## 4 Solution
### OGS-Lp-FAST Model
\begin{array}{l}\varepsilon_{i}^{(k+1)}=\frac{1+\sqrt{ 1+4{(c_{i}^{(k)})}^{2}}}{2}(i=1,2)\\ \mathbf{U}_{\mathbf{i}}^{(k+1)}=\mathbf{Z}_{\mathbf{i}}^{(k+1)}+\frac{\varepsilon_{i}^{(k)}-1 }{\varepsilon_{i}^{(k+1)}}(\mathbf{Z}_{\mathbf{i}}^{(k+1)}-\mathbf{Z}_{\mathbf{i}}^{(k)})(i=1,2) \\ \widetilde{\mathbf{U}}_{\mathbf{i}}^{(k+1)}=\widetilde{\mathbf{Z}}_{\mathbf{i}}^{(k+1)}+\frac{ \varepsilon_{i}^{(k)}-1}{\varepsilon_{i}^{(k+1)}}(\widetilde{\mathbf{Z}}_{\mathbf{i}}^{(k +1)}-\widetilde{\mathbf{Z}}_{\mathbf{i}}^{(k)})(i=1,2)\end{array} \right.. \tag{37}\]  
which is updated according to the equations below upon restart:  
\[\left\{ \begin{array}{l}\varepsilon_{i}^{(k+1)}=1(i=1,2)\\ \mathbf{U}_{\mathbf{i}}^{(k+1)}=\mathbf{Z}_{\mathbf{i}}^{(k+1)}(i=1,2)\\ \widetilde{\mathbf{U}}_{\mathbf{i}}^{(k+1)}=\widetilde{\mathbf{Z}}_{\mathbf{i}}^{(k+1)}(i=1,2) \\ c_{i}^{(k+1)}=\eta^{-1}c_{i}^{(k)}(i=1,2)\end{array} \right.. \tag{38}\]Up to this point, all subproblems of the proposed model are solved. The OGS-Lp-FAST algorithm just described is summarized as Algorithm 1. The \(\boldsymbol{F}\) sub-problem should be updated as  
\[\boldsymbol{F}^{(k+1)}=\] \[\mathcal{F}^{-1}\left(\frac{\beta_{1}\overline{\boldsymbol{k}}_{ \boldsymbol{k}}^{*}{}_{*}\left(\overline{\boldsymbol{u}}_{1}^{(k+1)}- \overline{\boldsymbol{\bar{u}}}_{1}^{(k+1)}\right)+\beta_{2}\overline{ \boldsymbol{k}}_{\boldsymbol{v}}^{*}{}_{*}\left(\overline{\boldsymbol{u}}_{2}^ {(k+1)}-\overline{\boldsymbol{\bar{u}}}_{2}^{(k+1)}\right)+\beta_{3}\left( \overline{\boldsymbol{Z}}_{3}^{(k+1)}+\overline{G}-\overline{\boldsymbol{ \bar{Z}}}_{3}^{(k+1)}\right)+\beta_{4}\left(\overline{\boldsymbol{Z}}_{4}^{(k +1)}-\overline{\boldsymbol{\bar{Z}}}_{4}^{(k+1)}\right)}{\beta_{1}\overline{ \boldsymbol{k}}_{\boldsymbol{u}}^{*}\overline{\boldsymbol{K}}_{\boldsymbol{u }}+\beta_{2}\overline{\boldsymbol{k}}_{\boldsymbol{v}}^{*}\overline{ \boldsymbol{K}}_{\boldsymbol{v}}+\beta_{3}\boldsymbol{1}+\beta_{4}\boldsymbol{ 1}}\right). \tag{39}\]  
```
Input: image \(\mathbf{G}\) with noise Output: denoised image \(\mathbf{F}\) Initialize: \(k=1\), \(n=0\), \(\boldsymbol{Z}_{i}^{(k)}=\boldsymbol{0}\), \(\boldsymbol{\bar{Z}}_{i}^{(k)}=\boldsymbol{0}(i=1,2,\cdots,4)\), \(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\), \(\beta_{4}\), \(\mu,p,\eta\), \(\boldsymbol{U}_{j}^{(k)}=\boldsymbol{0}\), \(\boldsymbol{\bar{U}}_{j}^{(k)}=\boldsymbol{0}(j=1,2)\), \(\gamma,tol,c^{1}=inf,\boldsymbol{F}^{(k)}=\boldsymbol{G}\), Max
1: for \(k=1:Max\)
2: If \(c_{i}^{(k+1)}<\eta c_{i}^{(k)}:\)
3: \(\boldsymbol{Z}_{i}^{(k+1)}\), \((i=1,2,3,4)\), are updated with Equations (33) and (34)
4: \(\boldsymbol{\bar{Z}}_{i}^{(k+1)}\)\((i=1,2,3,4)\) are updated with Equation (35)
5: \(\epsilon_{i}^{(k+1)},\boldsymbol{U}_{j}^{(k+1)}\), \(\boldsymbol{\bar{U}}_{j}^{(k+1)}\)\((j=1,2)\) is updated with Equation (37)
6: \(\boldsymbol{F}^{(k+1)}\) are updated with Equation (39)
7: \(E=\|\boldsymbol{F}^{(k+1)}-\boldsymbol{F}^{(k)}\|_{2}/\|\boldsymbol{F}^{(k)}\|_ {2}\)
8 Else
9: Restart as in Equation (38)
10: End If
11: If \(E<tol\) Break;
12: End For
13: Return \(\boldsymbol{F}^{(k)}\) as \(\boldsymbol{F}\)
```  
**Algorithm 1** OGS-Lp-FAST pseudo-code