24 August 2017 Edge-augmented Fourier partial sums with applications to Magnetic Resonance Imaging (MRI)
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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.
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Jade Larriva-Latt, Angela Morrison, Alison Radgowski, Joseph Tobin, Mark Iwen, Aditya Viswanathan, "Edge-augmented Fourier partial sums with applications to Magnetic Resonance Imaging (MRI)", Proc. SPIE 10394, Wavelets and Sparsity XVII, 1039414 (24 August 2017); doi: 10.1117/12.2271860; https://doi.org/10.1117/12.2271860
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