On the Role of the Double Fourier Sphere Method in Fast Algorithms on SO(3)

TL;DR

This paper investigates the Double Fourier Sphere method's role in enabling fast algorithms on the rotation group SO(3), highlighting its interpretation as a lifting technique and comparing implementation efficiencies.

Contribution

It reinterprets the Wigner transform as a DFS-based lifting from SO(3) to T^3, analyzing regularity loss and comparing implementation methods for improved stability and speed.

Findings

The direct Wigner transform method is often faster than polynomial transform approaches.

DFS-based lifting induces Sobolev regularity loss, affecting function smoothness.

Direct methods demonstrate better stability and efficiency in practical computations.

Abstract

We analyze the Double Fourier Sphere (DFS) method on the rotation group $\mathcal{SO}(3)$ in the frequency domain and demonstrate its central role in fast algorithms. Fast Fourier algorithms on $\mathcal{SO}(3)$ are commonly formulated as a Wigner transform - mapping harmonic to Fourier coefficients - followed by a Fourier transform. We revisit this formulation and interpret the Wigner transform as an explicit realization of the DFS method, lifting functions from $\mathcal{SO}(3)$ to $\mathbb{T}^3$. In this context, we analyze the Sobolev regularity loss induced by this lifting. Furthermore, we compare different Wigner transform implementations, examine additional symmetry enhancements, and observe that the direct method is often faster and more stable than the fast polynomial transform approaches.