A Numerical Algorithm for C-2-Splines on Symmetric Spaces
Journal article, 2018

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.

De Casteljau

cubic spline

Bezier curve

Riemannian symmetric space

Author

Geir Bogfjellmo

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Klas Modin

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Olivier Verdier

Royal Institute of Technology (KTH)

Western Norway University of Applied Sciences

SIAM Journal on Numerical Analysis

0036-1429 (ISSN) 1095-7170 (eISSN)

Vol. 56 4 2623-2647

Geometry and Computational Anatomy (GEOCA)

European Commission (Horizon 2020), 2015-03-01 -- 2017-06-30.

The Swedish Foundation for International Cooperation in Research and Higher Education (STINT), 2015-04-01 -- 2018-06-30.

Subject Categories

Computational Mathematics

Geometry

Mathematical Analysis

DOI

10.1137/17M1123353

More information

Latest update

10/7/2019