  We present two efficient numerical methods for susceptibility artifact
correction applicable in Echo Planar Imaging (EPI), an ultra fast Magnetic
Resonance Imaging (MRI) technique widely used in clinical applications. Both
methods address a major practical drawback of EPI, the so-called susceptibility
artifacts, which consist of geometrical transformations and intensity
modulations. We consider a tailored variational image registration problem that
is based on a physical distortion model and aims at minimizing the distance of
two oppositely distorted images subject to invertibility constraints. We follow
a discretize-then-optimize approach and present a novel face-staggered
discretization yielding a separable structure in the discretized distance
function and the invertibility constraints. The presence of a smoothness
regularizer renders the overall optimization problem non-separable, but we
present two optimization schemes that exploit the partial separability. First,
we derive a block-Jacobi preconditioner to be used in a Gauss-Newton-PCG
method. Second, we consider a splitting of the separable and non-separable part
and solve the resulting problem using the Alternating Direction Method of
Multipliers (ADMM). We provide a detailed convergence proof for ADMM for this
non-convex optimization problem. Both schemes are of essentially linear
complexity and are suitable for parallel computing. A considerable advantage of
the proposed schemes over established methods is the reduced time-to-solution.
In our numerical experiment using high-resolution 3D imaging data, our parallel
implementation of the ADMM method solves a 3D problem with more than 5 million
degrees of freedom in less than 50 seconds on a standard laptop, which is a
considerable improvement over existing methods.
