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Open AccessArticle

Reduction by Lie Group Symmetries in Diffeomorphic Image Registration and Deformation Modelling

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Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen E, Denmark
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Department of Mathematics, Imperial College, SW7 2AZ London, UK
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Authors to whom correspondence should be addressed.
Academic Editor: Zygmunt Pizlo
Symmetry 2015, 7(2), 599-624; https://doi.org/10.3390/sym7020599
Received: 30 November 2014 / Revised: 15 April 2015 / Accepted: 27 April 2015 / Published: 7 May 2015
(This article belongs to the Special Issue Symmetry: Theory and Applications in Vision)
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. View Full-Text
Keywords: image registration; reduction by symmetry; large deformation diffeomorphic metric mapping (LDDMM); isotropy subgroups; jet matching image registration; reduction by symmetry; large deformation diffeomorphic metric mapping (LDDMM); isotropy subgroups; jet matching
MDPI and ACS Style

Sommer, S.; Jacobs, H.O. Reduction by Lie Group Symmetries in Diffeomorphic Image Registration and Deformation Modelling. Symmetry 2015, 7, 599-624.

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