Sampling Sets and Quadrature Formulae on the Rotation Group

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 ² × 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 ² and ¹, we obtain uniformly sampling nodes on the rotation group SO(3). Furthermore, quadrature formulae on ² and 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 possess nonnegative quadrature weights. We propose an algorithm for computing the quadrature weights for scattered data on SO(3) based on fast algorithms. We confirm our theoretical results with examples and numerical tests.

Keywords:

• Quadrature rule
• Rotation group SO(3)
• Sampling sets
• Scattered data
• Spherical harmonics

AMS Subject Classification:

• 65T
• 33C55
• 42C10
• 42C15
• 65D32

ACKNOWLEDGMENTS

The authors gratefully acknowledge support by German Research Foundation within the project PO 711/9-1 and FI 883/3-1. Moreover, we would like to thank the referees for their valuable suggestions.