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19
A New Riemannian Setting for Surface Registration
, 2011
"... Abstract. We present a new approach for matching regular surfaces in a Riemannian setting. We use a Sobolev type metric on deformation vector fields which form the tangent bundle to the space of surfaces. In this article we compare our approach with the diffeomorphic matching framework. In the latte ..."
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Abstract. We present a new approach for matching regular surfaces in a Riemannian setting. We use a Sobolev type metric on deformation vector fields which form the tangent bundle to the space of surfaces. In this article we compare our approach with the diffeomorphic matching framework. In the latter approach a deformation is prescribed on the ambient space, which then drags along an embedded surface. In contrast our metric is defined directly on the deformation vector field and can therefore be called an inner metric. We also show how to discretize the corresponding geodesic equation and compute the gradient of the cost functional using finite elements.
EulerPoincaré formulation of hybrid plasma models
, 1012
"... Several different hybrid Vlasovfluid systems are formulated as EulerPoincaré systems and compared in the same framework. In particular, the discussion focuses on three major hybrid MHD models. These are the currentcoupling scheme and two different variants of the pressurecoupling scheme. The Kelv ..."
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Several different hybrid Vlasovfluid systems are formulated as EulerPoincaré systems and compared in the same framework. In particular, the discussion focuses on three major hybrid MHD models. These are the currentcoupling scheme and two different variants of the pressurecoupling scheme. The KelvinNoether theorem is presented explicitly for each scheme, together with the Poincaré invariants for its hot particle trajectories. Extensions of Ertel’s relation for the potential vorticity and for its gradient are also found for each hybrid MHD scheme, as well as new expressions of cross helicity invariants. Contents 1
Mixture of Kernels and Iterated Semidirect Product of Diffeomorphisms Groups. Multiscale Modeling and Simulation
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Sparse multiscale diffeomorphic registration: the kernel bundle framework
 JOURNAL OF MATHEMATICAL IMAGING AND VISION, SPRINGER VERLAG, 2013, 46 (3), PP.292308.
, 2013
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M.: Splines for diffeomorphic image regression
 In: MICCAI (2014
"... Abstract. This paper develops a method for splines on diffeomorphisms for image regression. In contrast to previously proposed methods to capture image changes over time, such as geodesic regression, the method can capture more complex spatiotemporal deformations. In particular, it is a first step ..."
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Abstract. This paper develops a method for splines on diffeomorphisms for image regression. In contrast to previously proposed methods to capture image changes over time, such as geodesic regression, the method can capture more complex spatiotemporal deformations. In particular, it is a first step towards capturing periodic motions for example of the heart or the lung. Starting from a variational formulation of splines the proposed approach allows for the use of temporal control points to control spline behavior. This necessitates the development of a shooting formulation for splines. Experimental results are shown for synthetic and real data. The performance of the method is compared to geodesic regression. 1
Geometry of image registration: The diffeomorphism group and momentum maps
, 2013
"... Abstract These lecture notes explain the geometry and discuss some of the analytical questions underlying image registration within the framework of large deformation diffeomorphic metric mapping (LDDMM) used in computational anatomy.1 1 ..."
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Abstract These lecture notes explain the geometry and discuss some of the analytical questions underlying image registration within the framework of large deformation diffeomorphic metric mapping (LDDMM) used in computational anatomy.1 1
F.X.: Invariant higherorder variational problems II
 J. Nonlinear Science
"... Fondly remembering our late friend Jerry Marsden Motivated by applications in computational anatomy, we consider a secondorder problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves know ..."
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Fondly remembering our late friend Jerry Marsden Motivated by applications in computational anatomy, we consider a secondorder problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of nonhorizontal geodesics on the group of transformations project to cubics. Finally, we apply secondorder Lagrange–Poincaré reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.
Higherorder Spatial Accuracy in Diffeomorphic Image Registration
 J. Geom. Imaging Comput
, 2015
"... We discretize a cost functional for image registration problems by deriving Taylor expansions for the matching term. Minima of the discretized cost functionals can be computed with no spatial discretization error, and the optimal solutions are equivalent to minimal energy curves in the space of k ..."
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We discretize a cost functional for image registration problems by deriving Taylor expansions for the matching term. Minima of the discretized cost functionals can be computed with no spatial discretization error, and the optimal solutions are equivalent to minimal energy curves in the space of kjets. We show that the solutions convergence to optimal solutions of the original cost functional as the number of particles increases with a convergence rate of O(hd+k) where h is a resolution parameter. The effect of this approach over traditional particle methods is illustrated on synthetic examples and real images. 1.
Geodesic Warps by Conformal Mappings
"... In recent years there has been considerable interest in methods for diffeomorphic warping of images, with applications e.g. in medical imaging and evolutionary biology. The original work generally cited is that of the evolutionary biologist D’Arcy Wentworth Thompson, who demonstrated warps to defor ..."
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In recent years there has been considerable interest in methods for diffeomorphic warping of images, with applications e.g. in medical imaging and evolutionary biology. The original work generally cited is that of the evolutionary biologist D’Arcy Wentworth Thompson, who demonstrated warps to deform images of one species into another. However, unlike the deformations in modern methods, which are drawn from the full set of diffeomorphism, he deliberately chose lowerdimensional sets of transformations, such as planar conformal mappings. In this paper we study warps of such conformal mappings. The approach is to equip the infinite dimensional manifold of conformal embeddings with a Riemannian metric, and then use the corresponding geodesic equation in order to obtain diffeomorphic warps. After deriving the geodesic equation, a numerical discretisation method is developed. Several examples of geodesic warps are then given. We also show that the equation admits totally geodesic solutions corresponding to scaling and translation, but not to affine transformations.