Decay properties of Riesz transforms and steerable wavelets

TL;DR

This paper investigates the decay properties of Riesz transforms and steerable wavelets, establishing conditions for decay and improving wavelet decay by modifying Fourier transforms to enhance steerability.

Contribution

It derives necessary conditions for decay based on smoothness, decay, and vanishing moments, and demonstrates how to improve wavelet decay for better steerable frames.

Findings

Necessary conditions for Riesz transform decay are established.

Wavelet decay can be improved by Fourier transform modifications.

Steerable frames with rapid decay are constructed.

Abstract

The Riesz transform is a natural multi-dimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of a wavelet often has slow decay. One can nevertheless overcome this problem by requiring the original wavelet to have sufficient smoothness, decay, and vanishing moments. In this paper, we derive necessary conditions in terms of these three properties that guarantee the decay of the Riesz transform and its variants, and as an application, we show how the decay of the popular Simoncelli wavelets can be improved by appropriately modifying their Fourier transforms. By applying the Riesz transform to these new wavelets, we obtain steerable frames with rapid decay.