# Vision-Based Damage Detection Method Using Multi-Scale Local Information Entropy and Data Fusion
## 2 Methodology
### A Novel Damage Index: MS-LIE  
Mode shape-based damage detection can be classified into two principal categories: change ratio analysis and singularity analysis. Singularity analysis is a reference-free damage detection approach that can detect local damage without the requirement for baseline data, and it is this approach that is attracting increasing attention. The damage-induced singularity in mode shapes or their spatial derivatives is analyzed to localize the damage. Traditionally, mode shape curvature has been selected as a sensitive feature to reflect damage locations sensitively. Nevertheless, the direct second-order difference operation on the mode shape renders it susceptible to measurement noise. Emerging signal processing tools such as the fractal dimension-based method [20] have been introduced, which aim to give a quantitative evaluation of damage-induced singularity in mode shapes. However  
The self-information \(I(a_{i})\) quantifying the total information in \(a_{i}\) can be calculated as  
\[I(a_{i})=log_{2}\frac{1}{p_{i}}=-log_{2}p_{i} \tag{7}\]  
Equation (7) demonstrates that the self-information is uniquely determined by the priori probability. As the probability approaches 0, the uncertainty increases and the information content becomes more significant. Furthermore, to characterize the information of the entire source \(X\), the information entropy \(H(X)\) is put forward, defined as the mathematical expectation of self-information, that is  
\[H(X)=I(a_{1})p(a_{1})+I(a_{2})p(a_{2})+...+I(a_{r})p(a_{r})=-\sum_{i=0}^{r}p_{ i}log_{2}p_{i} \tag{8}\]  
In the context of damage detection, each measurement point can be conceptualized as a component of the information source. Considering the localized nature of structural damage, a sliding window strategy is introduced to calculate the local probability, which reveals the local feature of the mode shapes rather than the overall characteristics:  
\[p_{i}=\frac{y_{i}}{\sum_{j=i-\frac{S-1}{2}}^{i+\frac{S+1}{2}}y_{j}} \tag{9}\]  
where \(y_{i}\) represents the obtained mode displacement of the \(i\)-th measurement point and \(S\) is the size of the sliding window. The process of calculating the local probability using a sliding window is depicted in Figure 4. By substituting Equation (9) into Equation (8), the information entropy gains the local sensitivity performance. Thus, the LIE for each measurement point is formulated as  
\[\mathrm{LIE}(i)=-\frac{y_{i}\bigg{(}log_{2}y_{i}-log_{2}\sum_{j=i-\frac{S-1}{ 2}}^{i+\frac{S+1}{2}}y_{j}\bigg{)}}{\sum_{j=i-\frac{S-1}{2}}^{i+\frac{S+1}{2} }y_{j}log_{2}S} \tag{10}\]  
In the healthy state of the structure, the mode shapes exhibit geometric smoothness, and the amplitude of the LIE remains small. Conversely, damage to the structure results in local irregularities within the mode shapes, which manifest as an increase in the amplitude of the LIE. Despite its sensitivity in detecting local damage, the LIE is not sufficiently robust against the measurement noise, making the direct application of the damage detection method without modifications a challenging endeavor. To suppress the measurement noise,  
Figure 4: The process of calculating the local probability. this paper utilizes the multi-scale space theory for discrete signals [41], mapping the mode shape into Gaussian multi-scale space:  
\[Y_{\sigma}(x)=G_{\sigma}(x)\otimes y(x) \tag{11}\]  
where \(Y_{\sigma}\) is the multi-scale representation of the mode shape, \(G_{\sigma}\) is the variable-scale Gaussian kernel, the subscript \(\sigma\) represents the scale parameter, and \(y(x)\) is the obtained mode shape. The form of variable-scale Gaussian kernel is written as  
\[G_{\sigma}(x)=\frac{1}{2\pi\sigma^{2}}e^{-\frac{x^{2}}{2\sigma^{2}}} \tag{12}\]  
Utilizing the mapping of the mode shape as described by Equations (11) and (12), the LIE of measurement points across all scales forms a novel damage index, termed MS-LIE, within the Gaussian multi-scale space.