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## Geometry of image registration: The diffeomorphism group and momentum maps (2013)

Citations: | 1 - 0 self |

### Citations

2821 |
Sobolev spaces
- Adams
- 1975
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Citation Context ... to prove the existence of minimizers for the image registration problem, that is for the energy E(u) = 1 2 ∫ 1 0 |ut |2L dt+ 1 2σ2 ‖I0 ◦ϕ−1− I1‖2L2 , where I0, I1 : R3 → R are grey-value images, u : =-=[0,1]-=-→ X(R3) a time-dependent vector field and ϕ1 its flow at time 1. There are two competing tendencies in the mathematical modelling for image registration. We want the diffeomorphism group to be an (inf... |

1287 |
Theory of reproducing kernels
- Aronszajn
- 1950
(Show Context)
Citation Context ...d derivatives, i.e. K ∈C2(Ω× Ω ,Rd×d) and |K|2,∞ < ∞. • K vanishes on the boundary of Ω ×Ω , i.e. K(x,y) = 0 whenever x ∈ ∂Ω or y ∈ ∂Ω . Further exposition of the theory of RKHS can be found, e.g. in =-=[3]-=-, [63]. Example 18. The Sobolev embedding theorem (see e.g. [1, Chapter 6]) states that for Ω ⊆ Rd there is an embedding Hk+m(Ω) ↪→Ck(Ω), m > d 2 , of the Sobolev space Hk+m(Ω) into the space of k-tim... |

791 | Multimodality image registration by maximization of mutual information
- Maes, Collignon, et al.
- 1997
(Show Context)
Citation Context ...he abstract formulation of LDDMM in Sect. 2 encompasses all these examples. Instead of the L2-norm one can use other similarity metrics to measure the distance between images, e.g. mutual information =-=[39]-=-. The biggest departure from LDDMM would be to change the way diffeomorphisms are generated. Possible approaches are stationary vector fields [5], free-form deformations [62], only affine transformati... |

556 |
A fast diffeomorphic image registration algorithm. Neuroimage 2007;38(1):95–113
- Ashburner
(Show Context)
Citation Context ... the distance between images, e.g. mutual information [39]. The biggest departure from LDDMM would be to change the way diffeomorphisms are generated. Possible approaches are stationary vector fields =-=[5]-=-, free-form deformations [62], only affine transformations [34] or demons [65, 69]. Common to all these methods however is the loss of geometric structure. Anatomical Shape and Function. Alzheimer’s d... |

550 |
A global optimisation method for robust affine registration of brain images
- Jenkinson, Smith
(Show Context)
Citation Context ...e biggest departure from LDDMM would be to change the way diffeomorphisms are generated. Possible approaches are stationary vector fields [5], free-form deformations [62], only affine transformations =-=[34]-=- or demons [65, 69]. Common to all these methods however is the loss of geometric structure. Anatomical Shape and Function. Alzheimer’s disease (AD) is a neurodegenerative disease and is the most freq... |

518 |
An integrable shallow water equation with peaked solitons, Phys
- Camassa, Holm
- 1993
(Show Context)
Citation Context ...group, also called EPDiff. It has the form ∂tm+Dm.u+DuT .m+div(u)m = 0 , m = u[ = Lu . (13) The EPDiff equation (13) first appeared in the context of unidirectional propagation of shallow water waves =-=[13]-=-. In the context of planar image registration, the crests of the shallow water waves correspond to the contour lines of the image [32]. To improve readability in (13), we have omitted the subscript t ... |

397 |
Whole brain segmentation: Automated labeling of neuroanatomical structures in the human brain.
- Fischl, DH, et al.
- 2002
(Show Context)
Citation Context ...here are N potentially contradicting segmentations of I1, that have to be combined using some classifier fusion technique, such as majority voting [4], Bayesian modelling [72] or Markov random fields =-=[21]-=-. For subcortical structures of the brain, this sort of atlas-based segmentation was shown to outperform other methods [7]. It is possible to use atlasbased registration with a variety of image regist... |

340 | Imagematching as a diffusion process: an analogy with Maxwell’s demons
- Thirion
- 1998
(Show Context)
Citation Context ...ture from LDDMM would be to change the way diffeomorphisms are generated. Possible approaches are stationary vector fields [5], free-form deformations [62], only affine transformations [34] or demons =-=[65, 69]-=-. Common to all these methods however is the loss of geometric structure. Anatomical Shape and Function. Alzheimer’s disease (AD) is a neurodegenerative disease and is the most frequent type of dement... |

