# Multi-Point Displacement Synchronous Monitoring Method for Bridges Based on Computer Vision
## 4 A Method for Multi-Point Displacement Mismatch Elimination
### Mismatch Elimination for Multi-Point Displacements  
The deflection \(y\) and rotation angle \(\theta\) are the two basic parameters of the displacement vector. The deflection \(y\) varies with the position of the section, and \(y=f(x)\) can be \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Load** & **Dial Indicator No.** & \
\begin{tabular}{c} **Dial Indicator** \\ **Value (mm)** \\ \end{tabular} &
\begin{tabular}{c} **Calculated** \\ **Value (mm)** \\ \end{tabular} & **Error (\%)** \\ \hline \multirow{8}{*}{100 kN} & 1 & 1.45 & 1.42 & 2.07\% \\  & 2 & 2.19 & 2.17 & 0.91\% \\  & 3 & 2.63 & 2.57 & 2.28\% \\  & 4 & 2.90 & 2.97 & 2.41\% \\  & 5 & 3.37 & 3.26 & 3.26\% \\  & 6 & 3.55 & 3.42 & 3.66\% \\  & 7 & 3.61 & 3.49 & 3.32\% \\  & 8 & 3.53 & 3.44 & 2.55\% \\  & 9 & 3.22 & 3.27 & 1.55\% \\  & 10 & 2.91 & 2.99 & 2.75\% \\  & 11 & 2.53 & 2.59 & 2.37\% \\  & 12 & 2.16 & 2.08 & 3.70\% \\  & 13 & 1.40 & 1.45 & 3.57\% \\ \hline \multirow{8}{*}{200 kN} & 1 & 2.34 & 2.32 & 0.85\% \\  & 2 & 3.55 & 3.63 & 2.25\% \\  & 3 & 4.24 & 4.12 & 2.83\% \\  & 4 & 4.73 & 4.72 & 0.21\% \\  & 5 & 5.29 & 5.19 & 1.89\% \\  & 6 & 5.34 & 5.24 & 1.87\% \\  & 7 & 5.49 & 5.34 & 2.73\% \\  & 8 & 5.51 & 5.32 & 3.45\% \\  & 9 & 5.33 & 5.26 & 1.31\% \\  & 10 & 5.03 & 4.88 & 2.98\% \\  & 11 & 4.07 & 4.16 & 2.21\% \\  & 12 & 3.04 & 3.13 & 2.96\% \\  & 13 & 2.11 & 2.01 & 4.74\% \\ \hline \multirow{8}{*}{300 kN} & 1 & 3.11 & 3.03 & 2.57\% \\  & 2 & 4.72 & 4.85 & 2.75\% \\  & 3 & 5.41 & 5.53 & 2.22\% \\  & 4 & 6.48 & 6.34 & 2.16\% \\  & 5 & 7.05 & 6.99 & 0.85\% \\  & 6 & 7.16 & 7.17 & 0.14\% \\  & 7 & 7.32 & 7.39 & 0.96\% \\  & 8 & 7.36 & 7.42 & 0.82\% \\  & 9 & 7.31 & 7.25 & 0.82\% \\  & 10 & 6.37 & 6.46 & 1.41\% \\  & 11 & 5.61 & 5.49 & 2.14\% \\  & 12 & 4.11 & 4.00 & 2.68\% \\  & 13 & 2.41 & 2.46 & 2.07\% \\ \hline \multirow{8}{*}{400 kN} & 1 & 3.73 & 3.71 & 0.54\% \\  & 2 & 5.88 & 6.11 & 3.91\% \\  & 3 & 7.16 & 7.06 & 1.40\% \\  & 4 & 8.19 & 8.03 & 1.95\% \\  & 5 & 8.96 & 8.78 & 2.01\% \\  & 6 & 9.13 & 9.09 & 0.44\% \\  & 7 & 9.47 & 9.32 & 1.58\% \\  & 8 & 9.22 & 9.37 & 1.63\% \\  & 9 & 8.94 & 9.13 & 2.13\% \\  & 10 & 8.35 & 8.07 & 3.35\% \\  & 11 & 6.64 & 6.79 & 2.26\% \\  & 12 & 4.65 & 4.74 & 1.94\% \\  & 13 & 2.76 & 2.86 & 3.62\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Validation of the beam’s edge deflection. expressed as a continuous function. This is the deflection curve equation of the beam, and the inclination of its \(y=f(x)\) tangent is the section rotation angle \(\theta\). For small deformations, \(\theta\) is very small, and the rotation angle equation can be expressed as:  
\[\theta\approx\tan\theta=\frac{dy}{dx}=f^{\prime}(x)\qquad y=f(x) \tag{4}\]  
Equation (4) can be used to solve the section rotation angle of the structural edge deflection (Figure 15b). Furthermore, the obtained section rotation angle can be the elimination condition for the multi-point displacement mismatch. Figure 16 shows the calculation results for the test beam's edge rotation angle. Figure 16 shows the test beam's angle curve under load levels 100-400 kN. The angle curve provides a rotation constraint for the multi-point displacement. If the angle of the feature point's displacement does not satisfy the angle curve, the displacement is considered a mismatch and can be eliminated. In addition to the angle curve elimination method, the beam's edge deflection can be used for multi-point displacement mismatch elimination. The maximum deflection value of the edge line (Figure 15b) can eliminate the displacements with incorrect lengths. Figure 17 shows the method of displacement mismatch elimination through the rotation angle \(\theta\) and deflection \(y\). Figure 16: Test beam’s edge rotation angle calculation results under various load levels. (**a**) Test beam’s edge rotation angle (100 kN). (**b**) Test beam’s edge rotation angle (200 kN). (**c**) Test beam’s edge rotation angle (300 kN). (**d**) Test beam’s edge rotation angle (400 kN).