Abstract
The main focus of this paper is the efficient approximation of the non-uniform Fourier transform (NUFFT). We reformulate the standard NUFFT approximation as a projection of the exact discrete Fourier transform onto a shift-invariant space. This reformulation enables the use of sophisticated tools, developed in the context of shift-invariant representations, to analyze the performance of the approximation. Using these techniques, we derive the optimal scale factors for a specified interpolator. Assuming these scale factors, we develop a worst-case error criterion that is only dependent on the interpolating function. We propose an iterative re-weighted optimization algorithm to derive the optimized least square (OLS) interpolator. This interpolator significantly reduces the approximation error in comparison to the standard methods. The improved performance of this scheme, for low oversampling factors, could lead to a memory efficient algorithm for non-Cartesian Fourier inversion.