Fast Diffeomorphic Image Registration via Fourier-Approximated Lie Algebras

This paper introduces Fourier-approximated Lie algebras for shooting (FLASH), a fast geodesic shooting algorithm for diffeomorphic image registration. We approximate the infinite-dimensional Lie algebra of smooth vector fields, i.e., the tangent space at the identity of the diffeomorphism group, with a low-dimensional, bandlimited space. We show that most of the computations for geodesic shooting can be carried out entirely in this low-dimensional space. Our algorithm results in dramatic savings in time and memory over traditional large-deformation diffeomorphic metric mapping algorithms, which require dense spatial discretizations of vector fields. To validate the effectiveness of FLASH, we run pairwise image registration on both 2D synthetic data and real 3D brain images and compare with the state-of-the-art geodesic shooting methods. Experimental results show that our algorithm dramatically reduces the computational cost and memory footprint of diffemorphic image registration with little or no loss of accuracy.