denoted by the bold faced lower case letters and all matrices will be denoted by the bold faced upper case letters. Note that a two dimensional images can also be identified with a vector whenever convenient. We denote by $\OO=\left\{-N/2,\cdots,N/2-1\right\}^2$ with $N\in2\N$ the set of $N\times N$ grid, and the space of complex valued functions on $\OO$ is denoted by $\mI_2\simeq\C^{|\OO|}$. Given two rectangular grids $\KK$ and $\MM$, the contraction $\KK:\MM$ is defined as $$\begin{aligned}
\KK:\MM=\left\{\bk\in\KK:\bk-\bsj\in\KK~\text{for all}~\bsj\in\MM\right\}.\end{aligned}$$ Let $\bsv\in\mI_2$ and let $\KK$ be a rectangular $K_1\times K_2$ grid. The corresponding Hankel matrix $\bmH\bsv$ is an $M_1\times M_2$ matrix ($M_1=|\OO:\KK|$ and $M_2=|\KK|$) generated by concatenating $K_1\times K_2$ patches of $\bsv$ into row vectors. With a slight abuse of notation, we also use $\bmH\w=\left[\begin{array}{cc}
\left(\bmH\w_1\right)^T&\left(\bmH\w_2\right)^T
\end{array}\right]^T$ to denote the $2M_1\times M_2$ two-fold Hankel matrix constructed from $\w=\left(\w_1,\w_2\right)\in\mI_2\times\mI_2$; see \[BlockHankelIllustrate\] for an illustration.

![Illustration of generating a two-fold Hankel matrix $\bmH\w$ from $\w=\left(\w_1,\w_2\right)\in\mI_2\times\mI_2$. Here, $K_1=K_2=2$.[]{data-label="BlockHankelIllustrate"}](BlockHankelIllustrate.pdf){width="90.00000%"}

The conventional CS-MRI approach aims to directly restore the MR image $\bu$ on $\OO$ from the undersampled k-space $\bsf$ satisfying the following two dimensional Fourier transform $$\begin{aligned}
\label{MRForward}
\bsf[\bk]=\wh{u}(L^{-1}\bk)=\int_{\R^2}u(\x)e^{-2\pi i\x\cdot\bk/L}\rd\x,~~~~~~\bk\in\MM,\end{aligned}$$ where $\MM$ is the sampling grid, $L>0$ is the length of field-of-view (FOV), and $u\in L_1(\R^2)$ is the 