Symmetry in Image Registration and Deformation Modeling

TL;DR

This paper reviews how symmetry principles simplify diffeomorphic registration of various data types, enabling compact shape representations and linking registration to geometric mechanics, with recent applications to jet-matching and higher-order models.

Contribution

It provides a unified theoretical framework for symmetry-based reduction in image registration, highlighting recent advances and potential new approaches for complex data types.

Findings

Reduction by symmetry simplifies registration models.

Symmetry links registration to geometric mechanics.

Recent applications include jet-matching and higher-order models.

Abstract

We survey the role of symmetry in diffeomorphic registration of landmarks, curves, surfaces, images and higher-order data. The infinite dimensional problem of finding correspondences between objects can for a range of concrete data types be reduced resulting in compact representations of shape and spatial structure. This reduction is possible because the available data is incomplete in encoding the full deformation model. Using reduction by symmetry, we describe the reduced models in a common theoretical framework that draws on links between the registration problem and geometric mechanics. Symmetry also arises in reduction to the Lie algebra using particle relabeling symmetry allowing the equations of motion to be written purely in terms of Eulerian velocity field. Reduction by symmetry has recently been applied for jet-matching and higher-order discrete approximations of the image matching problem. We outline these constructions and further cases where reduction by symmetry promises new approaches to registration of complex data types.