  Purpose: To reconstruct artifact-free images from measured k-space data, when
the actual k-space trajectory deviates from the nominal trajectory due to
gradient imperfections.
  Methods: Trajectory errors arising from eddy currents and gradient delays
introduce phase inconsistencies in several fast scanning MR pulse sequences,
resulting in image artifacts. The proposed algorithm provides a novel framework
to compensate for this phase distortion. The algorithm relies on the
construction of a multi-block Hankel matrix, where each block is constructed
from k-space segments with the same phase distortion. In the presence of
spatially smooth phase distortions between the segments, the complete
block-Hankel matrix is known to be highly low-rank. Since each k-space segment
is only acquiring part of the k-space data, the reconstruction of the phase
compensated image from their partially parallel measurements is posed as a
structured low-rank matrix optimization problem, assuming the coil
sensitivities to be known.
  Results: The proposed formulation is tested on radial acquisitions in several
settings including partial Fourier and golden-angle acquisitions. The
experiments demonstrate the ability of the algorithm to successfully remove the
artifacts arising from the trajectory errors, without the need for trajectory
or phase calibration. The quality of the reconstruction was comparable to
corrections achieved using the Trajectory Auto-Corrected Image Reconstruction
(TrACR) for radial acquisitions.
  Conclusion: The proposed method provides a general framework for the recovery
of artifact-free images from radial trajectories without the need for
trajectory calibration.
