in [@M.Fornasier2011; @K.Mohan2012; @G.Ongie2017], which can avoid the high computational cost of the SVD related to the rank minmization and the convex nuclear norm relaxation. Apart from these approaches, the nonconvex alternating projection methods based on the different parametrizations of the underlying low rank matrix structure are proposed and studied in [@J.F.Cai2018a; @J.F.Cai2019]. These nonconvex methods are reported to be superior to the other relaxation methods in terms of the computational efficiency while theoretical restoration guarantees are still available.

In this paper, we propose a new off-the-grid approach for the CS-MRI reconstruction based on the data driven tight frame [@J.F.Cai2014]. To this end, we first note that the SVD of a Hankel matrix induces a tight frame filter bank due to its underlying convolutional structure. More precisely, if we can associate a signal with a low rank Hankel matrix, its right singular vectors form a tight frame filter bank which allows us to represent the signal with a small number of nonzero canonical coefficients. Based on this observation, we develop the sparse regularization model via data driven tight frames [@J.F.Cai2014] for the CS-MRI restoration problem. Even though the data driven tight frame approach has been discussed for the sparse approximation 