# Image Restoration Using a Fixed-Point Method for a TVL2 Regularization Problem
## 3 Fixed-Point Method for TVL2 Problem (4)  
In this section, we propose a fixed-point method using CGLS for solving the new TVL2 regularization problem (4). From relation (12), TVL2 problem (4) can be expressed as  
\[\min_{u\in\mathbb{R}^{n^{2}}}\left\{\frac{1}{2}\|Au-f\|_{2}^{2}+\alpha\|u\|_{2 }+\beta(\varphi\circ B)(u)\right\}. \tag{13}\]  
Using Proposition 1, we can obtain the following property for a solution of TVL2 problem (13). **Theorem 1**. : _If \(\varphi\) is a real valued convex function on \(\mathbb{R}^{2n^{2}}\), \(B\) is a \(2n^{2}\times n^{2}\) matrix, \(A\) is an \(n^{2}\times n^{2}\) matrix, \(u\) is a solution to model (13), then, for any \(\gamma,\lambda>0\) there exist vectors \(a\in\mathbb{R}^{n^{2}}\) and \(b\in\mathbb{R}^{2n^{2}}\) such that_  
\[a =(I-\mathrm{prox}_{\frac{\delta}{\gamma}\|\cdot\|_{2}})(u+a), \tag{14}\] \[b =(I-\mathrm{prox}_{\frac{\delta}{\gamma}\varphi})(Bu+b),\] (15) \[(A^{T}A)u=A^{T}f-\alpha\gamma a-\lambda B^{T}b. \tag{16}\]  
_Conversely, if there exists \(\gamma\), \(\lambda>0\), \(a\in\mathbb{R}^{n^{2}}\), \(b\in\mathbb{R}^{2n^{2}}\) and \(u\in\mathbb{R}^{n^{2}}\) satisfying Equations (14)-(16), then \(u\) is a solution to model (13)._  
Proof. : Assume that \(u\in\mathbb{R}^{n^{2}}\) is a solution to the model (13). By [PERSON]'s rule in convex analysis for model (13), we obtain the following equivalent relation for the solution \(u\)  
\[ \begin{split} 0&\in A^{T}(Au-f)+\alpha\cdot\partial(\| \cdot\|_{2})(u)+\beta B^{T}\circ(\partial\varphi)\circ(Bu)\\ &=A^{T}(Au-f)+\alpha\cdot\partial(\|\cdot\|_{2})(u)+B^{T}\partial (\beta\varphi)(Bu).\end{split}  \tag{17}\]  
For any \(\gamma\), \(\lambda>0\), we can choose two vectors \(a\in\partial(\frac{1}{\gamma}\|\cdot\|_{2})(u)\) and \(b\in\partial(\frac{\beta}{\lambda}\varphi)(Bu)\) satisfying  
\[A^{T}(Au-f)+\alpha\gamma a+\lambda B^{T}b=0. \tag{18}\]Using Proposition 1, one obtains  
\[u=\text{prox}_{\frac{1}{\gamma}\|\cdot\|_{2}}(u+a)\quad\text{and}\quad Bu=\text{ prox}_{\frac{\delta}{\gamma}\varphi}(Bu+b). \tag{19}\]  
From Equation (19), we obtain Equations (14) and (15). In addition, from Equation (18), we obtain Equation (16). Conversely, suppose that there exist \(\gamma\), \(\lambda>0\), \(a\), \(b\) and \(u\in\mathbb{R}^{n^{2}}\) satisfying Equations (14)-(16). From Equation (16), we obtain Equation (18). By Proposition 1, Equations (14) and (15) ensure \(a\in\partial(\frac{1}{\gamma}\|\cdot\|_{2})(u)\) and \(b\in\partial(\frac{\delta}{\lambda}\varphi)(Bu)\), respectively. Using these relations and Equation (18), one obtains  
\[0 =A^{T}(Au-f)+\alpha\gamma a+\lambda B^{T}b\] \[\in A^{T}(Au-f)+\alpha\partial(\|\cdot\|_{2})(u)+\beta B^{T} \circ(\partial\varphi)\circ(Bu).\]  
Consequently, Equation (17) holds. Thus, \(u\) is a solution to model (13). From Equations (14)-(16) in Theorem 1, we can develop a fixed-point algorithm which converges to a solution to TVL2 problem (4). We now describe how to develop the fixed-point algorithm. Let \(u\) be an approximate solution to the ill-conditioned linear system (16) in Theorem 1. Then, \(u\) can be expressed as  
\[u=M(A^{T}f-\alpha\gamma a-\lambda B^{T}b), \tag{20}\]  
where \(M\) is a symmetric positive semi-definite matrix approximating an inverse of the linear system matrix in (16). For example, we can choose \(M=(A^{T}A)_{r}^{\dagger}\), which is a truncated pseudoinverse of \(A^{T}A\) using the \(r\) largest positive singular values of \(A^{T}A\). Substituting Equation (20) into Equations (14) and (15), one obtains  
\[a =(I-\text{prox}_{\frac{1}{\gamma}\|\cdot\|_{2}})\left((I_{n^{2}}- \alpha\gamma M)a-\lambda MB^{T}b+MA^{T}f\right), \tag{21}\] \[b =(I-\text{prox}_{\frac{\delta}{\gamma}\varphi})\left(-\alpha \gamma BMa+(I_{2n^{2}}-\lambda BMB^{T})b+BMA^{T}f\right). \tag{22}\]  
Let us define some operators.