260 |
General Pattern Theory
- Grenander
- 1993
(Show Context)
Citation Context ...solution in the elementary use of a certain method of the mathematician. This method is the Method of Coordinates, on which is based the Theory of Transformations. [66, p1032] More recently Grenander =-=[25, 26, 28]-=- generalized these ideas to encompass a diverse collection of real-world situations and formulated the principles of pattern theory. The following formulation is adapted from [51]: Martins Bruveris In... |

255 |
Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Internat
- Beg, Miller, et al.
- 2005
(Show Context)
Citation Context ... possible definitions of a distance d1(Id,ϕ) on the space of smooth invertible maps. We shall concentrate on the definition used in the large deformation diffeomorphic metric mapping (LDDMM) approach =-=[10, 49, 50, 67]-=-, which generates the transformation ϕ = ϕ1 as the flow of a time-dependent vector field. That is, t 7→ ut is a solution to the flow equation ∂tϕt(x) = ut(ϕt(x)), ϕ0(x) = x . The distance d1(Id,ϕ) is ... |

217 | The Euler–Poincaré equations and semidirect products with applications to continuum theories
- Holm, Marsden, et al.
- 1998
(Show Context)
Citation Context ...Poincaré equation is an evolution equation on the dual g∗ of the Lie algebra g, independent of I0, I1. Discussion of the history and some applications of the Euler-Poincaré equation can be found in =-=[31, 40]-=-. Now let us discuss the interplay between the group of transformations and the objects that are being matched. Let t 7→ ut be a solution of the matching problem. Then ut satisfies the Euler-Poincaré... |

192 |
Computational anatomy: An emerging discipline
- Grenander, Miller
- 1998
(Show Context)
Citation Context ...erns distorted and transfromed by a limited family of deformations. To have a specific example in mind, we will consider computational neuroanatomy; i.e., the study of the form and shape of the brain =-=[27]-=-. The observations in this case are the diagnostic tools accessible to the clinician; of particular interest to us are noninvasive imaging techniques like computed tomography (CT), magnetic resonance ... |

173 |
Landmark matching via large deformation diffeomorphisms
- Joshi, MI
(Show Context)
Citation Context ...ehind LDDMM helps illuminate relationships between anatomical shape and neurological functions. Data structures other than images can be registered within the LDDMM framework. These include landmarks =-=[35, 24]-=-, curves [23, 17], surfaces [68], tensor fields [2, 14] or functional data on a manifold [48, 55]. In fact the abstract formulation of LDDMM in Sect. 2 encompasses all these examples. Instead of the L... |

159 | Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration.
- Klein, A
- 2009
(Show Context)
Citation Context ...stration problem. This is not the only possible approach to image registration. In fact, a large literature about image registration exists. An overview of the available methods can be found, e.g. in =-=[29, 36]-=-. The LDDMM method, whilst being computationally more 4 Martins Bruveris and Darryl D. Holm expensive than others, is among the most accurate [6] registration methods. Here we will concentrate on the ... |

140 | A mathematical textbook of deformable neuroanatomies.
- Miller, Christensen, et al.
- 1993
(Show Context)
Citation Context ...els of I0 are assigned membership to a labelled anatomical structure, we can propagate this segmentation via ϕ to the image I1. This is the idea of registration-based or atlas-based segmentation, see =-=[15, 22, 47]-=-. To remove the bias inherent in the choice of the template I0, these multi-template registration techniques replace I0 with a collection (I j templ) j=1,...,N of several manually segmented images. Ea... |

136 |
On the metrics and EulerLagrange equations of computational anatomy
- Miller, Trouvé, et al.
- 2002
(Show Context)
Citation Context ... possible definitions of a distance d1(Id,ϕ) on the space of smooth invertible maps. We shall concentrate on the definition used in the large deformation diffeomorphic metric mapping (LDDMM) approach =-=[10, 49, 50, 67]-=-, which generates the transformation ϕ = ϕ1 as the flow of a time-dependent vector field. That is, t 7→ ut is a solution to the flow equation ∂tϕt(x) = ut(ϕt(x)), ϕ0(x) = x . The distance d1(Id,ϕ) is ... |

