## 3 Method
### Solution  
For the above model, Equation (8) is a non-convex and NP-hard problem since it contains a gradient \(L_{0}\)-norm. To resolve this problem efficiently, we first introduce two ad ditional parameters \(c_{1}\) and \(c_{2}\); Equation (8) could be redefined as a constrained optimization problem as follows:  
\[\underset{x,c_{1},c_{2}}{\text{argmin}}\frac{1}{2}||Ax-Y||_{2}^{2}+a||\
abla c_{1 }||_{0}+(1-a)||\
abla c_{2}||_{0}\text{ s.t. }c_{1}=x,c_{2}=x-x_{prior} \tag{9}\]  
Then, the above problem can be converted into an unconstrained problem by utilizing the scaled augmented Lagrangian function [44], mathematical model as follows:  
\[\underset{x,c_{1},c_{2},m_{1},m_{2}}{\text{argmin}}\frac{1}{2}||Ax-Y||_{2}^{2 }+a||\
abla c_{1}||_{0}+(1-a)||\
abla c_{2}||_{0}+\frac{\lambda_{1}}{2}||x-c_{ 1}-m_{1}||_{2}^{2}+\frac{\lambda_{2}}{2}||x-x_{prior}-c_{2}-m_{2}||_{2}^{2} \tag{10}\]  
where \(m_{1}\) and \(m_{2}\) are auxiliary variables, and \(\lambda_{1}\)and \(\lambda_{2}\) represent two Lagrangian multipliers. We utilize the split Bregman (SB) algorithm to divide Equation (10) into the following sub-problems:  
Sub-problem 1:  
\[x^{n+1}=\underset{x}{\text{argmin}}\frac{1}{2}||Ax-Y||_{2}^{2}+\frac{\lambda_{ 1}}{2}||x-c_{1}{}^{n}-m^{n}{}_{1}||_{2}^{2}+\frac{\lambda_{2}}{2}||x-x_{prior} -c^{n}{}_{2}-m^{n}{}_{2}||_{2}^{2} \tag{11}\]  
Sub-problem 2:  
\[c_{1}{}^{n+1}=\underset{c_{1}}{\text{argmin}}a||\
abla c_{1}||_{0}+\frac{ \lambda_{1}}{2}||x^{n+1}-c_{1}-m^{n}{}_{1}||_{2}^{2} \tag{12}\]  
\[c_{2}{}^{n+1}=\underset{c_{2}}{\text{argmin}}(1-a)||\
abla c_{2}||_{0}+\frac{ \lambda_{2}}{2}||x^{n+1}-x_{prior}-c_{2}-m^{n}{}_{2}||_{2}^{2} \tag{13}\]  
Sub-problem 3:  
\[m_{1}{}^{n+1}=m_{1}{}^{n}-(x^{n+1}-c_{1}{}^{n+1}) \tag{14}\]  
\[m_{2}{}^{n+1}=m_{2}{}^{n}-(x^{n+1}-x_{prior}-c_{2}{}^{n+1}) \tag{15}\]  
Equation (11) can be resolved via the separable surrogate algorithm [45] as follows:  
\[x_{ij}{}^{n+1}=x_{ij}{}^{n}-\left([A^{T}A]_{ij}+\lambda_{1}+\lambda_{2}\right) ^{-1}\left([A^{T}(Ax-Y)]_{ij}+\lambda_{1}(x^{n}-c_{1}{}^{n}-m_{1}{}^{n})_{ij} +\lambda_{2}(x^{n}-x_{prior}-c_{2}{}^{n}-m_{2}{}^{n})_{ij}\right) \tag{16}\]  
Because Equations (12) and (13) are similar, we only discuss the solution of (13). \(c_{2}\) is obtained by minimizing the \(L_{0}\)-norm, which is not easily solved directly. So, we reference an approximate method in the literature [46] to solve it. Firstly, we introduce two auxiliary variables (\(u_{2}{}^{k}\) and \(v_{2}{}^{k}\)) in place of the x- and y- direction of \(c_{2}\), and Equation (13) can be transformed as follows:  
\[\underset{c_{2},\{u_{2}{}^{k},v_{2}{}^{k}\}}{\text{argmin}}\left||x^{n+1}-x_{ prior}-c_{2}-m_{2}{}^{n}||_{2}^{2}+\gamma_{2}{\sum}_{k}\psi(|u_{2}^{k}|+|v_{2}^{k}|)+ \tau_{2}((\partial_{x}(c_{2}^{k})-u_{2}{}^{k})^{2}+(\partial_{y}(c_{2}^{k})- v_{2}{}^{k})^{2})\right. \tag{17}\]  
where \(\gamma_{2}\) = 2 \(\times\) (1 \(-\)\(a\))/\(\lambda_{2}\) affects the ability of image edge retention. \(\tau_{2}\) is utilized to trade off the proximity between (\(u_{2}{}^{k}\), \(v_{2}{}^{k}\)) and (\(\partial_{x}(c_{2}{}^{k}\)), \(\partial_{y}(c_{2}{}^{k}\))). Equation (17) has three parameters, and we still use the SB algorithm to compute. Equation (17) can be further converted into the two steps, as follows:  
Step 1:  
\[\{(u_{2}^{k})^{r+1},(v_{2}^{k})^{r+1}\}=\underset{\{u_{2}^{k},v_{2}^{k}\}}{ \text{argmin}}{\sum}_{k}\left((\partial_{x}(c_{2}^{k})^{r}-u_{2}^{k})^{2}+( \partial_{y}(c_{2}^{k})^{r}-v_{2}^{k})^{2}\right)+\frac{\tau_{2}}{\tau_{2}}{ \sum}_{k}\psi(|u_{\[\{c_{2}{}^{r+1}\}=\underset{\{c_{2}\}}{\mathrm{argmin}}||x^{n+1}-x_{prior}-c_{2}-m _{2}^{n}||_{2}^{2}+\tau_{2}{\sum}_{k}\left((\partial_{x}(c_{2}^{k})-{(u_{2}^{k})} ^{r+1}\right)^{2}+(\partial_{y}(c_{2}^{k})-{(v_{2}^{k})}^{r+1})^{2}\right) \tag{19}\]  
For Equation (18), the optimal condition is as follows:  
\[\left({(u_{2}^{k})}^{r+1},{(v_{2}^{k})}^{r+1}\right)=\left\{