# Image Restoration Using a Fixed-Point Method for a TVL2 Regularization Problem
## 1 Introduction  
Image restoration is the fundamental problem in image processing that recovers a true image from a blurry and noisy image. The problem of image restoration usually reduces to find the optimal solution \(u\in\mathbb{R}^{m}\) based on the following model:  
\[f=Au+\varepsilon, \tag{1}\]  
where \(A\in\mathbb{R}^{m\times m}\) is a blurring operator, \(\varepsilon\in\mathbb{R}^{m}\) is an unknown white Gaussian noise with variance \(\sigma\), and \(f\) and \(u\) denote the observed degraded image and the original image, respectively. Our purpose is to restore the original image \(u\) from blurred and noisy image \(f\) as well as possible. Over the past few decades, optimization techniques and various variation models [1; 2; 3] have been widely studied and applied in many image processing fields. The well-known ROF (Rudin-Osher-Fatemi) total variation model [3] produces the deblurred image given by the following minimization problem:  
\[\min_{u}\left\{\frac{1}{2}\|Au-f\|_{2}^{2}+\beta TV(u)\right\}, \tag{2}\]  
where \(TV(u)\) is the total variation (TV) of \(u\) and \(\beta>0\) is a regularization parameter. Many computational methods for solving problem (2) have been proposed in recent years. For example, the time-marching PDE method, the subgradient descent method, the Newton-like method, the second-order cone programming method, the lagged diffusivity fixed-point method, and the split Bregman method have been proposed by many researchers (see [1; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13] for details). Recently, [PERSON] et al. [14] proposed a fixed-point method for solving the following constrained TVL2 deblurring problem:  
\[\min_{u\in\mathbb{C}}\left\{\frac{1}{2}\|Au-f\|_{2}^{2}+\frac{\alpha}{2}\|u\|_{2 }^{2}+\beta\,TV(u)\right\}, \tag{3}\]  
where \(\alpha\) and \(\beta\) are positive constants, and \(C\) is a closed convex subset of \(\mathbb{R}^{n^{2}}\). In this paper, we only consider the case for \(C=\mathbb{R}^{n^{2}}\). That is, we only consider the unconstrained TVL2 problem (3) with \(C=\mathbb{R}^{n^{2}}\). This approach motivates us to propose a new TVL2 deblurring model  
\[\min_{u\in\mathbb{R}^{n^{2}}}\left\{\frac{1}{2}\|Au-f\|_{2}^{2}+\alpha\|u\|_{2 }+\beta\,TV(u)\right\}, \tag{4}\]  
where \(A\in\mathbb{R}^{n^{2}\times n^{2}}\) is a blurring matrix, \(u\in\mathbb{R}^{n^{2}}\) is an original image, \(f\in\mathbb{R}^{n^{2}}\) is a degraded image, \(\alpha\) and \(\beta\) are positive regularization parameters, and \(TV(u)\) stands for the isotropic TV of \(u\). Note that the new TVL2 model (4) uses a non-smooth term \(\|u\|_{2}\) instead of using a smooth term \(\|u\|_{2}^{2}\) for the purpose of better preserving the edges and corners in the restored images. The isotropic TV of \(u\) is defined by  
\[TV(u)=\sum_{i=1}^{n^{2}}\sqrt{|(\
abla u)_{i}^{x}|^{2}+|(\
abla u)_{i}^{y}|^{ 2}}=\sum_{i=1}^{n^{2}}\left\|\left( \begin{array}{c}(\
abla u)_{i}^{x}\\ (\
abla u)_{i}^{y}\end{array} \right)\right\|_{2},\]  
where the discrete gradient operator \(\
abla:\mathbb{R}^{n^{2}}\rightarrow\mathbb{R}^{2n^{2}}\) is defined by  
\[(\
abla u)_{i}=\left( \begin{array}{ll}(\
abla u)_{i}^{x},\,(\
abla u)_{i}^{ y}\end{array} \right),\quad i=1,2,\cdots,n^{2}\]  
with  
\[(\
abla u)_{i}^{x}=\left\{ \begin{array}{ll}0&\text{if }i\bmod n=1,\\ u_{i}-u_{i-1}&\text{if }i\bmod n\
eq 1,\end{array} \right.\quad\text{and}\quad( \
abla u)_{i}^{y}=\left\{ \begin{array}{ll}0&\text{if }i\leq n,\\ u_{i}-u_{i-n}&\text{if }i>n.\end{array} \right.\]  
Notice that TVL2 problems (3) have a unique solution since its objective function is strictly convex, while the new TVL2 problem (4) may not have a unique solution since its objective function is just convex, not strictly convex. The purpose of this paper is to propose two iterative methods, which are fixed-point and fixed-point-like methods, using CGLS (Conjugate Gradient Least Squares method [15]) for solving the new proposed TVL2 problem (4). This paper is organized as follows. In Section 2, we introduce some definitions and properties that are used in this paper.