135 |
Spatial transformations of diffusion tensor magnetic resonance images.
- Alexander, Pierpaoli, et al.
- 2001
(Show Context)
Citation Context ...1 (Image Registration via LDDMM). Given two images I0, I1 ∈V , find a time-dependent vector field t 7→ ut ∈ X(R3) that minimizes the energy E(u) = 1 2 ∫ 1 0 |ut |2 dt+ 12σ2 ∣∣I0 ◦ϕ−11 − I1∣∣2L2(R3) , =-=(2)-=- where ϕt ∈ Diff(R3) is the flow of ut , i.e. ∂tϕt(x) = ut(ϕt(x)), ϕ0(x) = x , The vector field t 7→ ut and the transformation ϕ1 are the solutions of the image registration problem. This is not the o... |

122 | Group actions, homeomorphisms, and matching: A general framework
- Miller, Younes
(Show Context)
Citation Context ... possible definitions of a distance d1(Id,ϕ) on the space of smooth invertible maps. We shall concentrate on the definition used in the large deformation diffeomorphic metric mapping (LDDMM) approach =-=[10, 49, 50, 67]-=-, which generates the transformation ϕ = ϕ1 as the flow of a time-dependent vector field. That is, t 7→ ut is a solution to the flow equation ∂tϕt(x) = ut(ϕt(x)), ϕ0(x) = x . The distance d1(Id,ϕ) is ... |

120 | Diffeomorphism groups and pattern matching in image analysis
- Trouvé
- 1998
(Show Context)
Citation Context |

112 |
Global prevalence of dementia: a Delphi consensus study,”
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- 2005
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Citation Context ...ese methods however is the loss of geometric structure. Anatomical Shape and Function. Alzheimer’s disease (AD) is a neurodegenerative disease and is the most frequent type of dementia in the elderly =-=[19]-=-. Related to AD is mild cognitive impairment (MCI), an intermediate cognitive state between healthy ageing and dementia. Although most patients who develop AD are first diagnosed with MCI, not all of ... |

103 | Surface matching via currents
- Vaillant, Glaunes
(Show Context)
Citation Context ...nships between anatomical shape and neurological functions. Data structures other than images can be registered within the LDDMM framework. These include landmarks [35, 24], curves [23, 17], surfaces =-=[68]-=-, tensor fields [2, 14] or functional data on a manifold [48, 55]. In fact the abstract formulation of LDDMM in Sect. 2 encompasses all these examples. Instead of the L2-norm one can use other similar... |

94 |
Theory of reproducing kernels and its applications
- Saitoh
- 1988
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Citation Context ...ivatives, i.e. K ∈C2(Ω× Ω ,Rd×d) and |K|2,∞ < ∞. • K vanishes on the boundary of Ω ×Ω , i.e. K(x,y) = 0 whenever x ∈ ∂Ω or y ∈ ∂Ω . Further exposition of the theory of RKHS can be found, e.g. in [3], =-=[63]-=-. Example 18. The Sobolev embedding theorem (see e.g. [1, Chapter 6]) states that for Ω ⊆ Rd there is an embedding Hk+m(Ω) ↪→Ck(Ω), m > d 2 , of the Sobolev space Hk+m(Ω) into the space of k-times con... |

79 | Population shape regression from random design data,” in Computer Vision,
- Davis, Fletcher, et al.
- 2007
(Show Context)
Citation Context ...ing the dimensionality of the anatomical shape variations [58]; generalizing geodesic regression to the anatomical shape manifold and computing the mean aging process of the brain across a population =-=[16]-=-; the use of parallel transport not only for longitudinal studies, but also to characterize the left-right asymetry of subcortical structures [57]; applications to other diseases like schizophrenia [6... |

76 | Symmetric log-domain diffeomorphic registration: a demons-based approach.
- Vercauteren, Pennec, et al.
- 2008
(Show Context)
Citation Context ...ture from LDDMM would be to change the way diffeomorphisms are generated. Possible approaches are stationary vector fields [5], free-form deformations [62], only affine transformations [34] or demons =-=[65, 69]-=-. Common to all these methods however is the loss of geometric structure. Anatomical Shape and Function. Alzheimer’s disease (AD) is a neurodegenerative disease and is the most frequent type of dement... |

65 | An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach”, Applied and Computational Harmonic Analysis
- Michor, Mumford
(Show Context)
Citation Context ...tails on the background from Riemannian geometry and the theory of group actions consult [43]. The Hamiltonian approach to Riemannian geometry, including the infinite dimensional case is described in =-=[44]-=-. Riemannian metrics induced by group actions, especially the diffeomorphism group, in the context of shape matching are discussed in [42] and [9]. 3 Existence of Solutions for Image Registration In t... |

