Gauss-Legendre Sampling on the Rotation Group

TL;DR

This paper introduces a Gauss-Legendre quadrature sampling method on the rotation group that enables exact Fourier transform computation of band-limited signals with high efficiency and stability, matching recent state-of-the-art schemes.

Contribution

It presents a novel Gauss-Legendre sampling scheme on the rotation group with efficient, stable algorithms for exact Fourier transforms of band-limited signals.

Findings

Sampling efficiency is asymptotically optimal.

Algorithms are stable, accurate, and require no pre-computation.

Numerical experiments confirm high accuracy with minimal errors.

Abstract

We propose a Gauss-Legendre quadrature based sampling on the rotation group for the representation of a band-limited signal such that the Fourier transform (FT) of a signal can be exactly computed from its samples. Our figure of merit is the sampling efficiency, which is defined as a ratio of the degrees of freedom required to represent a band-limited signal in harmonic domain to the number of samples required to accurately compute the FT. The proposed sampling scheme is asymptotically as efficient as the most efficient scheme developed very recently. For the computation of FT and inverse FT, we also develop fast algorithms of complexity similar to the complexity attained by the fast algorithms for the existing sampling schemes. The developed algorithms are stable, accurate and do not have any pre-computation requirements. We also analyse the computation time and numerical accuracy of the proposed algorithms and show, through numerical experiments, that the proposed Fourier transforms are accurate with errors on the order of numerical precision.