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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 of two two-dimensional spheres is determined as the solution of a minimization problem, which is iteratively solved using fast Fourier techniques for and SO(3). The favorable complexity and stability of the algorithm based on these techniques has been confirmed with numerical tests.