## 4 Solution
### OGS-Lp-FAST Model  
The denoising models based on OGS is more time consuming than the TV-based model. This is mainly because that OGS model considers the gradient information of the neighborhood in a reconstructed image, thus making the computation more complex. Thus, this is a shortcoming. [PERSON] et al. [12] proposed an accelerated ADMM algorithm with a restart that improves the convergence rate of the ADMM algorithm from \(O(\frac{1}{k})\) to \(O(\frac{1}{k^{2}})\). Inspired by them, we adopted this algorithm to improve the OGS-Lp model. This modified model is named OGS-Lp-FAST (\"Ours\" for short). The auxiliary variables \(\mathbf{U}_{i}(i=1,2)\) and \(\widetilde{\mathbf{U}}_{i}(i=1,2)\) are first adopted. Under this framework, \(\mathbf{Z_{i}}(i=1,2,3,4)\) is updated in the following way:  
\[\left\{ \begin{array}{l}\mathbf{Z}_{\mathbf{1}(n+1)}^{(k+1)}=\mathbf{mat}\Bigg{\{}\bigg{[} \mathbf{I}+\frac{\mu}{\beta_{1}}\mathbf{D}^{2}\Big{(}v^{(n)}\Big{)}^{2}\Big{(}\mathbf{Z}_{ \mathbf{1}(n+1)}^{(k+1)}\Big{)}\bigg{]}^{-1}\mathfrak{z}_{\mathbf{1}(0)}^{(k+1)} \Bigg{\}}\\ \mathbf{Z}_{\mathbf{2}(n+1)}^{(k+1)}=\mathbf{mat}\Bigg{\{}\Big{[}\mathbf{I}+\frac{\mu}{\beta_{2 }}\mathbf{D}^{2}\Big{(}v^{(n)}\Big{)}\Big{(}\mathbf{Z}_{\mathbf{2}(n+1)}^{(k+1)}\Big{)} \Big{]}^{-1}\mathfrak{z}_{\mathbf{2}(0)}^{(k+1)}\Bigg{\}}\\ \mathbf{Z}_{\mathbf{3}}^{(k+1)}=\textit{shrinkage}(\mathbf{F}^{(k)}-\mathbf{G}+\widetilde\mathbf{{Z}}_{\mathbf{3}}^{(k)},\frac{1}{\beta_{3}},p)\\ \mathbf{Z}_{\mathbf{4}}^{(k+1)}=\textit{min}\Big{(}1,\textit{max}\Big{(}\mathbf{F}^{(k)}+ \widetilde{\mathbf{Z}}_{\mathbf{4}}^{(k)},0\Big{)}\Big{)},\end{array} \right.. \tag{33}\]  
The initial values of \(\mathbf{Z}_{\mathbf{1}(0)}^{(k+1)},\mathbf{Z}_{\mathbf{2}(0)}^{(k+1)}\) in the above equation are:  
\[\left\{ \begin{array}{l}\mathbf{Z}_{\mathbf{1}(0)}^{(k+1)}=\mathbf{K}_{\mathbf{h}}*\mathbf{F}^{(k )}+\widetilde{\mathbf{U}}_{\mathbf{1}}^{(k)}\\ \mathbf{Z}_{\mathbf{2}(0)}^{(k+1)}=\mathbf{K}_{\mathbf{v}}*\mathbf{F}^{(k)}+\widetilde{\mathbf{U}}_\mathbf{{2}}^{(k)}\\ \end{array} \right.. \tag{34}\]  
The dual variable \(\widetilde{\mathbf{Z}}_{i}(i=1,2)\) can be updated as follows:  
\[\left\{ \begin{array}{l}\widetilde{\mathbf{Z}}_{\mathbf{1}}^{(k+1)}=\widetilde{\mathbf{ U}}_{\mathbf{1}}^{(k)}+\gamma\beta_{1}\Big{(}\mathbf{K}_{\mathbf{h}}*\mathbf{F}^{(k)}-\mathbf{Z}_{ \mathbf{1}}^{(k+1)}\Big{)}\\ \widetilde{\mathbf{Z}}_{\mathbf{2}}^{(k+1)}=\widetilde{\mathbf{U}}_{\mathbf{2}}^{(k)}+\gamma \beta_{2}\Big{(}\mathbf{K}_{\mathbf{v}}*\mathbf{F}^{(k)}-\mathbf{Z}_{\mathbf{2}}^{(k+1)}\Big{)}\\ \widetilde{\mathbf{Z}}_{\mathbf{3}}^{(k+1)}=\widetilde{\mathbf{Z}}_{\mathbf{3}}^{(k)}+\gamma \beta_{3}\Big{(}\mathbf{F}^{(k)}-\mathbf{G}-\mathbf{Z}_{\mathbf{3}}^{(k+1)}\Big{)}\\ \widetilde{\mathbf{Z}}_{\mathbf{4}}^{(k+1)}=\widetilde{\mathbf{Z}}_{\mathbf{4}}^{(k)}+\gamma \beta_{4}\Big{(}\mathbf{F}^{(k)}-\mathbf{Z}_{\mathbf{4}}^{(k+1)}\Big{)}\end{array} \right.. \tag{35}\]  
As image denoising is not a strong convex problem, the iteration needs to be restarted to ensure the convergence of the algorithm. When Equation (33) is not satisfied, the algorithm is restarted. \[c_{i}^{(k)}<\eta c_{i}^{(k-1)}(i=1,2), \tag{36}\]  
where \(c_{i}^{(k)}=\beta^{-1}\big{\|}\widetilde{\mathbf{Z}}_{\mathbf{i}}^{(k)}-\widetilde{ \mathbf{U}}_{\mathbf{i}}^{(k)}\big{\|}_{2}^{2}+\beta\big{\|}\mathbf{Z}_{\mathbf{i}}^{(k)}-\mathbf{U }_{\mathbf{i}}^{(k)}\big{\|}_{2}^{2}\) is the sum of the \((k)-th\) primal residuals \(\beta\big{\|}\mathbf{Z}_{\mathbf{i}}^{(k)}-\mathbf{U}_{\mathbf{i}}^{(k)}\big{\|}_{2}^{2}\) and the dual residuals \(\beta^{-1}\big{\|}\widetilde{\mathbf{Z}}_{\mathbf{i}}^{(k)}-\widetilde{\mathbf{U}}_{\mathbf{i}} ^{(k)}\big{\|}_{2}^{2}\), and \(\eta\) is a number close to 1. To prevent frequent restarts, \(\eta=0.97\) is set. When \(c_{i}^{(k)}<\eta c_{i}^{(k-1)}\), the acceleration step size is set to \(\varepsilon_{i}\), and the auxiliary variables \(\mathbf{U}_{\mathbf{i}}(i=1,2)\) and \(\widetilde{\mathbf{U}}_{\mathbf{i}}^{(k+1)}\)\((i=1,2)\) are updated as follows:  
\[\left\{