## 4 Solution  
### Solving the OSG-Lp Model  
The OSG-Lp model is treated as a minimization problem whose computation is given below. In real-life images, the values of all pixels usually fall in a limited interval \([\mathbf{a},\mathbf{b}]\). For calculation and verification convenience, the image data are normalized so that they are all within \([0,1]\). The operator \(P_{\Omega}\) is first defined with \(\Omega=\left\{\mathbf{F}\in\mathbb{R}^{M\times M}|0\leq\mathbf{F}_{i,j}\leq 1\right\}\). \[\Omega\left(\mathbf{F}_{i,j}\right)=\left\{ \begin{array}{ccc}0&,&\mathbf{F}_{i,j}<0, \\ \mathbf{F}_{i,j}&,&\mathbf{F}_{i,j}\in[0,1]\\ 1&,&\mathbf{F}_{i,j}>1,\end{array} \right. \tag{7}\]  
To solve the proposed model, we employed the ADMM framework and changed the complex problem into several subproblems. First, some intermediate variables were introduced to decouple the subproblems. That is, \(\mathbf{Z_{1}}=\mathbf{K_{h}}*\mathbf{F},\mathbf{Z_{2}}=\mathbf{K_{v}}*\mathbf{F},\mathbf{Z_{3}}=\mathbf{F}-\bm {G},\mathbf{Z_{4}}=\mathbf{F}\). then Equation (6) can be reformulated into the following constrained optimization problem:  
\[\mathbf{F}=\arg\min_{\mathbf{F}_{i,j}\in\Omega}\|\mathbf{Z_{3}}\|_{p}^{p}+ \mu[\varphi(\mathbf{Z_{1}})+\varphi(\mathbf{Z_{2}})] \tag{8}\] \[\mathbf{s.t.}\hskip 14.226378 pt\mathbf{Z_{1}}=\mathbf{K_{h}}*\mathbf{F},\mathbf{Z_{2}} =\mathbf{K_{v}}*\mathbf{F},\mathbf{Z_{3}}=\mathbf{F}-\mathbf{G},\mathbf{Z_{4}}=\mathbf{F}.\]  
According to the principle of the ADMM framework, the Lagrange multipliers and the quadratic penalty term are needed to establish the augmented function. Then we have the following:  
\[J(\mathbf{F},\mathbf{Z_{  
satisfy the form of \(\mathbf{a}^{2}-2\mathbf{a}\mathbf{b}+\mathbf{b}^{2}=\left(\mathbf{a}-\mathbf{b}\right)^{2}\), so Equation (10) can be written as Equation (12):  
\[ \begin{array}{l}J\big{(}F,\mathbf{Z_{1}},\mathbf{Z_{2}},\mathbf{Z_{3}},\mathbf{Z_{4}},\beta_{1 },\beta_{2},\beta_{3},\beta_{4}\big{)}\\ =\|\mathbf{Z_{3}}\|_{p}^{p}+\mu\varphi(\mathbf{Z_{1}})+\mu\varphi(\mathbf{Z_{2}})+\frac{ \beta_{1}}{2}\|\mathbf{Z_{1}}-K_{h}*F-\widetilde{\mathbf{Z}}_{1}\|_{2}^{2}\\ +\frac{\beta_{2}}{2}\|\mathbf{Z_{2}}-\mathbf{K_{v}}*F-\widetilde{\mathbf{Z}}_{2}\|_{2}^{2} +\frac{\beta_{2}}{2}\|\mathbf{Z_{3}}-F+G-\widetilde{\mathbf{Z}}_{3}\|_{2}^{2}\\ +\frac{\beta_{4}}{2}\|\mathbf{Z_{4}}-F-\widetilde{\mathbf{Z}}_{4}\|_{2}^{2}-\frac{ \beta_{1}}{2}\left(\widetilde{\mathbf{Z}}_{1}\right)^{2}-\frac{\beta_{2}}{2}\left( \widetilde{\mathbf{Z}}_{2}\right)^{2}-\frac{\beta_{3}}{2}\left(\widetilde{\mathbf{Z}} _{3}\right)^{2}-\frac{\delta_{4}}{2}\left(\widetilde{\mathbf{Z}}_{4}\right)^{2}. \end{array}  \tag{12}\]  
Because \(\mathbf{Z_{1}},\mathbf{Z_{2}},\mathbf{Z_{3}},\mathbf{Z_{4}},\widetilde{\mathbf{Z}}_{1},\widetilde{ \mathbf{Z}}_{2},\widetilde{\mathbf{Z}}_{3},\widetilde{\mathbf{Z}}_{4}\) are mutually independent, so, they can be solved as some independent subproblems according to the principle of the ADMM algorithm. These problems can be solved by the iterative algorithm for minimizing \(\mathbf{Z}_{i}\). Defining \(\mathbf{Z}_{i}^{(k)}(\mathbf{i}=1,2,3,4)\) represents the value of \(\mathbf{Z}_{i}\) after \((k)-th\) iterations. For a given \(\mathbf{Z}_{i}^{(k)}\), the next iteration \(\mathbf{Z}_{i}^{(k+1)}\) is generated as follows:  
\[\mathbf{Z_{1}^{(k+1)}}=\underset{\  
which means \(\varphi(v)\) can be solved by constructing the function \(S(v)\) of Equation (20):  
\[S(v,u)=\frac{1}{2}\!\sum_{i=1}^{N}\sum_{j=-N}^{N}\!\left[\left\|\widetilde{u}_{i, j,N,N}\right\|_{2}\left\|\widetilde{v}_{i,j,N,N}\right\|_{2}^{2}+\left\| \widetilde{u}_{i,j,N,N}\right\|_{2}\right]\geq\varphi(v)=\sum_{i=1}^{N}\sum_{ j=1}^{N}\left\|\widetilde{v}_{i,j,N,N}\right\|_{2},. \tag{20}\]  
After a simple calculation [31], \(S(v,u)\) is rewritten as  
\[S(v,u)=\frac{1}{2}\left\|\mathbf{D}(u)v\right\|_{2}^{2}+C(u), \tag{21}\]  
to facilitate future calculations.