A Semi-Lagrangian two-level preconditioned Newton-Krylov solver for constrained diffeomorphic image registration

TL;DR

This paper introduces a novel semi-Lagrangian preconditioned Newton-Krylov solver for diffeomorphic image registration, achieving significant speedups over previous methods through a two-level preconditioning strategy and efficient PDE discretization.

Contribution

The paper presents a new semi-Lagrangian two-level preconditioned Newton-Krylov algorithm for diffeomorphic image registration, improving computational efficiency and convergence.

Findings

Achieved up to 20x speedup in real-world medical image registration.

Demonstrated better grid convergence and efficiency compared to previous explicit schemes.

Validated the method on synthetic and real 2D image registration scenarios.

Abstract

We propose an efficient numerical algorithm for the solution of diffeomorphic image registration problems. We use a variational formulation constrained by a partial differential equation (PDE), where the constraints are a scalar transport equation. We use a pseudospectral discretization in space and second-order accurate semi-Lagrangian time stepping scheme for the transport equations. We solve for a stationary velocity field using a preconditioned, globalized, matrix-free Newton-Krylov scheme. We propose and test a two-level Hessian preconditioner. We consider two strategies for inverting the preconditioner on the coarse grid: a nested preconditioned conjugate gradient method (exact solve) and a nested Chebyshev iterative method (inexact solve) with a fixed number of iterations. We test the performance of our solver in different synthetic and real-world two-dimensional application scenarios. We study grid convergence and computational efficiency of our new scheme. We compare the performance of our solver against our initial implementation that uses the same spatial discretization but a standard, explicit, second-order Runge-Kutta scheme for the numerical time integration of the transport equations and a single-level preconditioner. Our improved scheme delivers significant speedups over our original implementation. As a highlight, we observe a 20$\times$ speedup for a two dimensional, real world multi-subject medical image registration problem.