Gräf, Manuel ; Potts, Daniel : Sampling sets and quadrature formulas on the rotation group

Author(s):

Gräf, Manuel
Potts, Daniel

Title:

Sampling sets and quadrature formulas on the rotation group

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 8, 2009

Mathematics Subject Classification:

65T40			[ Trigonometric approximation and interpolation ]
33C55			[ Spherical harmonics ]
42C10			[ Fourier series in special orthogonal functions ]
42C15			[ Series of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansions ]
65D32			[ Quadrature and cubature formulas ]

Abstract:

In this paper we construct sampling sets over the rotation group SO(3). The proposed construction is based on a parameterization, which reflects the product nature S^2xS^1 of SO(3) very well, and leads to a spherical Pythagorean-like formula in the parameter domain. We prove that by using uniformly distributed points on S^2 and S^1 we obtain uniformly sampling nodes on the rotation group SO(3). Furthermore, quadrature formulae on S^2 and S^1 lead to quadratures on SO(3), as well. For scattered data on SO(3) we give a necessary condition on the mesh norm such that the sampling nodes possesses nonnegative quadrature weights. We confirm our theoretical results with examples and numerical tests.

Keywords:

rotation group SO(3), spherical harmonics, sampling sets, quadrature rule, scattered data

Language:

English

Publication time:

4 / 2009