The Radon transform on SO(3): A Fourier slice theorem and numerical inversion.

The inversion of the one-dimensional Radon transform on the rotation group SO(3) is an ill-posed inverse problem which applies to x-ray tomography with polycrystalline materials. This paper presents a novel approach to the numerical inversion of the one-dimensional Radon transform on SO(3). Based on a Fourier slice theorem the discrete inverse Radon transform of a function sampled on the product space S-2 x S-2 of two two-dimensional spheres is determined as the solution of a minimization problem, which is iteratively solved using fast Fourier techniques for S-2 and SO(3). The favorable complexity and stability of the algorithm based on these techniques has been confirmed with numerical tests. (Preprint