The fast Fourier Transform and fast Wavelet Transform for Patterns on the Torus

TL;DR

This paper presents a fast Fourier transform algorithm for regular multidimensional lattices, enabling efficient, parallelizable multivariate wavelet decompositions with characterized directional properties.

Contribution

It introduces a novel fast Fourier transform on regular lattices and extends it to efficient multivariate wavelet decompositions with directional analysis.

Findings

Faster Fourier transform algorithms for multidimensional lattices.

Efficient parallelization of Fourier and wavelet transforms.

Characterization of preferred directions in wavelet decomposition.

Abstract

We introduce a fast Fourier transform on regular d-dimensional lattices. We investigate properties of congruence class representants, i.e. their ordering, to classify directions and derive a Cooley-Tukey-Algorithm. Despite the fast Fourier techniques itself, there is also the advantage of this transform to be parallelized efficiently, yielding faster versions than the one-dimensional Fourier transform. These properties of the lattice can further be used to perform a fast multivariate wavelet decomposition, where the wavelets are given as trigonometric polynomials. Furthermore the preferred directions of the decomposition itself can be characterised.