) of the shape function can be expressed as:  
\[w(\xi,\Delta p)=\left[ \begin{array}{ccc}1+\Delta u_{x}&\Delta u_{y}&\Delta u \\ \Delta v_{x}&1+\Delta v_{y}&\Delta v\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}\Delta x\\ \Delta y\\ 1\end{array} \right] \tag{17}\]Use the inverse Newton-Gauss iteration method (IC-GN), and its second order shape function is shown in formula (18). \[ \begin{split} x^{\prime}&=x+u+u_{x}\Delta x+u_{y} \Delta y+\frac{1}{2}u_{xx}\Delta x^{2}+\frac{1}{2}u_{yy}\Delta y^{2}+u_{xy} \Delta x\Delta y\\ y^{\prime}&=y+v+v_{x}\Delta x+v_{y}\Delta y+\frac{1 }{2}v_{xx}\Delta x^{2}+\frac{1}{2}v_{yy}\Delta y^{2}+v_{xy}\Delta x\Delta y \end{split}  \tag{18}\]  
The correlation function is a function that measures the similarity between the reference subregion and the target subregion. The selection of the correlation function directly affects the final accuracy of the calculation. When the function takes the minimum value of 0, it can be considered that there is no difference between the reference subregion and the target subregion at this time, that is, the best match. However, even under the most ideal test conditions, the digital images of stationary objects taken at different times are still different. The reasons can be attributed to camera noise, illumination changes, and surface changes of the specimen due to deformation, and so on. Therefore, the correlation function should have a certain degree of stability in addition to quantifying the difference between the subregions.