# Image Restoration Using a Fixed-Point Method for a TVL2 Regularization Problem
## 3 Fixed-Point Method for TVL2 Problem (4)
For \(\kappa\in(0,1)\), the \(\kappa\)_-averaged operator_\(S_{\kappa}\) of \(S\) is defined by  
\[S_{\kappa}=\kappa I+(1-\kappa)S.\]  
**Proposition 4** (Optial \(\kappa\)-averaged Theorem). : _Let \(C\) be a closed convex set in \(\mathbb{R}^{3n^{2}}\) and let \(S:C\to C\) be a non-expansive mapping with at least one fixed point. Then, for any \(w^{0}\in C\) and \(\kappa\in(0,1)\), the Picard iteration of \(S_{\kappa}\) converges to a fixed point of \(S\) (see [22])._  
**Theorem 2**. : _If \(\varphi\) is a convex function, \(A\) is an \(n^{2}\times n^{2}\) matrix, \(B\) is a \(2n^{2}\times n^{2}\) matrix and \(\alpha,\gamma,\lambda>0\) are positive constants such that \(\|I_{3n^{2}}-PMP^{T}R\|_{2}\leq 1\), then for any \(\kappa\in(0,1)\) the Picard iteration of \(G_{\kappa}\) converges to a fixed point of \(G\)._  
Proof. : From Propositions 2 and 3, we know that the opertor \(G\) has a fixed point and is non-expansive. Hence, for any \(w^{0}\in\mathbb{R}^{3n^{2}}\) and \(\kappa\in(0,1)\), the Picard iteration of \(G_{\kappa}\) converges to a fixed point of \(G\) by Proposition 4. For a square matrix \(K\), let \(\rho(K)\) denote the _spectral radius_ of \(K\). Then, the following lemma can be obtained. **Lemma 2**. : _Let \(\varphi\) and \(B\) be defined by Equations (10) and (11) respectively, and let \(A\) be a given \(n^{2}\times n^{2}\) blurring matrix. If we choose \(\alpha,\ \gamma\) and \(\lambda\) such that_  
\[0<\lambda=\alpha\gamma<\frac{2}{\rho(PMP^{T})},\]  
_then \(\|I_{3n^{2}}-PMP^{T}R\|_{2}\leq 1\) and thus the operator \(G\) is non-expansive._  
Proof. : Since \(\lambda=\alpha\gamma\) and thus \(R=\lambda I_{3n^{2}}\), one obtains  
\[I_{3n^{2}}-PMP^{T}R=I_{3n^{2}}-\lambda PMP^{T}.\]  
Note that \(PMP^{T}\) is a symmetric positive semi-definite matrix. Hence,  
\[\|G(w_{1})-G(w_{2})\|_{2} =\|T(Lw_{1})-T(Lw_{2})\|_{2}\]0 \[\|G(w_{1})-G(w_{2})\|_{2} =\|T(Lw_{1})-T(Lw_{2})\|_{2}\]1  
where \(\lambda_{\min}(PMP^{T})\) and \(\lambda_{\max}(PMP^{T})\) denote the minimum and maximaum eigenvalues of \(PMP^{T}\), respectively. Since \(0<\lambda=\alpha\gamma<\frac{2}{\rho(PMP^{T})}\), \(0\leq\lambda\cdot\lambda_{\min}(PMP^{T})<2\) and \(0<\lambda\cdot\lambda_{\max}(PMP^{T})<2\). Hence, one obtains  
\[\|G(w_{1})-G(w_{2})\|_{2} =\|T(Lw_{1})-T(Lw_{2})\|_{2}\]2  
Therefore, the operator \(G\) is non-expansive by Proposition 3. **Theorem 3**. : _If the assumptions of Lemma 2 hold and \(\kappa\in(0,1)\), then the Picard iteration of \(G_{k}\) converges to a fixed point of \(G\)._  
Proof. : The proof follows from Lemma 2 and Theorem 2. From Theorem 2 and the Picard iteration of the \(\kappa\)-averaged operator \(G_{\kappa}=\kappa I+(1-\kappa)G\), we can obtain a fixed-point method, called Algorithm 2, which converges to a solution to TVL2 problem (4). ```
1:Given : observed image \(f\), positive parameters \(\alpha,\ \beta,\ \gamma,\ \lambda\) and \(\kappa\in(0,1)\)
2:Initialization : \(a^{0}=0,b^{0}=0\) and \(u^{0}=f\)
3:for\(k=0\) to \(maxit\)do
4:\(\hat{a}^{k+1}=\left(\ I-\mathrm{prox}_{\frac{k}{\pi}\|.\|_{2}}\right)\left(u^{k} +a^{k}\right)\)
5:\(a^{k+1}=\kappa a^{k}+(1-\kappa)\hat{a}^{k+1}\)
6:\(\hat{b}^{k+1}=\left(\ I-\mathrm{prox}_{\frac{k}{\pi}\theta}\right)\left(Bu^{k} +b^{k}\right)\)
7:\(b^{k+1}=\kappa\hat{b}^{k}+(1-\kappa)\hat{b}^{k+1}\)
8: Solve \(A^{T}Au^{k+1}=A^{T}f-\alpha\gamma a^{k+1}-\lambda B^{T}b^{k+1}\) for \(u^{k+1}\)
9:if\(\frac{\|u^{k+1}-u^{k}\|_{2}}{\|u^{k+1}\|_{2}}<tol\)then
10: Stop
11:endif
12:endfor
```  
**Algorithm 2** Fixed-point method for TVL2 problem (4)  
The linear system in line 8 of Algorithm 2 is ill-conditioned, so we need to consider how to find an approximate solution to the ill-conditioned linear system. A typical method for finding an approximate solution to the linear system is  
\[\|G(w_{1})-G(w_{2})\|_{2} =\|T(Lw_{1})-T(Lw_{2})\|_{2}\]3  
where \(M=(A^{T}A)_{r}^{\dagger}\). However, computation of \((A^{T}A)_{r}^{\dagger}\) is very time-consuming when \(A\) is large.