## 3 Background  
### EIT Forward Model  
The principle of operation of EIT is demonstrated in Figure 1. When conducting scientific investigations, EIT is commonly studied from two primary perspectives: the forward problem and the inverse problem [29], as depicted in Figure 1c. The forward problem involves the computation of changes in the voltage observed on the object's surface by considering the distribution of conductivity within the object of interest and the applied excitation current. In medical imaging, the excitation current frequency typically falls within the range of 10-100 kHz for EIT. The objective of EIT image reconstruction centers on estimating the distribution of conductivity within a sensing region by administeringelectric currents and measuring the resulting variations in the boundary voltage [30]. The Dirichlet-to-Neumann (DtN) map, which forms a crucial part of the mathematical model utilized in EIT, is at the core. It is an essential element for investigating elliptic partial differential equations, thereby having a substantial impact on the classical Calderon problem [31].  
The physical meaning of EIT aligns with [PERSON]'s equations and electromagnetic field theory. The complete electrode model (CEM) of EIT is depicted below.  
\[\
abla\times(\sigma(x)\
abla\phi(x))=0,x\in\Omega \tag{1}\]  
Here, \(\
abla\) is a spin degree operator, \(\Omega\) represents the measuring field, \(\sigma(x)\) represents the conductivity distribution, and \(\phi(x)\) represents the potential distribution in the measuring field. The forward problem in EIT involves calculating the electric potential, denoted as \(\phi\), within a specific volume of interest, denoted as \(\Omega\). This calculation requires two pieces of information: First, the values of \(\phi\) on the boundary surface of \(\Omega\), denoted as \(\partial\Omega\). Second, the conductivity distribution, denoted as \(\sigma(x)\), for all points \(x\) within \(\Omega\). To solve this problem, FEM is commonly used to obtain the solution to the partial differential equation governing the electric potential in the forward model.  
\[\left\{\begin{aligned} \
abla\cdot(\sigma\
abla u)& =0,\quad\text{ in }\Omega\\ \sigma\frac{\partial u}{\partial n}\bigg{|}_{\partial\Omega}& =g\\ \int_{\partial\Omega}u&=0\end{aligned}\right. \tag{2}\]  
Let \(V_{meas}\in\mathbb{R}^{m}\) represent the voltage measurement value and \(\sigma\in\mathbb{R}^{n}\) denote the conductivity change. Then, \(n\gg m\) indicates that the number of pixels is significantly larger than the measurement value. Furthermore, the measurement vector b is always affected by noise. Hence, the problem of image reconstruction involves recovering the conductance image \(\sigma\) from measurements with \(V_{meas}\) noise [32].  
Figure 1: The principle of EIT. **(a)** The basic working principle of EIT involves taking voltage measurements at every electrode for each current injection. **(b)** Diagram illustrating the subdivision of finite elements on a 2D circular domain. **(c)** Typical schematic of forward and inverse problems in EIT.  
\[\arg\min_{\sigma}\left\{\frac{1}{2}\|J\sigma-V_{meas}\|_{2}^{2}+\lambda\|L\sigma\|_{ 2}^{2}\right\} \tag{3}\]  
where \(J\in\mathbb{R}^{m\times n}\) is the Jacobian matrix representing the linearization relationship between the measured theoretical value and the target conductance \(\sigma\), \(L\) is the regularization matrix, and \(\lambda\) is the hyperparameter. The second term, the regularization term, penalizes variations in \(\sigma\) based on prior knowledge. The hyperparameter evenly distributes the regularization term's and residual term's contributions to picture reconstruction. The aforementioned is the linear least squares approach according to the GN method  
\[\mathbf{\sigma}=\left(J^{\mathrm{T}}J+\lambda L^{\mathrm{T}}L\right)^{-1}J^{ \mathrm{T}}V_{meas} \tag{4}\]  
Different regularization methods are used for different \(\mathbf{L}\) outcomes. The EIDORS toolbox, citeeidors, may also calculate this Jacobian matrix of elements [33].