# Vision-Based Damage Detection Method Using Multi-Scale Local Information Entropy and Data Fusion
## 3 Numerical Investigations of the Methodology
### Noise Immunity Evaluations  
#### 3.2.1 Evaluations on Single Damage  
A steel cantilever beam with single damage is first considered. The parameters of this beam are as follows: mass density \(\rho\) = 7800 kg/m\({}^{3}\), elastic modulus \(E\) = 210 GPa, Poisson's ratio \(\mu\) = 0.32, length \(L\) = 600 mm, width \(b\) = 25 mm, and thickness \(h\) = 10 mm. A crack characterized by dimensionless parameters \(\alpha\) = 0.3 and \(\beta\) = 0.4 is introduced into the beam. To perform the noise immunity evaluation, white Gaussian noise (WGN) is introduced for this purpose. Once the mode shapes are obtained through the above analytical model, the WGN is added to the pure mode shapes, resulting in different noisy environments. The noise level is assessed by the signal-to-noise ratio (SNR). The selection of SNR in this paper is mainly based on empirical values for vision-based measurements, which is typically between 50 dB and 100 dB in our previous laboratory measurements. We have therefore considered two scenarios: one with lower noise level (SNR = 90 dB) and another with higher noise level (SNR = 60 dB).  
In the simulated scenario with an SNR of 90 dB, the MS-LIE at different scales obtained from the first, second, and third mode shapes are shown in Figure 7a, Figure 7b, Figure 7c, respectively, with the Gaussian multi-scale space spanning a scale range of 2-10 and a step size of 0.1. The corresponding damage detection results are demonstrated in Figure 7d-f. The figures reveal that the MS-LIE exhibits sharp amplitude changes at the location corresponding to a damage-induced singularity in the mode shape. The MS-LIE heat map derived from the first mode shape shows some undesired local amplitude fluctuations due to noise. However, as the order of the adopted mode shape increases, the MS-LIE heat maps become progressively smoother and more robust to noise. Furthermore, with the aid of data fusion, the precise location of the damage is identified at the peak of the final damage probability distribution.  
In the simulated scenario with an SNR of 60 dB, the MS-LIE across the scale range of 2-10 is displayed in Figure 8a-c. The corresponding damage detection results are shown in Figure 8d-f. An increase in the level of measurement noise results in more chaotic mode shapes, thereby making the noise-induced interference more visible. Specifically, a series of linear sidelobes appeared in the MS-LIE heat maps of the first and second mode shapes, posing challenges in detecting damage at a single scale. The larger scale parameter can reduce this phenomenon of noise interference enhancement, but it leads to increased ambiguity in locating the damage. Nevertheless, since the data fusion technique developed in this paper harnesses damage evidence at all scales, the final damage probability distributions obtained still accurately indicate damage.  
Figure 7: Evaluations on single damage with SNR = 90 dB: (**a–c**) the MS-LIE heat map obtained from the 1 st, 2 nd and 3 rd mode shapes, (**d–f**) damage localization results of the 1 st, 2 nd and 3 rd mode shapes.