## 4 Results  
### Phantom Simulation Results  
The reconstructed conductivity images of the cylindrical phantom were compared with the corresponding target image, as shown in Figure 2. Under the noise-free condition, the conductivity distribution in the homogeneous region is accurate for all the methods in Figure 2a. The distinct difference between these methods depends on the boundary artifacts in the transition region between different contrast components. The boundary artifacts are obviously around the small cylindrical for the CM, with origins from the locally homogeneous assumption. In the DRM conductivity image, the artifacts are reduced due to the adoption of gradient information in the critical equation. However, several artifacts remain around the transition region because of the Laplacian operator. By contrast, the proposed method does not require the calculation of the Laplacian operator, yielding a more precise result than the DRM. Figure 2b depicts the conductivity profiles along the red line in Figure 2a. The proposed method represents a smoother result that is close to the target values. Figure 3 shows the mean and SD of the reconstructed conductivity in each region for different methods. In most areas, the proposed method leads to accurate values and lower SD values than other methods.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**DRM**} & \multicolumn{3}{c}{**Proposed Method**} \\ \cline{3-8}  & & \(A_{1}\) & \(A_{2}\) & \(\alpha_{2}\) & \(\beta_{2}\) & \(\alpha_{1}\) & \(\beta_{1}\) \\ \hline \multirow{3}{*}{Simulation} & Phantom & 1 & 800 & 1 & 1 & 40 & 2 \\ \cline{2-8}  & Brain & 2 & 500 & 1 & 1 & 10 & 1 \\ \hline \multirow{2}{*}{Experiment} & Phantom & 1 & 800 & 1 & 1 & 40 & 2 \\ \cline{2-8}  & Human & 5 & 500 & 1 & 1 & 10 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the double regularization method (DRM) and the proposed method.  
Figure 4 illustrates the conductivity images under different noise conditions by using three reconstruction methods. The CM conductivity results are severely affected by noise because the Laplacian operator is sensitive to noise. The use of a larger Laplacian kernel can alleviate this phenomenon, but it can also over smoothen the results. The DRM results demonstrate better suppression of the noise effect. In contrast to DRM, the proposed method shows superior conductivity images. The reconstructed results display a clear phantom structure, and the small inner regions retain a sharper edge than the other methods as the noise increase. Thus, the proposed method can alleviate noise effects.  
Figure 3: Bar graph of reconstructed conductivity value using conventional method (CM), double regularization method (DRM), and proposed method (PM). It corresponds to seven regions.  
Figure 2: Comparison of reconstructed conductivity images using different methods under the noise-free condition. (**a**) The target image and reconstructed conductivity images using the conventional method (CM), double regularization method (DRM), and proposed method (PM). (**b**) The reconstructed conductivity profiles along the red line in the target image in (**a**).  
The RMSE was used to assess the accuracy of three reconstruction methods with various noise levels. Figure 5 shows the results; the \(x\)-axis presents the SD of zero-mean Gaussian noise, in which the SD is \(1/\big{(}\sqrt{2}SNR\big{)}\); the SNR is the signal to noise ratio in the MR magnitude image. Notably, the RMSE of the CM increases with noise and is much larger than that of other methods. For DRM and the proposed method, the change of RMSE in low noise stays small, and the DRM RMSE is larger than the proposed method. These two methods similarly show an increasing tendency of RMSE as the noise increases. The magnitude of the DRM is larger than that of the proposed method. The RMSE of the proposed method stays at a low level, even in severe noise corruption. Thus, the proposed method is robust for noisy phase data.