Notes on direct and gridding-based Fourier inversion methods

Numerous imaging methods, including MRI and CT, lead to the problem of reconstructing a square-integrable function. The problem of reconstructing a function f from a non-equidistantly sampled version of its Fourier transform ℱ f may be tackled with direct and gridding-based Fourier inversion methods. Both approaches require a weighting of the data. We present a favorable and generally applicable weighting strategy and show that gridding-based Fourier inversion provides a fast way of computing a highly accurate approximation to the result that would be produced by direct Fourier inversion.