Gridding-based three-dimensional image reconstruction from projections with arbitrary directions

We describe a fast and accurate method for reconstructing a function of three variables from a finite number of its parallel beam projections. The high computational efficiency of the method is achieved by using gridding-based Fourier techniques whenever possible. The gridding technique provides an efficient means to compute a uniformly sampled version of a function from a non-uniformly sampled version of its Fourier transform and vice versa. We apply the two-dimensional reverse gridding method to each projection to obtain its Fourier transform on a special spherical grid. Then we use the three-dimensional gridding method to reconstruct the three-dimensional function on a uniform grid from this sampled version of its Fourier transform. The proper weights required for the compensation for the non-uniform distribution of samples are obtained by constructing the Voronoi diagram on the unit sphere for a set of nodes derived from the directions of projections. The main application of the method is cryo-electron microscopy, where the number of two-dimensional projections is very large and their angular distribution is highly non-uniform. We demonstrate the excellent speed and accuracy of our reconstruction method using simulated data.