Decay Properties of Riesz Transforms and Steerable Wavelets

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.

Solutions to the 2D Euler equations with velocity unbounded at infinity

“…Finally, note that R i ∇χ also integrates to zero on R 2 (for a proof of this, see Theorem 3.4 of [13]). The moment condition on R i ∇χ, combined with (2.3) and (2.4), imply that R i ∇χ satisfies the conditions of Theorem 2 with N = 1 and = δ.…”

Divergence-Free Wavelet Frames

“…Then the components of the divergence-free wavelets have the same number of vanishing moments and similar decay to , i.e., for (15) for some . Proof: This result follows from the analysis of singular-integral operators that was presented in [15]. As each component is similar, we only consider .…”

A Sinusoidal Image Model Derived from the Circular Harmonic Vector

“…Scaling by 2 i allows for sub-sampling to create pyramidal decompositions, although alternative partitions that are more narrowly spaced can be used, such as in [17]. The third condition requires the primary wavelet to have at least N vanishing moments to account for the singularity of the RT at the origin, and thus for the wavelets to have sufficient spatial decay [51,56]. If reconstruction or pyramidal decompositions are not of interest, the second condition can be abandoned, and an image can be analysed using an isotropic filter bank that preferably satisfies the third condition.…”

“…Wavelets such as the Simoncelli, Papadakis and variance optimised wavelets (VOW) [37] satisfy these conditions; however, they contain discontinuities in the frequency domain that lead to a slow decay in the spatial domain [56]. In contrast, the second Meyer wavelet (Example 2, Sect.…”