Polynomial curvelets on higher-dimensional spheres

TL;DR

This paper introduces polynomial curvelets on higher-dimensional spheres, providing a new multiscale, directionally sensitive frame that effectively captures localized anisotropic features like edges in spherical data.

Contribution

The paper develops polynomial curvelets forming Parseval frames on spheres, with sharp localization bounds and enhanced directional resolution for analyzing localized features.

Findings

Frame elements decay rapidly away from their center

Curvelet coefficients peak at boundary segments of discontinuities

The construction improves analysis of localized anisotropic features

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.