Symmetry in Image Registration and Deformation Modeling

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.