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

翻译：众所周知的离散傅里叶变换（DFT）可以轻松推广到空间域中的任意节点。这种推广的快速算法称为非等距快速傅里叶变换（NFFT）。在磁共振成像、偏微分方程求解等多种应用中，需要解决逆问题，即从给定的非等距数据中计算傅里叶系数。本文综述了解决该问题的各类方法。与需要多次迭代步骤才能求解的迭代过程不同，我们重点关注所谓的直接求逆方法。我们回顾了密度补偿技术，并引入了一种新方案，该方案可实现三角多项式的精确重构。此外，我们还考虑了一种利用弗罗贝尼乌斯范数最小化的矩阵优化方法，以得到逆NFFT。