Fourier Methods for Nonparametric Image Registration

Nonparametric image registration algorithms use deformation fields to define nonrigid transformations relating two images. Typically, these algorithms operate by successively solving linear systems of partial differential equations. These PDE systems arise by linearizing the Euler-Lagrange equations associated with the minimization of a functional defined to contain an image similarity term and a regularizer. Iterative linear system solvers can be used to solve the linear PDE systems, but they can be extremely slow. Some faster techniques based on Fourier methods, multigrid methods, and additive operator splitting, exist for solving the linear PDE systems for specific combinations of regularizers and boundary conditions. In this paper, we show that Fourier methods can be employed to quickly solve the linear PDE systems for every combination of standard regularizers (diffusion, curvature, elastic, and fluid) and boundary conditions (Dirichlet, Neumann, and periodic).