## 2 EIT Imaging Basic Principles  
### Mathematical Model of EIT Imaging  
EIT technology is based on electromagnetic field theory, and the mathematical model of the electrical impedance tomography sensor, based on the differential form of [PERSON]'s equations, can be described as follows:  
\[\left\{ \begin{array}{l}\
abla\times H=J+\frac{\partial D}{\partial t}\\ J=\sigma E\\ D=\varepsilon E\\ E=-\
abla\phi\end{array} \right. \tag{1}\]  
In the above equation, \(E\) represents the electric field intensity, \(H\) represents the magnetic field intensity, \(J\) represents the current density, \(D\) represents the charge density, \(\phi\) represents the electric potential distribution function, \(\sigma\) represents the conductivity, and \(\varepsilon\) represents the absolute permittivity. When an excitation signal with a frequency of \(\omega\) is applied to the measured field domain \(\Omega\), the electric field intensity can be expressed as \(E=\hat{E}\,e^{i\omega t}\). Substituting this into Equation (1) yields:  
\[\
abla\times H=-(\sigma+j\omega\varepsilon)\
abla\phi \tag{2}\]  
Taking the divergence of both sides of Equation (2), we can obtain the following:  
\[\
abla\cdot[(\sigma+j\omega\varepsilon)\
abla\phi]=0 \tag{3}\]  
For the mathematical model of the sensitivity field in electrical impedance tomography (EIT), a unique solution exists when boundary conditions are applied to the field domain. The boundary condition utilized in this article is as follows:  
\[\left\{ \begin{array}{l}\sigma\cdot\frac{\partial\phi}{\partial t}=J\\ \int_{\partial\Omega}J\mathrm{ds}=0\end{array} \right. \tag{4}\]  
In the above equation, \(n\) represents the outward normal unit vector of the boundary \(\partial\Omega\), and \(J\) denotes the injected current density on the boundary \(\partial\Omega\) of the measured field \(\Omega\). For a given conductivity distribution \(\sigma\), the specification of the boundary current density determines the measured voltage. The relationship between the conductivity distribution, current density, and boundary voltage measurements can be described as a nonlinear equation as follows:  
\[U=\phi(\sigma,J) \tag{5}\]  
where \(U\) is the boundary voltage measurement at the electrodes. The expression for the change in boundary voltage measurements under the perturbation of conductivity distribution \(\Delta\sigma\) can be given by the following equation:  
\[\Delta U =\frac{\mathrm{d}\phi(\sigma,J)}{\mathrm{d}\sigma}\Delta\sigma+O \Big{(}(\Delta\sigma)^{2}\Big{)} \tag{6}\]  
By ignoring higher-order terms \(O((\Delta\sigma)^{2})\), the equation can be simplified to a linear form as follows:  
\[\Delta U =\frac{\mathrm{d}\phi(\sigma,J)}{\mathrm{d}\sigma}\Delta\sigma \tag{7}\]  
Discretization of the above equation can further yield the following:  
\[F =Sg \tag{8}\]  
In the above equation, \(F\) is the \(n\times 1\) normalized boundary measurement difference, \(g\) is the \(m\times 1\) reconstructed image grayscale vector, and \(S\) is the \(n\times m\) sensitivity matrix calculated based on [PERSON]'s sensitivity theorem [25]. m is the number of discretizedelements, and n is the number of independent boundary measurements. \(S_{nm}\) is the element at position (\(n\),\(m\)) in \(S\), with its calculation formula given as follows:  
\[S_{nm}=\int\frac{\
abla\phi_{m}(l_{i})}{I_{i}}\cdot\frac{\
abla\phi_{n}(I_{j})}{ I_{j}}\mathrm{d}x\mathrm{d}y \tag{9}\]  
where \(\
abla\phi_{m}(l_{i})\) is the potential at the \(m\)th pixel when the \(i\)th pair of electrodes injects current \(I_{i}\), and \(\
abla\phi_{m}(I_{j})\) is the potential at the \(m\)th pixel when the \(j\)th pair of electrodes injects current \(I_{j}\). This paper uses the finite element method to solve the forward problem of simulated boundary voltages and calculates the sensitivity matrix using COMSOL Multiphysics 5.6 software and MATLAB 2019 software. COMSOL Multiphysics, developed by Sweden's COMSOL Group (Stockholm, Sweden), is an advanced numerical simulation software based on the finite element method.