**Figure 2.5.6.7**
Partitions of the sampling Fourier space using Voronoi diagrams in direct inversion algorithms. (*a*) Central section of a 3D Fourier volume with sampling points originating from 2D Fourier transforms of projections. Although 2D projections are sampled on a uniform (Cartesian) grid, the arbitrary rotations of projections in 3D space yields a nonuniform distribution of points in three dimensions. In effect, the 3D reconstruction by direct inversion using 3D FFT is not possible. (*b*) Voronoi diagram on a sphere in the GDFR algorithm. Using the reverse gridding method, the 2D Fourier transforms of projections are resampled onto 1D central lines using a constant angular step. In 3D Fourier space, they are located on central sections and their angular directions are evenly distributed on grand circles. However, since central sections have nonuniform distributions, the distribution of angular directions (sampling points on the unitary sphere) is also nonuniform and effectively random. |