Higher-order Spatial Accuracy in Diffeomorphic Image Registration

TL;DR

This paper introduces a higher-order discretization method for diffeomorphic image registration that achieves spatial accuracy by using Taylor expansions, leading to convergence to optimal solutions as particle resolution increases.

Contribution

It develops a novel discretization approach based on Taylor expansions that improves spatial accuracy in diffeomorphic image registration and demonstrates convergence properties.

Findings

Discretized cost functionals can be minimized without spatial discretization error.

Solutions converge to the original problem's solutions at a rate of O(h^{d+k}).

The method outperforms traditional particle methods on synthetic and real images.

Abstract

We discretize a cost functional for image registration problems by deriving Taylor expansions for the matching term. Minima of the discretized cost functionals can be computed with no spatial discretization error, and the optimal solutions are equivalent to minimal energy curves in the space of $k$-jets. We show that the solutions convergence to optimal solutions of the original cost functional as the number of particles increases with a convergence rate of $O(h^{d+k})$ where $h$ is a resolution parameter. The effect of this approach over traditional particle methods is illustrated on synthetic examples and real images.