## 2 Methodology
### TV Regularization  
In the field of digital image processing, an image is commonly regarded as a function that requires appropriate modeling. This involves finding the most-suitable function for accurately describing and representing the image. Images often consist of various components such as edges, textures, and noise, each with distinct characteristics. Therefore, it is essential to identify function spaces that effectively capture these different characteristics. Researchers utilize functions within these function spaces to model the various components of an image. The TV regularization method, initially proposed by [PERSON], has been widely applied in image denoising [27]. Due to the excellent edge-preserving performance of this method, it has been noticed and improved by a wide range of scholars in the field of image processing. Fully variational regularization has had a significant impact on image processing and has greatly advanced the use of variational regularization methods in this field. However, it also has limitations. Fully variational minimization tends to provide a minimal solution for the binning constant, which makes it less effective in terms of preserving 'oscillating' details such as image textures. Additionally, it is prone to'step effects' in the grayscale gradient regions of the given image. To address the texture-preservation issue, [PERSON] introduced the nonlocal total variation (NLTV) regularization method, incorporating the concept of nonlocal methods [28]. In response to the problem that the Tikhonov regularization method smooths edges, resulting in a lower spatial resolution for the reconstructed image, a TV regularization method based on L1 parametrization was introduced [29]. The TV regularization method can enhance the stability of the solution when solving the EIT inverse problem, so as to significantly enhance the resolution of the discrete medium, thus to preserve the discontinuity of the boundary during the reconstruction process and enable the reconstruction of sharp edges to produce sharper images with good edge-preservation performance. Additionally, this method has a better temporal resolution for satisfying the real-time requirements of EIT. In the measured region \(\Omega\), the TV regularization term for the parameter \(\sigma\) can be defined as:  
\[TV(\sigma)=\int_{\Omega}|\
abla\sigma|\mathrm{d}\Omega \tag{7}\]  
Figure 2 depicts the edge-preserving principle of TV regularization, which can help to understand the edge-preservation properties highlighted by the above equation. Points A and B shown in Figure 2 are connected by three curves \(f_{1}(x)\), \(f_{2}(x)\), and \(f_{3}(x)\), and the TVs of the equations of the three curves have the same magnitude, which can be expressed as:  
\[TV(f)=\int_{A}^{B}f^{\prime}(x)dx=f(B)-f(A) \tag{8}\]  
It can be seen that the same TV value allows for different expressions of the function \(f(x)\), so when a discrete medium distribution is present in the measurement area, the TV regularization method can effectively reconstruct the edge information and maintain sharp edge characteristics. During EIT reconstruction, the total variance of the change in the conductivity distribution \(g\) is used as a TV regularization function:  
\[R(g)=TV(g)=\int_{\Omega}|\
abla g|\mathrm{d}\Omega \tag{9}\]  
To derive the above equation:  
\[R^{\prime}(g)=TV(g)=\
abla\cdot\left(\frac{\
abla g}{|\
abla g|}\right) \tag{10}\]  
To prevent the case in which \(R(g)\) is not differentiable when \(\
abla g=0\), a smooth approximation is used to ensure the differentiability of \(R(g)\) and is expressed as:  
\[R(g)=TV_{\beta}(g)=\int_{\Omega}\sqrt{|\
abla g|+\beta}\mathrm{d}\Omega \tag{11}\]  
where \(\beta\) is a positive constant, whose discrete form can be expressed as:  
\[TV_{\beta}(g)=\sum_{i}\sqrt{\|L_{i\mathcal{G}}\|_{2}^{2}+\beta} \tag{12}\]  
\(L\) is the sparse matrix corresponding to the change in conductivity distribution \(g\).