Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform

TL;DR

This paper reviews direct inversion methods for the nonequispaced fast Fourier transform (NFFT), introducing a new scheme for exact reconstruction and exploring matrix optimization techniques to improve inverse computations.

Contribution

It presents a novel direct inversion scheme for NFFT that guarantees exact reconstruction of trigonometric polynomials and analyzes matrix optimization approaches for inverse NFFT.

Findings

New exact reconstruction scheme for trigonometric polynomials

Comparison of density compensation and matrix optimization methods

Enhanced understanding of inverse NFFT techniques

Abstract

The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI, solution of PDEs, etc., are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this paper we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT.