## 2 Methodology
Only the image coordinates of the CPs are measured to determine the ROPs from the coplanarity condition in Equation (1), which shows that the two camera centers--i.e., any target point with the corresponding image points on the stereo-pair lie on a common plane, as shown in Figure 2. One coplanarity equation is described for one object point, which appears in the stereo images. The equation contains image coordinates of the corresponding points without object space coordinates, as shown in Equations (1)-(3). The ROPs are determined using the conjugate image coordinates of some more CPs to the first stereo frames. These parameters can be calculated according to the iterative least-squares method using Equation (1). As a second step, scale adjustment is performed to accurately measure the subject using ROPs that have been determined solely from the image location without the spatial location of the CPs. It should be noted that the distance (\(b\)) between the centers of the left and right camera lenses should be directly measured in the coplanarity of Figure 2, but this would not be feasible. Therefore, an accurate scale for the ROPs should be taken into account while considering the inaccurate lens-to-lens distance. In this study, the  
Figure 2: Geometry of the coplanarity for relative orientation. scale is determined using the ratio of the distance measured in an elaborate manner versus the distance computed by ROPs for two attached targets. The ROPs component under the coplanarity condition can be adjusted using the determined scale shown in Equations (4) and (5):  
\[\big{[}\;X^{\prime}_{O}\;Y^{\prime}_{O}\;Z^{\prime}_{O}\;\big{]}^{T}_{i} = s\cdot[\;X_{O}\;Y_{O}\;Z_{O}\;]^{T}_{i} \tag{4}\] \[M^{\prime}_{i} = s\cdot M_{i} \tag{5}\]  
where \(i\) =1, 2 (left and right camera); \(s\) is a scale factor; and \(X^{\prime}_{O}\), \(Y^{\prime}_{O}\), \(Z^{\prime}_{O}\), and \(M^{\prime}_{i}\) are ground coordinates of the camera centers and rotation matrices by the scale adjustment, respectively. As a third step, to measure the instantaneous displacement of the bridge, it is necessary to obtain 3D coordinates of all the targets on the stereo sequence images using the left and right cameras. These are determined based on the space intersection with the pre-identified IOPs and the scale-adjusted ROPs, and the corresponding image coordinates \(x_{ijk}\) and \(y_{ijk}\) (\(i\) = number of cameras, \(j\) = number of frames, \(k\) = number of targets). The target's image coordinates of the first stereo frames are obtained manually. From the second stereo frames, the target-matching technique automatically traces the target's image coordinates for all frames (Figure 3). In this study, target-matching based on the normalized cross correlation and the least-squares-technique was carried out with the first frames as a reference, and the new location of targets was obtained from the next frames [39]. The target matching based on the two matching method is for better sub-pixel image matching. The least square matching can overcome the brightness difference and geometric image distortion between the target and reference images. Finally, using IOPs and scaled ROPs which are pre-determined, the corresponding image coordinates (\(x_{ijk},y_{ijk}\)) are identified, and the 3D spatial coordinates of targets attached to the bridge in each stereo frame can be determined from the space intersection, which is based on the collinearity condition (Equation (6)):  
\[\big{[}\;x_{ijk}y_{ijk}-f_{i}\big{]}^{T}=\lambda_{i}\cdot M^{\prime}_{i}\cdot \Big{[}\;X_{jk}-X^{\prime}_{O1}Y_{jk}-Y^{\prime}_{O1}Z_{jk}-Z^{\prime}_{O1} \Big{]}^{T} \tag{6}\]  
where \(i\) = 1, 2 (left and right camera); \(j\) is the number of frames; \(k\) = the number of targets; is a scale factor between image and object space; and \(X,\;Y,\;\) and \(Z\) are the ground coordinates of the targets on the bridge. This methodology makes it possible to measure the entire span of a bridge because there is no limit to the number of targets with which to detect the 3D displacement.