FPS-SFT: A Multi-dimensional Sparse Fourier Transform Based on the Fourier Projection-slice Theorem

TL;DR

FPS-SFT is a novel multi-dimensional sparse Fourier transform method that uses the Fourier projection-slice theorem to efficiently reconstruct signals with sparse frequency components from line samples.

Contribution

The paper introduces FPS-SFT, a new multi-dimensional sparse Fourier transform algorithm leveraging the Fourier projection-slice theorem for low-complexity signal reconstruction.

Findings

Theoretical analysis confirms low sample complexity for sparse signals.

Numerical experiments demonstrate effective reconstruction with reduced computation.

Application to sparse image reconstruction shows practical utility.

Abstract

We propose a multi-dimensional (M-D) sparse Fourier transform inspired by the idea of the Fourier projection-slice theorem, called FPS-SFT. FPS-SFT extracts samples along lines (1-dimensional slices from an M-D data cube), which are parameterized by random slopes and offsets. The discrete Fourier transform (DFT) along those lines represents projections of M-D DFT of the M-D data onto those lines. The M-D sinusoids that are contained in the signal can be reconstructed from the DFT along lines with a low sample and computational complexity provided that the signal is sparse in the frequency domain and the lines are appropriately designed. The performance of FPS-SFT is demonstrated both theoretically and numerically. A sparse image reconstruction application is illustrated, which shows the capability of the FPS-SFT in solving practical problems.