Efficient Numerical Optimization For Susceptibility Artifact Correction   Of EPI-MRI
  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. 
Jan Macdonald and Lars Ruthotto
math.OC