A frame theoretic approach to the Non-Uniform Fast Fourier Transform

TL;DR

This paper analyzes the convergence of the non-uniform fast Fourier transform (NFFT) using frame theory, providing conditions for convergence and applications to feature detection in non-uniform Fourier data.

Contribution

It introduces a frame theoretic analysis of NFFT, enabling parameter design for convergence and extending its application to feature detection.

Findings

Established convergence conditions for NFFT via frame theory

Designed a frame-based convolutional gridding algorithm

Applied the method to detect features like edges in non-uniform Fourier data

Abstract

Nonuniform Fourier data are routinely collected in applications such as magnetic resonance imaging, synthetic aperture radar, and synthetic imaging in radio astronomy. To acquire a fast reconstruction that does not require an online inverse process, the non-uniform fast Fourier transform (NFFT), also called convolutional gridding, is frequently employed. While various investigations have led to improvements in accuracy, efficiency, and robustness of the NFFT, not much attention has been paid to the fundamental analysis of the scheme, and in particular its convergence properties. This paper analyzes the convergence of the NFFT by casting it as a Fourier frame approximation. In so doing, we are able to design parameters for the method that satisfy conditions for numerical convergence. Our so called frame theoretic convolutional gridding algorithm can also be applied to detect features (such as edges) from non-uniform Fourier samples of piecewise smooth functions.