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## Higher-order Spatial Accuracy in Diffeomorphic Image Registration (2015)

Venue: | J. Geom. Imaging Comput |

Citations: | 1 - 1 self |

### Citations

7417 | Convex optimization
- Boyd, Vandenberghe
- 2004
(Show Context)
Citation Context ... · q makes this a constrained optimization problem. We may take the dual of this constrained optimization problem by using Lagrange multipliers to get an equivalent unconstrained optimization problem =-=[BV04]-=-. In our case, the dual problem is that of extremizing the (unconstrained) curve energy E3 : C 1([0, 1]; g× T ∗Q)→ R given by E3[ξ, q, p] = ∫ 1 0 `(ξ(t)) + 〈p(t), q̇(t)− ξ(t) · q(t)〉dt+ f(q(1)). Using... |

829 | Introduction to Mechanics and Symmetry,
- Marsden, Ratiu
- 1999
(Show Context)
Citation Context ...he Euler-Lagrange equations pulled to the space V . For a generic `, the Euler-Poincaré equations take the form d dt ( δ` δv ) + ad∗v ( δ` δv ) = 0.(2) 6 Henry O. Jacobs and Stefan Sommer We suggest =-=[MR99]-=- for further information on the Euler-Poincaré equations. Equation (2) is an evolution equation, which allows us to search over the space of initial conditions (i.e. V ) in place of optimizing over s... |

518 |
An integrable shallow water equation with peaked solitons, Phys
- Camassa, Holm
- 1993
(Show Context)
Citation Context ...the cost functional of the ndimensional Camassa-Holm equation (see [HM05] and references therein). In 1-dimension, the particle solutions in [JM00] are identical to the peakon solutions discovered in =-=[CH93]-=-, and the numerical scheme reduces to that of [HR06]. The convergence of [JM00] was proven using geometric techniques in [HR06] in the one-dimensional case. The same proof was used in [CDTM12] for arb... |

356 |
Deformable templates using large deformation kinematics
- Christensen, Rabbitt, et al.
- 1996
(Show Context)
Citation Context ...y of the diffeomorphisms was to consider the linear space of vector fields. Given a time-dependent vector field v(t), one can integrate it to obtain a diffeomorphism ϕt, which is called the flow of v =-=[CRM96]-=-. This insight was used to obtain diffeomorphisms for imaging applications by posing an optimal control problem on the space of vector-fields, and then integrating the flow of the optimal vector field... |

324 | The local structure of Poisson manifolds - Weinstein - 1983 |

255 |
Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Internat
- Beg, Miller, et al.
- 2005
(Show Context)
Citation Context ...e the image data was allowed to be of a fairly general type (i.e. any entity upon which diffeomorphisms act smoothly). The analytic safe-guards provided by [DGM98] and [TY05] where then excercised in =-=[BMTY05]-=-, where a number of examples were numerically investigated. 2.2. Connections with geometric mechanics Following these early investigations, connections with geometric mechanics began to form. The cost... |

173 |
Landmark matching via large deformation diffeomorphisms
- Joshi, MI
(Show Context)
Citation Context ...a fundamental choice in ensuring well-posedness and controlling properties of the resulting diffeomorphisms. A particle method based upon [DGM98] was implemented for the purpose of medical imaging in =-=[JM00]-=-. The completeness of the Euler-Lagrange equations in [DGM98] was studied thoroughly in [TY05], where the image data was allowed to be of a fairly general type (i.e. any entity upon which diffeomorphi... |

159 | Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration. - Klein, A - 2009 |

150 | Natural operations in differential geometry
- Kolář, Michor, et al.
- 1993
(Show Context)
Citation Context ...he functor, J (k)Λ , is the principal bundle projection from G to Q(k) = G/G (k) Λ . Proof. This is merely the definition of J (k)Λ , and a more thorough description of this statement can be found in =-=[KMS93]-=-. Nonetheless, we will attempt a skeletal proof here. If ϕ2 = ϕ1 ◦ ψ for some ψ ∈ G(k)Λ then J kΛ(ϕ2) = J kΛ(ϕ1 ◦ ψ). However, ψ has absolutely no impact on the kth order Taylor expansion because the ... |

