Algorithm 2 Super-resolution using RED-HOGS4

Initialize:$\rho$, $\mu$, N While${\parallel \mathrm{PSNR}\left({\mathbf{F}}^{\left(k+1\right)},\mathbf{G}\right)-\mathrm{PSNR}\left({\mathbf{F}}^{\left(k\right)},\mathbf{G}\right)\parallel }_2^2>tol$ compute ${\mathbf{F}}^{\left(k+1\right)}=\underset{\mathbf{F}}{\arg min}\frac{1}{2}{\parallel \mathbf{WF}-\mathbf{G}\parallel }_2^2+\frac{\rho }{2}{\parallel \mathbf{F}-{\mathbf{Z}}^{\left(k\right)}+{\mathbf{Y}}^{\left(k\right)}\parallel }_2^2$ compute ${\tilde{\mathbf{Z}}}^{\left(1\right)}={\mathbf{Z}}^{\left(k\right)}$ $\mathbf{for}$$j=1,2,\dots ,\mathrm{N}$ compute ${\widehat{\mathbf{Z}}}^{\left(j\right)}=f_{\mathrm{HOGS}4}\left({\tilde{\mathbf{Z}}}^{\left(j\right)}\right)$ compute ${\tilde{\mathbf{Z}}}^{\left(j+1\right)}=\frac{\mu }{\mu +\rho }{\widehat{\mathbf{Z}}}^{\left(j\right)}+\frac{\rho }{\mu +\rho }\left({\mathbf{F}}^{\left(k+1\right)}+{\mathbf{Y}}^{\left(k\right)}\right)$ $\mathbf{end}\mathbf{for}$ compute ${\mathbf{Z}}^{\left(k+1\right)}={\tilde{\mathbf{Z}}}^{\left(\mathrm{N}\right)}$ compute ${\mathbf{Y}}^{\left(k+1\right)}={\mathbf{Y}}^{\left(k\right)}+\gamma \rho \left({\mathbf{F}}^{\left(k+1\right)}-{\mathbf{Z}}^{\left(k+1\right)}\right)$ End While