  In this survey, we provide a detailed review of recent advances in the
recovery of continuous domain multidimensional signals from their few
non-uniform (multichannel) measurements using structured low-rank matrix
completion formulation. This framework is centered on the fundamental duality
between the compactness (e.g., sparsity) of the continuous signal and the rank
of a structured matrix, whose entries are functions of the signal. This
property enables the reformulation of the signal recovery as a low-rank
structured matrix completion, which comes with performance guarantees. We will
also review fast algorithms that are comparable in complexity to current
compressed sensing methods, which enables the application of the framework to
large-scale magnetic resonance (MR) recovery problems. The remarkable
flexibility of the formulation can be used to exploit signal properties that
are difficult to capture by current sparse and low-rank optimization
strategies. We demonstrate the utility of the framework in a wide range of MR
imaging (MRI) applications, including highly accelerated imaging,
calibration-free acquisition, MR artifact correction, and ungated dynamic MRI.
