Abstract
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.
Highlights
Registration, the task of establishing correspondences between multiple instances of objects, such as images, landmarks, curves and surfaces, plays a fundamental role in a range of computer visionSymmetry 2015, 7 applications, including shape modelling [1], motion compensation and optical flow [2], remote sensing [3] and medical imaging [4]
We focus on large deformations modelled in subgroups of the group of diffeomorphic mappings on the spatial domain in the context of large deformation diffeomorphic metric mapping (LDDMM) [1,9,10,11]
A degree of comfort with differential geometry will be assumed, it is the aim of this paper to make the role of symmetry in registration and deformation modelling clear to the non-expert in geometric mechanics and symmetry groups in image registration
Summary
Registration, the task of establishing correspondences between multiple instances of objects, such as images, landmarks, curves and surfaces, plays a fundamental role in a range of computer vision. Symmetry 2015, 7 applications, including shape modelling [1], motion compensation and optical flow [2], remote sensing [3] and medical imaging [4]. Examples of the fundamental role of registration include quantifying developing Alzheimer’s disease by establishing correspondences between brain tissue at different stages of the disease [6]; measuring the effect of chronic obstructive pulmonary disease on lung tissue after removing variability caused by the respiratory process [7]; and correlating the shape of the hippocampus to schizophrenia after inter-subject registration [8]. We survey the role of reduction by symmetry in diffeomorphic registration and deformation modelling, linking symmetry as seen from the field of geometric mechanics with the image registration problem. We wish to describe these connections in a form that highlights the role of symmetry and points towards future applications of the ideas
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