## 4 Applications
### Application to the Image Restoration Problem  
The general image recovery problem can be formulated by the inversion of the observation model defined by  
\[z=Dp+h, \tag{75}\]  
where \(h\) is the unknown additive random noise, \(p\in\mathbb{R}^{k}\) is the known original image, \(D\) is the linear operator involved in the considered image recovery problem, and \(z\) is the known degraded observation. Model (75) is closely equivalent to different concepts of optimization problems. In recent years, the \(l_{1}\) norm has been widely used by many authors in these kinds of problems. The \(l_{1}\) regularization, which can be employed to remove noise in the recovery process, is defined by  
\[\min_{p\in\mathbb{R}^{k}}\{\lambda_{k}\|p\|_{1}+\frac{1}{2}\|z-Dp\|_{2}^{2}\}, \tag{76}\]where \(p\in\mathbb{R}^{k}\), \(z\in\mathbb{R}^{m}\), \(D\) is a \(m\times k\) matrix and \(\lambda_{k}>0\). Next, we use various algorithms, as listed above, to find the solution to the following CMP:  
\[\text{find}\ p\in\text{Argmin}_{p\in\mathbb{R}^{k}}\{\lambda_{k}\|p\|_{1}+\frac{ 1}{2}\|z-Dp\|_{2}^{2}\}, \tag{77}\]  
where \(D\) is an operator representing the mask and \(g\) is the degraded image. In this numerical experiment, \(g(p)=\|p\|_{1}\), \(h(p)=\frac{1}{2}\|z-Dp\|_{2}^{2}\), and in all the algorithms, set \(D=\triangledown h\) and \(E=\partial g\). We define the gradient by  
\[\triangledown h(p)=D^{*}(z-Dp).\]  
Moreover, we compare the image recovery efficiency of method Algorithms 5 (in short, OAURN3) and 6 (in short, OAURN4) with Algorithm 3.3 by [PERSON] et al. [36] (in short, ADIA Alg. 3.3) and Algorithm 3.1 by [PERSON] et al. [15] (in short, AOM Alg. 3.1). For all algorithms, we use the stopping criterion \(E_{k}=\|p_{k+1}-p_{k}\|<10^{-6}\) and choose the following parameters for all the methods:  
* In OAURN2 and OAURN3, we set \(\phi=0.73\), \(\lambda_{1}=3.5\), \(s=0.66\), \(c_{1}=0.99\), \(\theta_{k}=0.68\), \(\alpha_{k}=\frac{1}{k+1}\), \(\epsilon_{k}=\frac{100}{(k+1)^{2}}\), \(s_{k}=\frac{1}{k}\), \(\varrho=\frac{1.8\eta}{L_{1}^{2}}\) and \(q_{k}=\frac{1}{(k+1)^{1.1}}\). * In ADIA Alg. 3.3, we set \(\tau_{k}=\frac{1}{k+1}\), \(\sigma_{k}=\mu_{k}=0.5(1-\tau_{k})\), \(\epsilon_{k}=\gamma_{k}=s_{k}=\frac{100}{(k+1)^{2}}\), \(u=\frac{1}{2}\), \(\lambda_{k}=\frac{1}{4}\) and \(a=0.8\). * In AOM Alg. 3.1, we set \(\alpha_{k}=\frac{1}{k+1}\), \(\delta_{k}=\xi_{k}=0.5(1-\alpha_{k})\), \(\epsilon_{k}=\frac{100}{(k+1)^{2}}\), \(\theta=0.89\), \(\lambda_{1}=3.5\), \(\phi=0.89\), \(\alpha=0.145\), \(\beta=0.895\), \(Sp=\frac{2}{3}p\), \(Tp=\frac{3}{4}p\), \(f(p)=\frac{1}{3}p\) and \(\phi_{k}=\frac{1}{(k+1)^{1.1}}\). The test images are a hand X-ray and an apple. The performances of the algorithms are measured via the signal-to-noise ratio (SNR), defined by  
\[SNR=20\log_{10}\biggl{(}\frac{\|p\|_{2}}{\|p-p^{*}\|_{2}}\biggr{)}, \tag{78}\]  
where \(p^{*}\) is the restored image and \(p\) is the original image. We consider the blur function in MATLAB \"special ('motion', 40, 70)\" and add random noise. All numerical simulations were performed using MATLAB R2020b and carried out on a PC Desktop with an Intel(r) Core(tm) i7-3540M CPU @ 3.00 GHz x 4 and 400.00 GB memory. The numerical results are presented in Figures 1-4 and Table 1.