64 | A review of geometric transformations for nonrigid body registration,”
- Holden
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Citation Context ...stration problem. This is not the only possible approach to image registration. In fact, a large literature about image registration exists. An overview of the available methods can be found, e.g. in =-=[29, 36]-=-. The LDDMM method, whilst being computationally more 4 Martins Bruveris and Darryl D. Holm expensive than others, is among the most accurate [6] registration methods. Here we will concentrate on the ... |

57 |
D.J.: Nonrigid registration using free-form deformations: Application to breast MR images.
- Rueckert, Sonoda, et al.
- 1999
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Citation Context ..., e.g. mutual information [39]. The biggest departure from LDDMM would be to change the way diffeomorphisms are generated. Possible approaches are stationary vector fields [5], free-form deformations =-=[62]-=-, only affine transformations [34] or demons [65, 69]. Common to all these methods however is the loss of geometric structure. Anatomical Shape and Function. Alzheimer’s disease (AD) is a neurodegener... |

55 |
Geodesic shooting for computational anatomy
- Miller, Trouvé, et al.
(Show Context)
Citation Context ...F (R3) } , elements of which are specified using only one real-valued function P, while the vector field u0 or equivalently the momentum Lu0 needs 3 functions. This reduction strategy was employed in =-=[46, 71]-=- to solve the matching problem by estimating the initial momentum and using the EPDiff equation to reconstruct the path. The momentum map also allows for an intuitive interpretation. Equation (14) tel... |

42 |
Landmark matching via large deformation diffeomorphisms on the sphere
- Glaunès, Vaillant, et al.
(Show Context)
Citation Context ...ehind LDDMM helps illuminate relationships between anatomical shape and neurological functions. Data structures other than images can be registered within the LDDMM framework. These include landmarks =-=[35, 24]-=-, curves [23, 17], surfaces [68], tensor fields [2, 14] or functional data on a manifold [48, 55]. In fact the abstract formulation of LDDMM in Sect. 2 encompasses all these examples. Instead of the L... |

41 |
Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace-Beltrami operator
- Qiu, Bitouk, et al.
- 2006
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Citation Context ...ata structures other than images can be registered within the LDDMM framework. These include landmarks [35, 24], curves [23, 17], surfaces [68], tensor fields [2, 14] or functional data on a manifold =-=[48, 55]-=-. In fact the abstract formulation of LDDMM in Sect. 2 encompasses all these examples. Instead of the L2-norm one can use other similarity metrics to measure the distance between images, e.g. mutual i... |

40 | Large deformation diffeomorphic metric mapping of vector fields
- Cao, Miller, et al.
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Citation Context ...mical shape and neurological functions. Data structures other than images can be registered within the LDDMM framework. These include landmarks [35, 24], curves [23, 17], surfaces [68], tensor fields =-=[2, 14]-=- or functional data on a manifold [48, 55]. In fact the abstract formulation of LDDMM in Sect. 2 encompasses all these examples. Instead of the L2-norm one can use other similarity metrics to measure ... |

40 |
Pattern Theory: From Representation to Inference,
- Grenander, Miller
- 2007
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Citation Context ...solution in the elementary use of a certain method of the mathematician. This method is the Method of Coordinates, on which is based the Theory of Transformations. [66, p1032] More recently Grenander =-=[25, 26, 28]-=- generalized these ideas to encompass a diverse collection of real-world situations and formulated the principles of pattern theory. The following formulation is adapted from [51]: Martins Bruveris In... |

32 |
Statistical models of sets of curves and surfaces based on currents. Medical Image Analysis
- Durrleman, Trouvé, et al.
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Citation Context ... illuminate relationships between anatomical shape and neurological functions. Data structures other than images can be registered within the LDDMM framework. These include landmarks [35, 24], curves =-=[23, 17]-=-, surfaces [68], tensor fields [2, 14] or functional data on a manifold [48, 55]. In fact the abstract formulation of LDDMM in Sect. 2 encompasses all these examples. Instead of the L2-norm one can us... |

31 |
3D Model-Based Segmentation of Individual Brain Structures from Magnetic Resonance Imaging Data. Thesis, McGill Univ.,
- Collins
- 1994
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Citation Context ...els of I0 are assigned membership to a labelled anatomical structure, we can propagate this segmentation via ϕ to the image I1. This is the idea of registration-based or atlas-based segmentation, see =-=[15, 22, 47]-=-. To remove the bias inherent in the choice of the template I0, these multi-template registration techniques replace I0 with a collection (I j templ) j=1,...,N of several manually segmented images. Ea... |

