Recovering Exponential Accuracy from Non-harmonic Fourier Data Through Spectral Reprojection

Authors:
Anne Gelb;Taylor Hines
Affiliations:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, USA 85287-1804;Department of Mathematics, Purdue University, West Lafayette, USA 47907-2067
Venue:
Journal of Scientific Computing
Year:
2012

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Abstract

Spectral reprojection techniques make possible the recovery of exponential accuracy from the partial Fourier sum of a piecewise-analytic function, essentially conquering the Gibbs phenomenon for this class of functions. This paper extends this result to non-harmonic partial sums, proving that spectral reprojection can reduce the Gibbs phenomenon in non-harmonic reconstruction as well as remove reconstruction artifacts due to erratic sampling. We are particularly interested in the case where the Fourier samples form a frame. These techniques are motivated by a desire to improve the quality of images reconstructed from non-uniform Fourier data, such as magnetic resonance (MR) images.