## 2 Methods  
### Weighted Total Difference Minimization  
Because the projection data used in sparse-view CT reconstruction are incomplete, it is an ill-posed inverse problem. At present, a large number of documents realize the solution to such ill-posed problems by introducing the regularization method into the process of reconstruction. Finally, a small amount of projection data can be used to reconstruct high-quality CT images. The regularization method is used to solve this kind of ill-posed problem, which can be equivalent to the solution process of the optimization problem as shown in Equation (1). \[f=\underset{f}{\text{argmin}}\|Af-p\|_{2}^{2}+\lambda\cdot R(f) \tag{1}\]  
where \(f\) is the image to be restored, \(A\) is the projection matrix, \(p\) is the projected image which is captured, \(\lambda\) is a regularization parameter, \(\|Af-p\|_{2}^{2}\) is a fidelity item, and \(R(f)\) is the regular term of the iterative reconstruction algorithm. According to the literature [30], the weighted total difference minimization considers the sparsity constraint of the image gradient and the continuity constraint of the image gradient, which can effectively protect the gradient information in different directions at the edge of the image. Therefore, we take WTDM as a regularization constraint for the iterative reconstruction of sparse-view neutron images. According to WTD theory, the WTD of an image \(f\) can be defined according to Equation (2):  
\[\text{WTD}(f)=\left\|G_{\text{x}}f_{i,j}\right\|_{1}+\left\|G_{\text{y}}f_{i, j}\right\|_{1}+\varphi(\left\|G_{\text{xy}}f_{i,j}\right\|_{1}+\left\|G_{\text{yx}}f_{i,j }\right\|_{1}) \tag{2}\]  
where  
\[ \begin{array}{c}\left\|G_{\text{x}}f_{i,j}\right\|_{1}=\left\|f_{i+1,j}-f_{i,j}\right\|_{1}\\ \left\|G_{\text{y}}f_{i,j}\right\|_{1}=\left\|f_{i,j+1}-f_{i,j}\right\|_{1}\\ \left\|G_{\text{xy}}f_{i,j}\right\|_{1}=\left\|f_{i,j+1}-f_{i+1}\right\|_{1} \\ \left\|G_{\text{xy}}f_{i,j}\right\|_{1}=\left\|f_{i+1,j+1}-f_{i,j}\right\|_{1} \end{array}  \tag{3}\]  
where \(f_{i,j}\) represents the pixel value in the coordinates \((i,j)\) of the reconstructed image. \(\varphi\) is the weight value between gradient continuity and gradient sparsity constraints. When \(\varphi=0\), it only constrains the sparsity of the image gradient. When \(\varphi=1\), the same degree of constraint for gradient sparsity and continuity can be achieved. If we substitute the regularization term \(\text{WTD}(f)\) into the iterative reconstruction Equation (1), the image iterative reconstruction model is transformed into Equation (4). \[f=\underset{f}{\text{argmin}}\|Af-g\|_{2}^{2}+\lambda\cdot\sum_{i}\sum_{j} \bigl{[}\left|f_{i+1,j}-f_{i,j}\right|+\left|f_{i,j+1}-f_{i,j}\right|+\varphi \bigl{(}\left|f_{i+1,j+1}-f_{i,j}\right|+\left|f_{i,j+1}-f_{i+1,j}\right| \bigr{)}\bigr{]} \tag{4}\]  
Therefore, the solution of the weighted total difference minimization CT reconstruction model is equivalent to the solution of Equation (5). There are many methods for solving Equation (5), such as the soft threshold filtering method and the split Bregman method [31]. The convergence and effectiveness of the soft threshold filtering method have been proved in theory, and the soft threshold filtering method has been successfully applied to the field of CT reconstruction [26]. Therefore, the soft threshold filtering method is also used in this paper to solve Equation (5). In the framework of soft threshold filtering, a pseudoinverse equation is constructed to solve the reconstruction model based on weighted total difference minimization. \[f_{ij}^{n+1}=\frac{1}{4+4\cdot\varphi}\times\left\{