We describe a fast and accurate direct Fourier method for reconstructing a function <i>f</i> 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 <i>f</i> 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 <i>g</i> from a nonuniformly sampled version of <i>Fg</i>, the Fourier transform of <i>g</i>, or vice versa. We apply the two-dimensional reverse gridding method to each available projection of <i>f</i>, the function to be reconstructed, in order to obtain <i>Ff</i> on a special spherical grid. Then we use the three-dimensional gridding method to reconstruct <i>f</i> from this sampled version of <i>Ff</i>. This stage requires a proper weighting of the samples of <i>Ff</i> 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 <i>Ff</i> 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.
© 2004 Optical Society of America
(100.3010) Image processing : Image reconstruction techniques
(100.3190) Image processing : Inverse problems
(100.6890) Image processing : Three-dimensional image processing
(100.6950) Image processing : Tomographic image processing
Pawel A. Penczek, Robert Renka, and Hermann Schomberg, "Gridding-based direct Fourier inversion of the three-dimensional ray transform," J. Opt. Soc. Am. A 21, 499-509 (2004)