A FAST AND ACCURATE FOURIER ALGORITHM FOR ITERATIVE PARALLEL-BEAM TOMOGRAPHY

We use a series-expansion approach and an operator framework to derive a new, fast, and accurate Fourier algorithm for iterative tomographic reconstruction. This algorithm is applicable for parallel-ray project ions collected at a finite number of arbitrary view angles and radiall y sampled at a rate high enough that aliasing errors are small. The co njugate gradient (CG) algorithm is used to minimize a regularized, spe ctrally weighted least-squares criterion, and we prove that the main s tep in each iteration is equivalent to a 2-D discrete convolution, whi ch can be cheaply and exactly implemented via the fast Fourier transfo rm (FFT). The proposed algorithm requires O(N-2 log N) floating-point operations per iteration to reconstruct an N x N image from P view ang les, as compared to O(N-2 P) floating-point operations per iteration f or iterative convolution-backprojection algorithms or general algebrai c algorithms that are based on a matrix formulation of the tomography problem. Numerical examples using simulated data demonstrate the effec tiveness of the algorithm for sparse- and limited-angle tomography und er realistic sampling scenarios. Although the proposed algorithm canno t explicitly account for noise with nonstationary statistics, addition al simulations demonstrate that for low to moderate levels of nonstati onary noise, the quality of reconstruction is almost unaffected by ass uming that the noise is stationary.