Figure 1.
Parallel beam geometry: a detector measures an integral of attenuation along the line at angle θ and distance r to the iso-center
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2022, 2(2): 121-139.
doi: 10.3934/steme.2022009
Fast non-uniform Fourier transform based regularization for sparse-view large-size CT reconstruction
School of Engineering and Technology, Central Queensland University, Bruce Highway, North Rockhampton, QLD 4702, Australia |
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Spare-view CT imaging is advantageous to decrease the radiation exposure, acquisition time and computational cost, but suffers from severe streak noise in reconstruction if the classical filter back projection method is employed. Although a few compressed sensing based algorithms have recently been proposed to remedy the insufficiency of projections and have achieved remarkable improvement in reconstruction quality, they face computational challenges for large-scale CT images (e.g., larger than 2000℅2000 pixels). In this paper, we present a fast non-uniform Fourier transform based reconstruction method, targeting at under-sampling high resolution Synchrotron-based micro-CT imaging. The proposed method manipulates the Fourier slice theorem to avoid the involvement of large-scale system matrices, and the reconstruction process is performed in the Fourier domain. With a total variation penalty term, the proposed method can be formulated into an unconstrained minimization problem, which is able to be efficiently solved by the limited-memory BFGS algorithm. Moreover, direct non-uniform Fourier transform is computationally costly, so the developed NUFFT algorithm is adopted to approximate it with little loss of quality. Numerical simulation is implemented to compare the proposed method with some other competing approaches, and then real data obtained from the Australia Synchrotron facility are tested to demonstrate the practical applications of the proposed approach. In short, the significance of the proposed approach includes (1) that it can handle high-resolution CT images with millions of pixels while several other contemporary methods fail; (2) that it can achieve much better reconstruction quality than other methods when the projections are insufficient.
Keywords:
non-uniform Fourier transform,
sparse-view CT imaging,
Fourier slice theorem,
regularization.
Citation:
Zhenglin Wang. Fast non-uniform Fourier transform based regularization for sparse-view large-size CT reconstruction. STEM Education,
2022, 2
(2)
: 121-139.
doi: 10.3934/steme.2022009
![]()
References:
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Ginat, D.T. and Gupta, R., Advances in computed tomography imaging technology. Annual review of biomedical engineering, 2014, 16: 431-453. https://doi.org/10.1146/annurev-bioeng-121813-113601
doi: 10.1146/annurev-bioeng-121813-113601. |
| [2] |
Cnudde, V. and Boone, M.N., High-resolution X-ray computed tomography in geosciences: A review of the current technology and applications. Earth-Science Reviews, 2013, 123: 1-17. https://doi.org/10.1016/j.earscirev.2013.04.003
doi: 10.1016/j.earscirev.2013.04.003. |
| [3] |
Bonse, U. and Busch, F., X-ray computed microtomography (μCT) using synchrotron radiation (SR). Progress in biophysics and molecular biology, 1996, 65(1): 133-169. https://doi.org/10.1016/S0079-6107(96)00011-9
doi: 10.1016/S0079-6107(96)00011-9. |
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Murrie, R.P., Morgan, K.S., Maksimenko, A., Fouras, A., Paganin, D.M., Hall, C., et al., Live small-animal X-ray lung velocimetry and lung micro-tomography at the Australian Synchrotron Imaging and Medical Beamline. Journal of Synchrotron Radiation, 2015, 22(4): 1049-1055. https://doi.org/10.1107/S1600577515006001
doi: 10.1107/S1600577515006001. |
| [5] |