31 |
Ratiu T.S., Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems
- Marsden
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Citation Context ...he system under consideration is invariant under the action of G, then the generalized momentum I pi is a constant of motion. This theorem enables one generate conservation laws from symmetries. See =-=[33, 40]-=- for more details on momentum maps and geometric mechanics. Tangent and Cotangent Lifted Actions. The action of a Lie group G on the vector space V is a map ` : G×V→V . Fixing an element g∈G we obtain... |

28 | Large deformation diffeomorphic metric curve mapping
- Glaunès, Qiu, et al.
- 2008
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Citation Context ... illuminate relationships between anatomical shape and neurological functions. Data structures other than images can be registered within the LDDMM framework. These include landmarks [35, 24], curves =-=[23, 17]-=-, surfaces [68], tensor fields [2, 14] or functional data on a manifold [48, 55]. In fact the abstract formulation of LDDMM in Sect. 2 encompasses all these examples. Instead of the L2-norm one can us... |

24 | L.: Soliton dynamics in computational anatomy
- Holm, Rananather, et al.
(Show Context)
Citation Context ... the context of unidirectional propagation of shallow water waves [13]. In the context of planar image registration, the crests of the shallow water waves correspond to the contour lines of the image =-=[32]-=-. To improve readability in (13), we have omitted the subscript t for the time-dependence. For the sake of completeness, we also include the coadjoint action, Ad∗ϕ m = (detDϕ)Dϕ T .(m◦ϕ) . 20 Martins ... |

23 |
Geometric mechanics and symmetry. From finite to infinite dimensions. With solutions to selected exercises by
- Holm, Schmah, et al.
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Citation Context ...he system under consideration is invariant under the action of G, then the generalized momentum I pi is a constant of motion. This theorem enables one generate conservation laws from symmetries. See =-=[33, 40]-=- for more details on momentum maps and geometric mechanics. Tangent and Cotangent Lifted Actions. The action of a Lie group G on the vector space V is a map ` : G×V→V . Fixing an element g∈G we obtain... |

22 |
Pattern theory: The stochastic analysis of real world signals.
- Mumford, Desolneux
- 2010
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Citation Context ...ly Grenander [25, 26, 28] generalized these ideas to encompass a diverse collection of real-world situations and formulated the principles of pattern theory. The following formulation is adapted from =-=[51]-=-: Martins Bruveris Institut de Mathématiques, EPFL, Lausanne 1015, Switzerland, e-mail: martins.bruveris@ epfl.ch Darryl D. Holm Department of Mathematics, Imperial College London, London SW7 2AZ, Un... |

19 |
de Solorzano. Combination strategies in multi-atlas image segmentation: Application to brain MR data
- Artaechevarria, Munoz-Barrutia, et al.
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Citation Context ...o propagate the segmentation of I jtempl to I1. Now there are N potentially contradicting segmentations of I1, that have to be combined using some classifier fusion technique, such as majority voting =-=[4]-=-, Bayesian modelling [72] or Markov random fields [21]. For subcortical structures of the brain, this sort of atlas-based segmentation was shown to outperform other methods [7]. It is possible to use ... |

18 | The momentum map representation of images
- Bruveris, Gay-Balmaz, et al.
(Show Context)
Citation Context ...stration as follows: a group of transformations acts on a space of objects and we are searching for the transformation that deforms a template object to a target object. The presentation here follows =-=[12]-=-. Let us model the group of transformations by a Lie group G and the space of objects by a vector space V . We will in this section assume that both G and V are finite-dimensional in order to avoid qu... |

18 | Elastically deforming a three-dimensional atlas to match anatomical brain images
- Gee, Reivich, et al.
- 1993
(Show Context)
Citation Context ...els of I0 are assigned membership to a labelled anatomical structure, we can propagate this segmentation via ϕ to the image I1. This is the idea of registration-based or atlas-based segmentation, see =-=[15, 22, 47]-=-. To remove the bias inherent in the choice of the template I0, these multi-template registration techniques replace I0 with a collection (I j templ) j=1,...,N of several manually segmented images. Ea... |

