Harmonic Singular Integrals and Steerable Wavelets in $L_2(\mathbb{R}^d)$

TL;DR

This paper introduces a unified method for constructing steerable wavelet frames in $L_2( ^d)$ using spherical harmonics, enabling better orientation analysis with theoretical and computational advantages.

Contribution

The authors develop a general, unified approach to steerable wavelet construction based on spherical harmonics and harmonic analysis, extending previous methods.

Findings

The proposed method maintains frame bounds through isometric angular decomposition.

Fourier multipliers with spherical harmonics correspond to singular integrals, enabling efficient computation.

The approach unifies and generalizes existing steerable wavelet constructions.

Abstract

Here we present a method of constructing steerable wavelet frames in $L_2(\mathbb{R}^d)$ that generalizes and unifies previous approaches, including Simoncelli's pyramid and Riesz wavelets. The motivation for steerable wavelets is the need to more accurately account for the orientation of data. Such wavelets can be constructed by decomposing an isotropic mother wavelet into a finite collection of oriented mother wavelets. The key to this construction is that the angular decomposition is an isometry, whereby the new collection of wavelets maintains the frame bounds of the original one. The general method that we propose here is based on partitions of unity involving spherical harmonics. A fundamental aspect of this construction is that Fourier multipliers composed of spherical harmonics correspond to singular integrals in the spatial domain. Such transforms have been studied extensively in the field of harmonic analysis, and we take advantage of this wealth of knowledge to make the proposed construction practically feasible and computationally efficient.