Liu, Y., Nelson, J., Holzner, C., Andrews, J.C. and Pianetta, P., Recent advances in synchrotron-based hard x-ray phase contrast imaging. Journal of Physics D: Applied Physics, 2013, 46(49): 494001. https://doi.org/10.1088/0022-3727/46/49/494001
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| [6] |
Bian, J., Siewerdsen, J.H., Han, X., Sidky, E.Y., Prince, J.L., Pelizzari, C.A., et al., Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT. Physics in medicine and biology, 2010, 55(22): 6575. https://doi.org/10.1088/0031-9155/55/22/001
doi: 10.1088/0031-9155/55/22/001. |
| [7] |
Zhu, Z., Wahid, K., Babyn, P., Cooper, D., Pratt, I. and Carter, Y., Improved Compressed Sensing-Based Algorithm for Sparse-View CT Image Reconstruction. Computational and Mathematical Methods in Medicine, 2013, 2013: 15. https://doi.org/10.1155/2013/185750
doi: 10.1155/2013/185750. |
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Hashemi, S., Beheshti, S., Gill, P.R., Paul, N.S. and Cobbold, R.S., Fast fan/parallel beam CS-based low-dose CT reconstruction. Acoustics, Speech and Signal Processing (ICASSP), 2013 IEEE International Conference on. 2013. https://doi.org/10.1109/ICASSP.2013.6637820
doi: 10.1109/ICASSP.2013.6637820. |
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Pan, X., Sidky, E.Y. and Vannier, M., Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction? Inverse problems, 2009, 25(12): 1230009-1230009. https://doi.org/10.1088/0266-5611/25/12/123009
doi: 10.1088/0266-5611/25/12/123009. |
| [10] |
Gordon, R., Bender, R. and Herman, G.T., Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography. Journal of Theoretical Biology, 1970, 29(3): 471-481. https://doi.org/10.1016/0022-5193(70)90109-8
doi: 10.1016/0022-5193(70)90109-8. |
| [11] |
Min, J. and Ge, W., Convergence of the simultaneous algebraic reconstruction technique (SART). Image Processing, IEEE Transactions on, 2003, 12(8): 957-961. https://doi.org/10.1109/TIP.2003.815295
doi: 10.1109/TIP.2003.815295. |
| [12] |
Andersen, A.H. and Kak, A.C., Simultaneous Algebraic Reconstruction Technique (SART): A superior implementation of the ART algorithm. Ultrasonic Imaging, 1984, 6(1): 81-94. https://doi.org/10.1177/016173468400600107
doi: 10.1177/016173468400600107. |
| [13] |
Donoho, D.L., Tsaig, Y., Drori, I. and Starck, J.L., Sparse Solution of Underdetermined Systems of Linear Equations by Stagewise Orthogonal Matching Pursuit. IEEE Transactions on Information Theory, 2012, 58(2): 1094-1121. https://doi.org/10.1109/TIT.2011.2173241
doi: 10.1109/TIT.2011.2173241. |
| [14] |
Candès, E.J., Romberg, J. and Tao, T., Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 2006, 52(2): 489-509. https://doi.org/10.1109/TIT.2005.862083
doi: 10.1109/TIT.2005.862083. |
| [15] |
Lustig, M., Donoho, D. and Pauly, J.M., Sparse MRI: The application of compressed sensing for rapid MR imaging. Magnetic Resonance in Medicine, 2007, 58(6): 1182-1195. https://doi.org/10.1002/mrm.21391
doi: 10.1002/mrm.21391. |
| [16] |
Lauzier, P.T., Tang, J. and Chen, G.-H., Prior image constrained compressed sensing: Implementation and performance evaluation. Medical Physics, 2012, 39(1): 66-80. https://doi.org/10.1118/1.3666946
doi: 10.1118/1.3666946. |
| [17] |
Li, X. and Luo, S., A compressed sensing-based iterative algorithm for CT reconstruction and its possible application to phase contrast imaging. Biomedical Engineering Online, 2011, 10(1): 1-14. https://doi.org/10.1186/1475-925X-10-73
doi: 10.1186/1475-925X-10-73. |
| [18] |
Sidky, E.Y. and Pan, X., Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization. Physics in medicine and biology, 2008, 53(17): 4777. https://doi.org/10.1088/0031-9155/53/17/021
doi: 10.1088/0031-9155/53/17/021. |
| [19] |
Emil, Y.S., Jakob, H.J.r. and Xiaochuan, P., Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle Pock algorithm. Physics in Medicine and Biology, 2012, 57(10): 3065. https://doi.org/10.1088/0031-9155/57/10/3065
doi: 10.1088/0031-9155/57/10/3065. |
| [20] |
Hsieh, J., Computed tomography: principles, design, artifacts, and recent advances. 2009, SPIE Press: Bellingham, WA. |