17 |
Michor P.W., The convenient setting of global analysis
- Kriegl
- 1997
(Show Context)
Citation Context ...= H∞(R3) functions with squareintegrable derivatives or F (R3) = C∞c (R3) compactly supported functions. Then the action ` : Diffc(R3)×C∞c (R3)→C∞c F(R3) is smooth in the sense of convenient calculus =-=[37]-=- and all the operations described below can be interpreted in that framework. See [45] for details on diffeomorphism groups with other decay properties. The Lie algebra of Diff(R3) is X(R3), the space... |

16 |
Diffeomorphic registration using geodesic shooting and gauss-newton optimisation
- Ashburner, Friston
- 2011
(Show Context)
Citation Context ...ew of the available methods can be found, e.g. in [29, 36]. The LDDMM method, whilst being computationally more 4 Martins Bruveris and Darryl D. Holm expensive than others, is among the most accurate =-=[6]-=- registration methods. Here we will concentrate on the geometric structure of the LDDMM solutions. In particular, we will sketch some applications in which the geometry behind LDDMM helps illuminate r... |

14 | Time sequence diffeomorphic metric mapping and parallel transport track time-dependent shape changes. NeuroImage 45:S51-S60
- Qui, Albert, et al.
- 2009
(Show Context)
Citation Context ...rding the transport operation several methods have been proposed. From a geometrical point of view, parallel transport from Riemannian geometry is the most natural operation and this has been used in =-=[54, 74, 77]-=-. Since computing the parallel transport of the momentum along geodesics is numerically quite challenging, a first-order approximation called Schild’s ladder was proposed in [38] as an alternative. Ot... |

12 |
A.: The emerging discipline of computational functional anatomy. NeuroImage 45
- Miller, Qiu
- 2009
(Show Context)
Citation Context ...ata structures other than images can be registered within the LDDMM framework. These include landmarks [35, 24], curves [23, 17], surfaces [68], tensor fields [2, 14] or functional data on a manifold =-=[48, 55]-=-. In fact the abstract formulation of LDDMM in Sect. 2 encompasses all these examples. Instead of the L2-norm one can use other similarity metrics to measure the distance between images, e.g. mutual i... |

12 |
Deformation-based nuclear morphometry: capturing nuclear shape variation in HeLa cells.,” Cytometry A, in press
- Rohde, Ribeiro, et al.
- 2008
(Show Context)
Citation Context ...s like schizophrenia [60] or cerebral palsy [18]; addition of functional data to anatomical shapes [48]. There are also applications outside the medical field to the study of variations of cell shape =-=[61]-=- and to construct generative models for cells [53]. 2 Geometry of Matching Problems In order to better see the geometric properties of image registration with LDDMM, we will first formulate an abstrac... |

10 |
D.: An evaluation of four automatic methods of segmenting the subcortical structures in the brain
- Babalola, Patenaude, et al.
- 2009
(Show Context)
Citation Context ...such as majority voting [4], Bayesian modelling [72] or Markov random fields [21]. For subcortical structures of the brain, this sort of atlas-based segmentation was shown to outperform other methods =-=[7]-=-. It is possible to use atlasbased registration with a variety of image registration methods. However, a study that involved segmenting brain scans of mice [8] has shown that the choice of the registr... |

10 |
X.: Schild’s Ladder for the parallel transport of deformations in time series of images
- Lorenzi, Ayache, et al.
- 2011
(Show Context)
Citation Context ... been used in [54, 74, 77]. Since computing the parallel transport of the momentum along geodesics is numerically quite challenging, a first-order approximation called Schild’s ladder was proposed in =-=[38]-=- as an alternative. Other methods that depend only on the end-deformation and not on the whole geodesic path were considered and compared in [20]. From the viewpoint of applications, there is, as of n... |

10 | Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds
- Micheli, Michor, et al.
(Show Context)
Citation Context ...etry, including the infinite dimensional case is described in [44]. Riemannian metrics induced by group actions, especially the diffeomorphism group, in the context of shape matching are discussed in =-=[42]-=- and [9]. 3 Existence of Solutions for Image Registration In this section we want to present a framework that allows us to prove the existence of minimizers for the image registration problem, that is... |

9 |
Overview of the geometries of shape spaces and diffeomorphism groups
- Bauer, Bruveris, et al.
- 2014
(Show Context)
Citation Context ...luding the infinite dimensional case is described in [44]. Riemannian metrics induced by group actions, especially the diffeomorphism group, in the context of shape matching are discussed in [42] and =-=[9]-=-. 3 Existence of Solutions for Image Registration In this section we want to present a framework that allows us to prove the existence of minimizers for the image registration problem, that is for the... |

