Continuous Shearlet Frames and Resolution of the Wavefront Set

TL;DR

This paper extends the theoretical understanding of shearlet transforms by showing they can resolve the wavefront set under weaker conditions and introduces a new Radon transform-based approach for frame construction.

Contribution

It demonstrates that shearlet frames can be constructed from functions with weaker assumptions and provides a novel Radon transform-based method for analyzing shearlet properties.

Findings

Shearlet coefficients can resolve the wavefront set with weaker assumptions.

Frames for L^2(R^2) can be built from functions with anisotropic vanishing moments.

A new Radon transform-based approach is developed for shearlet analysis.

Abstract

In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transforms are - unlike more traditional transforms like wavelets - able to efficiently handle data with features along edges. The main result in [G. Kutyniok, D. Labate. Resolution of the Wavefront Set using continuous Shearlets, Trans. AMS 361 (2009), 2719-2754] confirming this property for shearlets is due to Kutyniok and Labate where it is shown that for very special functions $\psi$ with frequency support in a compact conical wegde the decay rate of the shearlet coefficients of a tempered distribution $f$ with respect to the shearlet $\psi$ can resolve the Wavefront Set of $f$. We demonstrate that the same result can be verified under much weaker assumptions on $\psi$, namely to possess sufficiently many anisotropic vanishing moments. We also show how to build frames for $L^2(\mathbb{R}^2)$ from any such function. To prove our statements we develop a new approach based on an adaption of the Radon transform to the shearlet structure.