# New Four Points Initialization for Digital Image Correlation in Metal-Sheet Strain Measurements
## 3 Materials and Methods
### Four-Points Initialization  
The construction of the 3D centroid mesh \(CM=C_{i,j}\) of the deformed circles pattern begins with the process of obtaining the binarized image, which is detailed in Appendix A. After this, two clicks given on two consecutive ellipses of the left binarized image and two on the same circles of the right binarized image (Figure 5), allows for a knowledge of the distance and slope between two centroids of pseudo-ellipses. The clicks do not necessarily have to be done on the centroids; it is only necessary that they are done on the two consecutives pseudo-ellipses. The selected points are referred to as \(\left(x_{0,0}^{left},\ y_{0,0}^{left}\right)\) and \(\left(x_{1,0}^{left},\ y_{1,0}^{left}\right)\), while in the right image the points are \(\left(x_{0,0}^{right},\ y_{0,0}^{right}\right)\) and \(\left(x_{1,0}^{right},\ y_{1,0}^{right}\right)\). To work with discrete values, the values of each point, in each image, are rounded up because the computational functions give fractions of the pixels where the mouse is clicked via the computer. From the rounded coordinates provided by the user, the blob number \(nb_{i}^{s}\) (considering a blob as a group of pixels with a connected connection equal to four) is retrieved to provide the coordinates of the centroid, also rounded; as an index of the tagged image (7). Figure 5: The user with blue (+) and the centroids of each blob in red (*). The engraving of light reference circles on the metal sheet resulted from an equidistance of 1.55 mm between the centroids when the metal sheet had not been deformed. The measurement made with a microscope is shown in Figure 8, where the before (Figure 8a) and after (Figure 8b) deformation is appreciated. Figure 8: (**a**) Image captured before the forming mechanical process of the metal sheet; and (**b**) image captured after the forming mechanical process of the metal sheet. Figure 6: Pairing of points with typical initialization of the Digital Image Correlation. Figure 7: Correlation between a left image ellipse versus the overall possible \(21\times 21\) facets of the right image. On the other hand, the calibration parameters of the right and left cameras, as well as the translation vector between both cameras are the following:  
Intrinsic parameters of the left camera:  
Focal length: fc_left = [ 4139.77293 4125.08567 ] + [ 262.92862 287.85915 ]  
Principal point: cc_left = [ 1386.44233 1290.81589 ] + [ 145.82339 108.75101 ]  
Skew: alpha_c_left = [ 0.00000 ] + [ 0.00000 ] => pixel angle = 90.00000 + 0.00000 degrees  
Distortion: kc_left = [ -0.03826 -0.81032 -0.00495 -0.00231 0.00000 ] + [ 0.15686 2.09125 0.00925 0.01010 0.00000 ]  
Intrinsic parameters of the right camera:  
Focal length: fc_right = [ 4009.97898 4233.47842 ] + [ 271.16800 339.84438 ]  
Principal point: cc_right = [ 1149.40996 962.27029 ] + [ 198.63016 96.89436 ]  
Skew: alpha_c_right = [ 0.00000 ] + [ 0.00000 ] => pixel angle = 90.00000 + 0.00000 degrees  
Distortion: kc_right = [ -0.12949 0.25429 0.00496 0.02934 0.00000 ] + [ 0.13731 0.93124 0.00825 0.01424 0.00000 ]  
Extrinsic parameters (position of the right chamber with respect to the one on the left):  
Rotation (In Rodrigues format): om = [ -0.00856 0.37660 0.00093 ] + [ 0.03213 0.04287 0.00594 ]  
Translation vector: T = [ -85.14492 33.53035 24.13032 ] + [ 2.97387 1.09323 11.19968 ]  
The results of the image processing step are shown in Figure 9. In the top row, the pair of stereo images before the area filter is presented, while in the row below, the images after the area filter application are reported. We selected one hundred points to show the method's accuracy in a space of \(100\times 100\) marks. To verify if each point matching was right, the local DIC was calculated for the proposed points estimated in the initialization stage. The best correspondence was in the same position projected by the four-point method. Figure 10 shows a case of the correlation using a window of \(31\times 31\) pixels on a zone of \(21\times 21\) pixels.