期刊
DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS
卷 39, 期 -, 页码 166-183出版社
ELSEVIER
DOI: 10.1016/j.difgeo.2014.12.008
关键词
Curve matching; Shape space; Sobolev type metric; Reparametrization group; Riemannian shape analysis; Infinite dimensional geometry
资金
- FWF Project [P24625]
- Austrian Science Fund (FWF) [P 24625] Funding Source: researchfish
- Austrian Science Fund (FWF) [P24625] Funding Source: Austrian Science Fund (FWF)
We study metrics on shape space of immersions that have a particularly simple horizontal bundle. More specifically, we consider reparametrization invariant Sobolev metrics G on the space Imm(M, N) of immersions of a compact manifold M in a Riemannian manifold (N, (g) over bar). The tangent space T-f IMM(M N) at each immersion f has two natural splittings: one into components that are tangential/normal to the surface f (with respect to g) and another one into vertical/horizontal components (with respect to the projection onto the shape space B-i(M, N) = Imm(M, N) / Diff (M) of unparametrized immersions and with respect to the metric G). The first splitting can be easily calculated numerically, while the second splitting is important because it mirrors the geometry of shape space and geodesics thereon. Motivated by facilitating the numerical calculation of geodesics on shape space, we characterize all metrics G such that the two splittings coincide. In the special case of planar curves, we show that the regularity of curves in the metric completion can be controlled by choosing a strong enough metric within this class. We demonstrate in several examples that our approach allows us to efficiently calculate numerical solutions of the boundary value problem for geodesics on shape space. (C) 2015 Elsevier B.V. All rights reserved.
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