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Polynomial curvelets on higher-dimensional spheres

Frederic Schoppert

Abstract

In this article, we introduce and investigate polynomial curvelets on spheres, which form a class of Parseval frames for $L^2(\mathbb{S}^{d-1})$, $d \geq 3$. The proposed construction offers a directionally sensitive multiscale decomposition and provides a sparse representation of spherical data. As a main result, we derive a sharp pointwise localization bound which shows that the frame elements decay rapidly away from their center of mass, making them a powerful tool for position-based analyses. In contrast to previous constructions, polynomial curvelets are not limited in their directional resolution. Consequently, the frames established in this article are particularly powerful when it comes to the analysis of localized anisotropic features, such as edges. To illustrate this point, we show that, given a suitable test signal that exhibits (higher-order) discontinuities at the boundary $\partial A$ of a spherical cap $A\subset \mathbb{S}^{d-1}$, the corresponding curvelet coefficients peak precisely when the analysis function matches some segment of the boundary $\partial A$, both in terms of position and orientation. Otherwise, the coefficients decay rapidly.

Polynomial curvelets on higher-dimensional spheres

Abstract

In this article, we introduce and investigate polynomial curvelets on spheres, which form a class of Parseval frames for , . The proposed construction offers a directionally sensitive multiscale decomposition and provides a sparse representation of spherical data. As a main result, we derive a sharp pointwise localization bound which shows that the frame elements decay rapidly away from their center of mass, making them a powerful tool for position-based analyses. In contrast to previous constructions, polynomial curvelets are not limited in their directional resolution. Consequently, the frames established in this article are particularly powerful when it comes to the analysis of localized anisotropic features, such as edges. To illustrate this point, we show that, given a suitable test signal that exhibits (higher-order) discontinuities at the boundary of a spherical cap , the corresponding curvelet coefficients peak precisely when the analysis function matches some segment of the boundary , both in terms of position and orientation. Otherwise, the coefficients decay rapidly.
Paper Structure (8 sections, 9 theorems, 165 equations, 2 figures)

This paper contains 8 sections, 9 theorems, 165 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of polynomial curvelets
  4. Localization
  5. Directional sensitivity
  6. Detection of singularities
  7. Some auxiliary results
  8. Main results

Key Result

Proposition 3.1

The system $\mathcal{X}[(\Psi_{ \mathrm{C}}^j)_{j=0}^\infty]$ is a Parseval frame for $L^2(\mathbb{S}^{d-1})$. In particular, converges unconditionally in $L^2(\mathbb{S}^{d-1})$, for each $f \in L^2(\mathbb{S}^{d-1})$.

Figures (2)

  • Figure 1: Re-scaled polynomial 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
  • Figure 2: Auto-correlation function $\mathrm{A}_{\mathrm{corr}}^j$ for $j=4, 5, 6, 7$

Theorems & Definitions (18)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Proposition 5.1
  • proof
  • ...and 8 more