A Local Fourier Slice Theorem

TL;DR

This paper introduces a local Fourier slice theorem that allows for efficient, sparse, and localized projections of signals using wavelets, significantly reducing computational costs in tomographic reconstructions.

Contribution

It develops a novel local Fourier slice equation that enables analytical projection of wavelets in spherical coordinates, improving efficiency and sparsity in signal processing.

Findings

Reduces computation time in tomographic reconstruction

Enables sparse and localized signal projections

Demonstrates effectiveness on synthetic data

Abstract

We present a local Fourier slice equation that enables local and sparse projection of a signal. Our result exploits that a slice in frequency space is an iso-parameter set in spherical coordinates. Therefore, the projection of suitable wavelets defined separably in these coordinates can be computed analytically, yielding a sequence of wavelets closed under projection. Our local Fourier slice equation then realizes projection as reconstruction with "sliced" wavelets with computational costs that scale linearly in the complexity of the projected signal. We numerically evaluate the performance of our local Fourier slice equation for synthetic test data and tomographic reconstruction, demonstrating that locality and sparsity can significantly reduce computation times and memory requirements.