## 2 Materials and Methods
### The Solution of the Lp_HOGS Model
\begin{array}{ll}\mathbf{J}_{\mathbf{F}}=\underset{\mathbf{F}}{min}&\{\frac{\beta_{0}}{2 }\|(\mathbf{\overline{H}}\circ\mathbf{\overline{F}}-\mathbf{\overline{G}})+\mathbf{\overline{ \Lambda}}_{0}-\mathbf{\overline{Z}}_{0}\|_{2}^{2}+\frac{\beta_{1}}{2}\|\mathbf{\overline {K}}_{h}\circ\mathbf{\overline{F}}+\mathbf{\overline{\Lambda}}_{1}-\mathbf{\overline{Z}}_{ 1}\|_{2}^{2}\\ &+\frac{\beta_{2}}{2}\|\mathbf{\overline{K}}_{o}\circ\mathbf{\overline{F}}+\mathbf{ \overline{\Lambda}}_{2}-\mathbf{\overline{Z}}_{2}\|_{2}^{2}+\frac{\beta_{2}}{2}\| \mathbf{\overline{K}}_{h}\circ\mathbf{\overline{K}}_{h}\circ\mathbf{\overline{F}}+\mathbf{ \overline{\Lambda}}_{3}-\mathbf{\overline{Z}}_{3}\|_{2}^{2}\\ &+\frac{\beta_{4}}{2}\|\mathbf{\overline{K}}_{o}\circ\mathbf{\overline{K}}_{o}\circ \mathbf{\overline{F}}+\mathbf{\overline{\Lambda}}_{4}-\mathbf{\overline{Z}}_{4}\|_{2}^{2}+ \frac{\beta_{5}}{2}\|\mathbf{\overline{K}}_{o}\circ\mathbf{\overline{K}}_{h}\circ\mathbf{ \overline{F}}+\mathbf{\overline{\Lambda}}_{5}-\mathbf{\overline{Z}}_{5}\|_{2}^{2}\}, \end{array}  \tag{12}\]  
where \(\circ\) denotes the dot product operator and \(\mathbf{\overline{X}}\) is the Fourier transform of the \(\mathbf{X}\) matrix. The partial derivative of \(\mathbf{\overline{F}}\) can be calculated using:  
\[\begin{array}{ll}\frac{\partial\mathbf{f}_{\mathbf{F}}}{\partial\mathbf{\overline{F}}}&= \beta_{0}\mathbf{\overline{H}}^{*}\circ((\mathbf{\overline{H}}\circ\mathbf{\overline{F}}- \mathbf{\overline{G}})+\mathbf{\overline{\Lambda}}_{0}-\mathbf{\overline{Z}}_{0})+\beta_{1} \mathbf{\overline{K}}_{h}^{*}\circ(\mathbf{\overline{K}}_{h}\circ\mathbf{\overline{F}}+\mathbf{ \overline{\Lambda}}_{1}-\mathbf{\overline{Z}}_{1})\\ &+\beta_{2}\mathbf{\overline{K}}_{o}*\circ(\mathbf{\overline{K}}_{o}\circ\mathbf{\overline{F}} _{o}\circ\mathbf{\overline{F}}_{o}\mathbf{\overline{F}}_{o}\mathbf{\overline{F}}_{o}\mathbf{ \overline{F}}_{o}\mathbf{\overline{F}}_{o}\mathbf{\overline{F}}_{h}\mathbf{\overline{F}}_{o}\mathbf{ \overline{where./ represents the dot division operator and \(\mathcal{F}^{-1}(\mathbf{X})\) is the inverse Fourier transform of \(\mathbf{X}\).  
The objective function to solve the subproblem \(\mathbf{Z}_{0}\) is expressed as:  
\[\mathbf{J}_{\mathbf{Z}_{0}}=\underset{\mathbf{Z}_{0}}{min}\{\ \|\mathbf{Z}_{0}\|_{p}^{p}+\frac{ \beta_{0}}{2}\|\mathbf{Z}_{0}-\left(\mathbf{H}*\mathbf{F}-\mathbf{G}\right)-\mathbf{\Lambda}_{0}\|_ {2}^{2}\ \} \tag{18}\]  
According to the Lp shrinkage operator:  
\[\mathit{shrink}_{p}(\mathbf{\upvarepsilon},\tau)=max\{|\mathbf{\upvarepsilon}|-\tau ^{2-p}|\mathbf{\upvarepsilon}|^{p-1},0\}\frac{\mathbf{\upvarepsilon}}{|\mathbf{\upvarepsilon }|} \tag{19}\]  
The updated formula of \(\mathbf{Z}_{0}\) is expressed as:  
\[ \begin{array}{ll}\mathbf{Z}_{0}&=\mathit{shrink}_{p}\Big{(}(\mathbf{H}*\mathbf{F}-\mathbf{ G})+\mathbf{\Lambda}_{0},\frac{1}{\beta_{0}}\Big{)}\\ &=\mathit{max}\Big{\{}|(\mathbf{H}*\mathbf{F}-\mathbf{G})+\mathbf{\Lambda}_{0}|-\beta_{0}p^{- 2}|(\mathbf{H}*\mathbf{F}-\mathbf{G})+\mathbf{\Lambda}_{0}|^{p-1},0\Big{\}}\frac{(\mathbf{H}*\mathbf{ F}-\mathbf{G})+\mathbf{\Lambda}_{0}}{|(\mathbf{H}*\mathbf{F}-\mathbf{G})+\mathbf{\Lambda}_{0}|}.\end{array}