Fast Finite Shearlet Transform

TL;DR

This paper introduces a fast, FFT-based finite shearlet transform that efficiently captures directional features in data, with a focus on practical implementation and inverse transform computation.

Contribution

It presents a novel fast finite shearlet transform algorithm based solely on FFT, including a Parseval frame and straightforward inverse transform.

Findings

Constructed a Parseval frame using band-limited shearlets

Developed a simple inverse shearlet transform

Provided proofs and implementation details

Abstract

In recent years it has turned out that shearlets have the potential to retrieve directional information so that they became interesting for many applications. Moreover the continuous shearlet transform has the outstanding property to stem from a square integrable group representation. However, to use shearlets and the shearlet transform for reasonable applications one needs fast algorithms to compute a discrete shearlet transform. In this tutorial we present the steps towards an implementation of a fast and finite shearlet transform that is only based on the FFT. Using band-limited shearlets we construct a Parseval frame that provides a simple and straightforward inverse shearlet transform. We provide all proofs and discuss several aspects of our implementation.