  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.
