literature, the famous compressed sensing (CS) MRI can be viewed as a sub-Nyquist sampling method which exploits the sparse representation of an image to compensate the undersampled k-space data [@E.J.Candes2006; @Donoho2006; @C.M.Hyun2018; @M.Lustig2007]. A typical $\ell_1$ norm based CS-MRI model restores the MR image $\bu$ on the grid $\OO\subseteq\Z^2$ via $$\begin{aligned}
\label{CSType}
\min_{\bu}~\f{1}{2}\left\|\bmP_{\MM}\bmsF\bu-\bsf\right\|_2^2+\lambda\left\|\Ph\bu\right\|_1,\end{aligned}$$ where $\bmsF$ is the discrete Fourier transform (DFT), $\bsf$ is the undersampled k-space data on $\MM\subseteq\OO$, and $\bmP_{\MM}$ is the projection onto $\MM$. In addition, $\Ph$ is some sparsifying linear transformation, for example, by computing the discrete gradient [@M.Lustig2007] or the wavelet frame coefficients [@Y.Liu2015] of an image. The “on-the-grid” restoration model is widely used for regularizing smooth image components while preserving image singularities such as edges, ridges, and corners.

Even though the CS-MRI model \[CSType\] has shown strong ability to reduce the data acquisition time and thus received a lot of attention over the past few years [@Y.Liu2015], it has to be further improved as the k-space data is a discrete (and truncated) sampling of a Fourier transform of an underlying function in the *continuous domain* [@E.M.Haacke1999]. This means that, \[CSType\] will work well when the singularities of the image $\bu$ are well aligned with the 