# Vision-Based Damage Detection Method Using Multi-Scale Local Information Entropy and Data Fusion
## 2 Methodology
### Data Fusion for Multi-Scale Damage Information  
In the above-established Gaussian multi-scale space, the detection of local singularity peaks with MS-LIE at a specific scale has the potential to identify the damage in theory. However, in practical applications, the selection of scale parameters presents a challenging task. At a smaller scale, the mode shape exhibits a pronounced damage-induced singularity, which is detectable yet may be obscured by noise interference. Conversely, at a larger scale, noise is markedly reduced, but the damage-induced singularity diminishes due to the mode shape's increased smoothness. To address these deficiencies, in this study, a data fusion algorithm based on D-S evidence theory is developed to fuse the MS-LIE at all scales, which eliminates noise-induced uncertainty and leads to the final determination of damage locations.  
Let \(\Omega\) be a finite set called frame of discernment, whose physical significance is the set of measurement points at which damage may occur:  
\[\Omega=\{e_{1},\;e_{2},\;...,\;e_{N}\} \tag{15}\]  
where \(e_{i}\) (\(i=1,\,2,\ldots,\,N\)) represents measurement points composing the structure, and \(N\) is the number of measurement points. All damage localization results can be represented as a subset of \(\Omega\), and \(2^{\Omega}\) is defined as the power set including all the subsets of \(\Omega\) and empty set (\(\varnothing\)). Furthermore, the basic probability \(m\) is assigned in the mapping of the power set:  
\[m:2^{\Omega}\rightarrow[0,\ 1] \tag{16}\]  
where \(m\) denotes the basic probability distribution, which must ensure that the assigned probabilities sum to unity and there is no belief in the empty set:  
\[\sum_{A\subseteq\Omega}m(A)=1,\ m(\varnothing)=0 \tag{17}\]  
Considering that the MS-LIE at each scale should be valid evidence of damage localization, the basic damage probability of the \(i\)-th measurement point at each scale is defined as  
\[m_{\sigma}(i)=\frac{\text{MS-LIE}_{\sigma}^{\text{norm}}(i)}{\sum_{j=0}^{N} \text{MS-LIE}_{\sigma}^{\text{norm}}(j)} \tag{18}\]  
where \(m_{\sigma}\) is an \(N\)-dimensional vector containing the damage probability distribution at the scale \(\sigma\).  
According to the D-S evidence theory [42], the basic probability assignments can be combined. The combination rule for the two basic damage probability distributions, i.e., \(m_{1}\) and \(m_{2}\), is shown as follows:  
\[F(i)=\frac{\sum_{j\cap k=i}m_{1}(j)m_{2}(k)}{1-\sum_{j\cap k=\varnothing}m_{1 }(j)m_{2}(k)} \tag{19}\]  
where \(F\) denotes the fused probability distribution.  
Based on Equation (19), the damage probability distributions of two adjacent scales are fused, starting at the lowest scale and progressing upwards to the highest. This fusion operation continues until the final damage probability distribution is obtained, as shown in Figure 5.  
Without loss of generality, and briefly, the data fusion process is represented as  
\[F^{n}(i)=\frac{\sum_{j\cap k=i}F^{n-1}(j)m_{\sigma_{n+1}}(k)}{1-\sum_{j\cap k= \varnothing}F^{n-1}(j)m_{\sigma_{n+1}}(k)}\ (n\geq 2) \tag{20}\]  
where \(F^{n}\) denotes the fused damage probability distribution after \(n\) times of data fusion, and \(\sigma_{n}\) represents the \(n\)-th scale. With the above steps, the imprecision in damage lo  
Figure 5: Flow diagram for data fusion.  
calization caused by measurement noise is eliminated, and the damage detection can be user independent.