## 5 Discussion  
In the present study, a novel approach was proposed for imaging the conductivity distribution from the MR transceive phase. On the basis of the gradient-based method, the proposed method \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Measured** & **Conventional Method** & **Double Regularizations Method** & **Proposed Method** \\ \hline Region 1 & 0.97 & 0.71 \(\pm\) 0.31 & 0.84 \(\pm\) 0.13 & 0.92 \(\pm\) 0.08 \\ Region 2 & 2.01 & 1.56 \(\pm\) 0.59 & 1.92 \(\pm\) 0.25 & 1.98 \(\pm\) 0.19 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Reconstructed conductivity values of phantom experiment. Figure 10: Magnitude, phase, and reconstructed conductivity images of head experiment. **(a)** Spin-echo (SE) magnitude and phase images; the SNR of the magnitude image is 75. **(b)** Conductivity images using the conventional method (CM), the double regularization method (DRM), and the proposed method (PM). Figure 9: Phantom reconstruction result. **(a)** Magnitude and phase images of the phantom; the SNR of the magnitude image is 132. **(b)** The reconstructed conductivity images using the conventional method (CM), double regularization method (DRM), and the proposed method (PM). reformulates the central equation through divergence properties and then leverages the finite difference method to construct an inverse problem. The resultant optimization problem is regularized by TGV and solved by ADMM. The proposed method does not require the Laplacian operator of the transceive phase, which is the origin of the dominant errors of general approaches. A systematic investigation using numerical simulations and experiments at 3T illustrate that the proposed method can improve the reconstruction accuracy and is more robust to the noise than the CM and DRM. The CM, which relies on the locally homogeneous assumption, is applicable for those objects with a simple structure but not for complicated objects such as the brain, as shown in Figure 5. The pre-segmentation of objects can partially mitigate these adverse effects before reconstruction. However, this method introduces extra work, and the assumption remains. Therefore, it cannot solve the problem fundamentally. By contrast, the proposed method uses conductivity gradient information and thus in principle is immune to the effects of the above-mentioned assumption. The Ella model simulation results indicate that the proposed approach is favorable for objects with complicated structures. The CM and DRM require the calculation of the Laplacian operator, which is sensitive to noise. The results of the CM show the effect of noise, while those of the DRM show the impact of the Laplacian operator despite applying conductivity gradient information. A noise-robust Laplacian kernel was used to alleviate the effect of noise [49]. However, despite the noise suppression, this leads to over-smoothened reconstruction results. Another method reconstructs conductivity profiles by using the first-order differencing of the RF filed data [50]. However, the induced current density image must be determined to obtain the EPs. By contrast, the proposed method transforms the equation to remove the necessity of the Laplacian operator of the transceive phase and directly retrieve the conductivity. The simulation and experiments show that the proposed method decreases the impact of noise. The proposed method depends on the central equation of the gradient-based method [36]. The following three assumptions are made. The first assumption is that the spatial derivatives of \(B_{1}^{+}\) and \(B_{1}^{-}\) are negligible. This assumption is valid in the quadrature birdcage coil central region when the coil is used to transmit and receive. However, this assumption appears untrue at high field strengths (>3T). The RF shimming technique may provide a resolution to improve the homogeneity of the RF field. The second assumption is \(\sigma^{2}>>\left(\omega\varepsilon\right)^{2}\), which is sufficient for common tissues at field strengths \(\leq\)3T.