$\bsW$ defined as in \[AnalConv\], which is generated by a set of finitely supported $p\times p$ filters $\{\bq_1,\cdots,\bq_{p^2}\}$ satisfying , is constructed via the following minimization $$\begin{aligned}
\label{DDTFModel}
\min_{\bc,\bsW}~\|\bc-\bsW\bu\|_2^2+\lambda^2\|\bc\|_0~~~~~\text{subject to}~~~~\bsW^T\bsW=\bsI,\end{aligned}$$ with the $\ell_0$ norm $\|\bc\|_0$ encoding the number of nonzero entries in the coefficient vector $\bc$.

Given that each filter is supported on a $p\times p$ grid, we can reformulate \[DDTFModel\] the following way. Reshape all $p\times p$ patches of $\bu$ into $\bU\in\R^{p^2\times m}$ where $m$ denotes the number of total patches. Let $\bsD\in\R^{p^2\times p^2}$ be the matrix generated by concatenating filters $\{\bq_1,\cdots,\bq_{p^2}\}$ into $p^2\times 1$ column vectors $\{\vec{\bq}_1,\cdots,\vec{\bq}_{p^2}\}$. Denote by $\bC\in\R^{p^2\times m}$ the frame coefficients. Hence, we have $$\begin{aligned}
\bu&\Leftrightarrow\bU=(\vec{\bu}_1,\cdots\,\vec{\bu}_m)\in\R^{p^2\times m},\\
\bsW&\Leftrightarrow\bsD=\left(\vec{\bq}_1,\cdots,\vec{\bq}_{p^2}\right)\in\R^{p^2\times p^2},\\
\bc&\Leftrightarrow\bC=(\vec{\bc}_1,\cdots,\vec{\bc}_m)\in\R^{p^2\times m}.\end{aligned}$$ After making these substitutions and relaxing \[SomeUEP\] into $\bsD\bsD^T=p^{-2}\bsI$, \[DDTFModel\] can be reformulated as $$\begin{aligned}
\label{DDTFModelModi}
\min_{\bC,\bsD}~\|\bC-\bsD^T\bU\|_F^2+\lambda^2\|\bC\|_0~~~~~\text{subject to}~~~~\bsD\bsD^T=p^{-2}\bsI,\end{aligned}$$ where $\|\cdot\|_F$ is the Frobenius norm of a matrix. To solve \[DDTFModelModi\], the alternating minimization method with closed form solutions for each stage is presented in [@J.F.Cai2014]. In addition, the proximal alternating minimization (PAM) scheme with global convergence to critical points is proposed in [@C.Bao2015].

Structured low rank matrix approaches in CS-MRI {#SRLACSMRI}
-----------------------------------------------

We begin with introducing some notation. All two dimensional images will be 