## 4 Indoor Dynamic Beam Deflection Measurement Experiment and Validation
### DIC Processing Result Comparison and Validation  
Digital Image Correlation (DIC) is a classic machine-vision-based full-field displacement and strain measurement method involving speckles, and several commercial software has been developed. However, its application is limited by the requirement for preinstalled high-quality speckles on the measurement object. The GOM Correlation Professional software was used to measure and validate the DFEM-DSMR method proposed in this paper; to be specific, five points on the left end, 1/4 length, mid-span, 3/4 length, and the right end of the beam were selected as shown in Figure 14.  
First, the temporal displacement of the measurement was measured by the DIC method of GOM software and the DFEM-DSMR method, and the comparison of two displacement-time curves is shown in Figure 15, with the maximum errors being enlarged and shown in the bottom right box. It can be seen that the two curves correlate well with each other. Figure 14: Measurement points on the beam. Figure 13: Deflection curve after polynomial fitting. Figure 15: Comparison of the results of deflection between the two methods. To quantitatively demonstrate the temporal displacement measurement accuracy of the DFEM-DSMR method, the maximum error, average error, and root mean squared error (\(RMSE\)) of the dynamic deflection-time curves at the five measurement points were calculated and are listed in Table 3. The \(RMSE\) is defined as:  
\[RMSE=\sqrt{\frac{\sum\limits_{i=1}^{n}{{{(X_{ei}-X_{ti})}^{2}}}}{n}} \tag{11}\]  
where \(X_{e}\) is the deflection data obtained by the DFEM-DSMR method; \(X_{t}\) is the data calculated by the GOM software, both in mm; and \(i\) is the data number (\(i\) = 1, 2, 3,..., \(n\), \(n\) is the total image frame number). Table 3 shows that the maximum errors range from 0.025 to 0.053 mm for the five measured points, the maximum average error is 0.013 mm, and the RMSE is 0.01 mm, the latter two of which are sub-pixel errors. From the perspective of temporal displacement measurement, the DFEM-DSMR method demonstrates high measurement accuracy and is consistent with the measured results of existing commercial software. Then, the spatial displacement of the measurement was also measured by the two methods, and the pixel-by-pixel deflection distribution curves at the times 0.4, 0.8, 1.2, 1.6, 2 s along the beam were plotted as shown in Figure 16.  
For clearer comparison, the spatial deflection curve of all measurement point at \(t\) = 1.2 s was plotted as shown in Figure 17, and the maximum errors were enlarged and are shown in the bottom left box. It shows that spatial deflection curves obtained by the DFEM-DSMR method are smoother and indicate better accuracy. Figure 16: Five spatial deflection curves of DIC and DFEM-DSMR. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Time**} & **Maximum Error** & **Average Error** & \multicolumn{2}{c}{**RMSE/mm**} \\  & **/mm** & **/mm** & **DFEM-DSMR** & **References** \\ \hline left end & 0.032 & 0.005 & 0.007 & \\
1/4 point & 0.053 & 0.009 & 0.012 & 0.1–0.2 [27] \\ mid-span & 0.049 & 0.010 & 0.015 & 0.1–0.3 [28] \\
3/4 point & 0.025 & 0.006 & 0.007 & \\ right end & 0.034 & 0.013 & 0.015 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Temporal displacement measurement error of DIC and DFEM-DSMR. Furthermore, the maximum error, average error, and correlation coefficient \(\rho\) of the two spatial deflection curves were calculated and are listed in Table 4, and the correlation coefficient \(\rho\) is:  
\[\rho=\frac{\sum\limits_{i=1}^{n}\left(X_{ei}-\overline{X_{e}}\right)(X_{ii}- \overline{X_{t}})}{\sqrt{\sum\limits_{i=1}^{n}\left(X_{ei}-\overline{X_{e}} \right)\sum\limits_{i=1}^{n}\left(X_{ti}-\overline{X_{t}}\right)^{2}}} \tag{12}\]  
where \(\overline{X_{e}}\) and \(\overline{X_{t}}\) represent the mean values of the two sets of data.