# Vision-Based Damage Detection Method Using Multi-Scale Local Information Entropy and Data Fusion
## 4 Experimental Verifications of the Methodology
### The Accuracy of Vibration Measurement  
The schematic diagram of the vision-based experimental system is shown in Figure 17a, while the set-up for the experimental platform is illustrated in Figure 17b. The experimental system primarily comprises a high-speed camera, a laser vibrometer, a pair of lamps, a vibration test rig and a laptop. The cantilever beam employed in the experiment is made of aluminum alloy 6061 with dimensions of 400 \(\times\) 40 \(\times\) 5 mm, whose mass density, elastic modulus and Poisson's ratio are 2700 kg/m\({}^{-3}\), 69 GPa, and 0.33, respectively. During the experiment, we capture the digital video of the structural vibration at 3000 frames per second (fps) using the high-speed camera (Revealer 5F04, full resolution 2320 \(\times\) 1718 pixels, pixel size 7 \(\times\) 7 \(\upmu\)m, maximum frame rate 52,800 fps, responsivity ISO 6400). Figure 17c displays a screenshot of the motion video. As a reference, the laser vibrometer (Keyence LK-H155, resolution 0.1 \(\upmu\)m, maximum sampling frequency 390 kHz) is used to acquire the response of the vibrating beam.  
Figure 16: Computational complexity analysis: (**a**) impact of total number of scales, (**b**) impact of total number of measurement points.  
The comparison of displacement responses obtained from the vision-based system (the scale factor between the image plane and the beam's physical plane is 31.6 pixels per millimeter) and the laser vibrometer is shown in Figure 18. To enable quantitative analysis of the similarity between the two time-domain responses, we introduce the Pearson correlation coefficient (PCC), calculated as  
\[\text{PCC}(x_{V},x_{L})=\frac{\sum_{t=1}^{T}\left(x_{V}(t)-\overline{x}_{V} \right)\left(x_{L}(t)-\overline{x}_{L}\right)}{\sqrt{\sum_{t=1}^{T}\left(x_{V} (t)-\overline{x}_{V}\right)^{2}}\sqrt{\sum_{t=1}^{T}\left(x_{L}(t)-\overline{ x}_{L}\right)^{2}}} \tag{27}\]  
where \(x_{V}(t)\) and \(x_{L}(t)\) are the responses at time \(t\) obtained from the vision-based system and the laser vibrometer, respectively, and \(T\) is the calculation period.  
A PCC value approaching 1 suggests a strong correlation between the two data. It is worth noting that the laser vibrometer operates at a higher sampling frequency than the video frame rate, resulting in inconsistent sample sizes. Thus, we down sample the laser vibrometer data to enable the calculation of PCC, yielding a PCC of 0.988. Furthermore, we use the fast Fourier transform (FFT) technology to extract the first three frequencies from the responses and compare these with the results of the Finite Element Method (FEM), as detailed in Table 1. The findings confirm the accuracy of the vision-based system in vibration measurement, utilizing the phase-based optical flow algorithm.  
Figure 17: The experimental set-up: **(a)** the vision-based system, **(b)** the experimental platform, **(c)** a screenshot of the motion video.  
Figure 18: The comparison of displacement responses obtained from the vision-based system and laser vibrometer.  
Based on the captured vibration responses, the high-spatial-resolution mode shapes are identified by the BSS and ICA technique, which is shown in Figure 19a. To quantitatively assess the accuracy of the mode shapes obtained, we use the modal assurance criterion (MAC), defined as  
\[\text{MAC}(\varphi_{V},\varphi_{F})=\frac{\left\|\varphi_{V}^{T}\varphi_{F} \right\|^{2}}{\left(\varphi_{V}^{T}\varphi_{V}\right)\left(\varphi_{F}^{T} \varphi_{F}\right)} \tag{28}\]  
where \(\varphi_{V}\) and \(\varphi_{F}\) are the mode shapes obtained from the vision-based system and FEM, respectively.  
A high correlation between the two mode shapes is indicated by a MAC value close to 1. The bar chart in Figure 19b depicts the MAC values for the first three mode shapes, validating the strong correspondence between the mode shapes obtained from the vision-based system and those from the FEM. This means that the high-spatial-resolution mode shapes given by vision-based method are reliable.