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

Ralf Hielscher, Erik Wuensche

TL;DR

This paper provides a frequency-domain analysis of the Double Fourier Sphere (DFS) method on $\mathrm{SO}(3)$, showing that DFS is essentially the Wigner transform operating in Fourier space and enabling fast harmonic evaluations via torus methods. It establishes Sobolev-regularity requirements for the DFS lift, derives explicit band-limited and non-band-limited formulations, and demonstrates how symmetry can be exploited to reduce storage and computation. It then compares two practical realizations of the Wigner transform—direct Wigner and fast polynomial transform (FPT)—and finds that the direct approach is typically faster and more stable for moderate bandwidths, with symmetry-based speedups further enhancing performance. The numerical results highlight the efficiency and stability trade-offs, emphasizing that, for many applications, DFS-based Wigner computations on $\mathbb{T}^3$ provide a robust, scalable route to NSOFT on $\mathrm{SO}(3)$.

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.

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

TL;DR

This paper provides a frequency-domain analysis of the Double Fourier Sphere (DFS) method on $\mathrm{SO}(3)$, showing that DFS is essentially the Wigner transform operating in Fourier space and enabling fast harmonic evaluations via torus methods. It establishes Sobolev-regularity requirements for the DFS lift, derives explicit band-limited and non-band-limited formulations, and demonstrates how symmetry can be exploited to reduce storage and computation. It then compares two practical realizations of the Wigner transform—direct Wigner and fast polynomial transform (FPT)—and finds that the direct approach is typically faster and more stable for moderate bandwidths, with symmetry-based speedups further enhancing performance. The numerical results highlight the efficiency and stability trade-offs, emphasizing that, for many applications, DFS-based Wigner computations on $\mathbb{T}^3$ provide a robust, scalable route to NSOFT on $\mathrm{SO}(3)$.

Abstract

We analyze the Double Fourier Sphere (DFS) method on the rotation group in the frequency domain and demonstrate its central role in fast algorithms. Fast Fourier algorithms on 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 to . 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.
Paper Structure (14 sections, 10 theorems, 80 equations, 2 figures, 1 table)

This paper contains 14 sections, 10 theorems, 80 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Double Fourier Sphere Method
  4. The DFS Method on SO(3)
  5. The DFS Method for Band-Limited Functions on SO(3)
  6. The DFS-Method for Non-Band-Limited Functions on SO(3)
  7. Fast Algorithms for Harmonic Series on SO(3)
  8. Factorization of the SO(3)-Fourier Transform
  9. Computation of the SO(3)-Fourier Coefficients
  10. Symmetry Properties on SO(3)
  11. Fast Realizations of the W_N-Operator
  12. Numerical Experiments
  13. Running Time
  14. Accuracy

Key Result

Lemma 3

For $(k,j,l)\in{\mathbb{Z}}^{3}$, the Fourier coefficients of the BMC function $g\in\mathrm{L}_{2}({\mathbb{T}}^{3})$ satisfy $\IfNoValueTF{k} {\bm{\hat{g}}} {\hat{g}_{k,j,l}} = (-1)^{k+l}\, \IfNoValueTF{k} {\bm{\hat{g}}} {\hat{g}_{k,-j,l}}$.

Figures (2)

  • Figure 1: Comparison of the CPU times, that are required to compute the direct Wigner transform, the Wigner transform via FPT and the NFFT (oversampling factor $\sigma=1.5$, cut-off parameter $m=4$, Kaiser-Bessel window function, $N^3$ nodes).
  • Figure 2: Investigation of the accuracy of different Wigner transform implementations depending on the bandwidth. The left panel shows the relative error $E_{\ell_{1}\to\ell_{2}}$ for a randomly chosen harmonic coefficient vector $\IfNoValueTF{-NoValue-} {\bm{\hat{f}}} {{\hat{f}}_{-NoValue-}^{-NoValue-,-NoValue-}}$, while the right panel displays the variance of this error, estimated from 100 independent random coefficient vectors.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Remark 5
  • Definition 6
  • Remark 7
  • Lemma 8
  • ...and 17 more