  Magnetic Resonance Imaging (MRI) has excellent soft tissue contrast but is
hindered by an inherently slow data acquisition process. Compressed sensing,
which reconstructs sparse signals from incoherently sampled data, has been
widely applied to accelerate MRI acquisitions. Compressed sensing MRI requires
one or more model parameters to be tuned, which is usually done by hand, giving
sub-optimal tuning in general. To address this issue, we build on previous work
by the authors on the single-coil Variable Density Approximate Message Passing
(VDAMP) algorithm, extending the framework to multiple receiver coils to
propose the Parallel VDAMP (P-VDAMP) algorithm. For Bernoulli random variable
density sampling, P-VDAMP obeys a "state evolution", where the intermediate
per-iteration image estimate is distributed according to the ground truth
corrupted by a zero-mean Gaussian vector with approximately known covariance.
To our knowledge, P-VDAMP is the first algorithm for multi-coil MRI data that
obeys a state evolution with accurately tracked parameters. We leverage state
evolution to automatically tune sparse parameters on-the-fly with Stein's
Unbiased Risk Estimate (SURE). P-VDAMP is evaluated on brain, knee and
angiogram datasets and compared with four variants of the Fast Iterative
Shrinkage-Thresholding algorithm (FISTA), including two tuning-free variants
from the literature. The proposed method is found to have a similar
reconstruction quality and time to convergence as FISTA with an optimally tuned
sparse weighting and offers substantial robustness and reconstruction quality
improvements over competing tuning-free methods.
