# New Four Points Initialization for Digital Image Correlation in Metal-Sheet Strain Measurements
## 2 Theoretical Bases  
The dimensional estimation using stereoscopic vision systems was described in detail by Tsai in [31], who quantitatively concluded that the accuracy and precision relied on the distance between  
Figure 1: Global correlation with an error of image registration transform. Figure 2: The surface view of a metal sheet from a metallographic microscope AX ZEISS @ 200X. the camera and the object to be measured, as well as the resolution of the cameras used. Stereoscopic vision is a technique frequently used to locate points in three dimensions (3D) based on points in two or more 2D images [6; 32]. To achieve a stereoscopic point triangulation, it is necessary to calibrate both cameras. In this paper, we used the PinHole model, which is described by Equation (1) [33]:  
\[s\overline{m}=A[R|t]\widetilde{M}=\left[ \begin{array}{cccc}f_{x}&c&u_{0}\\ 0&f_{y}&v_{0}\\ 0&0&1\\ \end{array} \right]\left[ \begin{array}{cccc}r_{11}&r_{12}&r_{13}\\ r_{21}&r_{22}&r_{23}\\ r_{31}&r_{32}&r_{33}\\ \end{array} \right]\left[ \begin{array}{cccc}t_{x}\\ t_{y}\\ t_{z}\\ \end{array} \right]\widetilde{M}=\left[ \begin{array}{cccc}m_{11}&m_{12}&m_{13 }&m_{14}\\ m_{21}&m_{22}&m_{23}&m_{24}\\ m_{31}&m_{32}&m_{33}&m_{34}\\ \end{array} \right]\left[ \begin{array}{cccc}x\\ y\\ z\\ 1\\ \end{array} \right] \tag{1}\]  
since \(s\) is the number that defines the scale of the objects with respect to their real size in the image, and \([R|t]\) is the matrix of extrinsic parameters of the camera that describe the rigid transformation (rotation and translation) between the coordinate system of the camera and the coordinate system of an object outside the camera. \(A\) is the matrix of intrinsic parameters that describes the position of the center of the image in pixels \((u_{0},v_{0})\), and the ratio of the size of the pixel \((fx,\,fy)\) has units expressed in \(\frac{pixels}{meters}\) in the axes \(x\) and \(y\) regarding the focal distance between the entrance hole of the light and the matrix of the light sensors of the camera. Parameter \(c\) describes the asymmetry of the two axes of a pixel where a zero expresses an angle of 90 degrees. On the other hand, the entry \(M=\widetilde{M}=\left[ \begin{array}{c}x\\ y\\ z\\ 1\\ \end{array} \right]\) is a 3D point \((x,y,z)\) in homogeneous coordinates of the scene or object expressed in meters in its own coordinate system, and \(\overline{m}=\left[ \begin{array}{c}u\\ v\\ 1\\ \end{array} \right]\) is the corresponding 2D point in the image expressed in pixels \((u,v)\). Finally, the correction of the lens distortion is made with a polynomial based on the idea that the distortion changes as a circumference, as in Equations (2) and (3):  
\[\ddot{u}=u+(u-u_{0})\left[k_{1}\left(x^{2}+y^{2}\right)+k_{2}\left(x^{2}+y^{2} \right)^{2}\right] \tag{2}\]  
\[\ddot{v}=v+(v-v_{0})\left[k_{1}\left(x^{2}+y^{2}\right)+k_{2}\left(x^{2}+y^{2} \right)^{2}\right] \tag{3}\]  
Subsequently, to achieve a stereoscopic calibration, the translation vector that joins each camera reference system is calculated [34; 35]. The point position in three dimensions can be estimated from the coordinates in two dimensions and the Equation (4) [36; 37; 38; 36]:  
\[\left[ \begin{array}{cccc}um_{31}-m_{11}&um_{32}-m_{12}&um_{33}-m_{13}\\ vm_{31}-m_{21}&vm_{32}-m_{22}&vm_{33}-m_{23}\\ u^{\prime}m_{31}^{\prime}-m_{11}^{\prime}&u^{\prime}m_{32}^{\prime}-m_{12}^{ \prime}&u^{\prime}m_{33}^{\prime}-m_{13}^{\prime}\\ v^{\prime}m_{31}^{\prime}-m_{21}^{\prime}&v^{\prime}m_{32}^{\prime}-m_{22}^{ \prime}&v^{\prime}m_{33}^{\prime}-m_{23}^{\prime}\\ \end{array} \right]\left[ \begin{array}{c}\hat{x}\\ \hat{y}\\ \hat{z}\\ \end{array} \right]=\left[ \begin{array}{c}m_{14}-um_{34}\\ m_{24}-vm_{34}\\ m_{14}^{\prime}-u^{\prime}m_{34}^{\prime}\\ m_{24}^{\prime}-v^{\prime}m_{34}^{\prime}\\ \end{array} \right] \tag{4}\]  
where \((u,v)\) and \((u^{\prime},v^{\prime})\) are the coordinates of the paired points of the left and right cameras that correspond to the 3D point to be reconstructed.