Radon transforms, alignment, and 3d-reconstruction from random projections

Abstract

Since their inception three-dimensional reconstruction techniques have been based on the theory of Radon transforms. Only much later have Radon transforms been recognized as powerful tools for image processing and pattern recognition. Techniques like the “common lines ” technique for finding the orientation of projections of highly symmetrical particles, which had been developed using Fourier transforms, can easily be translated into a technique that uses Radon transforms. In contrast to Fourier transforms Radon transforms have the advantage of being real valued which simplifies many interpolation steps. The central section theorem known for Fourier transforms also applies to Radon transforms. The two- or three-dimensional Fourier transform on a polar grid can be obtained from the two- or three-dimensional Radon transform by a one-dimensional (radial) Fourier transforms and vice versa. Fourier transforms obtained by a one-dimensional transformation of the Radon transform will be referred to a Fourier/Radon transforms.