[Paper Review] Efficient Numerical Optimization For Susceptibility Artifact Correction Of EPI-MRI

Jan Macdonald, Lars Ruthotto|arXiv (Cornell University)|Jul 2, 2016

Advanced MRI Techniques and Applications23 references3 citations

TL;DR

This paper proposes two efficient numerical optimization methods—Gauss-Newton with block-Jacobi preconditioning and ADMM—for correcting susceptibility artifacts in EPI-MRI using a reversed-phase-encoding approach. By introducing a face-staggered discretization that exploits directional displacement constraints, the methods achieve near-linear complexity and enable parallel computation, reducing solution time to under 50 seconds for 3D problems with 5M+ degrees of freedom on a standard laptop.

ABSTRACT

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.

Motivation & Objective

• Address the high computational cost of existing susceptibility artifact correction methods in EPI-MRI, which limits real-time and high-resolution applications.
• Overcome the inefficiency of standard optimization schemes in solving the nonlinear, large-scale image registration problem arising from the physical distortion model.
• Develop a numerical discretization that exploits the directional nature of EPI distortions (limited to phase-encoding direction) to enable separable, parallelizable optimization.
• Design scalable, fast solvers with provable convergence for non-convex, large-scale optimization problems arising in MRI artifact correction.
• Achieve significant speedups over existing methods while maintaining correction quality comparable to state-of-the-art techniques.

Proposed method

• Adopt a discretize-then-optimize approach using a face-staggered finite difference discretization that aligns with the physical constraint that distortions are primarily along the phase-encoding direction.
• Formulate a variational image registration problem minimizing the distance between oppositely distorted EPI images, subject to invertibility and smoothness constraints.
• Introduce a block-Jacobi preconditioner for Gauss-Newton methods that exploits the block-diagonal structure of the Hessian, derived from the separable distance function and constraints.
• Apply the Alternating Direction Method of Multipliers (ADMM) by splitting the objective into separable (data fidelity) and non-separable (smoothness) parts, enabling parallel subproblem solving.
• Solve the separable subproblems via sequential quadratic programming (SQP) on small, independent image slices, and the quadratic subproblem using DCT-based solvers with BTTB structure.
• Provide a convergence proof for ADMM in the non-convex setting by leveraging problem smoothness and derive a theoretical lower bound for the augmentation parameter.

Experimental results

Research questions

• RQ1 Can a tailored discretization strategy exploiting the directional nature of EPI distortions lead to a separable optimization structure that enables efficient parallel computation?
• RQ2 To what extent can a block-Jacobi preconditioner improve convergence and reduce iteration count in Gauss-Newton methods for large-scale EPI artifact correction?
• RQ3 Can ADMM be effectively applied to non-convex MRI registration problems with provable convergence and high scalability?
• RQ4 What is the achievable speedup in time-to-solution compared to existing methods like HySCO, especially for high-resolution 3D EPI data?
• RQ5 How do adaptive versus fixed augmentation parameters in ADMM affect convergence and solution quality in this non-convex setting?

Key findings

• The proposed face-staggered discretization yields a separable structure in the distance function and constraints, enabling efficient parallelization and reducing computational complexity.
• The block-Jacobi preconditioner in the Gauss-Newton-PCG method reduces the number of PCG iterations and improves eigenvalue clustering, leading to a 15x speedup over default HySCO settings.
• The ADMM-based method achieves a 50x speedup over HySCO on a 3D EPI dataset with over 5 million degrees of freedom, solving the problem in under 50 seconds on a standard laptop.
• The ADMM subproblems are highly parallelizable: the separable data fidelity subproblem is solved slice-wise via SQP, and the quadratic subproblem uses DCT-based solvers with BTTB structure.
• A theoretical lower bound for the ADMM augmentation parameter is derived, and numerical experiments show that using a sufficiently large lower bound ensures smoothness and convergence.
• Both proposed methods achieve correction quality comparable to state-of-the-art methods, as validated on 2D and 3D EPI data, while drastically reducing time-to-solution.