# Image Deblurring under Impulse Noise via Total Generalized Variation and Non-Convex Shrinkage
## 3 Model Proposal and Solution Method
### Solution Method  
To solve the model described by Equation (4), the complex optimization problem is decomposed into several sub-problems that are solved using the ADMM framework. Then, alternating iterations of the solutions of the sub-problems are performed until the solutions converge to the optimal solution of the original optimization problem. To this end, a series of intermediate variables \(\mathbf{X}_{i}\) are introduced, with \(\mathbf{X}_{i}\in\mathbb{R}^{N\times N}(i=0\sim 5)\), and they are defined as  
\[\left\{ \begin{array}{l}\mathbf{X}_{0}=\mathbf{H\ast F-G}\\ \mathbf{X_{1}}=\mathbf{K_{h}\ast F-V_{h}}\\ \mathbf{X_{2}}=\mathbf{K_{v}\ast F-V_{v}}\\ \mathbf{X_{3}}=\mathbf{K_{h}\ast V_{h}}\\ \mathbf{X_{4}}=\mathbf{K_{v}\ast V_{v}}\\ \mathbf{X_{5}}=\mathbf{K_{v}\ast V_{h}+K_{h}\ast V_{v}}\end{array} \right.. \tag{5}\]  
The original problem is now transformed into the following constraint problem:  
\[ \begin{split}&\underset{F,\mathbf{V_{h}},\mathbf{V_{v}}}{min}\left\|\bm {X_{0}}\right\|_{p}^{p}+\mu[\alpha_{0}(\left\|\mathbf{X_{1}}\right\|_{1}+\left\| \mathbf{X_{2}}\right\|_{1})+\alpha_{1}(\left\|\mathbf{X_{3}}\right\|_{1}+\left\|\mathbf{X _{4}}\right\|_{1}+\left\|\mathbf{X_{5}}\right\|_{1})]\\ & s.t.\mathbf{X_{0}}=\mathbf{H\ast F-G},\mathbf{X_{1}}=\mathbf{K_{h}\ast F-V_{h}},\mathbf{X_{2 }}=\mathbf{K_{v}\ast F-V_{v}},\\ &\mathbf{X_{3}}=\mathbf{K_{h}\ast V_{h}},\mathbf{X_{4}}=\mathbf{K_{v}\ast V_{v}},\mathbf{X_{5} }=\mathbf{K_{v}\ast V_{h}+K_{h}\ast V_{v}}.\end{split}  \tag{6}\]  
The augmented Lagrangian multiplier (ALM) method [20] is used to transform the constraint problem described by Equation (6) into an unconstrained augmented Lagrangian function as follows:  
\[ \begin{split} J=\underset{\mathbf{\Lambda_{0}}\sim\mathbf{\Lambda_{5}}}{max} \left\{\begin{array}{l}\underset{F,\mathbf{V_{h}},\mathbf{V_{v}}}{min}\left\|\mathbf{X_{ 0}}\right\|_{p}^{p}+\mu[\alpha_{0}(\left\|\mathbf{X_{1}}\right\|_{1}+\left\|\mathbf{X _{2}}\right\|_{1})+\alpha_{1}(\left\|\mathbf{X_{3}}\right\|_{1}+\left\|\mathbf{X_{4}} \right\|_{1}+\left\|\mathbf{X_{5}}\right\|_{1})]\\ &\mathbf{X_{0}\sim X_{5}}\end{split} \right.\\ &-\left\langle\beta_{0}\mathbf{\Lambda_{0}},\mathbf{X_{0}}-(H\ast F-G) \right\rangle-\left\langle\beta_{1}\mathbf{\Lambda_{1}},\mathbf{X_{1}}-(K_{h}\ast F-V_{ h})\right\rangle-\left\langle\beta_{1}\mathbf{\Lambda_{2}},\mathbf{X_{2}}-(K_{v}\ast F-V_{ v})\right\rangle\\ &-\left\langle\beta_{2}\mathbf{\Lambda_{3}},\mathbf{X_{3}}-K_{h}\ast V_{ h}\right\rangle-\left\langle\beta_{2}\mathbf{\Lambda_{4}},\mathbf{X_{4}}-K_{v}\ast V_{ v}\right\rangle-\left\langle\beta_{2}\mathbf{\Lambda_{5}},\mathbf{X_{5}}-(K_{v}\ast V _{v}+K_{h}\ast V_{v})\right\rangle\\ &+\frac{\beta_{0}}{2}(\left\|\mathbf{X_{0}}-(H\ast F-G)\right\|_{2}^{2}+ \frac{\beta_{1}}{2}(\left\|\mathbf{X_{1}}-(K_{h}\ast F-V_{h})\right\|_{2}^{2}+ \left\|\mathbf{X_{2}}-(K_{v}\ast F-V_{v})\right\|_{2}^{2})\\ &+\frac{\beta_{2}}{2}(\left\|\mathbf{X_{3}}-K_{h}\ast V_{h}\right\|_{2}^ {2}+\left\|\mathbf{X_{4}}-K_{v}\ast V_{v}\right\|_{2}^{2}+\left\|\mathbf{X_{5}}-(K_{v} \ast V_{h}+K_{h}\ast V_{v})\right\|_{2}^{2}))\end{split} \tag{7}\]  
problem described by Equation (9). To this end, \(\mathbf{F}\), \(\mathbf{V_{h}}\), and \(\mathbf{V_{v}}\) in Equation (9) are re-expressed in the frequency domain as follows:  
\[\left\{