147 | M.I.: Variational problems on flows of diffeomorphisms for image matching
- Dupuis, Grenander, et al.
- 1998
(Show Context)
Citation Context ..., where the cost functional (i.e. the norm) was identified as a fundamental choice in ensuring well-posedness and controlling properties of the resulting diffeomorphisms. A particle method based upon =-=[DGM98]-=- was implemented for the purpose of medical imaging in [JM00]. The completeness of the Euler-Lagrange equations in [DGM98] was studied thoroughly in [TY05], where the image data was allowed to be of a... |

123 | Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids - Marsden, Weinstein - 1983 |

64 |
A multivariate Faà di Bruno formula with applications
- Constantine, Savits
- 1996
(Show Context)
Citation Context ...ty. For arbitrary k, the relationship between the momenta (q, p) ∈ T ∗Q(k) and the velocity field u invokes the multivariate Faà di Bruno formula, and so the algebra rises in complexity very quickly =-=[CS96]-=-. In the case of k = 2, the Hamiltonian is substantially more complex but still tractable (see (12) in Appendix A). However, beyond k = 2, a symbolic algebra package is advised. The examples we will b... |

38 | Local geometry of deformable templates
- Trouvé, Younes
- 2005
(Show Context)
Citation Context ...ffeomorphisms. A particle method based upon [DGM98] was implemented for the purpose of medical imaging in [JM00]. The completeness of the Euler-Lagrange equations in [DGM98] was studied thoroughly in =-=[TY05]-=-, where the image data was allowed to be of a fairly general type (i.e. any entity upon which diffeomorphisms act smoothly). The analytic safe-guards provided by [DGM98] and [TY05] where then excercis... |

35 | Deformable Medical Image Registration: A Survey,” - Sotiras, Davatzikos, et al. - 2013 |

27 |
Momentum Maps and Measure-valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation
- Holm, Marsden
(Show Context)
Citation Context ...eometric mechanics began to form. The cost functional chosen in [JM00] was the H1-norm of the vector-fields. Coincidentally, this is the cost functional of the ndimensional Camassa-Holm equation (see =-=[HM05]-=- and references therein). In 1-dimension, the particle solutions in [JM00] are identical to the peakon solutions discovered in [CH93], and the numerical scheme reduces to that of [HR06]. The convergen... |

27 | A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Discrete Contin
- Holden, Raynaud
(Show Context)
Citation Context ... equation (see [HM05] and references therein). In 1-dimension, the particle solutions in [JM00] are identical to the peakon solutions discovered in [CH93], and the numerical scheme reduces to that of =-=[HR06]-=-. The convergence of [JM00] was proven using geometric techniques in [HR06] in the one-dimensional case. The same proof was used in [CDTM12] for arbitrary dimensions. As images appear as advected quan... |

18 | The momentum map representation of images
- Bruveris, Gay-Balmaz, et al.
(Show Context)
Citation Context ...maps became a useful conceptual technique for geometers to understand the numerical scheme of [BMTY05]. The identification of numerous mathematical terms in [BMTY05] as momentum maps was performed in =-=[BGBHR11]-=-. 4 Henry O. Jacobs and Stefan Sommer 2.3. Jet particles The particle method implemented in [JM00] allowed only for deformations that acted as “local translations” (see Figure 1(a)). Motivated by a de... |

10 |
Integration of the EPDiff equation by particle methods,
- CHERTOCK, TOIT, et al.
- 2012
(Show Context)
Citation Context ...overed in [CH93], and the numerical scheme reduces to that of [HR06]. The convergence of [JM00] was proven using geometric techniques in [HR06] in the one-dimensional case. The same proof was used in =-=[CDTM12]-=- for arbitrary dimensions. As images appear as advected quantities, the use of momentum maps became a useful conceptual technique for geometers to understand the numerical scheme of [BMTY05]. The iden... |

