## 2 Digital Image Processing for Tracking Target Point Displacement
### Displacement Tracking of the Tracking Point (TP) Using DIP
The equation for finding the coordinates \(\left(X_{k}^{\mathbf{B}},\,Y_{k}^{\mathbf{B}}\right)\) of the \(k\)-th point constituting the ETP is as follows:  
\[Y_{k}^{\mathbf{B}}\ =\ \frac{Y^{TP}-Y_{k}^{ETP}}{X^{TP}-X_{k}^{ETP}}(X_{k}^{ \mathbf{B}}-X^{TP})+Y^{TP}, \tag{4}\]  
where \(\left(X^{TP},\,Y^{TP}\right)\) and \(\left(X_{k}^{ETP},\,Y_{k}^{ETP}\right)\) are the coordinates of each \(TP\) and each \(ETP\), respectively, and \(\left(X_{k}^{B},\,Y_{k}^{\mathbf{B}}\right)\) denotes the coordinates of the \(k\)-th point included in vector \(\mathbf{B}\) constituting the boundary points of the target. In this study, only a one-layer target space surrounding the reference target was considered to generate the ETP; to this end, the analysis area was limited by inputting the distance information of the points constituting the TP set. To track the deformation of the specimen during loading, the displacement change or history of the TPs extracted from the images acquired at regular intervals must be tracked. To this end, the TPs were labeled by aligning the coordinates of the X- and Y-axes in order, and the serial number was assigned automatically. However, a few images yielded errors because the initial label could not be maintained because of the excessive movement of the TP caused by the relatively large deformation of the specimen. For example, target no. 25 of image no. 2 was mislabeled as no. 27 of image no. 3, as shown in Figure 6. In this case, it was difficult to apply the automatic labeling method, because the error occurred in the TP history, disallowing the displacement tracking. Figure 5: Two target tracking methods: (**a**) TP, (**b**) expanded TPs. In this study, the label of the TP remained constant, even if it exhibited excessive movement because of the relatively large deformation of the specimen. To prevent the TP labeling errors, the bounding box, which is a set of pixels comprising the border of the TP in the previous image, was synchronized with the TP at the same position in the current image when labeling the TP. In the improved labeling method, the initial image of the image set uses the number of TPs generated according to Equation (5). From the second image to the last, the TP of the current image included in the bounding box of the previous image TP was found, and it inherited the label of TPs of the previous image, as shown in Equation (6). \[targetNum_{k}(i)\ =\ TP_{k}(i)\quad in\ k\ =\ 1, \tag{5}\]  
\[if\ boundingBox_{k-1}(i)\supset TP_{k}(j),\ TP_{k}(j)\to TP_{k-1}(i)\quad in\ k>1, \tag{6}\]  
where \(k\) denotes the number of images included in the image set; \(i\) denotes the target label automatically generated from the first image or the target label of the previous image; and \(j\) denotes the target label automatically extracted from the current image. Consequently, as TP no. 27 of image no. 3 is inside the bounding box of TP no. 25 of image no. 2, as shown on the right of Figure 6, no. 27, which was assigned automatically, was changed to no. 25 based on the process described in Equations (5) and (6). Through this process, the existing labeling error was improved, and the overall result is shown in Figure 7.