  In phase-contrast magnetic resonance imaging (PC-MRI), spin velocity
contributes to the phase measured at each voxel. Therefore, estimating velocity
from potentially wrapped phase measurements is the task of solving a system of
noisy congruence equations. We propose Phase Recovery from Multiple Wrapped
Measurements (PRoM) as a fast, approximate maximum likelihood estimator of
velocity from multi-coil data with possible amplitude attenuation due to
dephasing. The estimator can recover the fullest possible extent of unambiguous
velocities, which can greatly exceed twice the highest venc. The estimator uses
all pairwise phase differences and the inherent correlations among them to
minimize the estimation error. Correlations are directly estimated from
multi-coil data without requiring knowledge of coil sensitivity maps, dephasing
factors, or the actual per-voxel signal-to-noise ratio. Derivation of the
estimator yields explicit probabilities of unwrapping errors and the
probability distribution for the velocity estimate; this, in turn, allows for
optimized design of the phase-encoded acquisition. These probabilities are also
incorporated into spatial post-processing to further mitigate wrapping errors.
Simulation, phantom, and in vivo results for three-point PC-MRI acquisitions
validate the benefits of reduced estimation error, increased recovered velocity
range, optimized acquisition, and fast computation. A phantom study at 1.5T
demonstrates 48.5% decrease in root mean squared error using PRoM with
post-processing versus a conventional "dual-venc" technique. Simulation and 3T
in vivo results likewise demonstrate the proposed benefits.
