## 2 Materials and Methods
### The Solution of the Lp_HOGS Model
 \tag{20}\]  
Additionally, the objective function of the subproblem \(\mathbf{Z}_{1}\) is expressed as:  
\[\mathbf{J}_{\mathbf{Z}_{1}}=\underset{\mathbf{Z}_{1}}{min}\{\ \mu_{1}[\varphi(\mathbf{Z}_{1})]+ \frac{\beta_{1}}{2}\|\mathbf{Z}_{1}-\mathbf{K}_{h}*\mathbf{F}-\mathbf{\Lambda}_{1}\|_{2}^{2}\} \tag{21}\]  
According to the ADMM algorithm, the updated formula of \(\mathbf{Z}_{1}\) is expressed as:  
\[\mathbf{Z}_{1}^{(k+1)}=\textbf{mat}\{[\mathbf{I}+\frac{\mu_{1}}{\beta_{1}}\mathbf{D}^{2}( \mathbf{Z}_{1}^{(k+1)})]^{-1}\mathit{vec}\Big{(}\mathbf{Z}_{1}^{(k+1)}\Big{)}\}\ \ (\mathbf{Z}_{1}^{(k+1)})=\mathbf{K}_{h}*\mathbf{F}+\mathbf{\Lambda}_{1}^{(k)}) \tag{22}\]  
where **mat** is the vector matricization operator, \(\mathit{vec}\) is the matrix vectorization operator, \(\mathbf{I}\in\mathbb{R}^{N^{2}\times N^{2}}\) is the identity matrix, and \(\mathbf{D}\in\mathbb{R}^{N^{2}\times N^{2}}\) is the diagonal matrix with diagonal elements. \[\left[\mathbf{D}(\mathbf{U})\right]_{m,m}=\sqrt{\sum_{i=-K_{i}}^{K_{r}}\sum_{j=-K_{i} }^{K_{r}}\left\{\sum_{k_{1}=-K_{i}}^{K_{r}}\sum_{k_{2}=-K_{i}}^{K_{r}}\left| \mathbf{U}_{m-i+k_{1},m-j+k_{2}}\right|^{2}\right\}^{-\frac{1}{2}}} \tag{23}\]  
where \(\mathbf{U}\) represents an arbitrary matrix. Similarly, the updated formulas of \(\mathbf{Z}_{2}\sim\mathbf{Z}_{5}\) can be obtained, which are expressed as:  
\[\left\{ \begin{array}{ll}\mathbf{Z}_{2}^{(k+1)}=\textbf{mat}\{[\mathbf{I}+\frac{\mu _{2}}{\beta_{2}}\mathbf{D}^{2}(\mathbf{Z}_{2}^{(k+1)})]^{-1}\mathit{vec}\Big{(}\mathbf{Z} _{2}^{(k+1)}\Big{)}\}&\Big{(}\mathbf{Z}_{2}^{(k+1)}=\mathbf{K}_{p}*\mathbf{F}+\mathbf{\Lambda} _{2}^{(k)}\Big{)}\\ \mathbf{Z}_{3}^{(k+1)}=\textbf{mat}\{[\mathbf{I}+\frac{\mu_{3}}{\beta_{3}}\mathbf{D}^{2}( \mathbf{Z}_{3}^{(k+1)})]^{-1}\mathit{vec}\Big{(}\mathbf{Z}_{3}^{(k+1)}\Big{)}\}&\Big{(} \mathbf{Z}_{3}^{(k+1)}=\mathbf{K}_{h}*\mathbf{K}_{h}*\mathbf{F}+\mathbf{\Lambda}_{3}^{(k)}\Big{)} \\ \mathbf{Z}_{4}^{(k+1)}=\textbf{mat}\{[\mathbf{I}+\frac{\mu_{4}}{\beta_{4}}\mathbf{D}^{2}( \mathbf{Z}_{4}^{(k+1)})]^{-1}\mathit{vec}\Big{(}\mathbf{Z}_{4}^{(k+1)}\Big{)}\}&\Big{(} \mathbf{Z}_{4}^{(k+1)}=\mathbf{K}_{p}*\mathbf{K}_{p}*\mathbf{F}+\mathbf{\Lambda}_{4}^{(k)}\Big{)} \\ \mathbf{Z}_{5}^{(k+1)}=\textbf{mat}\{[\mathbf{I}+\frac{\mu_{5}}{\beta_{5}}\mathbf{D}^{2}( \mathbf{Z}_{5}^{(k+1)})]^{-1}\mathit{vec}\Big{(}\mathbf{Z}_{5}^{(k+1)}\Big{)}\}&\Big{(} \mathbf{Z}_{5}^{(k+1)}=\mathbf{K}_{p}*\mathbf{K}_{h}*\mathbf{F}+\mathbf{\Lambda}_{5}^{(k)}\Big{)}.\end{array} \right. \tag{24}\]  
The objective function to solve subproblem \(\mathbf{\Lambda}_{0}\) is expressed as:  
\[\mathbf{J}_{\mathbf{\Lambda}_{0}}=\underset{\mathbf{\Lambda}_{0}}{max}\{\ -\left\langle\beta_{0}\mathbf{\Lambda}_{0},\mathbf{Z}_{0}-(\mathbf{H}*\mathbf{F}-\mathbf{G})\right\rangle\ \} \tag{25}\]  
According to the gradient ascent method, the updated formula of \(\mathbf{\Lambda}_{0}\) is expressed as:  
\[\mathbf{J}_{\mathbf{\Lambda}_{0}}=\underset{\mathbf{\Lambda}_{0}}{max}\{\ -\left\langle\beta_{0}\mathbf{\Lambda}_{0},\mathbf{Z}_{0}-(\mathbf{H}*\mathbf{F}-\mathbf{G})\right\rangle\ \} \tag{26}\]  
where \(\gamma\) is the learning rate. The objective function to solve subproblem \(\mathbf{\Lambda}_{1}\) is expressed as:  
\[\mathbf{J}_{\mathbf{\Lambda}_{1}}=\underset{\mathbf{\Lambda}_{1}}{max}\{\ -\ \langle\beta_{1}\mathbf{\Lambda}_{1},\mathbf{Z}_{1}-\mathbf{K}_{h}*\mathbf{F}\rangle\ \} \tag{27}\]  
According to the gradient ascent method, the updated formula of \(\mathbf{\Lambda}_{1}\) is expressed as:  
\[\mathbf{\Lambda}_{1}{}^{(k+1)}=\mathbf{\Lambda}_{1}{}^{(k)}+\gamma\beta_{1}( \mathbf{K}_{h}*\mathbf{F}-\mathbf{Z}_{1}{}^{(k+1)}) \tag{28}\]  
Similarly, the updated formulas of \(\mathbf{\Lambda}_{2}\sim\mathbf{\Lambda}_{5}\) can be obtained, which are expressed as:  
\[\left\{