We propose a variational method for reconstructing divergence-free velocity fields from 4D Flow MRI data with compensation of spatial displacement artifacts. A standard 4D Flow MRI sequence to measure flow velocities in an expanding flow phantom with Reynolds number of 4000 was used. A variational formulation allows to reconstruct a divergence-free velocity field used to estimate the displacement artefact associated with spin motion between the time point of motion encoding and echo time. The field was used to apply a spatial shift of the original MRI data and compute new divergence-free flow velocities corrected for spatial motion artifacts.
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