Gridding-based direct Fourier inversion of the three-dimensional ray transform

Abstract

We describe a fast and accurate direct Fourier method for reconstructing a function f of three variables from a number of its parallel beam projections. The main application of our method is in single particle analysis, where the goal is to reconstruct the mass density of a biological macromolecule. Typically, the number of projections is extremely large, and each projection is extremely noisy. The projection directions are random and initially unknown. However, it is possible to determine both the directions and f by an iterative procedure; during each stage of the iteration, one has to solve a reconstruction problem of the type considered here. Our reconstruction algorithm is distinguished from other direct Fourier methods by the use of gridding techniques that provide an efficient means to compute a uniformly sampled version of a function g from a nonuniformly sampled version of Fg, the Fourier transform of g, or vice versa. We apply the two-dimensional reverse gridding method to each available projection of f, the function to be reconstructed, in order to obtain Ff on a special spherical grid. Then we use the three-dimensional gridding method to reconstruct f from this sampled version of Ff. This stage requires a proper weighting of the samples of Ff to compensate for their nonuniform distribution. We use a fast method for computing appropriate weights that exploits the special properties of the spherical sampling grid for Ff and involves the computation of a Voronoi diagram on the unit sphere. We demonstrate the excellent speed and accuracy of our method by using simulated data.

Cryo-EM image alignment based on nonuniform fast Fourier transform

“…Matey and Fessler used NUFFTs in an iterative Fourier-based 2-D tomographic image reconstruction algorithm [26]. Penczek and co-workers [8] used the gridding method to devise a fast reconstruction algorithm to be used in single particle analysis. Gridding method has also been applied to the resampling of discrete data [11].…”

“…The application of gridding method to 3-D reconstruction from projections is described in [8], where data g is a 2-D projection available on a Cartesian grid, but arbitrarily oriented in 3-D space. Thus, there is a preparatory step required, in which g is padded with zeroes to twice the size and its FFT computed to yield g^.…”

“…The convolution in step 1 recovers from g^ samples of the target 3-D function on a Cartesian (uniform) 3-D grid. The implementation of gridding method described in [8] is significantly more complex than it would follow from the simple schematic given above. In application to 3-D reconstruction from projections nonuniform distribution of the input samples has to be accounted for by additional geometrical weighting.…”

“…The method was introduced [8] and it corresponds to the reverse gridding method described in Section II of this work (also see Fig.1), although now the real and Fourier space of the Section II schematic are reversed. Given uniformly distributed samples of g, the steps are:…”

“…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].…”