Fast computation of 3D radon transform via a direct Fourier method

Abstract

Motivation: Arrays of three-dimensional (3D) data are ubiquitous in structural biology, biomedicine and clinical imaging. The Radon transform can be implied in their manipulation mainly for the solution of the inverse tomographic problem, since experimental data are often collected as projections or as samples of the Radon space. In electron tomography, new applications of the transform may become convenient if a fast and accurate transformation algorithm is adopted. Results: A direct Fourier method (DFM) is proposed to compute the 3D Radon transform from a sampled function with compact support. This paper describes an already known two-step algorithm and illustrates its DFM implementation by coordinate transformations in 2D Fourier space. The algorithm is easily inverted to obtain a density distribution from the Radon transform. The main applications are in the field of electron tomography, especially in processes of angular refinement, since whatever projection of a structure can be retrieved from its Radon transform in a fast and accurate way. The times required to compute a number of projections with use of the Radon transform are compared with those required by other algorithms. Further uses of the Radon transform can be foreseen in applications based on 'projection onto convex sets' (POCS). Availability: Software is available free of charge upon request to the authors. Contact: salvator@@@csmtbo.mi.cnr.it.