flexibility in sparse approximation. Namely, given that $\bsW$ is a tight frame, we have $$\begin{aligned}
\left\|\bsW\left(\La\bsv\right)-\bc\right\|_2^2=\left\|\bsW^*\bc-\La\bsv\right\|_2^2+\left\|\left(\bsI-\bsW\bsW^*\right)\bc\right\|_2^2,\end{aligned}$$ which yields a balanced approach (e.g. [@J.F.Cai2008; @R.H.Chan2003]). In this work, we choose this balanced approach for the following two reasons. Firstly, the k-space data is in general corrupted by the (thermal) noise [@E.M.Haacke1999] and $\La$ amplifies the noise in the high frequency, which always hampers $\bsW\left(\La\bsv\right)$ from being sparse (or $\bmH\left(\La\bsv\right)$ being low rank). Secondly, the direct penalization on the canonical coefficient $\bsW\left(\La\bsv\right)$ requires an iterative solver, which will be too expensive in constructing an adaptive tight frame [@J.F.Cai2014]. Therefore, by promoting the distance between $\bsW\left(\La\bsv\right)$ and $\bc$, we expect to achieve a better sparse approximation of $\La\bsv$ (and thus a better low rank approximation of $\bmH\left(\La\bsv\right)$) with a low computational complexity.

To solve \[ProposedCSMRIModel\], we use the proximal alternating minimization (PAM) algorithm introduced in [@H.Attouch2010]. We initialize $\bsv_0=\bmP_{\mC}\left(\bmP_{\MM}\bsf\right)$ where $$\begin{aligned}
\label{ProjectOntoC}
\bmP_{\mC}\left(\bsv\right)[\bk]=\min\left\{\left|\bsv[\bk]\right|,R\right\}\exp\left\{i\arg\bsv[\bk]\right\}.\end{aligned}$$ The initializations of $\bsW_0$ and $\bc_0$ are obtained from the SVD of $\bmH\left(\La\bsv_0\right)=\bX_0\Sig_0\bY_0^*$: $$\begin{aligned}
\label{Initialize}
\bsW_0=M_2^{-1/2}\bY_0~~~~~\text{and}~~~~~\bc_{0,j}=\left\{\begin{array}{cl}
M_2^{-1/2}\left(\La\bsv_0\right)\ast\bY_0^{(:,j)}[-\cdot]~&j=1,\cdots,r,\vspace{0.4em}\\
\0~&j=r+1,\cdots,M_2,
\end{array}\right.\end{aligned}$$ where $r$ is an estimated rank. After the initializations, we optimize $(\bsv,\bc,\bsW)$ alternatively, as summarized in \[Alg1\].

It is easy to see that each subproblem in \[Alg1\] has a closed form 