Reduction by Lie Group symmetries in diffeomorphic image registration and deformation modelling

TY - JOUR

T1 - Reduction by Lie Group symmetries in diffeomorphic image registration and deformation modelling

AU - Sommer, Stefan Horst

AU - Jacobs, Henry O.

N1 - OA

PY - 2015

Y1 - 2015

N2 - We survey the role of reduction by symmetry in the large deformation diffeomorphic metric mapping framework for registration of a variety of data types (landmarks, curves, surfaces, images and higher-order derivative data). Particle relabelling symmetry allows the equations of motion to be reduced to the Lie algebra allowing the equations to be written purely in terms of the Eulerian velocity field. As a second use of symmetry, the infinite dimensional problem of finding correspondences between objects can be reduced for a range of concrete data types, resulting in compact representations of shape and spatial structure. Using reduction by symmetry, we describe these models in a common theoretical framework that draws on links between the registration problem and geometric mechanics. We outline these constructions and further cases where reduction by symmetry promises new approaches to the registration of complex data types.

AB - We survey the role of reduction by symmetry in the large deformation diffeomorphic metric mapping framework for registration of a variety of data types (landmarks, curves, surfaces, images and higher-order derivative data). Particle relabelling symmetry allows the equations of motion to be reduced to the Lie algebra allowing the equations to be written purely in terms of the Eulerian velocity field. As a second use of symmetry, the infinite dimensional problem of finding correspondences between objects can be reduced for a range of concrete data types, resulting in compact representations of shape and spatial structure. Using reduction by symmetry, we describe these models in a common theoretical framework that draws on links between the registration problem and geometric mechanics. We outline these constructions and further cases where reduction by symmetry promises new approaches to the registration of complex data types.

U2 - 10.3390/sym7020599

DO - 10.3390/sym7020599

M3 - Journal article

SN - 2073-8994

VL - 7

SP - 599

EP - 624

JO - Symmetry

JF - Symmetry

IS - 2

ER -