proton spin density distribution in $\R^2$ [@E.M.Haacke1999]. Note that when $\MM=\OO$, i.e. the fully sampled case, $\bu$ is obtained from the inverse DFT $$\begin{aligned}
\label{MRInverse}
\bu[\bp]=\left(\bmsF^{-1}\bsf\right)[\bp]=\f{1}{N^2}\sum_{\bk\in\OO}\bsf[\bk]e^{2\pi i\bp\cdot\bk/N},~~~~~~~\bp\in\OO.\end{aligned}$$ Hence, the conventional CS-MRI approach is based on the DFT $$\begin{aligned}
\label{DFT}
\bsf[\bk]=\left(\bmsF\bu\right)[\bk]=\sum_{\bp\in\OO}\bu[\bp]e^{2\pi i\bp\cdot\bk/N},~~~~~~~\bk\in\MM.
 \end{aligned}$$ Note that this approach is the on-the-grid scheme, as we only consider the information on $\OO$. In other words, we can apply the sparse regularization effectively provided that the singularities of the image $\bu$ are well aligned with $\OO$. However, since the k-space data actually comes from the Fourier transform of a continuous domain function, there exists a basis mismatch between the singularities of $u$ in the continuous domain and the discrete grid $\OO$. Such a basis mismatch would destroy the sparsity structure, leading to a degradation of the restoration performance.

Instead of directly restoring the discrete MR image, the off-the-grid approaches attempt to first restore the fully sampled k-space data $\bsv$ from its undersampled version $\bsf$ modeled as in \[MRForward\]. In other words, we first $$\begin{aligned}
\label{OurProblem}
\text{find}~~\bsv~~~\text{such that}~~\bsv=\msF(u)\big|_{L^{-1}\OO}~~\text{and}~~\bmP_{\MM}\bsv=\bsf,\end{aligned}$$ and then compute $\bu=\bmsF^{-1}\bsv$; see \[CSMRIComparisons\] for the schematic comparison.

![Comparison between the conventional CS-MRI and the off-the-grid CS-MRI approach.[]{data-label="CSMRIComparisons"}](CSMRIComparisons.pdf){width="95.00000%"}

Generally, it is impossible to directly solve \[OurProblem\] 