# Image Deblurring under Impulse Noise via Total Generalized Variation and Non-Convex Shrinkage
## 3 Model Proposal and Solution Method
### Solution Method
\begin{array}{ll}\frac{\partial\overline{J}_{\overline{F}}}{\partial \overline{F}}=\beta_{0}(\overline{\mathbf{H}})^{*}\circ[\overline{\mathbf{H}}\circ \overline{F}-\overline{G}-(\overline{\mathbf{X}}_{\mathbf{0}}^{(k)}-\overline{\mathbf{ \Lambda}}_{\mathbf{0}}^{(k)})]+\beta_{1}[\overline{K}_{h}{}^{*}\circ[\overline{K}_ {h}\circ\overline{F}-\overline{\mathbf{V}}_{h}-(\overline{\mathbf{X}}_{\mathbf{1}}^{(k)}- \overline{\mathbf{\Lambda}}_{\mathbf{1}}^{(k)})]\\ +\overline{\mathbf{K}}_{\mathrm{p}}{}^{*}\circ[\overline{K}_{\mathrm{p}}\circ \overline{F}-\overline{\mathbf{V}}_{\mathrm{p}}-(\overline{\mathbf{X}}_{\mathbf{2}}^{(k)}- \overline{\mathbf{\Lambda}}_{\mathbf{2}}^{(k)})]\Big{\}}=0\\ \frac{\partial\overline{J}_{\Psi_{h}}}{\partial\overline{V}_{h}}-\beta_{1}[ \overline{K}_{h}\circ\overline{F}-\overline{\mathbf{V}}_{h}-(\overline{\mathbf{X}}_{ \mathbf{1}}^{(k)}-\overline{\mathbf{\Lambda}}_{\mathbf{1}}^{(k)})]+\beta_{2}[\overline{( \mathbf{K}}_{h})^{*}\circ[\overline{K}_{h}\circ\overline{\mathbf{V}}_{h}-(\overline{ \mathbf{X}}_{\mathbf{3}}^{(k)}-\overline{\mathbf{\Lambda}}_{\mathbf{3}}^{(k)})]\\ +(\overline{K}_{\mathrm{p}})^{*}\circ[\overline{K}_{\mathrm{p}}\circ \overline{V}_{h}+\overline{K}_{h}\circ\overline{V}_{\mathrm{p}}-(\overline{ \mathbf{X}}_{\mathbf{5}}^{(k)}-\overline{\mathbf{\Lambda}}_{\mathbf{5}}^{(k)})]\Big{\}}=0\\ \frac{\partial\overline{J}_{\Psi_{\mathrm{p}}}}{\partial\overline{V}_{\mathrm{p} }}-\beta_{1}[(\overline{K}_{\mathrm{p}}\circ\overline{F}-\overline{\mathbf{V}}_{ \mathrm{p}}-(\overline{\mathbf{X}}_{\mathbf{2}}^{(k)}-\overline{\mathbf{\Lambda}}_{\mathbf{2}}^ {(k)})]+\beta_{2}[\overline{(\mathbf{K}}_{\mathrm{p}})^{*}\circ[\overline{K}_{ \mathrm{p}}\circ\overline{\mathbf{V}}_{\mathrm{p}}-(\overline{\mathbf{X}}_{\mathbf{4}}^{(k)} -\overline{\mathbf{\Lambda}}_{\mathbf{4}}^{(k)})]\\ +(\overline{K}_{h})^{*}\circ[\overline{K}_{\mathrm{p}}\circ\overline{\mathbf{V}}_{h} +\overline{K}_{h}\circ\overline{V}_{\mathrm{p}}-(\overline{\mathbf{X}}_{\mathbf{5}}^{(k)} -\overline{\mathbf{\Lambda}}_{\mathbf{5}}^{(k)})]\Big{\}}=0\\ \end{array} \right.. \tag{11}\]  
For the sake of reading convenience, Equation (12) is re-formulated into simultaneous equations of the three variables \(\overline{F}\), \(\overline{V}_{h}\), and \(\overline{V}_{v}\). \[\left\{ \begin{array}{ll}\mathbf{A_{11}}\circ\overline{F}+\mathbf{A_{12}}\circ \overline{V}_{h}+\mathbf{A_{13}}\circ\overline{V}_{\mathrm{p}}=\mathbf{B_{1}}\\ \mathbf{A_{21}}\circ\overline{F}+\mathbf{A_{22}}\circ\overline{V}_{h}+\mathbf{A_{23}}\circ \overline{V}_{\mathrm{p}}=\mathbf{B_{2}}\\ \mathbf{A_{31}}\circ\overline{F}+\mathbf{A_{32}}\circ\overline{V}_{h}+\mathbf{A_{33}}\circ \overline{V}_{\mathrm{p}}=\mathbf{B_{3}}\\ \end{array} \right.. \tag{12}\]  
By re-organizing Equation (11), we obtain  
\[\left\{ \begin{array}{ll}\mathbf{A_{11}}&=\beta_{0}\big{(}\overline{H} \big{)}^{*}\circ\overline{H}+\beta_{1}\big{(}\overline{K}_{h}\big{)}^{*}\circ \overline{K}_{h}+\beta_{1}\big{(}\overline{K}_{\mathrm{p}}\big{)}^{*}\circ \overline{K}_{v}\\ \mathbf{A_{12}}&=-\beta_{1}(\overline{K}_{h})^{*}\\ \mathbf{A_{13}}&=-\beta_{1}(\overline{K}_{v})^{*}\\ \mathbf{B_{1}}&=\beta_{0}\big{(}\overline{H}\big{)}^{*}\circ(\overline{\mathbf{X}}_{ 0}^{(k)}+\overline{G}-\overline{\mathbf{\Lambda}}_{\mathbf{0}}^{(k)})+\beta_{1}( \overline{K}_{h})^{*}\circ(\overline{\mathbf{X}}_{\mathbf{1}}^{(k)}-\overline{\mathbf{ \Lambda}}_{\mathbf{1}}^{(k)})\\ &+\beta_{1}(\overline{K}_{v})^{*}\circ(\overline{\mathbf{X}}_{2}^{(k)}-\overline{\mathbf{ \Lambda}}_{2}^{(k)})\\ \end{array} \right. \tag{13}\]  
To solve the sub-problem of \(\mathbf{X_{0}}\) in Equation (17), Lp shrinkage [16,17] is used, with the Lp shrinkage operator defined as \(\textit{shrink}_{p}(\xi,\tau)=\text{sgn}(\xi)max(|\xi|-\tau^{2-p}|\xi|^{p-1},0)\). In the case of \(p=1\), the Lp shrinkage operator degenerates into a soft-threshold shrinkage operator. Soft-threshold shrinkage of the sub-problems of \(X_{1}\sim X_{5}\) gives  
\[\left\{