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Aliasing Effects for Samples of Spin Random Fields on the Sphere

Claudio Durastanti

TL;DR

This paper addresses aliasing that arises when reconstructing spin spherical random fields on $\mathbb{S}^2$ from discrete samples. It introduces a spin aliasing function $\tau_s(\ell,m;u,v)$ to identify where aliases occur in the harmonic coefficients and analyzes how popular spherical sampling schemes—Gauss-Jacobi (with trapezoidal) and equiangular—affect the angular power spectrum. The authors prove that band-limited spin fields are alias-free if the sampling rule uses sufficiently many nodes (e.g., $N$ and $Q$ exceeding the bandwidth), and they provide explicit expressions and a practical example to illustrate alias behavior under these schemes. Rigorous proofs accompany the main results, highlighting the role of parity, Jacobi polynomials, and Wigner $d$-matrices in the alias structure.

Abstract

This paper investigates aliasing effects emerging from the reconstruction from discrete samples of spin spherical random fields defined on the two-dimensional sphere. We determine the location in the frequency domain and the intensity of the aliases of the harmonic coefficients in the Fourier decomposition of the spin random field and evaluate the consequences of aliasing errors in the angular power spectrum when the samples of the random field are obtained by using some very popular sampling procedures on the sphere, the equiangular and the Gauss-Jacobi sampling schemes. Finally, we demonstrate that band-limited spin random fields are free from aliases, provided that a sufficiently large number of nodes is used in the selected quadrature rule.

Aliasing Effects for Samples of Spin Random Fields on the Sphere

TL;DR

This paper addresses aliasing that arises when reconstructing spin spherical random fields on from discrete samples. It introduces a spin aliasing function to identify where aliases occur in the harmonic coefficients and analyzes how popular spherical sampling schemes—Gauss-Jacobi (with trapezoidal) and equiangular—affect the angular power spectrum. The authors prove that band-limited spin fields are alias-free if the sampling rule uses sufficiently many nodes (e.g., and exceeding the bandwidth), and they provide explicit expressions and a practical example to illustrate alias behavior under these schemes. Rigorous proofs accompany the main results, highlighting the role of parity, Jacobi polynomials, and Wigner -matrices in the alias structure.

Abstract

This paper investigates aliasing effects emerging from the reconstruction from discrete samples of spin spherical random fields defined on the two-dimensional sphere. We determine the location in the frequency domain and the intensity of the aliases of the harmonic coefficients in the Fourier decomposition of the spin random field and evaluate the consequences of aliasing errors in the angular power spectrum when the samples of the random field are obtained by using some very popular sampling procedures on the sphere, the equiangular and the Gauss-Jacobi sampling schemes. Finally, we demonstrate that band-limited spin random fields are free from aliases, provided that a sufficiently large number of nodes is used in the selected quadrature rule.
Paper Structure (18 sections, 5 theorems, 141 equations, 3 figures, 2 tables)

This paper contains 18 sections, 5 theorems, 141 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivations
  3. Aim of the paper
  4. Plan of the paper
  5. Background
  6. Spin elements on the sphere
  7. Spin spherical harmonics and the Jacobi polynomials
  8. Spin spherical random fields
  9. Spherical sampling methods for spin objects
  10. Separability of the sampling scheme
  11. Spin spherical uniform sampling
  12. The equiangular sampling approach
  13. The parity of quadrature points
  14. Measuring aliasing effects for spin random fields
  15. Introducing the spin aliasing function
  16. ...and 3 more sections

Key Result

Theorem 1

Let us assume that Condition cond:sep holds. Then it follows that where and

Figures (3)

  • Figure 1: Sampled $\vartheta$ for both the sampling schemes (GJ: Gauss-Jacobi, EA: equiangular)
  • Figure 2: Sampling points for Gauss-Jacobi (Figures \ref{['fig:subfig1']} and \ref{['fig:subfig2']}) and equiangular scheme (Figures \ref{['fig:subfig3']} and \ref{['fig:subfig4']}) for $Q=1$ and $Q=2$.
  • Figure 3: Aliases locations ($N=6$, $s=2$, $Q=1,2$)

Theorems & Definitions (15)

  • Remark 1: Jacobi polynomials and other classes of orthonormal polynomials
  • Remark 2: Other types of Gauss quadrature rules
  • Remark 3: Alternative equiangular sampling schemes
  • Remark 4: Locations, intensity and distance of the aliases
  • Theorem 1
  • Theorem 2
  • Remark 5: Comparison with the scalar case
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • ...and 5 more