## 3 Results and Discussion
### The Discussion of the Overall Reconstruction Error and the Hotspot Tracking Performance  
Subsequently, the basis method, the AHK method, the AGES and the AGES-AHK method were used to reconstruct the three temperature phantoms under four \(\text{coef}_{\text{HE}}\) values, and a quantitative analysis was conducted. The overall reconstruction error and the condition number of coefficient matrix \(\mathbf{B}\) (denoted as \(\text{Cond}(\mathbf{B})\)) are shown in Figures 11 and 12, respectively. The hotspot tracking performance is represented by the hotspot temperature error and the hotspot position deviation. The hotspot temperature error \(E_{\text{h}}\) is shown in Figure 13. And the hotspot position deviation \(D_{\text{h}}\) is shown in Tables 3-5. Figure 11: The reconstruct errors of three temperature phantoms under four \(\text{coef}_{\text{HE}}\) values. Figure 12: The Cond(**B**) of three temperature phantoms under four \(\text{coef}_{\text{HE}}\) values. Figure 13: The hotspot temperature error of three temperature phantoms under four \(\text{coef}_{\text{HE}}\) values. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**coef\({}_{\text{HE}}\)**} & \multicolumn{4}{c}{\(\text{\boldmath$D_{\text{h}}$/$\%}\)} \\ \cline{2-5}  & **Basis** & **AHK** & **AGES** & **AGES-AHK** \\ \hline
0.1 & 0 & 0 & 0 & 0 \\
0.2 & 0 & 0 & 0 & 0 \\
0.5 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The hotspot position deviation of _Tfield\(1\)_. As shown in Figure 11, the NRMSE of all four methods exhibits a marked increase with elevated \(\text{coef}_{\text{HE}}\) values in the three temperature phantoms, verifying the effect of the intrinsic characteristics of the reconstructed temperature field on the difficulty of reconstruction. For the basis method, when \(\text{coef}_{\text{HE}}\) values were maintained at 0.1 and 0.2, all three temperature phantoms exhibited relatively uniform temperature distributions with NRMSE values below 7%, indicating good reconstruction performance. However, at \(\text{coef}_{\text{HE}}\) values of 0.5 and 1, these phantoms demonstrated a sharp escalation in temperature gradients accompanied by a significant NRMSE increase. The implementation of both AHK and AGES effectively reduced the NRMSE across all twelve cases compared to the basis method. AGES demonstrated superior NRMSE reduction capability over the AHK method under \(\text{coef}_{\text{HE}}\) conditions of 0.1, 0.2, and 0.5. Notably, the two methodologies exhibited comparable efficacy at \(\text{coef}_{\text{HE}}=1\). Ultimately, the integrated AGES-AHK method achieved the optimal overall reconstruction performance, with NRMSE less than 6.7%. As demonstrated in Figure 11, the AGES-AHK method exhibits the lowest NRMSE, improving reconstruction accuracy by more than 33% compared to the basis method (traditional LSQR algorithm, incorporated with MQ-RBF approximation and uniform domain discretization). Figure 12 shows the condition number of the coefficient matrix \(\mathbf{B}\) of Equation (10) (\(\text{Cond}(\mathbf{B})\)) for the four reconstruction methods. \(\text{Cond}(\mathbf{B})\) is directly related to the ill-conditioned nature of the AP problem. An increasing \(\text{Cond}(\mathbf{B})\) indicates the exacerbating ill-conditioned nature of Equation (10) and the increased reconstruction difficulty. As shown in Figure 12, the implementation of AHK substantially reduces \(\text{Cond}(\mathbf{B})\) across all three temperature phantoms. This suggests that modifications to the kernel function through AHK effectively mitigate the ill-conditioned nature of Equation (10), thereby enhancing reconstruction accuracy and reducing the NRMSE. The implementation of AGES reduces \(\text{Cond}(\mathbf{B})\) in _Tfeld1_ and _Tfeld2_, which reduces the ill-conditioned nature of the problem and thus improves the reconstruction performance.