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The well-known discrete Fourier transform (DFT) can easily be generalized toarbitrary nodes in the spatial domain. The fast procedure for thisgeneralization is referred to as nonequispaced fast Fourier transform (NFFT).Various applications such as MRI, solution of PDEs, etc., are interested in theinverse problem, i.e., computing Fourier coefficients from given nonequispaceddata. In this paper we survey different kinds of approaches to tackle thisproblem. In contrast to iterative procedures, where multiple iteration stepsare needed for computing a solution, we focus especially on so-called directinversion methods. We review density compensation techniques and introduce anew scheme that leads to an exact reconstruction for trigonometric polynomials.In addition, we consider a matrix optimization approach using Frobenius normminimization to obtain an inverse NFFT.

Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform