The gridding method for image reconstruction by Fourier transformation

The authors explore a computational method for reconstructing an n-dimensional signal f from a sampled version of its Fourier transform fˆ. The method involves a window function wˆ and proceeds in three steps. First, the convolution gˆ=wˆ*fˆ is computed numerically on a Cartesian grid, using the available samples of fˆ. Then, g=wf is computed via the inverse discrete Fourier transform, and finally f is obtained as g/w. Due to the smoothing effect of the convolution, evaluating wˆ*fˆ is much less error prone than merely interpolating fˆ. The method was originally devised for image reconstruction in radio astronomy, but is actually applicable to a broad range of reconstructive imaging methods, including magnetic resonance imaging and computed tomography. In particular, it provides a fast and accurate alternative to the filtered backprojection. The basic method has several variants with other applications, such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the Radon (Hough) transform