Algorithm 1 Image Restoration Using SPG-SC Model.

Require:  The observed image $\mathit{y}$ and measurement matrix $\mathit{H}$.    1:Initial ${{\mathit{X}}_G}_0=\mathbf{0}$, ${{\mathit{A}}_G}_0=\mathbf{0}$, ${\overline{\mathit{B}}}_G=\mathbf{0}$, $\mathit{C}=\mathbf{0}$ and $\mathit{J}=\mathbf{0}$.    2:Set parameters b, m, L, ${\mu }_1$, ${\mu }_2$, $\eta$, $\tau$, p, t, ${\sigma }_n$, ${\epsilon }_P$, ${\epsilon }_G$, ${ϵ}_P$ and ${ϵ}_G$.    3:for$t=0$ Max-Iter do    4: Update ${{\mathit{X}}_G}^{(t+1)}$ by Equation (20);    5: ${\mathit{R}}^{(t+1)}={{\mathit{X}}_G}^{(t+1)}-{\mathit{C}}^{\left(t\right)}$;    6: Construct dictionary $\mathit{D}$ by ${\mathit{R}}^{(t+1)}$ using K-means algorithm and PCA.    7: for Each patch ${\mathit{r}}_i$ do    8:  Choose the best match PCA dictionary ${\mathit{D}}_i$ for ${\mathit{r}}_i$;    9:  Compute ${\mathit{\alpha }}_i^{\left(t\right)}$ by ${\mathit{D}}_i^{-1}{\mathit{r}}_i$;  10:  Update ${\mathit{w}}_i^{(t+1)}$ by computing Equation (30);  11:  Update $\lambda$ by computing Equation (32);  12:  Update ${\mathit{\alpha }}_i^{t+1}$ by computing Equation (22);  13: end for  14: ${\mathit{R}}_G^{(t+1)}={\mathit{X}}_G^{(t+1)}-{\mathit{J}}^{\left(t\right)}$;  15: for Each patch-group ${\mathit{R}}_{G_i}$ do  16:  Construct dictionary ${\mathit{D}}_{G_i}$ by computing Equation (27);  17:  Compute ${\mathit{B}}_{G_i}^{\left(t\right)}$ by ${\mathit{D}}_{G_i}^{-1}{\mathit{R}}_{G_i}$;  18:  Update ${\mathit{K}}_{G_i}^{(t+1)}$ by computing Equation (31);  19:  Update $\rho$ by computing Equation (33);  20:  Update ${\mathit{B}}_{G_i}^{(t+1)}$ by computing Equation (29);  21: end for  22: Update ${{\mathit{A}}_G}^{(t+1)}$ by concatenating all ${\mathit{\alpha }}_i$;  23: Update ${\mathit{D}}^{(t+1)}$ by concatenating all ${\mathit{D}}_i$;  24: Update ${{\overline{\mathit{B}}}_G}^{(t+1)}$ by concatenating all ${\mathit{B}}_{G_i}$;  25: Update ${\mathit{D}}_G^{(t+1)}$ by concatenating all ${\mathit{D}}_{G_i}$;  26: Update ${\mathit{C}}^{(t+1)}$ by Equation (17);  27: Update ${\mathit{J}}^{(t+1)}$ by Equation (18);  28:end for  29:$\mathbf{Output}:$ The final restored image $\widehat{\mathit{x}}$ by aggregating patches in ${\mathit{X}}_G$.