# Image Restoration Using a Fixed-Point Method for a TVL2 Regularization Problem
## 3 Fixed-Point Method for TVL2 Problem (4)
For the given convex functions \(\frac{1}{\gamma}\|\cdot\|_{2}\) on \(\mathbb{R}^{n^{2}}\) and \(\frac{\delta}{\lambda}\varphi\) on \(\mathbb{R}^{2n^{2}}\), we define an operator \(T:\mathbb{R}^{3n^{2}}\rightarrow\mathbb{R}^{3n^{2}}\) at a vector \(\binom{u}{v}\in\mathbb{R}^{3n^{2}}\) with \(u\in\mathbb{R}^{n^{2}}\) and \(v\in\mathbb{R}^{2n^{2}}\) as follows:  
\[T\binom{u}{v}=\left( \begin{array}{c}(I-\text{prox}_{\frac{1}{\gamma}\|\cdot \|_{2}})\left(u\right)\\ (I-\text{prox}_{\frac{\delta}{\gamma}\varphi})\left(v\right)\end{array} \right). \tag{23}\]  
We also introduce an affine transformation \(L:\mathbb{R}^{3n^{2}}\rightarrow\mathbb{R}^{3n^{2}}\) defined, for all \(\binom{u}{b}\in\mathbb{R}^{3n^{2}}\) with \(a\in\mathbb{R}^{n^{2}}\) and \(b\in\mathbb{R}^{2n^{2}}\), by  
\[L\binom{a}{b}=\left( \begin{array}{cc}I_{n^{2}}-\alpha\gamma M&-\lambda MB^{T} \\ -\alpha\gamma BM&I_{2n^{2}}-\lambda BMB^{T}\end{array} \right)\binom{a}{b}+ \left( \begin{array}{c}MA^{T}f\\ BMA^{T}f\end{array} \right), \tag{24}\]  
and an operator \(G:\mathbb{R}^{3n^{2}}\rightarrow\mathbb{R}^{3n^{2}}\) defined by  
\[G=T\circ L. \tag{25}\]  
Then, Equations (21) and (22) can be expressed as  
\[w=Gw, \tag{26}\]  
where \(w=(\binom{a}{b})\). **Proposition 2**. : _The operator \(G\) defined by (25) has a fixed point._  
Proof. : Since a solution of TVL2 problem (13) exists, from Equation (26) and the first part of the proof of Theorem 1, \(G\) has a fixed point. **Lemma 1**. : _If the operator \(T\) is defined by Equation (23), then the operator \(T\) is non-expansive._  
Proof. : Note that \(I-\mathrm{prox}_{\frac{1}{7}\|\cdot\|_{2}}\) and \(I-\mathrm{prox}_{\frac{6}{7}\varphi}\) are firmly non-expansive and thus non-expansive [7]. For any vectors \(s=\binom{u}{v}\in\mathbb{R}^{3n^{2}}\) and \(t=\binom{a}{b}\in\mathbb{R}^{3n^{2}}\) with \(u\), \(a\in\mathbb{R}^{n^{2}}\) and \(v\), \(b\in\mathbb{R}^{2n^{2}}\)  
\[\|T(s)-T(t)\|_{2}^{2} =\left\|\left( \begin{array}{c}(I-\mathrm{prox}_{\frac{1}{7}\| \cdot\|_{2}})(u)-(I-\mathrm{prox}_{\frac{1}{7}\|\cdot\|_{2}})(a)\\ (I-\mathrm{prox}_{\frac{6}{7}\varphi})(v)-(I-\mathrm{prox}_{\frac{6}{7} \varphi})(b)\end{array} \right)\right\|_{2}^{2}\] \[=\left\|(I-\mathrm{prox}_{\frac{1}{7}\|\cdot\|_{2}})(u)-(I- \mathrm{prox}_{\frac{1}{7}\|\cdot\|_{2}})(a)\right\|_{2}^{2}\] \[\qquad+\left\|(I-\mathrm{prox}_{\frac{1}{7}\|\cdot\|_{2}})(v)-(I -\mathrm{prox}_{\frac{1}{7}\|\cdot\|_{2}})(b)\right\|_{2}^{2}\] \[\leq\|u-a\|_{2}^{2}+\|v-b\|_{2}^{2}\] \[=\left\|\left( \begin{array}{c}u-a\\ v-b\end{array} \right)\right\|_{2}^{2}=\|s-t\|_{2}^{2}.\]  
Hence, one obtains \(\|T(s)-T(t)\|_{2}\leq\|s-t\|_{2}\), which means that the operator \(T\) is non-expansive. Let  
\[c:=\left( \begin{array}{c}MA^{T}f\\ BMA^{T}f\end{array} \right)\in\mathbb{R}^{3n^{2}},\quad P:=\left( \begin{array}[] {c}I_{n^{2}}\\ B\end{array} \right),\quad R:=\left( \begin{array}{cc}\alpha\gamma I_{n^{2}}& 0\\ 0&\lambda I_{2n^{2}}\end{array} \right).\]  
Then, Label (24) can be expressed as  
\[Lw=(I_{3n^{2}}-PMP^{T}R)w+c. \tag{27}\]  
**Proposition 3**. : _If \(\varphi\) is a 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 and \(\alpha\), \(\gamma\), \(\lambda\) are positive constants such that \(\|I_{3n^{2}}-PMP^{T}R\|_{2}\leq 1\), then \(G\) is non-expansive._  
Proof. : Since the operator \(T\) is non-expansive by Lemma 1, for all \(w_{1}\), \(w_{2}\in\mathbb{R}^{3n^{2}}\), we have  
\[\|G(w_{1})-G(w_{2})\|_{2} =\|T(Lw_{1})-T(Lw_{2})\|_{2}\] \[\leq\|Lw_{1}-Lw_{2}\|_{2}.\]  
By the assumption \(\|I_{3n^{2}}-PMP^{T}R\|_{2}\leq 1\) and Equation (27), we obtain  
\[\|Lw_{1}-Lw_{2}\|_{2} =\|(I_{3n^{2}}-PMP^{T}R)(w_{1}-w_{2})\|_{2}\] \[\leq\|I_{3n^{2}}-PMP^{T}R\|_{2}\ \|w_{1}-w_{2}\|_{2}\] \[\leq\|w_{1}-w_{2}\|_{2}.\]  
Hence, \(G\) is non-expansive. Let \(S:\mathbb{R}^{N}\to\mathbb{R}^{N}\) be an operator. Then, the _Picard iteration_ of the operator \(S\) is defined by  
\[x_{i+1}=Sx_{i}\quad(i=0,1,2,\cdots)\]for a given vector \(x_{0}\in\mathbb{R}^{N}\).