## 2 Materials and Methods
### The Solution of the Lp_HOGS Model  
The alternating direction method of multipliers (ADMM) was used to solve the Lp_HOGS model. When solving this model, ADMM transformed the original complex problem into several relatively simple subproblems by introducing decoupling variables. According to the ADMM solution framework, the intermediate variables \(Z_{0}=H*F-G,Z_{1}=K_{h}*F,Z_{2}=K_{v}*F,Z_{3}=K_{h}*K_{h}*F,Z_{4}=K_{v}*K_{v}*F,Z_{5}=K_{v}*K_{h}*F\), were included and the original problem was transformed into an optimization problem with constraints, which is expressed as:  
\[ \begin{array}{l}J=\underset{Z_{0}\sim Z_{5}}{\text{min}}\{\|Z_{0}\|_{p}^{p} +\mu_{1}[\varphi(Z_{1})]+\mu_{2}[\varphi(Z_{2})]\\ +\mu_{3}[\varphi(Z_{3})]+\mu_{4}[\varphi(Z_{4})]+\mu_{5}[\varphi(Z_{5})]\}. \end{array}  \tag{7}\]  
The corresponding Lagrange multiplier \(\Lambda_{i}(i=0,1,2,3,4,5)\) and quadratic penalty coefficient \(\beta_{i}(i=0,1,2,3,4,5)\) were used to transform Equation (7) into an unconstrained optimization problem, i.e., the augmented Lagrange objective function of the original problem, which is expressed as:  
\[\begin{array}{l}J=\underset{\Lambda_{0}\sim\Lambda_{5}}{\text{max}}\{& \underset{F,Z_{0}\sim Z_{5}}{\text{min}}\{\|Z_{0}\|_{p}^{p}-\langle\beta_{0} \Lambda_{0},Z_{0}-(H*F-G)\rangle+\frac{\beta_{0}}{2}\|Z_{0}-(H*F-G)\|_{2}^{2} \\ +\mu_{1}[\varphi(Z_{1})]-\langle\beta_{1}\Lambda_{1},Z_{1}-K\[0=\frac{\beta_{i}}{2}\|\mathbf{\Lambda}_{i}\|_{2}^{2}-\frac{\beta_{i}}{2}\|\mathbf{\Lambda }_{i}\|_{2}^{2}(i=0,1,2,3,4,5) \tag{9}\]  
Subsequently, adding Equation (9) to the right side of Equation (8) obtains:  
\[ \begin{array}{ll}\mathbf{I}=\underset{\mathbf{\Lambda}_{0}\sim\mathbf{\Lambda}_{5}}{max}& \{\underset{\mathbf{F},\mathbf{Z}_{0}\sim\mathbf{Z}_{5}}{min}\{\|\mathbf{Z}_{0}\|_{p}^{p}+ \frac{\beta_{0}}{2}\|\mathbf{Z}_{0}-(\mathbf{H}*\mathbf{F}-\mathbf{G})-\mathbf{\Lambda}_{0}\|_{2}^ {2}-\frac{\beta_{0}}{2}\|\mathbf{\Lambda}_{0}\|_{2}^{2}\\ &+\mu_{1}[\varphi(\mathbf{Z}_{1})]+\frac{\beta_{1}}{2}\|\mathbf{Z}_{1}-\mathbf{K}_{h}*\bm {F}-\mathbf{\Lambda}_{1}\|_{2}^{2}-\frac{\beta_{1}}{2}\|\mathbf{\Lambda}_{1}\|_{2}^{2} \\ &+\mu_{2}[\varphi(\mathbf{Z}_{2})]+\frac{\beta_{2}}{2}\|\mathbf{Z}_{2}-\mathbf{K}_{o}*\bm {F}-\mathbf{\Lambda}_{2}\|_{2}^{2}-\frac{\beta_{2}}{2}\|\mathbf{\Lambda}_{2}\|_{2}^{2} \\ &+\mu_{3}[\varphi(\mathbf{Z}_{3})]+\frac{\beta_{2}}{2}\|\mathbf{Z}_{3}-\mathbf{K}_{h}*\bm {K}_{h}*\mathbf{F}-\mathbf{\Lambda}_{3}\|_{2}^{2}-\frac{\beta_{5}}{2}\|\mathbf{\Lambda}_{3} \|_{2}^{2}\\ &+\mu_{4}[\varphi(\mathbf{Z}_{4})]+\frac{\beta_{4}}{2}\|\mathbf{Z}_{4}-\mathbf{K}_{o}*\bm {K}_{o}*\mathbf{F}-\mathbf{\Lambda}_{4}\|_{2}^{2}-\frac{\beta_{4}}{2}\|\mathbf{\Lambda}_{4} \|_{2}^{2}\\ &+\mu_{5}[\varphi(\mathbf{Z}_{5})]+\frac{\beta_{5}}{2}\|\mathbf{Z}_{5}-\mathbf{K}_{o}*\bm {K}_{h}*\mathbf{F}-\mathbf{\Lambda}_{5}\|_{2}^{2}-\frac{\beta_{5}}{2}\|\mathbf{\Lambda}_{5 }\|_{2}^{2}\}\}.\end{array}  \tag{10}\]  
Each subproblem must first be solved to solve the objective function. As the introduced variables \(\mathbf{Z}_{i}(i=0,1,2,3,4,5)\), \(\mathbf{\Lambda}_{i}(i=0,1,2,3,4,5)\), and \(\mathbf{F}\) were decoupled, the objective function corresponding to the subproblems of \(\mathbf{F}\) became:  
\[ \begin{array}{ll}\mathbf{J}_{\mathbf{F}}=\underset{\mathbf{F}}{min}&\{\frac{\beta_{0}}{2 }\|\mathbf{Z}_{0}-(\mathbf{H}*\mathbf{F}-\mathbf{G})-\mathbf{\Lambda}_{0}\|_{2}^{2}+\frac{\beta_{1} }{2}\|\mathbf{Z}_{1}-\mathbf{K}_{h}*\mathbf{F}-\mathbf{\Lambda}_{1}\|_{2}^{2}\\ &+\frac{\beta_{2}}{2}\|\mathbf{Z}_{2}-\mathbf{K}_{o}*\mathbf{F}-\mathbf{\Lambda}_{2}\|_{2}^{2} +\frac{\beta_{2}}{2}\|\mathbf{Z}_{3}-\mathbf{K}_{h}*\mathbf{K}_{h}*\mathbf{F}-\mathbf{\Lambda}_{3} \|_{2}^{2}\\ &+\frac{\beta_{4}}{2}\|\mathbf{Z}_{4}-\mathbf{K}_{o}*\mathbf{K}_{o}*\mathbf{F}-\mathbf{\Lambda}_{4 }\|_{2}^{2}+\frac{\beta_{5}}{2}\|\mathbf{Z}_{5}-\mathbf{K}_{o}*\mathbf{K}_{h}*\mathbf{F}-\mathbf{ \Lambda}_{5}\|_{2}^{2}\}.\end{array}  \tag{11}\]  
The convolution theorem was used to apply the Fourier transform to both sides of Equation (11), which is expressed as:  
\[