  Compressed Sensing Magnetic Resonance Imaging (CS-MRI) significantly
accelerates MR data acquisition at a sampling rate much lower than the Nyquist
criterion. A major challenge for CS-MRI lies in solving the severely ill-posed
inverse problem to reconstruct aliasing-free MR images from the sparse k-space
data. Conventional methods typically optimize an energy function, producing
reconstruction of high quality, but their iterative numerical solvers
unavoidably bring extremely slow processing. Recent data-driven techniques are
able to provide fast restoration by either learning direct prediction to final
reconstruction or plugging learned modules into the energy optimizer.
Nevertheless, these data-driven predictors cannot guarantee the reconstruction
following constraints underlying the regularizers of conventional methods so
that the reliability of their reconstruction results are questionable. In this
paper, we propose a converged deep framework assembling principled modules for
CS-MRI that fuses learning strategy with the iterative solver of a conventional
reconstruction energy. This framework embeds an optimal condition checking
mechanism, fostering \emph{efficient} and \emph{reliable} reconstruction. We
also apply the framework to two practical tasks, \emph{i.e.}, parallel imaging
and reconstruction with Rician noise. Extensive experiments on both benchmark
and manufacturer-testing images demonstrate that the proposed method reliably
converges to the optimal solution more efficiently and accurately than the
state-of-the-art in various scenarios.
