A fast and accurate Fourier algorithm for iterative parallel-beam tomography

• Published in: IEEE transactions on image processing Vol. 5; no. 5; pp. 740 - 753
• Main Authors: Delaney, A.H., Bresler, Y.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 1996; Institute of Electrical and Electronics Engineers
• Subjects: Applied sciences; Character generation; Convolution; Exact sciences and technology; Fast Fourier transforms; Image processing; Image reconstruction; Information, signal and communications theory; Iterative algorithms; Iterative methods; Noise level; Numerical simulation; Sampling methods; Signal processing; Telecommunications and information theory; Tomography; Conjugate gradient method; Series expansion; Tomography; Image processing; Fast Fourier transformation; Image reconstruction
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/83.495957

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 O(N/sup 2/logN) floating-point operations per iteration to reconstruct an N/spl times/N image from P view angles, as compared to O(N/sup 2/P) 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.