## 1 Introduction  
The electrical properties (EPs) of biological tissues, which include electrical conductivity \(\sigma\) and permittivity \(\varepsilon\), are fundamental properties that quantify the ability to transfer electrical current inside the medium and characterize the effect of electric polarization. EPs rely on the frequency of the applied electromagnetic field and are related to multiple tissue characteristics, which involve ion concentration, volume fraction, cellular membrane permeability, and pathological condition [1; 2; 3]. It has been reported that cancerous tissues show a significantly higher value of EPs than normal tissues according to the ex vivo experimental results; for example, the EP value of breast cancer is approximately \(>\)200% higher than normal tissue and that of bladder cancer is approximately \(>\)100% higher [4; 5; 6]. Thus, mapping EPs has the potential for cancer detection and diagnosis. Moreover, tissue EPs are critical for the application of transcranial magnetic stimulation and transcranial direct current stimulation, which provides potential therapeutic applications for a wide variety of disorders,including [PERSON]'s disease, stroke, depression, and schizophrenia [7; 8; 9]. Furthermore, the estimation of the specific absorption rate (SAR), which is a crucial problem in high and ultra-high field magnetic resonance imaging (MRI) systems, requires accurate conductivity distribution [10; 11]. This evidence suggests that imaging the distribution of EPs can provide essential information for both research and medical purposes. Many efforts have been made to image in vivo EPs based on several distinct measuring techniques [12; 13; 14]. Among them, magnetic resonance electrical properties tomography (MREPT) is a non-invasive method to simultaneously map conductivity and permittivity using the MRI system [15; 16]. MREPT exploits the function between EPs and electromagnetic field produced by the radiofrequency (RF) coil system, the \(B_{1}^{+}\) field. The \(B_{1}^{+}\) field can be measured by MR scanning. Therefore, MREPT reconstructs the distribution of tissue EPs from the \(B_{1}^{+}\) measurement [17]. MREPT reconstruction algorithms can be divided into three categories, which are direct methods, inverse methods, and learning-based approaches. The direct methods reconstruct EPs by applying Maxwell's equations to the measured RF field. The frequently used method of direct methods is the conventional method (referred to as CM); it depends on the assumption that the EPs distribution is locally homogeneous. This method has reconstructed EPs successfully [15], but it includes certain disadvantages. Firstly, the true EPs distribution in the human body, which is spatially heterogeneous, violates the above-mentioned assumption. Thus, the boundary artifacts are brought around the boundary between different tissues [18]. Several studies have attempted to address this problem by introducing the gradient information of EPs to the reconstruction method [19; 20]. Another problem of the CM originates from obtaining phase information. MREPT reconstruction requires the magnitude and phase of \(B_{1}^{+}\). The magnitude information can be obtained using the B1 mapping methods [21; 22; 23], but phase information cannot be explicitly determined using existing technology. The transceive phase assumption (TPA) is used [24]. TPA approximates the \(B_{1}^{+}\) phase as half of the transceive phase, namely the sum of the transmit and receive phase. [PERSON] et al. found that the TPA is valid in low field strength, low permittivity, and symmetric objects [25]. Lastly, the Laplacian operator is sensitive to noise and then introduces numerical errors. [PERSON] et al. provided the theoretical relationship between the signal to noise ratio (SNR) of MREPT and the input \(B_{1}^{+}\) data [26].