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A Numerical Algorithm for C-2-Splines on Symmetric Spaces

Journal article
Authors Geir Bogfjellmo
Klas Modin
O. Verdier
Published in SIAM Journal on Numerical Analysis
Volume 56
Issue 4
Pages 2623-2647
ISSN 0036-1429
Publication year 2018
Published at Department of Mathematical Sciences
Pages 2623-2647
Language en
Links dx.doi.org/10.1137/17m1123353
Subject categories Computational Mathematics

Abstract

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 Bézier 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.

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