# Three-Dimensional Reconstruction-Based Vibration Measurement of Bridge Model Using UAVs
## 5 Discussion and Conclusions  
### Discussion  
Our experimental results suggested that the two methods work well in the situation that the reference points are on the same plane as the target points. However, for the situation that the reference points and the target points are in different planes, the homography-based correction method is invalid, while the 3D reconstruction is still feasible. As the plane homography is defined as a projection mapping from one plane to another, the homography matrix calculated from the fixed object plane and image plane cannot be used for the points on the third plane. This is why the homography transformation fails if the fixed object plane is non-coplanar with the bridge model plane.  
Although the proposed method shows a promising measurement result, there are still several problems that should be addressed in the future study:  
1. For the monocular camera, the value of \(Z_{w}\) is assumed to be constant, that is, the out-of-plane displacement (\(Z_{w}\)-direction) is ignored. In some structures with mainly in-plane displacement, the small change of \(Z_{w}\) can be ignored. Nevertheless, there will be an obvious error for the structures with large out-of-plane displacement if the change of \(Z_{w}\) is ignored. The structure-from-motion (SFM) technique can restore a bridge's 3D model coordinates [31], including the \(Z_{w}\)-direction, by processing high-resolution stereo-photogrammetric photos; it has been used for slow deformation monitoring. An image splitter system, which consisted of four fixed mirrors, is used  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{\(Z_{w}\) **(m)**} & **Homography** & **3D Reconstruction** & **Homography** & **3D Reconstruction** \\ \cline{2-5}  & **Frequency (Hz)** & **Frequency (Hz)** & **Relative Error (\%)** & **Relative Error (\%)** \\ \hline
0 & 3.940 & 3.940 & 0 & 0 \\
1 & 0.103 & 3.927 & 97.4 & 0.33 \\ \(-1\) & 0.044 & 3.966 & 98.9 & 0.66 \\ \hline \end{tabular}
\end{table}
Table 1: Frequencies obtained by two methods.  
Figure 19: Comparison of displacement corrected by 3D reconstruction and homography methods (\(Z_{w}=-1000\)).  
Figure 20: Comparison of PSD from 3D reconstruction and homography methods (\(Z_{w}=-1000\)).  
to mimic four different views by using a single camera with a 45-degree horizontal angle with respect to the target [43]. However, it needs a large splitter and mirror to measure large bridges from a sufficient distance. Using two UAVs combined with a binocular vision principle to measure three-dimensional displacement needs further investigation.
2. The experiment is carried out in a laboratory environment. Its conditions, including light, weather, reference points, etc. are in the ideal state. It is, however, inevitable that some negative factors may occur in the real bridge measurement, for example, difficulty in finding fixed reference objects. An artificial fixed object needs to be deployed under this situation. It is expected that this method can be realized in the real bridge measurement in the near future.
3. In the current algorithm, the theory of planar homography is used to calculate the camera extrinsic matrix R and t. Hence, the four fixed reference points must be on the same plane with fixed-\(Z_{w}\). However, in some measurement circumstances, it is hard to guarantee that all four points are coplanar. Whether or not the reference points can be on different planes needs further study.