Soliton dynamics in computational anatomy

Holm, Darryl D.; Ratnanather, J. Tilak; Trouvé, Alain; Younes, Laurent

doi:10.1016/j.neuroimage.2004.07.017

Cited by 82 publications

(100 citation statements)

References 55 publications

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“…Of course these different interpretations also provide opportunities: for example, to see to what extent the singular solutions found in the EPDiff equations are applicable, either for shallow water wave interactions, or for computer vision applications. A recent combination of these ideas in which image processing imitates soliton interactions appears in Holm, Trouvé and Younes [2003].…”

Section: The Epdiff Equationmentioning

confidence: 99%

Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation

Holm

Marsden

Progress in Mathematics

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115

253

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We study the dynamics of measure-valued solutions of what we call the EPDiff equations, standing for the Euler-Poincaré equations associated with the diffeomorphism group (of R n or an n-dimensional manifold M ). Our main focus will be on the case of quadratic Lagrangians; that is, on geodesic motion on the diffeomorphism group with respect to the right invariant Sobolev H 1 metric. The corresponding Euler-Poincaré (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa-Holm (CH) equations in one dimension. The corresponding equations for the volume preserving diffeomorphism group are the well-known LAE (Lagrangian averaged Euler) equations for incompressible fluids.We first show that the EPDiff equations are generated by a smooth vector field on the diffeomorphism group for sufficiently smooth solutions. This is analogous to known results for incompressible fluids-both the Euler equations and the LAE equations-and it shows that for sufficiently smooth solutions, the equations are well-posed for short time. In fact, numerical evidence suggests 1 CONTENTS 2 that, as time progresses, these smooth solutions break up into singular solutions which, at least in one dimension, exhibit soliton behavior.With regard to these non-smooth solutions, we study measure-valued solutions that generalize to higher dimensions the peakon solutions of the (CH) equation in one dimension. One of the main purposes of this paper is to show that many of the properties of these measure-valued solutions may be understood through the fact that their solution ansatz is a momentum map. Some additional geometry is also pointed out, for example, that this momentum map is one leg of a natural dual pair.

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Section: The Epdiff Equationmentioning

confidence: 99%

Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation

Holm

Marsden

Progress in Mathematics

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115

253

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“…Numerous methods have been proposed to address those questions for different shape modalities. Some of these methods, [24,7,22,21] to mention only a few, rely on the extraction of a limited number of feature points (or landmarks). Other methods like [12,32,4,3,36] instead consider shapes as entire 2D or 3D images.…”

Section: Introductionmentioning

confidence: 99%

A General Framework for Curve and Surface Comparison and Registration with Oriented Varifolds

Kaltenmark

Charlier

Charon

2017

2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)

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“…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]. Connections with geometric mechanics [12,13] have highlighted the role of symmetry, and properties that were previously known to be connected with the specific data types have been described in a common theoretical framework [14]. We wish to describe these connections in a form that highlights the role of symmetry and points towards future applications of the ideas.…”

Section: Introductionmentioning

confidence: 99%

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

Sommer

Jacobs

2015

Symmetry

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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.

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Soliton dynamics in computational anatomy

Abstract: Abstract-

Cited by 82 publications

References 55 publications

Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation

Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation

A General Framework for Curve and Surface Comparison and Registration with Oriented Varifolds

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

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