## 5 Structural Displacement Extraction Validation
### Structural Displacement Tracking Results and Error Characteristics  
In the proposed UAV vision-based earthquake-induced structural displacement monitoring framework, the image points are reprojected to the 3-D world space based on the world-to-image point correspondence at each video frame. Then, the displacement of each reference target is computed as the coordinate changes with respect to its initial world coordinates in the first video frame. The extracted displacements are compared to ground truth measurements based on the comparison plan illustrated in Section 5.1.2. Since the ground truth measurements are the filtered results as described in Section 5.1.1, the same fourth-order bandpass Butterworth filter with cutoff frequencies of 0.1 and 50 Hz is also applied to the extracted displacements to avoid the possible discrepancy caused by the filtering. For brevity, comparisons of the displacement results of a Type 2 target (4-tile black-and-white checkerboard pattern) from two Ferndale earthquake tests (MID 12 and 15) and a Type 4 target (black-and-white concentric circular pattern) from two Northridge earthquake tests (MID 7 and 8) are presented in Figure 12. These results indicate an overall root-mean-square error (RMSE) between the UAV vision-based analysis results and ground truth measurements of less than 1 cm. In addition, the percentage error at the peak displacement amplitude is also calculated as follows:  
\[\text{Error}^{\%}_{\text{0}}=\frac{\Delta_{\text{UAV}}-\Delta_{\text{GT,max} }}{\Delta_{\text{GT,max}}}\times 100\% \tag{10}\]  
where \(\Delta_{\text{GT,max}}\) is the maximum displacement from ground truth measurement, and \(\Delta_{\text{UAV}}\) is the displacement value at the timestamp corresponding to \(\Delta_{\text{GT,max}}\). The overall percentage error per this computation at the peak displacement amplitudes is less than 10%. The results indicate that the displacements extracted from the proposed method show good alignment with the ground truth measurements for both the overall trend and the peak displacements regardless of locations of measurements, reference target patterns, displacement directions, earthquake motion input intensities, and earthquake events. Figure 11: Reference target distribution in five regions of the roof plan and the selected analog sensors utilized as ground truth measurements. Figure 13 summarizes the average RMSEs for tracking the roof-level displacements across the various reference target patterns for each test. Each light-colored data point in this plot represents the average RMSE calculated from multiple displacement time series comparisons in the X or Y direction of one test (results from each single target forms a comparison) with a specific pattern. The average RMSEs from all seven patterns are also presented as filled symbols within the plots. For earthquake inputs in the given direction from a specific earthquake event (e.g., X-direction inputs of the Ferndale earthquake), larger average RMSEs are observed in the test with a greater earthquake input motion intensity, i.e., when the achieved peak input acceleration (PIA) is larger. To provide a generalized quantitative illustration of the effect of earthquake input motion intensity on structural displacement tracking errors, the PIAs in the X and Y directions for each test, along with the corresponding average RMSEs over all reference target patterns, are normalized by the achieved PIA and average RMSE values from the 100% MCE\({}_{\text{R}}\) test in the given direction of that specific earthquake event. Mathematical expressions of this normalization are presented in Equations (11) and (12), where PIA\({}_{\text{MCER}}\) and RMSE\({}_{\text{MCER}}\) in the denominator are the achieved PIA and average RMSE from the 100% MCE\({}_{\text{R}}\) test for the same earthquake event and input direction as the corresponding achieved PIA and average RMSE in the numerator.