  Recently, the finite-rate-of-innovation (FRI) based continuous domain
regularization is emerging as an alternative to the conventional on-the-grid
sparse regularization for the compressed sensing (CS) due to its ability to
alleviate the basis mismatch between the true support of the shape in the
continuous domain and the discrete grid. In this paper, we propose a new
off-the-grid regularization for the CS-MRI reconstruction. Following the recent
works on two dimensional FRI, we assume that the discontinuities/edges of the
image are localized in the zero level set of a band-limited periodic function.
This assumption induces the linear dependencies among the Fourier samples of
the gradient of the image, which leads to a low rank two-fold Hankel matrix. We
further observe that the singular value decomposition of a low rank Hankel
matrix corresponds to an adaptive tight frame system which can represent the
image with sparse canonical coefficients. Based on this observation, we propose
a data driven tight frame based off-the-grid regularization model for the
CS-MRI reconstruction. To solve the nonconvex and nonsmooth model, a proximal
alternating minimization algorithm with a guaranteed global convergence is
adopted. Finally, the numerical experiments show that our proposed data driven
tight frame based approach outperforms the existing approaches.