10 | On Euler’s equation and ’EPDiff
- Mumford, Michor
(Show Context)
Citation Context ...nvented for the purpose of incompressible fluid modelling in [DJR13]. Solutions to this fluid model were numerically computed in [CHJM14] based upon the regularized Euler fluid equations developed in =-=[MM13]-=- and expressions for matrix-valued reproducing kernels derived in [MG14]. The final section of [CHJM14] provides formulas which illustrate how jet-particles in the kth level of the hierarchy yield def... |

8 | The Morse Lemma in infinite dimensions via singularity theory - Golubitsky, Marsden - 1983 |

6 | On the coupling between an ideal fluid and immersed particles”, arXiv:1208.6561 [math-ph
- Jacobs, Ratiu, et al.
- 2012
(Show Context)
Citation Context ...g the case for jet-particles as multi-scale representations of diffeomorphisms. Independently, an incompressible version of this idea was invented for the purpose of incompressible fluid modelling in =-=[DJR13]-=-. Solutions to this fluid model were numerically computed in [CHJM14] based upon the regularized Euler fluid equations developed in [MM13] and expressions for matrix-valued reproducing kernels derived... |

6 | X.: Higher-order momentum distributions and locally affine LDDMM registration
- Sommer, Nielsen, et al.
(Show Context)
Citation Context .... Jet particles The particle method implemented in [JM00] allowed only for deformations that acted as “local translations” (see Figure 1(a)). Motivated by a desire to create more general deformations =-=[SNDP13]-=- introduced a hierarchy of particles which advect jet-data. We call the particles jet-particles in this paper. The first order jet-particles modify the Jacobian matrix at the particle locations and al... |

4 | A sufficient condition for a critical point of a functional to be a minimum and its application to Plateau’s problem - Tromba - 1983 |

2 |
A jetlet hierarchy for ideal fluid dynamics
- Cotter, Holm, et al.
(Show Context)
Citation Context ...omorphisms. Independently, an incompressible version of this idea was invented for the purpose of incompressible fluid modelling in [DJR13]. Solutions to this fluid model were numerically computed in =-=[CHJM14]-=- based upon the regularized Euler fluid equations developed in [MM13] and expressions for matrix-valued reproducing kernels derived in [MG14]. The final section of [CHJM14] provides formulas which ill... |

2 | Matrix-valued Kernels for Shape Deformation Analysis
- Micheli, Glaunès
(Show Context)
Citation Context ...lutions to this fluid model were numerically computed in [CHJM14] based upon the regularized Euler fluid equations developed in [MM13] and expressions for matrix-valued reproducing kernels derived in =-=[MG14]-=-. The final section of [CHJM14] provides formulas which illustrate how jet-particles in the kth level of the hierarchy yield deformations which are approximated by particles in the (k − 1)th level of ... |

1 |
Symmetries of higher-order momentum distributions in the LDDMM formalism, Mathematical foundations of computational anatomy, Sep 2013, a MICCAI workshop
- Jacobs
(Show Context)
Citation Context ...rticles allow for deformations which are “locally quadratic” (i.e. transformations with nontrivial Hessians (see Figure 1(f-h)). The geometric and hierarchal structure of [SNDP13] was investigated in =-=[Jac13]-=- where the Lie groupoid structure of jet-particles was linked to the Lie group structure of the diffeomorphism group, thus making the case for jet-particles as multi-scale representations of diffeomor... |

1 |
Reduction by Lie Groups
- Sommer, Jacobs
(Show Context)
Citation Context ...t is this later perspective which we shall take in this paper, since problem setup is stated in Lagrangian form. A more thorough overview of the role of reduction by symmetry in LDDMM can be found in =-=[SJ15]-=-. Let G be a Lie group and Gs ⊂ G be a Lie subgroup with Lie algebras g and gs respectively. We will denote the homogenous space of right cosets by Q = G/Gs, and we will denote the corresponding princ... |

1 | An Infinite Dimensional Group Approach for Physics Based Models - Trouv - 1995 |

1 | E-mail address: sommer@di.ku.dk - Springer |