A novel sampling theorem on the rotation group

TL;DR

This paper introduces a new sampling theorem for functions on the rotation group SO(3), reducing sample requirements and providing fast algorithms for Fourier transforms, enhancing efficiency in directional wavelet computations.

Contribution

It presents a novel sampling theorem connecting SO(3) to the three-torus, reducing samples needed and developing fast Fourier transform algorithms for SO(3).

Findings

Requires 4L^3 samples for band-limited signals, halving previous requirements.

Develops algorithms with O(L^4) complexity, reduced to O(N L^3) for low directional band-limits.

Provides publicly available code for the algorithms.

Abstract

We develop a novel sampling theorem for functions defined on the three-dimensional rotation group SO(3) by connecting the rotation group to the three-torus through a periodic extension. Our sampling theorem requires $4L^3$ samples to capture all of the information content of a signal band-limited at $L$, reducing the number of required samples by a factor of two compared to other equiangular sampling theorems. We present fast algorithms to compute the associated Fourier transform on the rotation group, the so-called Wigner transform, which scale as $O(L^4)$, compared to the naive scaling of $O(L^6)$. For the common case of a low directional band-limit $N$, complexity is reduced to $O(N L^3)$. Our fast algorithms will be of direct use in speeding up the computation of directional wavelet transforms on the sphere. We make our SO3 code implementing these algorithms publicly available.