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Directional polynomial frames on spheres

Marzieh Hasannasab, Larissa Kaldewey, Frederic Schoppert

TL;DR

The paper develops a unified framework for constructing polynomial frames on the sphere $\\mathbb{S}^{d-1}$ by rotating a sequence of polynomials and discretizing rotations with quadrature rules, bridging zonal and directional systems. Frame existence and duality are characterized solely by Fourier coefficients of the generators $\\Psi^j$, and the canonical dual frame shares the same structural form. Under mild spectral and steerability conditions, the frame functions achieve optimal spatial localization in the sense of the spherical uncertainty principle, with localization scaling as $\\mathrm{Var}_{S}(\\Psi^j) \sim N_j^{-2}$. The authors illustrate the framework with well-localized, highly directional examples, including directional wavelets and higher-dimensional curvelets, highlighting potential for efficient computation and robust position-frequency analysis on spheres.

Abstract

We introduce a general framework for the construction of polynomial frames in $L^2(\mathbb{S}^{d-1})$, $d \geq 3$, where the frame functions are obtained as rotated versions of an initial sequence of polynomials $Ψ^j$, $j\in \mathbb{N}_0$. The rotations involved are discretized using suitable quadrature rules. This framework includes classical constructions such as spherical needlets and directional wavelet systems, and at the same time permits the systematic design of new frames with adjustable spatial localization, directional sensitivity, and computational complexity. We show that a number of frame properties can be characterized in terms of simple, easily verifiable conditions on the Fourier coefficients of the functions $Ψ^j$. Extending an earlier result for zonal systems, we establish sufficient conditions under which the frame functions are optimally localized in space with respect to a spherical uncertainty principle, thus making the corresponding systems a viable tool for position-frequency analyses. To conclude this article, we explicitly discuss examples of well-localized and highly directional polynomial frames.

Directional polynomial frames on spheres

TL;DR

The paper develops a unified framework for constructing polynomial frames on the sphere by rotating a sequence of polynomials and discretizing rotations with quadrature rules, bridging zonal and directional systems. Frame existence and duality are characterized solely by Fourier coefficients of the generators , and the canonical dual frame shares the same structural form. Under mild spectral and steerability conditions, the frame functions achieve optimal spatial localization in the sense of the spherical uncertainty principle, with localization scaling as . The authors illustrate the framework with well-localized, highly directional examples, including directional wavelets and higher-dimensional curvelets, highlighting potential for efficient computation and robust position-frequency analysis on spheres.

Abstract

We introduce a general framework for the construction of polynomial frames in , , where the frame functions are obtained as rotated versions of an initial sequence of polynomials , . The rotations involved are discretized using suitable quadrature rules. This framework includes classical constructions such as spherical needlets and directional wavelet systems, and at the same time permits the systematic design of new frames with adjustable spatial localization, directional sensitivity, and computational complexity. We show that a number of frame properties can be characterized in terms of simple, easily verifiable conditions on the Fourier coefficients of the functions . Extending an earlier result for zonal systems, we establish sufficient conditions under which the frame functions are optimally localized in space with respect to a spherical uncertainty principle, thus making the corresponding systems a viable tool for position-frequency analyses. To conclude this article, we explicitly discuss examples of well-localized and highly directional polynomial frames.
Paper Structure (16 sections, 9 theorems, 163 equations, 3 figures)

This paper contains 16 sections, 9 theorems, 163 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polynomial Frames
  4. Properties
  5. Steerability
  6. Rotational Symmetries
  7. Localization
  8. Directional Sensitivity
  9. A Note on the Quadrature Formulas
  10. The General Case
  11. Steerable Frames
  12. Rotation-Invariant Frames
  13. Steerable and Rotation-Invariant Frames
  14. Examples
  15. Directional Wavelets
  16. ...and 1 more sections

Key Result

Lemma 2.1

With respect to the orthonormal basis of spherical harmonics given in appendix A, the matrix functions satisfy as well as where $h(\gamma)\in \mathbb{R}^{3\times 3}$ is a positive rotation by $\gamma$ in the $(x_1, x_2)$-plane.

Figures (3)

  • Figure 1: Re-scaled directional wavelets $\psi_1^j (t, \varphi)$, $(t, \varphi) \in [0, 1]\times[0, 2\pi)$, for $K=4,9$ from top to bottom and $j=5, 6, 7$ from left to right
  • Figure 2: Re-scaled directional wavelets $\psi_2^j (t, \varphi)$, $(t, \varphi) \in [0, 1]\times[0, 2\pi)$, for $K=4,9$ from top to bottom and $j=5, 6, 7$ from left to right
  • Figure 3: Re-scaled curvelets $\psi_{ \mathrm{C}}^j (t, \varphi)$, $(t, \varphi) \in [0, 1]\times[0, 2\pi)$, for $j=5, 6, 7$ from left to right

Theorems & Definitions (19)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 4.1
  • Proposition 4.2
  • ...and 9 more