# Image Restoration Using a Fixed-Point Method for a TVL2 Regularization Problem
## 2 Preliminaries  
In this section, we briefly refer to some definitions and important properties which will be the foundation for the development of algorithms proposed in this paper.  
**Definition 1**.: _Let \(\varphi:\mathbb{R}^{n}\rightarrow\mathbb{R}\cup\{+\infty\}\) be a proper, convex, and lower semi-continuous function (see [16]). The proximity operator of \(\varphi\) at \(x\in\mathbb{R}^{n}\) is defined by_  
\[\text{prox}_{\varphi}(x)=\text{argmin}\left\{\frac{1}{2}\|u-x\|_{2}^{2}+\varphi (u):u\in\mathbb{R}^{n}\right\}. \tag{5}\]  
**Definition 2**.: _Let \(\varphi:\mathbb{R}^{n}\rightarrow\mathbb{R}\cup\{+\infty\}\) be a proper, convex, and lower semi-continuous function. The subdifferential of \(\varphi\) at \(x\in\mathbb{R}^{n}\) is defined by_  
\[\partial\varphi(x)=\{y\in\mathbb{R}^{n}:\varphi(z)\geq\varphi(x)+\langle y,z- x\rangle,\,\forall z\in\mathbb{R}^{n}\}. \tag{6}\]  
**Definition 3**.: _A nonlinear operator \(H:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is called non-expansive if for any \(x,y\in\mathbb{R}^{n}\),_  
\[\|H(x)-H(y)\|_{2}\leq\|x-y\|_{2}. \tag{7}\]  
**Definition 4**.: _A nonlinear operator \(H:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is called firmly non-expansive if for any \(x,y\in\mathbb{R}^{n}\),_  
\[\|H(x)-H(y)\|_{2}^{2}\leq\langle x-y,H(x)-H(y)\rangle. \tag{8}\]  
It is easy to show that a firmly non-expansive operator is non-expansive. The following proposition shows a relationship between the proximity operator and the subdifferential of a convex function.  
**Proposition 1**.: _If \(\varphi\) is a convex function defined on \(\mathbb{R}^{n}\) and \(x,y\in\mathbb{R}^{n}\), then (see [16; 17; 18])_  
\[y\in\partial\varphi(x)\;\Leftrightarrow\;x=\text{prox}_{\varphi}(x+y). \tag{9}\]  
Let \(\varphi:\mathbb{R}^{2n^{2}}\rightarrow\mathbb{R}\) be a convex function defined by  
\[\varphi(d)=\sum_{i=1}^{n^{2}}\left\|\left(\begin{array}{c}d_{i}\\ d_{n^{2}+i}\end{array}\right)\right\|_{2}\;\;\text{for each }d=(d_{i})\in \mathbb{R}^{2n^{2}}, \tag{10}\]  
and let \(B\) be a \(2n^{2}\times n^{2}\) matrix that represents a discrete gradient operator \(\
abla\), which is  
\[B=\left(\begin{array}{c}I_{n}\otimes D\\ D\otimes I_{n}\end{array}\right), \tag{11}\]  
where \(I_{n}\) is the \(n\times n\) identity matrix, \(\otimes\) denotes the Kronecker product, and \(D\) is the first order finite difference matrix of order \(n\)  
\[D=\left(\begin{array}{cccccc}0&0&\cdots&\cdots&0\\ -1&1&\cdots&\cdots&0\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ 0&\cdots&-1&1&0\\ 0&\cdots&0&-1&1\end{array}\right).\]  
Then, the isotropic TV of \(u\in\mathbb{R}^{n^{2}}\) can be expressed as  
\[TV(u)=(\varphi\circ B)(u). \tag{12}\]  
We now provide the fixed-point method, called Algorithm 1, for solving TVL2 problem (3), which was proposed by [PERSON] et al. [14].  
Notice that line 4 of Algorithm 1 is solved using CGLS instead of using CG (Conjugate Gradient method [19; 20; 21]) since the linear system in line 4 is equivalent to solving the following least squares problem  
\[\min_{u}\left\|\left(\begin{array}{c}f\\ -\frac{\lambda}{\sqrt{a}}B^{T}v^{k}\end{array}\right)-\left(\begin{array}{c} A\\ \sqrt{\alpha}I\end{array}\right)u\right\|_{2}^{2}.\]  
For all algorithms presented in this paper, _maxit_ denotes the maximum number of iterations, and _tol_ denotes the tolerance value of the stopping criterion.