Technical Report: Improved Fourier Reconstruction using Jump Information with Applications to MRI

TL;DR

This paper introduces an edge-augmented Fourier reconstruction method that uses jump information to improve the accuracy of Fourier-based function approximation, especially for piecewise-smooth functions in applications like MRI.

Contribution

The paper presents a novel edge-augmented Fourier reconstruction technique that accurately estimates jump discontinuities from limited Fourier data and enhances approximation quality.

Findings

Improved reconstruction accuracy demonstrated through theoretical analysis.

Empirical results show superior performance over existing methods.

Method extends to multivariable functions with preliminary success.

Abstract

Certain applications such as Magnetic Resonance Imaging (MRI) require the reconstruction of functions from Fourier spectral data. When the underlying functions are piecewise-smooth, standard Fourier approximation methods suffer from the Gibbs phenomenon - with associated oscillatory artifacts in the vicinity of edges and an overall reduced order of convergence in the approximation. This paper proposes an edge-augmented Fourier reconstruction procedure which uses only the first few Fourier coefficients of an underlying piecewise-smooth function to accurately estimate jump information and then incorporate it into a Fourier partial sum approximation. We provide both theoretical and empirical results showing the improved accuracy of the proposed method, as well as comparisons demonstrating superior performance over existing state-of-the-art sparse optimization-based methods. Extensions of the proposed techniques to functions of several variables are also addressed preliminarily. All code used to generate the results in this report are made publicly available.