Reconstruction of Piecewise Smooth Functions from Non-uniform Fourier Data
We discuss the reconstruction of compactly supported piecewise smooth functions from non-uniform samples of their Fourier transform. This problem is relevant in applications such as magnetic resonance imaging (MRI). We summarize two standard techniques, convolutional gridding and uniform resampling, and address the issue of non-uniform sampling density and its effect on reconstruction quality. We compare these classical reconstruction approaches with alternative methods such as spectral re-projection and methods incorporating jump information.