its SVD constructs a tight frame such that $\La\bsv$ has only $r$ *nonzero* canonical coefficients.

Based on these observations, we can consider constructing an *adaptive* tight frame under which $\La\bsv$ has a small number of canonical nonzero coefficients, instead of directly minimizing $\rank\left(\bmH\left(\La\bsv\right)\right)$. However, it is in general difficult to directly impose such an idea on the reconstruction model for the following reasons: 1) it is an NP-hard problem to minimize the number of nonzero coefficients that represents/approximates $\La\bsv$ even under a given tight frame; 2) the penalization on canonical coefficients in general requires an iterative solver, which will be too expensive in constructing an adaptive tight frame [@J.F.Cai2014]; 3) the available k-space data is always corrupted by the noise in practice, which makes \[LowRankHankelTightFrame\] only approximately true in practice. Nevertheless, \[LowRankHankelTightFrame\] implies that there exists an adaptive tight frame under which $\La\bsv$ can be *sparsely approximated*, so we can adopt the idea of data driven tight frames [@J.F.Cai2014]. In particular, we propose the following data driven tight frame (DDTF) based continuous domain CS-MRI reconstruction model $$\begin{aligned}
\label{ProposedCSMRIModel}
\begin{split}
&~~~\min_{\bsv,\bc,\bsW}~\f{1}{2}\left\|\bmP_{\MM}\bsv-\bsf\right\|_2^2+\f{\mu}{2}\left\|\bsW\left(\La\bsv\right)-\bc\right\|_2^2+\gamma\left\|\bc\right\|_0\\
&\text{subject to}~~\bsv\in\mC~~\text{and}~~\bsW^*\bsW=\bsI
\end{split}\end{aligned}$$ to reconstruct the fully sampled k-space data $\bsv$ and then obtain the MR image by $\bu=\bmsF^{-1}\bsv$. Here, 