The gridding method for image reconstruction by Fourier transformation

Abstract

The authors explore a computational method for reconstructing an n-dimensional signal f from a sampled version of its Fourier transform f;. The method involves a window function w; and proceeds in three steps. First, the convolution g;=w;*f; is computed numerically on a Cartesian grid, using the available samples of f;. 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 w;*f; is much less error prone than merely interpolating f;. 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.

Diffusion-weighted imaging of the spine using radialk-space trajectories

“…Regridding is based on the algorithm described in [33] involving a window function that is convolved with the k-space data during regridding. The kernel size used was 4 × 4.…”

Cryo-EM image alignment based on nonuniform fast Fourier transform

“…There are various forms of NUFFTs, depending on whether the input grid or the output grid is nonuniform. Besides, NUFFTs are known under various other names, such as nonequispaced FFT [23], unequally spaced FFT [24], or gridding method [9,10,25]. We tend to use the term gridding method here.…”

“…Nonuniform input or output grids are often encountered in the field of image reconstruction. The gridding method has been used in Computed Tomography [10] and is widely used in Magnetic Resonance Imaging, see e.g. [9,25].…”

“…As T we wish to compare, then we define the maximum pixel error as: (10) where V r is a sphere with radius r and error ε T 1 , T 2 (v) is given by (11) Unlike in the 2-D case, in which the maximum pixel error can be conveniently expressed by a closed-form formula, in the 3-D case this formula would be rather complicated and not particularly informative. Therefore, we evaluate (11) numerically.…”

“…To some extent, this was demonstrated in our recent work on a 3-D reconstruction algorithm that is capable of recovering a virtually perfect map almost to the Nyquist frequency [8]. It was made possible by taking advantage of a novel interpolation scheme based on the nonuniform fast Fourier transform (NUFFT) [9,10]. This interpolation method, also known as gridding method, is known for its accuracy [11].…”