Symmetry in Image Registration and Deformation Modeling
This foundational work addresses the challenge of efficient and theoretically grounded registration for complex data in fields like medical imaging and computer vision, though it is incremental as it builds on existing symmetry concepts.
The paper surveys how symmetry principles can reduce the infinite-dimensional problem of diffeomorphic registration for various data types like landmarks and images, leading to compact representations of shape and spatial structure by leveraging geometric mechanics and Eulerian velocity fields.
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.