A fast but ill-conditioned formal inverse to Radon transforms in 2D and 3D

Abstract

We present a formal inversion of the multiscale discrete Radon trasform, valid both for 2D and 3D. With the transformed data from just one of the four quadrants of the direct 2D Radon transform, or one of the twelve dodecants, in case of 3D Radon transform, we can invert exactly and directly, with no iterations, the whole domain. The computational complexity of the proposed algorithms will be O(N log N). With N the total size of the problem, either square or cubic. But this inverse transforms are extremely ill conditioned, so the presence of noise in the transformed domain turns them useless. Still we present both algorithms, and characterize its weakness against noise.