## 3 Principles of Image Method
According to Equation (32), the magnitude of the synthesized signal \(v_{s}(t,x,y)\) can be obtained, and this magnitude is used as the corresponding pixel value at position \(s(x,y)\). By following a specific step size and the previously mentioned calculation process, the magnitude at each position within the region is obtained. Then, the corresponding pixel values are presented in the form of an image, representing the imaging of the sound source. The areas in the image with larger and more concentrated pixel values indicate the location of the sound source. In the computation for sound source localization mentioned above, obtaining the phase velocity \(c_{\rho A_{0}}\) corresponding to each frequency component of Lamb wave propagation is necessary. This necessitates performing Equations (30) and (31), which involve Fourier and inverse Fourier transformations. When dealing with wide-ranging monitoring, a large number of piezoelectric sensors is required, which imposes a significant challenge. According to the relationship of the absolute value inequality, Equation (34) satisfies the following:  
\[ \begin{split}&|v_{s}(t,x,y)|\approx\left|\sum\limits_{i=1}^{n}e_{i} \bigg{(}\tau-t+\frac{r_{i}}{c_{gA_{0}}}\bigg{)}\right|\leq\sum\limits_{i=1}^{ n}\left|e_{i}\bigg{(}\tau-t+\frac{r_{i}}{c_{gA_{0}}}\bigg{)}\right|\end{split}  \tag{33}\]  
where the frequency component with a center frequency of \(\omega_{c}\) from the narrowband signal \(E_{i}(\omega)\) is extracted, with a phase velocity of \(c_{\rho\omega_{c}A_{0}}\). This simplification reduces a certain amount of computation, but it still requires Fourier transformation and inverse Fourier transformation calculations during the solving process. This simplification decreases resolution, particularly when the Lamb wave frequency is low and the frequency band is narrow, leading to reduced resolution. Figure 4: Schematic diagram illustrating time-reversal imaging. Based on the dispersion characteristics of narrowband Lamb waves, neglecting the influence of the amplitude coefficient \(1/2\pi\), Equation (32) can be simplified as follows:  
\[\left|v_{s}(t,x,y)\right|\approx\left|\sum_{i=1}^{n}e_{i}\left(\tau-t+\frac{r_{i }}{c_{gA_{0}}}\right)\right| \tag{34}\]  
where \(e_{i}\) represents the narrowband signal extracted through the Shannon continuous complex wavelet transform, and \(c_{gA_{0}}\) represents the group velocity of this narrowband signal. This simplification method eliminates the need for Fourier transformation and inverse Fourier transformation calculations, significantly reducing computational requirements. By performing time-domain superposition focusing on the narrowband signals extracted from Lamb response signals using the Shannon continuous complex wavelet transform, it greatly contributes to enhancing resolution. According to the relationship of the absolute value inequality, Equation (34) satisfies the following:  
\[\left|v_{s}(t,x,y)\right|\approx\left|\sum_{i=1}^{n}e_{i}\left(\tau-t+\frac{r_ {i}}{c_{gA_{0}}}\right)\right|\leq\sum_{i=1}^{n}\left|e_{i}\left(\tau-t+\frac{r _{i}}{c_{gA_{0}}}\right)\right| \tag{35}\]  
Therefore, the expression for signal phase synthesis is as follows:  
\[\left|v_{s}^{\prime}(t,x,y)\right|=\sum_{i=1}^{n}\left|e_{i}\left(\tau-t+ \frac{r_{i}}{c_{gA_{0}}}\right)\right| \tag{36}\]  
Equation (34) represents the need to first compute the composite signal by summing the focused signals and then calculate the modulus of the composite signal. Equation (36) represents the approach of first computing the modulus of each signal and then performing a focused summation of these moduli. During the signal-focusing process, the resolution of the resulting image is affected by factors such as the propagation distance and velocity of the signal within the structure, as well as the size of the imaging search step. These factors pose challenges to accurately localizing the source of the signal. After obtaining the pixel values for all target points, normalization is performed.