9 | Combining anatomical manifold information via diffeomorphic metric mappings for studying cortical thinning of the cingulate gyrus in schizophrenia,”
- Qiu, Younes, et al.
- 2007
(Show Context)
Citation Context ...6]; the use of parallel transport not only for longitudinal studies, but also to characterize the left-right asymetry of subcortical structures [57]; applications to other diseases like schizophrenia =-=[60]-=- or cerebral palsy [18]; addition of functional data to anatomical shapes [48]. There are also applications outside the medical field to the study of variations of cell shape [61] and to construct gen... |

7 |
Regular Structure
- Grenander
- 1981
(Show Context)
Citation Context ...solution in the elementary use of a certain method of the mathematician. This method is the Method of Coordinates, on which is based the Theory of Transformations. [66, p1032] More recently Grenander =-=[25, 26, 28]-=- generalized these ideas to encompass a diverse collection of real-world situations and formulated the principles of pattern theory. The following formulation is adapted from [51]: Martins Bruveris In... |

7 |
Regional shape abnormalities in mild cognitive impairment and Alzheimer’s disease. Neuroimage 45, 656–661. doi: 10.1016/j.neuroimage.2009.01.013
- Qiu, Fennema-Notestine, et al.
- 2009
(Show Context)
Citation Context ... clinical measures of cognitive status that are used to diagnose and distinguish between AD, MCI and normal cognitive state (NCS) [73]. Alzheimer’s Disease and the Shape of Subcortical Structures. In =-=[56]-=- a population (I j)1≤ j≤383 of 383 subjects, both healthy and diseased, was registered to a common template Itempl; i.e., for each pair I j, Itempl a deformation ϕ j, satisfying Itempl ◦ ϕ−1j ≈ I j, w... |

5 |
On Banach-Lie groups acting on finite dimensional manifolds
- Omori
- 1978
(Show Context)
Citation Context ... Lie algebra of the diffeomorphism group with the norm 〈., .〉L to be a Hilbert space, so that we can use completeness to show the existence of minimizers. Unfortunately the following theorem by Omori =-=[52]-=- shows that these two requirements are incompatible. 26 Martins Bruveris and Darryl D. Holm Theorem 15 (Omori, 1978). If a connected Banach-Lie group G acts effectively, transitively and smoothly on a... |

4 |
Atlas-based automatic mouse brain image segmentation revisited: model complexity vs. image registration. Magn Reson Imaging
- Bai, Trinh, et al.
- 2012
(Show Context)
Citation Context ...on was shown to outperform other methods [7]. It is possible to use atlasbased registration with a variety of image registration methods. However, a study that involved segmenting brain scans of mice =-=[8]-=- has shown that the choice of the registration method is more important than the choice of fusion method. Thus in applications where accuracy is important, LDDMM may be preferred, despite having highe... |

4 |
Quantitative analysis of brain pathology based on MRI and brain atlases—applications for cerebral palsy. Neuroimage 54: 1854–1861
- AV, Hoon, et al.
- 2011
(Show Context)
Citation Context ... transport not only for longitudinal studies, but also to characterize the left-right asymetry of subcortical structures [57]; applications to other diseases like schizophrenia [60] or cerebral palsy =-=[18]-=-; addition of functional data to anatomical shapes [48]. There are also applications outside the medical field to the study of variations of cell shape [61] and to construct generative models for cell... |

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Methodology for patient-specific modeling of atrial fibrosis as a substrate for atrial fibrillation
- McDowell, Vadakkumpadan, et al.
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Citation Context ... rate of expansion per region of interest?” The ability to automatically answer these questions is a key step toward automating the diagnostic process. Patient-Specific Models for Atrial Fibrillation =-=[41]-=-. Atrial fibrillation is a cardiac arrhythmia, characterized by the irregular propagation of electrocardial waves across the atrium. Advances in late-gadolinium enhanced MRI, allows the in-vivo locali... |

4 | R.: Instance-based generative biological shape modeling
- Peng, Wang, et al.
- 2009
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Citation Context ...addition of functional data to anatomical shapes [48]. There are also applications outside the medical field to the study of variations of cell shape [61] and to construct generative models for cells =-=[53]-=-. 2 Geometry of Matching Problems In order to better see the geometric properties of image registration with LDDMM, we will first formulate an abstract version of it. As we study this abstract problem... |

