## 4 Reduced-Order Tikhonov Regularization Imaging Based on Soft-Threshold Segmentation  
During the process of electrical impedance tomography (EIT) image reconstruction, the number of image pixels is much larger than the number of boundary measurements. To obtain higher-quality reconstructed images, increasing the number of image pixels is necessary, which further increases the ill-posedness of the image reconstruction process. However, if the pixels belonging to the background region in the image are removed from the solution process, it can partially improve the ill-posedness of the problem-solving process. Moreover, removing more background regions while ensuring the preservation of the region of interest leads to a better improvement in image reconstruction. The reduced-order Tikhonov algorithm considers grayscale value of 0 pixels in the reconstructed image as the background, which helps reduce the number of unknowns. In this study, based on the results of the image segmentation algorithm, the pixels belonging to the background region are removed from the solution process. This reduction in unknowns is done while maintaining the same number of equations in the inverse problem formulation. After solving for the pixel values in the region of interest, they are merged with the background region obtained through image segmentation to obtain a new reconstructed image. The specific steps are as follows:  
1. Firstly, an existing image reconstruction algorithm is used to obtain an initial distribution map \(g\) of the medium field, which serves as the initial image for subsequent image processing. The initial algorithm used in this study is the Tikhonov regularization algorithm. 2. An image segmentation algorithm is used to segment the initial distribution map \(g\) into three parts: the inner region \(q_{1}\), the edge region \(q_{2}\), and the background region \(q_{3}\). The segmented medium distribution image is denoted as \(g^{*}_{1}\). 3. The background region is removed from the image reconstruction process to reduce the number of unknowns. The corresponding rows of the sensitivity matrix \(S\) are reduced based on the image segmentation result. The reduced sensitivity matrix is denoted as \(S^{*}\). The process of obtaining the reduced sensitivity matrix is shown as follows:  
\[S^{*}=\left\{ \begin{array}{cl}\left[\right]&i\in q_{3}\\ S(;i)&i\in q_{1}\cup q_{2}\end{array} \right. \tag{22}\]
4. Based on the obtained reduced sensitivity matrix \(S\), the Tikhonov regularization imaging algorithm is applied to reconstruct the inner region \(q_{1}\) and the edge region \(q_{2}\). The reconstructed image after dimension reduction is denoted as \(g_{2}^{*}\). 5. For the reconstructed image \(g_{2}^{*}\) after dimension reduction, it is restored to the complete image \(g_{3}^{*}\) based on the segmented medium distribution \(g_{1}^{*}\) and the image segmentation result. The mathematical expression for the image restoration process is shown as follows: \[g_{3}^{*}= \begin{cases}g_{1}^{*}(i)&i\in q_{3}\\ g_{2}^{*}(i)&i\in q_{1}\cup q_{2}\end{cases} \] (23)
6. For the obtained complete reconstructed image \(g_{3}^{*}\), an image segmentation algorithm is applied once again to perform image segmentation and extract the regions of interest. The algorithm flowchart is shown in Figure 6. Different relaxation factors should be selected for different application conditions. For obtaining the reduced-order Tikhonov regularization, a larger relaxation factor is used to preserve the medium information that may exist at the segmentation edges and ensure that the regions of interest, including the inner region \(q_{1}\) and the edge region \(q_{2}\), are not mistakenly eliminated when excluding the background region \(q_{3}\). In order to achieve a clearer representation of the regions of interest, a smaller relaxation factor is chosen to mitigate potential artifacts in the reconstructed image.