Ah. Delaney et Y. Bresler, A FAST AND ACCURATE FOURIER ALGORITHM FOR ITERATIVE PARALLEL-BEAM TOMOGRAPHY, IEEE transactions on image processing, 5(5), 1996, pp. 740-753
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.