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SOBOLEV METRICS ON DIFFEOMORPHISM GROUPS AND THE DERIVED GEOMETRY OF SPACES OF SUBMANIFOLDS
"... Abstract. Given a finite dimensional manifold N, the group DiffS(N) of diffeomorphism of N which fall suitably rapidly to the identity, acts on the manifold B(M,N) of submanifolds on N of diffeomorphism type M where M is a compact manifold with dimM < dimN. For a right invariant weak Riemannian m ..."
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Abstract. Given a finite dimensional manifold N, the group DiffS(N) of diffeomorphism of N which fall suitably rapidly to the identity, acts on the manifold B(M,N) of submanifolds on N of diffeomorphism type M where M is a compact manifold with dimM < dimN. For a right invariant weak Riemannian metric on DiffS(N) induced by a quite general operator L: XS(N) → Γ(T ∗ N⊗vol(N)), we consider the induced weak Riemannian metric on B(M,N) and we compute its geodesics and sectional curvature. For that we derive a covariant formula for curvature in finite and infinite dimensions, we show how it makes O’Neill’s formula very transparent, and we use it finally to compute sectional curvature on B(M,N). 1.
Matrixvalued Kernels for Shape Deformation Analysis
, 2013
"... The main purpose of this paper is providing a systematic study and classification of nonscalar kernels for Reproducing Kernel Hilbert Spaces (RKHS), to be used in the analysis of deformation in shape spaces endowed with metrics induced by the action of groups of diffeomorphisms. After providing an ..."
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The main purpose of this paper is providing a systematic study and classification of nonscalar kernels for Reproducing Kernel Hilbert Spaces (RKHS), to be used in the analysis of deformation in shape spaces endowed with metrics induced by the action of groups of diffeomorphisms. After providing an introduction to matrixvalued kernels and their relevant differential properties, we explore extensively those, that we call TRI kernels, that induce a metric on the corresponding Hilbert spaces of vector fields that is both translation and rotationinvariant. These are analyzed in an effective manner in the Fourier domain, where the characterization of RKHS of curlfree and divergencefree vector fields is particularly natural. A simple technique for constructing generic matrixvalued kernels from scalar kernels is also developed. We accompany the exposition of the theory with several examples, and provide numerical results that show the dynamics induced by different choices of TRI kernels on the manifold of labeled landmark points.
UDC 514.83, 517.988.24
"... Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds Given a finitedimensional manifold N, the group DiffS(N) of diffeomorphisms of N which decrease suitably rapidly to the identity, acts on the manifold B(M, N) of submanifolds of N of diffeomorphismtype M, w ..."
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Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds Given a finitedimensional manifold N, the group DiffS(N) of diffeomorphisms of N which decrease suitably rapidly to the identity, acts on the manifold B(M, N) of submanifolds of N of diffeomorphismtype M, where M is a compact manifold with dim M < dim N. Given the rightinvariant weak Riemannian metric on DiffS(N) induced by a quite general operator L: XS(N) → Γ(T ∗ N ⊗ vol(N)), we consider the induced weak Riemannian metric on B(M, N) and compute its geodesics and sectional curvature. To do this, we derive a covariant formula for the curvature in finite and infinite dimensions, we show how it makes O’Neill’s formula very transparent, and we finally use it to compute the sectional curvature on B(M, N).
Convergent Stochastic Expectation Maximization
, 2013
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doi:10.3934/jgm.?????? pp. X–XX ON EULER’S EQUATION AND ‘EPDIFF’
"... (Communicated by JuanPablo Ortega) Abstract. We study a family of approximations to Euler’s equation depending on two parameters ε, η ≥ 0. When ε = η = 0 we have Euler’s equation and when both are positive we have instances of the class of integrodifferential equations called EPDiff in imaging sci ..."
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(Communicated by JuanPablo Ortega) Abstract. We study a family of approximations to Euler’s equation depending on two parameters ε, η ≥ 0. When ε = η = 0 we have Euler’s equation and when both are positive we have instances of the class of integrodifferential equations called EPDiff in imaging science. These are all geodesic equations on either the full diffeomorphism group DiffH ∞ (Rn) or, if ε = 0, its volume preserving subgroup. They are defined by the right invariant metric induced by the norm on vector fields given by ‖v‖ε,η = Rn 〈Lε,ηv, v 〉 dx where Lε,η = (I − η2 p △)p ◦ (I − 1 ε2 ∇ ◦ div). All geodesic equations are locally wellposed, and the Lε,ηequation admits solutions for all time if η> 0 and p ≥ (n + 3)/2. We tie together solutions of all these equations by estimates which, however, are only local in time. This approach leads to a new notion of momentum which is transported by the flow and serves as a generalization of vorticity. We also discuss how delta distribution momenta lead to“vortexsolitons”, also called “landmarks ” in imaging science, and to new numeric approximations to fluids. In Arnold’s famous 1966 paper [2], he showed that Euler’s equation in R n for incompressible, nonviscous flow was identical to the geodesic equation on the group of volume preserving diffeomorphisms for the right invariant L 2metric. This raises the question, what are the equations for geodesic flow on the full group of diffeomorphisms in various right invariant metrics? Arnold also gave the general recipe for writing down these equations but, as far as we know, geodesics of this sort were not specifically studied beyond the 1dimensional case, until Miller and Grenander and coworkers introduced them into medical imaging applications. In 1993 they laid out a program for comparing individual medical scans with standard human body templates [17]. Subsequently they introduced a large class of rightinvariant metrics on the group of (suitably smooth) diffeomorphisms using norms on vector