## 3 Our Proposed Method
\begin{array}{c}L(u,\mathbf{p},\mathbf{y},\mathbf{z},\mathbf{v},\mathbf{w})=\alpha_{1}\|\mathbf{y}\| _{1}+\alpha_{0}\|z\|_{1}+\frac{\lambda}{2}\|f-Au\|_{2}^{2}\\ +\frac{\mu_{1}}{2}\|\mathbf{y}-(\
abla u-\mathbf{p})+\mathbf{v}\|_{2}^{2}-\frac{\mu_{1}}{2 }\|\mathbf{v}\|_{2}^{2}+\frac{\mu_{2}}{2}\|z-\overline{\varepsilon}(p)+\mathbf{w}\|_{ F}^{2}-\frac{\mu_{2}}{2}\|w\|_{F}^{2}\end{array}  \tag{34}\]  
The augmented Lagrange function \(L(u,\mathbf{p},\mathbf{y},z,\mathbf{v},\mathbf{w})\) of the given formula is minimized using the improved ADMM. This method yields the following iterative format:  
\[\left\{ \begin{array}{c}y^{k+1}=\arg\min_{\gamma}\alpha_{1}\|\mathbf{y}\|_{1}+ \frac{\mu_{1}}{2}\Big{\|}\mathbf{y}-\Big{(}\
abla u^{k}-\mathbf{p}^{k}\Big{)}+\mathbf{v}^ {k}\Big{\|}_{F}^{2}\\ z^{k+1}=\arg\min_{z}\alpha_{0}\|z\|_{1}+\frac{\mu_{2}}{2}\Big{\|}z- \varepsilon\Big{(}\mathbf{p}^{k}\Big{)}+\mathbf{w}^{k}\Big{\|}_{F}^{2}\\ u^{k+1}=\arg\min_{u}\frac{\lambda}{2}\|f-Au\|_{2}^{2}+\frac{\mu_{1}}{2}\Big{\|} \mathbf{y}^{k+1}-\Big{(}\
abla u-\mathbf{p}^{k}\Big{)}+\mathbf{v}^{k}\Big{\|}_{F}^{2}\\ p^{k+1}=\arg\min_{p}\frac{\mu_{1}}{2}\Big{\|}\mathbf{y}^{k+1}-\Big{(}\
abla u^{k+1} -\mathbf{p}\Big{)}+\mathbf{v}^{k}\Big{\|}_{2}^{2}+\frac{\mu_{2}}{2}\Big{\|}\mathbf{z}^{k+1 }-\overline{\varepsilon}(p)+\mathbf{w}^{k}\Big{\|}_{F}^{2}\\ \mathbf{v}^{k+1}=\mathbf{v}^{k}+\mathbf{y}^{k+1}-\Big{(}\
abla u^{k+1}-\mathbf{p}^{k+1}\Big{)} \\ \mathbf{w}^{k+1}=\mathbf{w}^{k}+\mathbf{z}^{k+1}-\varepsilon\Big{(}\mathbf{p}^{k+1}\Big{)}\end{array} \right. \tag{35}\]  
Solve the \(y\) subproblem, via the nonlinear contraction operator, as follows,  
\[y^{k+1}=\mathrm{Shrink}_{2}\Big{(}\
abla u^{k}-\mathbf{p}^{k}-\mathbf{v}^{k},\alpha_{ 1}/\mu_{1}\Big{)} \tag{36}\]  
where \(\mathrm{Shrink}_{2}(a,\mu)=\max\{\|a\|_{2}-\mu,0\}\frac{a}{\|a\|_{2}}\). For the \(z\) subproblem, the closed-form solution can be obtained similarly:  
\[z^{k+1}=\mathrm{Shink}_{F}\Big{(}\varepsilon\Big{(}p^{k}\Big{)}-w^{k},\alpha_{ 0}/\mu_{2}\Big{)} \tag{37}\]  
where \(\mathrm{Shrink}_{F}(b,\mu)=\max\{\|b\|_{F}-\mu,0\}\frac{b}{\|b\|_{F}}\). For the \(u\) subproblem, it is necessary to solve its Euler-Lagrange equation as shown below:  
\[ \begin{split}\Big{(}\lambda A^{\mathrm{T}}\mathbf{A}+\mu_{1}\mathbf{\mathrm{Q }}_{x}^{*}\mathbf{\mathrm{Q}}_{x}+\mu_{1}\mathbf{\mathrm{Q}}_{y}^{*}\mathbf{\mathrm{Q}}_{y} \Big{)}u^{k+1}=&\lambda A^{\mathrm{T}}f+\mu_{1}\mathbf{\mathrm{Q}}_{x}^ {*}\Big{(}y_{1}^{k+1}+p_{1}+v_{1}^{k}\Big{)}+\\ &\mu_{1}\mathbf{\mathrm{Q}}_{y}^{*}\Big{(}y_{2}^{k+1}+p_{2}+v_{2}^{k} \Big{)}\end{split}  \tag{38}\]  
To obtain the solution of the \(p\) subproblem, the following equation must be solved:  
\[ \begin{split}\begin{cases}\Big{(}\mu_{1}I+\mu_{2}\mathbf{\mathrm{Q}}_ {x}^{*}\mathbf{\mathrm{Q}}_{x}+\mu_{2}\mathbf{\mathrm{Q}}_{y}^{*}\mathbf{\mathrm{Q}}_{y} \Big{)}p_{1}^{k+1}+\mu_{2}\mathbf{\mathrm{Q}}_{y}^{*}\mathbf{\mathrm{Q}}_{x}p_{2}^{k+1 }\\ =&\mu_{1}\Big{(}\mathbf{\mathrm{Q}}_{x}u^{k+1}-y_{1}^{k+1}-v_ {1}^{k}\Big{)}+\mu_{2}\mathbf{\mathrm{Q}}_{x}^{*}\Big{(}x_{1}^{k+1}+w_{1}^{k} \Big{)}+\mu_{2}\mathbf{\mathrm{Q}}_{y}^{*}\Big{(}x_{3}^{k+1}+w_{3}^{k}\Big{)}\\ &\Big{(}\mu_{1}I+\mu_{2}\mathbf{\mathrm{Q}}_{x}^{*}\mathbf{\mathrm{Q}}_{x}+\mu _{2}\mathbf{\mathrm{Q}}_{y}^{*}\mathbf{\mathrm{Q}}_{y}\Big{)}p_{2}^{k+1}+\mu_{2}\mathbf{ \mathrm{Q}}_{x}^{*}\Big{(}x_{3}^{k+1}+w_{3}^{k}\Big{)}\\ =&\mu_{1}\Big{(}\mathbf{\mathrm{Q}}_{y}u^{k+1}-y_{2}^{k+1}-v_ {2}^{k}\Big{)}+\mu_{2}\mathbf{\mathrm{Q}}_{y}^{*}\Big{(}x_{2}^{k+1}+w_{2}^{k} \Big{)}+\mu_{2}\mathbf{\mathrm{Q}}_{x}^{*}\Big{(}x_{3}^{k+1}+w_{3}^{k}\Big{)}\end{cases} \end{split}  \tag{39}\]  
Since both the differential operation and its conjugate operation can be calculated using convolution, the above formula can be computed in the Fourier transform domain by leveraging the fast Fourier transform [39].