## 3 Results and Discussion
### Parametric Investigation of the Initial Reconstruction Result  
The accuracy of the reconstructed field is affected [16] by the selection of grid collocation, shape parameter \(\varepsilon\) and RBF types, etc. Using the simulation settings in Section 3.1 and a uniform grid collocation of \(n\)\(\times\)\(n\) (\(n\) = 7-10), the \(\text{coef}_{\text{HE}}\) of 0.1 was selected and \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**coef\({}_{\text{HE}}\)**} & \multicolumn{3}{c}{\(\text{C}_{\text{HE}}\)} \\ \cline{2-4}  & _Tfield1_ & _Tfield2_ & _Tfield3_ \\ \hline
0.1 & 3.26 & 2.23 & 2.34 \\
0.2 & 6.51 & 5.22 & 4.08 \\
0.5 & 16.53 & 14.96 & 15.28 \\
1 & 32.04 & 31.33 & 34.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The \(\text{C}_{\text{HE}}\) values of the three temperature phantoms under four \(\text{coef}_{\text{HE}}\) values. three temperature phantoms in Equation (12) were reconstructed. Furthermore, G and MQ in Equation (8) were chosen with 501 shape parameters equally spaced between 0 and 1. The NRMSE was exclusively employed to evaluate the overall reconstruction error during the research process. The NRMSEs of the reconstructed temperature field under different conditions are shown in Figure 8, and the optimal parameters can be determined using the results obtained. The trend of the variation in the reconstruction error with different shape parameters for two distinct RBF types is displayed in Figure 8. The overall trend of the reconstruction error does not significantly change with different grids, but the NRMSE varies greatly with the shape parameters. The effect of the shape parameter \(\varepsilon\) on the reconstruction results has been investigated, and the NRMSE is critically influenced by the RBF type and shape parameters [49; 50; 51], as demonstrated in Figure 8 For the G-RBF, the NRMSE can be less than 1% at the optimal \(\varepsilon\) in the preset temperature phantoms. However, the G-RBF has a limited optimal \(\varepsilon\) range of approximately 0.1 in _Tfield_1 and 0.3 in _Tfield_2 and _Tfield_3, as demonstrated in Figure 8a-c. This is attributed to the fact that _Tfield_1 possesses the simplest single-peak structure, devoid of the influence of other heat sources, and consequently can be fitted with the sharper G-RBF. Conversely, _Tfield_2 and _Tfield_3 exhibit a more intricate structure characterized by interactions among heat sources, necessitating the use of a less sharp G-RBF for adequate fitting. In contrast, the MQ-RBF exhibits a wider range of optimal \(\varepsilon\) values than G-RBF, and reaches a stable low NRMSE value when \(\varepsilon\) approaches 1, with the lowest NRMSE stabilizing at 2%, as shown in Figure 8d-f. In summary, the MQ-RBF exhibits a wide range of optimal \(\varepsilon\) values and good global characterization. Therefore, based on preliminary experiments and parametric investigation, the MQ-RBF with \(\varepsilon\) set to 1 was utilized as the basis method in this paper. Next, the three temperature phantoms at \(\mathrm{coef_{HE}}\) = 0.1 were reconstructed with a uniform grid collocation of \(n\times n\) (\(n\) = 7-19) using the basis method, and their reconstruction performances are shown in Figure 9.  
Figure 8: Parametric investigation of the influence of the \(\varepsilon\) and RBF types on the reconstruction results. The red region represents the local zoom position. As shown in Figure 9, the NRMSEs fluctuate downward as the number of grids increases. When the number of grids increases to a certain extent, the NRMSE tends to converge. Increasing the number of grids provides a more detailed description of the reconstructed field and reduces the error. However, owing to the nature of the ill-posed problem, obtaining an unbiased estimator of the true value is impossible. With the increase in the number of grids, as does the increase in the number of unknown parameters, the problem's ill-conditioned nature then exacerbates, which results in an increase in the reconstruction error.