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Neuroanatomical asymmetry patterns in individuals with schizophrenia and their non-psychotic siblings. Neuroimage
- Qiu, Wang, et al.
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Citation Context ...e mean aging process of the brain across a population [16]; the use of parallel transport not only for longitudinal studies, but also to characterize the left-right asymetry of subcortical structures =-=[57]-=-; applications to other diseases like schizophrenia [60] or cerebral palsy [18]; addition of functional data to anatomical shapes [48]. There are also applications outside the medical field to the stu... |

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J.G.: Parallel transport in diffeomorphisms distinguishes the time-dependent pattern of hippocampal surface deformation due to healthy aging and the dementia of the alzheimer’s type. NeuroImage 40(1), 68–76 (2008) Geometry of Image Registration 33
- Qiu, Younes, et al.
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Citation Context ...s, there is, as of now, no consensus about which is the most appropriate method for the transport of deformation momenta. Longitudinal Study of the Shape of Hippocampi. Parallel transport was used in =-=[59]-=- as the transport method to compare deformations of the hippocampus in subjects with early AD and healthy controls across a time span of two years. It was shown that the conversion from normal cogniti... |

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The Breadth of Symplectic and Poisson Geometry, volume 232
- Holm, Marsden
- 2005
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Citation Context ...x = ∫ R3 div(Pu)(x)I(x)dx = 〈div(Pu), I〉F (R3)∗×F (R3) . Thus the evolution equations have the form ∂t It +∇It ·ut = 0 ∂tPt +div(Ptut) = 0 Lut =−Pt∇It . (17) See also [76] for a direct derivation and =-=[30]-=- for an explanation and classification of the cotangent lift momentum maps associated with EPDiff. 22 Martins Bruveris and Darryl D. Holm Matching via Initial Momentum. The evolution equations in (17)... |

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Principal component based diffeomorphic surface mapping
- Qiu, Younes, et al.
- 2012
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Citation Context ...n of all the applications of LDDMM and its associated geometry to computational medicine in these notes. Among the omitted topics are: estimating the dimensionality of the anatomical shape variations =-=[58]-=-; generalizing geodesic regression to the anatomical shape manifold and computing the mean aging process of the brain across a population [16]; the use of parallel transport not only for longitudinal ... |

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Geometry of diffeomorphism groups and shape matching
- Bruveris
- 2012
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Citation Context ...ge I : Ω → R is defined. We consider a certain class of spaces of vector fields, called admissible vector spaces, to serve as the equivalent of a Lie algebra. The following introduction is taken from =-=[11]-=-. Definition 16. A Hilbert space H , consisting of vector fields on the domain Ω , is called admissible, if it is continuously embedded in C10(Ω ,Rd), i.e. there exists a constant C > 0 such that |u|1... |

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F.X.: Local vs global descriptors of hippocampus shape evolution for Alzheimers longitudinal population analysis
- Fiot, Risser, et al.
- 2012
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Citation Context ...r approximation called Schild’s ladder was proposed in [38] as an alternative. Other methods that depend only on the end-deformation and not on the whole geodesic path were considered and compared in =-=[20]-=-. From the viewpoint of applications, there is, as of now, no consensus about which is the most appropriate method for the transport of deformation momenta. Longitudinal Study of the Shape of Hippocam... |

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D.: A zoo of diffeomorphism groups on Rn. arXiv:1211.5704
- Michor, Mumford
- 2012
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Citation Context ...pported functions. Then the action ` : Diffc(R3)×C∞c (R3)→C∞c F(R3) is smooth in the sense of convenient calculus [37] and all the operations described below can be interpreted in that framework. See =-=[45]-=- for details on diffeomorphism groups with other decay properties. The Lie algebra of Diff(R3) is X(R3), the space of vector fields on R3. Given a time-dependent vector field t 7→ ut ∈ X(R3) its flow ... |

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J.T.: Ontological labels for automated location of anatomical shape differences
- Steinert-Threlkeld, Ardekani, et al.
- 2012
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Citation Context ... is important, LDDMM may be preferred, despite having higher computational cost than some other registration methods. Automatic Labelling via Ontologies. In the same spirit, Steinert-Threlkeld et al. =-=[64]-=- combined an ex-vivo scan of the left ventricle, manually parcellated and labelled, with the LDDMM registration method and an ontology query language to allow the medical practitioner to obtain quanti... |