Algorithm 3 ADMM for the $L_1$-TGV${}^2$ Model (20).

1:Input: Blurred image $\mathbf{B}$, blur kernel k (i.e., $\mathcal{K}$ in Section 3), ${\rho }_1=50$, ${\rho }_2=0.5$, ${\rho }_3=5$, ${\alpha }_1=1$, ${\alpha }_2=1.5$, $\tau =1.618$, $T_{max}=10$ and $ϵ=5\times {10}^{-5}$. 2:Initialize: ${\mathbf{L}}^0=\mathbf{B}$, ${\mathbf{V}}^0=0$, ${\xi }^0=0$, ${\zeta }^0=0$, ${\eta }^0=0$ and $t=0$. 3:while (not converged and $t\le T_{max}$) do 4:  Compute ${\mathbf{W}}^{t+1}$ according to ${\mathbf{W}}^{t+1}=\mathbf{shrinkage}\left(\mathcal{K}{\mathbf{L}}^t-\mathbf{B}+{\xi }^t/{\rho }_1,1/{\rho }_1\right)$. 5:  Compute ${\mathbf{Y}}^{t+1}$ according to ${\mathbf{Y}}^{t+1}=\mathbf{shrinkage}\left(\nabla {\mathbf{L}}^t-{\mathbf{V}}^t+{\zeta }^t/{\rho }_2,\lambda {\alpha }_1/{\rho }_2\right)$. 6:  Compute ${\mathbf{Z}}^{t+1}$ according to ${\mathbf{Z}}^{t+1}=\mathbf{shrinkage}\left(\mathcal{E}\left({\mathbf{V}}^t\right)+{\eta }^t/{\rho }_3,\lambda {\alpha }_0/{\rho }_3\right)$. 7:  Compute $\left({\mathbf{L}}^{t+1},{\mathbf{V}}^{t+1}\right)$ according to ${\mathbf{L}}^{t+1}={\mathcal{F}}^{-1}\left(\frac{{\mathbf{det}}_{\mathbf{L}}}{{\mathbf{det}}_{\mathbf{T}}}\right),{\mathbf{V}}_1^{t+1}={\mathcal{F}}^{-1}\left(\frac{{\mathbf{det}}_{{\mathbf{V}}_1}}{{\mathbf{det}}_{\mathbf{T}}}\right)\mathrm{and}{\mathbf{V}}_2^{t+1}={\mathcal{F}}^{-1}\left(\frac{{\mathbf{det}}_{{\mathbf{V}}_2}}{{\mathbf{det}}_{\mathbf{T}}}\right).$ 8:  Update Lagrangian multipliers $\left(\xi ,\zeta ,\eta \right)$: ${\xi }^{t+1}={\xi }^t-\tau {\rho }_1\left({\mathbf{W}}^{t+1}-\left(\mathcal{K}{\mathbf{L}}^{t+1}-\mathbf{B}\right)\right).$ ${\zeta }^{t+1}={\zeta }^t-\tau {\rho }_2\left({\mathbf{Y}}^{t+1}-\left(\nabla {\mathbf{L}}^{t+1}-{\mathbf{V}}^{t+1}\right)\right).$ ${\eta }^{t+1}={\eta }^t-\tau {\rho }_3\left({\mathbf{Z}}^{t+1}-\mathcal{E}\left({\mathbf{V}}^{t+1}\right)\right).$ 9:  Check convergence condition: ${∥{\mathbf{L}}^{t+1}-{\mathbf{L}}^t∥}_{\infty }<ϵ$ 10:end while 11:Output: Deblurred image $\mathbf{L}$.