## 4 Methodology
### Damage Localization  
In this section, we apply a reconstruction algorithm to localize the damage on the plate. For this purpose, we only considered those exciter-sensor paths with maximum peak values; hence, these peak values were used as weights for the pixels. We first subtracted the damage signal from the pristine signal and then determined the signal envelope. Here, we exploited time-frequency transformation for this purpose. In this section, we implement a reconstruction algorithm to pinpoint the location of damage on the plate. To facilitate this, we limit our focus to the exciter-sensor paths that display the highest peak values. The first step in this process involves subtracting the pristine, undamaged signal from the damaged signal. Subsequently, we ascertain the signal envelope. We utilize a time-frequency transformation to leverage its capabilities to identify and analyze signal components effectively. In this study, we used smoothed pseudo-Wigner Ville distribution (SPWVD) to obtain the maximum energy distribution curve in time domain. SPWVD possesses two inherent properties; first, it exhibits an anti-noise characteristic, allowing filtered energy to pass through an exceptionally narrow passband. As a result, the impact of low SNR value can be effectively diminished. Secondly, unlike linear Time-Frequency (TF) distributions, which provide only an approximate energy distribution in the TF domain, SPWVD accurately represents the true TF energy distribution of the signal. This precise estimation enables the identification and assessment of damage-related features with high accuracy [29].  
\[W(t,\omega)\int z\Big{(}t+\frac{\tau}{2}\Big{)}z^{*}\Big{(}t-\frac{\tau}{2} \Big{)}e^{-i\omega\tau} \tag{10}\]  
\[Envelop=max(W(t,\omega)) \tag{11}\]  
In addition, we utilized the normal distribution to reduce the width of the generated signal envelope. This technique involves using the maximum peak value of the signal as the mean (average) of the normal distribution. The normal distribution, also known as the Gaussian distribution, is a symmetric probability distribution about the mean. By centering this around the maximum peak value using it as a windowing function, we effectively narrow the envelope's width, focusing more precisely on the region of interest within the signal. This technique provides a more focused and accurate understanding of where potential anomalies, such as damage, might occur within the signal. Figure 16 shows the pristine signal and its comparison with the damage signal D1 for a particular exciter-sensor path.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Damage ID** & **Average MAE Difference** \\ \hline D1 & 0.37 \\ D2 & 0.48 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Average MAE for both damage.  
Figure 15: MAE difference vs. Exciter-sensor paths for damage D2.  
The Figures 17 and 18 show the location of the damage D1 and D2, respectively. As we can observe, the position of D1 was determined by utilizing only two exciter-sensor paths, as the proposed autoencoder scheme accurately determined which exciter-sensor paths were optimal for localized damage. However, the result can be made more precise by adding more paths.