Fundamental Problems in the Theory of Infinite-Dimensional Lie Groups

Glökner, Helge

doi:10.7546/jgsp-5-2006-24-35

Cited by 5 publications

(6 citation statements)

References 19 publications

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“…Although it is not clear whether theoretically the space G is a BCH-Lie group (Glockner 2006), BCH composition of diffeomorphisms of G has experimentally shown promising results in terms of image registration and statistics on diffeomorphisms (Bossa et al 2007;Vercauteren et al 2008).…”

Section: Background: Log-domain Diffeomorphic Demonsmentioning

confidence: 99%

iLogDemons: A Demons-Based Registration Algorithm for Tracking Incompressible Elastic Biological Tissues

et al. 2010

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Tracking soft tissues in medical images using non-linear image registration algorithms requires methods that are fast and provide spatial transformations consistent with the biological characteristics of the tissues. LogDemons algorithm is a fast non-linear registration method that computes diffeomorphic transformations parameterised by stationary velocity fields. Although computationally efficient, its use for tissue tracking has been limited because of its ad-hoc Gaussian regularisation, which hampers the implementation of more biologically motivated regularisations. In this work, we improve the logDemons by integrating elasticity and incompressibility for soft-tissue tracking. To that end, a mathematical justification of demons Gaussian regularisation is proposed. Building on this result, we replace the Gaussian smoothing by an efficient elasticlike regulariser based on isotropic differential quadratic forms of vector fields. The registration energy functional is finally minimised under the divergence-free constraint to get incompressible deformations. As the elastic regulariser and the constraint are linear, the method remains computationally tractable and easy to implement. Tests on synthetic incompressible deformations showed that our approach outperforms the original logDemons in terms of elastic incompressible deformation recovery without reducing the image matching accuracy. As an application, we applied the proposed algorithm to estimate 3D myocardium strain on clinical cine MRI of two adult patients. Results showed that incompressibility constraint improves the cardiac motion re-T. Mansi ( ) · X. Pennec · M. Sermesant · H. Delingette · N. Ayache covery when compared to the ground truth provided by 3D tagged MRI.

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Section: Background: Log-domain Diffeomorphic Demonsmentioning

confidence: 99%

iLogDemons: A Demons-Based Registration Algorithm for Tracking Incompressible Elastic Biological Tissues

et al. 2010

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“…As far as we are aware, there might be a better behaved topology on HS(λ) that would turn it into a Lie group. While no counterexample is available, it is not known in the infinitedimensional case if a Lie group necessarily comes with an exponential map (see Section 2 of [36]).…”

Section: The Higher Spin Symmetry Groupmentioning

confidence: 99%

Finite Higher Spin Transformations from Exponentiation

Monnier

2014

Commun. Math. Phys.

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We study the exponentiation of elements of the gauge Lie algebras hs(λ) of three-dimensional higher spin theories. Exponentiable elements generate one-parameter groups of finite higher spin symmetries. We show that elements of hs(λ) in a dense set are exponentiable, when pictured in certain representations of hs(λ), induced from representations of SL(2, Ê) in the complementary series. We also provide a geometric picture of higher spin gauge transformations clarifying the physical origin of these representations. This allows us to construct an infinite-dimensional topological group HS(λ) of finite higher spin symmetries. Interestingly, this construction is possible only for 0 ≤ λ ≤ 1, which are the values for which the higher spin theory is believed to be unitary and for which the Gaberdiel-Gopakumar duality holds. We exponentiate explicitly various commutative subalgebras of hs(λ). Among those, we identify families of elements of hs(λ) exponentiating to the unit of HS(λ), generalizing the logarithms of the holonomies of BTZ black hole connections. Our techniques are generalizable to the Lie algebras relevant to higher spin theories in dimensions above three.

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“…Indeed if γ(t) is the curve corresponding to the constant path X(t) = X 0 for some X 0 ∈ g, then γ(1) = exp(X 0 ). All known Lie groups modeled on sequentially complete locally convex topological vector spaces are regular [3]. In the convenient setting for calculus, it has been shown [7] that all connected regular abelian Lie groups are of the form a/Γ for some discrete subgroup Γ ⊆ a of an abelian Lie algebra a.…”

Section: Definition 22 a Lie Groupmentioning

confidence: 99%

“…In particular, Lie's third theorem no longer holds and the question of integrability, i.e. whether a Lie algebra corresponds to a Lie group, becomes relevant [3].…”

Section: Introductionmentioning

confidence: 99%