$\bsW$ is a tight frame transform defined as in \[OurAnalysis\] whose filter banks have to be learned, and $\mC$ is a constraint set imposing the boundedness of $\bsv$. In this paper, we choose $$\begin{aligned}
\label{ConstraintSet}
\mC=\left\{\bsv\in\mI_2:|\bsv[\bk]|\leq R~~~\text{for all}~~\bk\in\OO\right\}\simeq\DD_R^{N^2},\end{aligned}$$ where $\DD_{R}=\left\{v\in\C:|v|\leq R\right\}$ with a sufficiently large $R>0$.

\[RK1\] The choice of the constraint set $\mC$ comes from the following: if $u\in L_1(\R^2)$ is modeled as in \[uModel\], then from \[FouGradu\], we have $$\begin{aligned}
\left|\msF(\na u)(\xxi)\right|\leq\sum_{j=1}^J|\alpha_j|\ssi(\Om_j)<\infty~~~~~\text{for all}~~\xxi\in\R^2.\end{aligned}$$ In other words, $\wh{u}(\xxi)$ decays at infinity in the sense that $$\begin{aligned}
\sup_{\xxi\in\R^2}\left|\xxi\right|\left|\wh{u}(\xxi)\right|<\infty,\end{aligned}$$ and this implies that $\wh{u}(\xxi)$ (and whence $\bsv[\bk]$ since we want $\bsv=\msF(u)\big|_{L^{-1}\OO}$) has to be bounded. In addition, as we shall see in \[ConvergenceAnalysis\], the boundedness constraint is required for the convergence analysis of the proximal alternating scheme to solve \[ProposedCSMRIModel\]. That being said, numerically, the constraint set has a very minor effect on the restoration results provided that $R>0$ is sufficiently large. In fact, the proximal alternating minimization seems to converge even without using $\mC$.

In \[ProposedCSMRIModel\], the sparsity promoting $\ell_0$ norm $\left\|\bc\right\|_0$ is used as a relaxation of \[TightFrameSparse\] (or equivalently, \[HankelSVDSparse\]), and the tight frame constraint $\bsW^*\bsW=\bsI$ is related to \[LowRankHankelTightFrame\]. Finally, the term $\left\|\bsW\left(\La\bsv\right)-\bc\right\|_2^2$ is to provide the 