Direct Fourier Reconstruction in Fan-Beam Tomography

We consider the problem of reconstructing tomographic imagery from fan-beam projections using the direct Fourier method (DFM). Previous DFM reconstructions from parallel-beam projections produced images of quality comparable to that of filtered convolution back-projection. Moreover, the number of operations using DFM in the parallel-beam case is proportional to N2 log N versus N3 for back projection [3]. The fan-beam case is more complicated because additional interpolation of the nonuniformly spaced rebinned data is required. We derive bounds on the detector spacing in fan-beam CT that enable direct Fourier reconstruction and describe the full algorithm necessary for processing the fan-beam data. The feasibility of the method is demonstrated with an example. A key result of this paper is that high-quality imagery can be reconstructed from fan-beam data using the DFM in 0 (N2 log N) operations.