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A Numerical Algorit...
A Numerical Algorithm for C-2-Splines on Symmetric Spaces
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- Bogfjellmo, Geir, 1987 (author)
- Chalmers tekniska högskola,Chalmers University of Technology,Chalmers & Univ Gothenburg, Math Sci, SE-41296 Gothenburg, Sweden.;CSIC, Inst Ciencias Matemat, Madrid, Spain.
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- Modin, Klas, 1979 (author)
- Chalmers tekniska högskola,Chalmers University of Technology,Chalmers & Univ Gothenburg, Math Sci, SE-41296 Gothenburg, Sweden.
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- Verdier, Olivier (author)
- KTH,Matematik (Inst.),Western Norway Univ Appl Sci, Dept Comp Math & Phys, N-5020 Bergen, Norway.,Kungliga Tekniska Högskolan (KTH),Royal Institute of Technology (KTH),Høgskulen på Vestlandet (HVL),Western Norway University of Applied Sciences
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Chalmers tekniska högskola Chalmers & Univ Gothenburg, Math Sci, SE-41296 Gothenburg, Sweden;CSIC, Inst Ciencias Matemat, Madrid, Spain. (creator_code:org_t)
- Siam Publications, 2018
- 2018
- English.
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In: SIAM Journal on Numerical Analysis. - : Siam Publications. - 1095-7170 .- 0036-1429. ; 56:4, s. 2623-2647
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Abstract
Subject headings
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- Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example, computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, requires the solution of a coupled set of nonlinear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De Casteljau's algorithm, which leads to generalized .Bezier curves. To construct C-2-splines from such curves is a complicated nonlinear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C-2-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel and thus suitable for multicore implementation. We demonstrate the algorithm for three geometries of interest: the n-sphere, complex projective space, and the real Grassmannian.
Subject headings
- NATURVETENSKAP -- Matematik -- Beräkningsmatematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Computational Mathematics (hsv//eng)
- NATURVETENSKAP -- Matematik -- Geometri (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Geometry (hsv//eng)
- NATURVETENSKAP -- Matematik -- Matematisk analys (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Mathematical Analysis (hsv//eng)
- NATURVETENSKAP -- Matematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics (hsv//eng)
Keyword
- De Casteljau
- cubic spline
- Bezier curve
- Riemannian symmetric space
Publication and Content Type
- art (subject category)
- ref (subject category)
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