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An interpolation method useful for reconstructing an image from its Fourier plane samples on a linear spiral scan trajectory is presented. This kind of sampling arises in NMR imaging. We first present a theorem that enables exact interpolation from spiral samples to a Cartesian lattice. We then investigate two practical implementations of the theorem in which a finite number of interpolating points are used to calculate the value at a new point. Our experimental results confirm the theorem's validity and also demonstrate that both practical implementations yield very good reconstructions. Thus, the theorem and/or its practical implementations suggest the possibility of using direct Fourier reconstruction from linear spiral-scan NMR imaging.