## 3 Results and Discussion
### The Reconstruction Results of the AGES-AHK Method  
The reconstruction results of the finalized AGES-AHK method is shown in Table 2.  
The coefficient of heating effect in Equation (12) is a critical parameter that affects the gradient and the complexity of the temperature phantoms. As shown in Table 2, the NRMSE and \(E_{\text{h}}\) exhibit a marked increase with elevated \(\text{coef}_{\text{HE}}\) values in the three temperature \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & **coef\({}_{\text{HE}}\)** & **Input Values** & **NRMSE/\%** & \(E_{\text{h}}\)/\%** & \(D_{\text{h}}\)/\% \\ \hline \multirow{6}{*}{_Tfield1_} & 0.1 & \(f_{\text{b}}\) = 0.3, \(\kappa\) = 1.5 \(\times\)\(10^{-4}\) & 0.60 & 1.19 & 0 \\  & 0.2 & \(f_{\text{b}}\) = 0.4, \(\kappa\) = 1.5 \(\times\)\(10^{-4}\) & 1.18 & 2.43 & 0 \\  & 0.5 & \(f_{\text{b}}\) = 0.2, \(\kappa\) = 1.5 \(\times\)\(10^{-4}\) & 2.90 & 1.39 & 0 \\  & 1 & \(f_{\text{b}}\) = 0.25, \(\kappa\) = 1 \(\times\)\(10^{-4}\) & 6.61 & 19.01 & 0 \\ \hline \multirow{6}{*}{_Tfield2_} & 0.1 & \(f_{\text{b}}\) = 0.1, \(\kappa\) = 1.5 \(\times\)\(10^{-4}\) & 0.79 & 0.58, 0.74 & 1.41, 1.41 \\  & 0.2 & \(f_{\text{b}}\) = 0.2, \(\kappa\) = 1.5 \(\times\)\(10^{-4}\) & 3.06 & 2.28, 0.70 & 1.41, 1.41 \\  & 0.5 & \(f_{\text{b}}\) = 0.5, \(\kappa\) = 9 \(\times\)\(10^{-5}\) & 2.92 & 2.91, 4.18 & 0, 0 \\  & 1 & \(f_{\text{b}}\) = 0.4, \(\kappa\) = 7 \(\times\)\(10^{-5}\) & 3.87 & 14.59, 13.49 & 0, 0 \\ \hline \multirow{6}{*}{_Tfield3_} & 0.1 & \(f_{\text{b}}\) = 0.8, \(\kappa\) = 1.5 \(\times\)\(10^{-4}\) & 1.58 & 2.10, 0.86, 0.89 & 1.41, 3, 2 \\  & 0.2 & \(f_{\text{b}}\) = 0.4, \(\kappa\) = 7.5 \(\times\)\(10^{-5}\) & 2.18 & 4.08, 0.95, 0.55 & 1.41, 2.41, 1 \\ \cline{1-1}  & 0.5 & \(f_{\text{b}}\) = 0.5, \(\kappa\) = 3 \(\times\)\(10^{-5}\) & 3.65 & 14.94, 12.09, 8.20 & 2.24, 0, 1.41 \\ \cline{1-1}  & 1 & \(f_{\text{b}}\) = 0.5, \(\kappa\) = 1 \(\times\)\(10^{-4}\) & 6.67 & 24.98, 20.12, 22.96 & 1.41, 1, 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The reconstruction results of the AGES-AHK method. Figure 9: The reconstruction errors and the total calculation time of the three temperature phantoms. phantoms. It can be contributed to the fact that, with the increase of \(\text{coef}_{\text{HE}}\), the temperature distribution becomes more concentrated in localized high-temperature regions, making it more challenging to accurately reconstruct the temperature field. In all three temperature phantoms, the AGES-AHK method achieves a maximum \(D_{\text{h}}\) of less than 3% and NRMSE values below 6.7%, representing the optimal performance in hotspot position tracking and overall reconstruction. At \(\text{coef}_{\text{HE}}\) values of 0.1, 0.2, and 0.5, the NRMSE is less than 3.7% and \(E_{\text{h}}\) is less than 15%, representing the superior overall reconstruction performance and peak temperature tracking ability of AGES-AHK method. Even at the highest \(\text{coef}_{\text{HE}}\) value of 1, the AGES-AHK method was still able to obtain meaningful results under extreme conditions. The temperature contour maps reconstructed by the finalized AGES-AHK method is shown in Figure 10.  
Figure 10: The temperature contour maps reconstructed by the finalized AGES-AHK method under four \(\text{coef}_{\text{HE}}\) values. As illustrated in Figure 10, when the \(\text{coef}_{\text{HE}}\) value is low, the temperature distribution of the reconstructed field is relatively uniform, and the interactions between the heat sources are pronounced. Conversely, when \(\text{coef}_{\text{HE}}\) increases, the reconstructed temperature attenuates faster with distance, and the temperature distribution is more concentrated in space. Furthermore, higher \(\text{coef}_{\text{HE}}\) values results in more pronounced localized high-temperature zones, with strong heat source independence, weaker superposition effects, and high-temperature zones confined to the center of each source.