A factorization method for the 3D X-ray transform

We consider the inversion of the three-dimensional (3D) X-ray transform with a limited data set containing the line integrals which have two intersections with the lateral surface of a cylindrical detector. The usual solution to this problem is based on 3D filtered-backprojection, but this method is slow. This paper presents a new algorithm which factors the 3D reconstruction problem into a set of independent 2D radon transforms for a stack of parallel slices. Each slice is then reconstructed using standard 2D filtered-backprojection. The algorithm is based on the application of the stationary-phase approximation to the 2D Fourier transform of the data, and is an extension to three dimensions of the frequency-distance relation derived by Edholm et al.(1986) for the 2D radon transform. Error estimates are also obtained.