## 3 Experiments and Results
### Results  
#### 3.2.1 No Noise Condition  
The experimental setup for the vibration testing of the MTS hydraulic system adopted a sine wave form. The amplitude was set at 10 mm, and the frequencies were set to 0.5 Hz and 1.5 Hz, respectively. The displacement curves of the objects were derived using MATLAB programs implementing both the traditional optical flow method and the improved method. These curves were then compared with the displacement curves output by the MTS. The test results for the vibration at a frequency of 0.5 Hz are depicted in \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Thresholds** & **1\(\times\sigma\)** & **0.5\(\times\sigma\)** & **0.3\(\times\sigma\)** \\ \hline reservations & 186 & 130 & 97 \\ removals & 37 & 93 & 126 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quantities of Retained and Removed Feature Corners Under Different Threshold Values. Figure 6: The motion trajectories of the detected feature points are represented by arrows. Red arrows signify that the coordinate points at the starting ends of these arrows are the retained feature points, while blue arrows indicate that the coordinate points at the starting ends of their arrows are the eliminated feature points. (**a**) Shows all the detected feature points along with their corresponding arrows. (**b**) Displays all the detected feature points whose angles are smaller than the threshold of 1\(\times\sigma\), together with their arrows. (**c**) Presents all the detected feature point arrows for which the angles are less than the threshold value of 0.5\(\times\sigma\). (**d**) Illustrates all the detected feature point arrows with angles smaller than the threshold value of 0.3\(\times\sigma\). As illustrated in Figure 7, the main computational error is concentrated in the peak region. The region of interest was defined as centered around the peak point, with a range spanning five frames of the picture before and after the peak as the area for calculating the average accuracy. The Normalized cross-correlation (NCC) method was used to calculate accuracy between two vibrations. And the how to calculate the NCC is shown in Equation (14). The accuracy for the vibration curve at different regions with a frequency of 0.5 Hz are presented in Table 2. The accuracy of the region of interest stabilizes at approximately 97.0% by using the proposed Improved LK Method, comparing with accuracy at 96% by using the traditional LK Method. \[NCC=\frac{\sum_{x}\big{[}f(x)-\overline{f}\big{]}\big{[}t(x)-\overline{t} \big{]}}{\big{\{}\sum_{x}\big{[}f(x)-\overline{f}\big{]}^{2}\big{[}t(x)-\overline {t}^{2}\big{]}^{2}\big{\}}^{0.5}} \tag{14}\]  
where \(f\) is displacement value of the first vibration; \(\overline{f}\) is the mean of the first vibration; \(t\) is displacement value of the second vibration; \(\overline{t}\) is the mean of the second vibration. In the vibration test with a frequency of 0.5 Hz, the average error of the calculation results for all frames is 2.0%. The distribution of the calculation errors for all consecutive frames is shown in Figure 8. Here, the errors are mainly concentrated at the peak positions. The distribution pattern of the calculation errors of the improved method is roughly the same as that of the traditional LK method. Figure 7: The calculated data show a high degree of synchronization with the real position at each instant. Among them, the displacement curve with a threshold of 0.3\(\times\)\(\sigma\) is the closest to the displacement curve output by the MTS, yielding the most accurate results, especially in the vicinity of the peak where the comparison is more pronounced. The test results for the vibration at a frequency of 1.5 Hz are presented in Figure 9. The overall waveform of the displacement curve obtained by the improved method generally aligns well with the actual displacement curve output by the MTS machine. Based on the assumptions of the optical flow method, it can be inferred that the object moves faster at 1.5 Hz compared to 0.5 Hz.