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 projections collected at a finite number of arbitrary view angles and radially sampled at a rate...
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| Published in | IEEE transactions on image processing Vol. 5; no. 5; pp. 740 - 753 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York, NY
IEEE
1996
Institute of Electrical and Electronics Engineers |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1057-7149 1941-0042 |
| DOI | 10.1109/83.495957 |
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| Abstract | 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. |
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| AbstractList | 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 super(2) log N) floating-point operations per iteration to reconstruct an N x N image from P view angles, as compared to O(N super(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. 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. 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(2)logN) floating-point operations per iteration to reconstruct an NxN image from P view angles, as compared to O(N (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 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(2)logN) floating-point operations per iteration to reconstruct an NxN image from P view angles, as compared to O(N (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.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(2)logN) floating-point operations per iteration to reconstruct an NxN image from P view angles, as compared to O(N (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. 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(2)logN) floating-point operations per iteration to reconstruct an NxN image from P view angles, as compared to O(N (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. |
| Author | Delaney, A.H. Bresler, Y. |
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| Cites_doi | 10.1109/TPAMI.1984.4767596 10.1364/AO.20.000395 10.1109/42.141651 10.1088/0031-9155/38/8/004 10.1117/12.7972766 10.1364/JOSA.73.001493 10.1109/83.236536 10.1109/ICASSP.1987.1169768 10.1109/42.222670 10.1109/42.52985 10.1109/34.120331 10.1007/978-3-663-01409-6 10.1109/78.193220 10.1364/AO.24.004028 10.1109/23.159774 10.1007/3-540-09417-2_2 10.1109/78.193196 10.1109/PROC.1983.12597 10.1109/83.287017 10.1088/0031-9155/37/3/015 10.1109/42.61759 10.1080/01630568008816043 10.1109/PROC.1983.12598 |
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| Keywords | Conjugate gradient method Series expansion Tomography Image processing Fast Fourier transformation Image reconstruction |
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| References | ref13 ref12 ref15 ref14 ref11 natterer (ref8) 1986 ref10 ref1 ref17 ref16 ref19 sezan (ref9) 1987 herman (ref4) 1980 golub (ref20) 1989 ref24 ref23 ref26 ref25 ref22 ref21 luenberger (ref18) 1984 ref28 ref27 ref29 rowland (ref2) 1979; 32 ref7 ref5 deans (ref3) 1983 medoff (ref6) 1987 |
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| SubjectTerms | 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 |
| Title | A fast and accurate Fourier algorithm for iterative parallel-beam tomography |
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