Three-dimensional reconstruction from cone-beam data in O(N3 log N) time.

We have used direct Fourier techniques to modify and implement the 3D reconstruction method from cone-beam projections proposed by Grangeat. In this way we manage to decrease the computational complexity from O(N4) to O(N3 log N). Just as Grangeat's original method is exact in the mathematical sense, so is our method, provided a complete set of projection data is acquired. Also in accordance with Grangeat, our algorithm consists of two distinct phases: phase 1, from cone-beam data to derivatives of Radon data; phase 2, from derivatives of Radon data to reconstructed 3D object. In phase 1 we use the direct Fourier method in reverse to obtain line integrals in the detector plane. In phase 2 the 2D linogram method is employed for reconstruction of vertical and horizontal planes in the Radon space.