The momentum map representation of images

TL;DR

This paper introduces a mathematical framework using momentum maps for large deformation image registration, unifying various algorithms and extending to complex data structures and manifold-based images.

Contribution

It proves a theorem linking diffeomorphism actions to momentum maps, enabling the design of registration algorithms for diverse data types and scales.

Findings

Recovering known algorithms for landmarks, scalar images, and vector fields

Extending methods to diffusion tensor images and multiscale registration

Applying momentum maps to images on manifolds

Abstract

This paper discusses the mathematical framework for designing methods of large deformation matching (LDM) for image registration in computational anatomy. After reviewing the geometrical framework of LDM image registration methods, a theorem is proved showing that these methods may be designed by using the actions of diffeomorphisms on the image data structure to define their associated momentum representations as (cotangent lift) momentum maps. To illustrate its use, the momentum map theorem is shown to recover the known algorithms for matching landmarks, scalar images and vector fields. After briefly discussing the use of this approach for Diffusion Tensor (DT) images, we explain how to use momentum maps in the design of registration algorithms for more general data structures. For example, we extend our methods to determine the corresponding momentum map for registration using semidirect product groups, for the purpose of matching images at two different length scales. Finally, we discuss the use of momentum maps in the design of image registration algorithms when the image data is defined on manifolds instead of vector spaces.