TY - JOUR
T1 - A fast and accurate Fourier algorithm for iterative parallel-beam tomography
AU - Delaney, Alexander H.
AU - Bresler, Yoram
PY - 1996
Y1 - 1996
N2 - 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 projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small. The conjugate gradient (CG) algorithm is used to minimize a regularized, spectrally weighted least-squares criterion, and we prove that the main step in each iteration is equivalent to a 2-D discrete convolution, which can be cheaply and exactly implemented via the fast Fourier transform (FFT). The proposed algorithm requires script O sign(N2 log N) floating-point operations per iteration to reconstruct an N x N image from P view angles, as compared to script O sign(N2P) floating-point operations per iteration for iterative convolution-backprojection algorithms or general algebraic algorithms that are based on a matrix formulation of the tomography problem. Numerical examples using simulated data demonstrate the effectiveness of the algorithm for sparse- and limited-angle tomography under realistic sampling scenarios. Although the proposed algorithm cannot explicitly account for noise with nonstationary statistics, additional simulations demonstrate that for low to moderate levels of nonstationary noise, the quality of reconstruction is almost unaffected by assuming that the noise is stationary.
AB - 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 projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small. The conjugate gradient (CG) algorithm is used to minimize a regularized, spectrally weighted least-squares criterion, and we prove that the main step in each iteration is equivalent to a 2-D discrete convolution, which can be cheaply and exactly implemented via the fast Fourier transform (FFT). The proposed algorithm requires script O sign(N2 log N) floating-point operations per iteration to reconstruct an N x N image from P view angles, as compared to script O sign(N2P) floating-point operations per iteration for iterative convolution-backprojection algorithms or general algebraic algorithms that are based on a matrix formulation of the tomography problem. Numerical examples using simulated data demonstrate the effectiveness of the algorithm for sparse- and limited-angle tomography under realistic sampling scenarios. Although the proposed algorithm cannot explicitly account for noise with nonstationary statistics, additional simulations demonstrate that for low to moderate levels of nonstationary noise, the quality of reconstruction is almost unaffected by assuming that the noise is stationary.
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U2 - 10.1109/83.495957
DO - 10.1109/83.495957
M3 - Article
C2 - 18285163
AN - SCOPUS:0030151068
SN - 1057-7149
VL - 5
SP - 740
EP - 753
JO - IEEE Transactions on Image Processing
JF - IEEE Transactions on Image Processing
IS - 5
ER -