dimension) and $\KK'$ is the assumed filter support strictly containing $\KK$, then we have $$\begin{aligned}
\label{RankUB}
\rank\left(\bmH\left(\msF(\na u)\big|_{L^{-1}\OO}\right)\right)\leq|\KK'|-|\KK':\KK|.\end{aligned}$$ Roughly speaking, \[RankUB\] means that if $\varphi$ in \[MajorAssumption\] is the minimal polynomial with $\a$ satisfying \[FouGradConvDiscrete,RankDeficientHankel\], so does the translation $\a[\cdot-\bm]$ for all $\bm\in\KK'\setminus\left(\KK':\KK\right)$, or equivalently, $e^{2\pi i\bm\cdot\x}\varphi(\x)$ is also an annihilating polynomial. See [@G.Ongie2015; @G.Ongie2016] for details.

Note that \[RankUB\] means the two-fold Hankel matrix constructed by $\msF(\na u)\big|_{L^{-1}\OO}$ is *rank deficient*. Based on this observation, we can formulate \[OurProblem\] as $$\begin{aligned}
\label{LRSMCompletionRevist}
\min~~\rank\left(\bmH\left(\La\bsv\right)\right)~~~~~~~\text{subject to}~~\bmP_{\MM}\bsv=\bsf.\end{aligned}$$ Here, $\La\in\C^{2|\OO|\times|\OO|}$ is a weight matrix defined as $$\begin{aligned}
\label{Lambda}
\La=\left[\begin{array}{cc}
\La_1&\La_2
\end{array}\right]^T=\left[\begin{array}{cc}
\operatorname{diag}\left(2\pi ik_1/L\right)&\operatorname{diag}\left(2\pi ik_2/L\right)
\end{array}\right]_{\bk=(k_1,k_2)\in\OO}^T,\end{aligned}$$ which is derived from $\msF\left(\na u\right)(\xxi)=2\pi i\xxi\wh{u}(\xxi)$. Hence, the piecewise constant property of $u$ in the continuous domain can be transformed to the low rank property of $\bmH\left(\La\bsv\right)$.

Proposed off-the-grid reconstruction model {#ModelandAlgorithm}
==========================================

Model and algorithm {#ProposedModelandAlgorithm}
-------------------

Denote by $\bsv\in\mI_2$ the k-space data on the grid $\OO$, which is to be restored from a given data $\bsf$ undersampled on $\MM$. Consider a two-fold Hankel matrix $\bmH\left(\La\bsv\right)\in\C^{2M_1\times M_2}$ with $\La$ defined as \[Lambda\]. Note that for $\bq\in\C^{M_2}$, we have $$\begin{aligned}
\bmH\left(\La\bsv\right)\bq=\left(\La\bsv\right)\ast\bq[-\cdot]:=\left[\begin{array}{c}
\left(\La_1\bsv\right)\ast\bq[-\cdot]\vspace{0.25em}\\
\left(\La_2\bsv\right)\ast\bq[-\cdot]
\end{array}\right],\end{aligned}$$ where the two dimensional discrete convolution $\ast$ is performed by reformulating $\bq\in\C^{M_2}$ into 