| [21] |
Gottleib, D., Gustafsson, B. and Forssen, P., On the direct Fourier method for computer tomography. Medical Imaging, IEEE Transactions on, 2000, 19(3): 223-232. https://doi.org/10.1109/42.845180
doi: 10.1109/42.845180. |
| [22] |
Averbuch, A., Coifman, R.R., Donoho, D.L., Elad, M. and Israeli, M., Fast and accurate Polar Fourier transform. Applied and Computational Harmonic Analysis, 2006, 21(2): 145-167. https://doi.org/10.1016/j.acha.2005.11.003
doi: 10.1016/j.acha.2005.11.003. |
| [23] |
Duijndam, A.J.W. and Schonewille, M.A., Nonuniform fast Fourier transform. GEOPHYSICS, 1999, 64(2): 539-551. https://doi.org/10.1190/1.1444560
doi: 10.1190/1.1444560. |
| [24] |
Fessler, J.A. and Sutton, B.P., Nonuniform fast Fourier transforms using min-max interpolation. Signal Processing, IEEE Transactions on, 2003, 51(2): 560-574. https://doi.org/10.1109/TSP.2002.807005
doi: 10.1109/TSP.2002.807005. |
| [25] |
Jacob, M., Optimized Least-Square Nonuniform Fast Fourier Transform. Signal Processing, IEEE Transactions on, 2009, 57(6): 2165-2177. https://doi.org/10.1109/TSP.2009.2014809
doi: 10.1109/TSP.2009.2014809. |
| [26] |
Yang, Z. and Jacob, M., Mean square optimal NUFFT approximation for efficient non-Cartesian MRI reconstruction. Journal of Magnetic Resonance, 2014, 242: 126-135. https://doi.org/10.1016/j.jmr.2014.01.016
doi: 10.1016/j.jmr.2014.01.016. |
| [27] |
Keiner, J., Kunis, S. and Potts, D., Using NFFT 3---A Software Library for Various Nonequispaced Fast Fourier Transforms. ACM Transactions on Mathematical Software, 2009, 36(4): 1-30. https://doi.org/10.1145/1555386.1555388
doi: 10.1145/1555386.1555388. |
| [28] |
Fahimian, B.P., Zhao, Y., Huang, Z., Fung, R., Mao, Y., Zhu, C., et al., Radiation dose reduction in medical x-ray CT via Fourier-based iterative reconstruction. Medical Physics, 2013, 40(3): 031914. https://doi.org/10.1118/1.4791644
doi: 10.1118/1.4791644. |
| [29] |
Matej, S., Fessler, J.A. and Kazantsev, I.G., Iterative tomographic image reconstruction using Fourier-based forward and back-projectors. Medical Imaging, IEEE Transactions on, 2004, 23(4): 401-412. https://doi.org/10.1109/TMI.2004.824233
doi: 10.1109/TMI.2004.824233. |
| [30] |
Yan, B., Jin, Z., Zhang, H., Li, L. and Cai, A., NUFFT-based iterative image reconstruction via alternating direction total variation minimization for sparse-view CT. Computational and Mathematical Methods in Medicine, 2015, 2015: 1-9. https://doi.org/10.1155/2015/691021
doi: 10.1155/2015/691021. |
| [31] |
Liu, D.C. and Nocedal, J., On the limited memory BFGS method for large scale optimization. Mathematical Programming, 1989, 45: 503-528. https://doi.org/10.1007/BF01589116
doi: 10.1007/BF01589116. |
| [32] |
Besson, G., CT image reconstruction from fan-parallel data. Medical Physics, 1999, 26(3): 415-426. https://doi.org/10.1118/1.598533
doi: 10.1118/1.598533. |
| [33] |
Swan, P., Discrete Fourier transforms of nonuniformly spaced data. The Astronomical Journal, 1982, 87: 1608. https://doi.org/10.1086/113252
doi: 10.1086/113252. |
| [34] |
Candès, E.J. and Romberg, J., 11-magic: Recovery of sparse signals via convex programming. 2005, Applied & Computational Mathematics, California Institute of Technology, Pasadena. |
| [35] |
Chen, G.-H., Tang, J. and Leng, S., Prior image constrained compressed sensing (PICCS): A method to accurately reconstruct dynamic CT images from highly undersampled projection data sets. Medical Physics, 2008, 35(2): 660-663. https://doi.org/10.1118/1.2836423
doi: 10.1118/1.2836423. |
| [36] |
Wang, Z., Bovik, A.C., Sheikh, H.R. and Simoncelli, E.P., Image quality assessment: from error visibility to structural similarity. Image Processing, IEEE Transactions on, 2004, 13(4): 600-612. https://doi.org/10.1109/TIP.2003.819861
doi: 10.1109/TIP.2003.819861. |
show all references
References:
| [1] |
Ginat, D.T. and Gupta, R., Advances in computed tomography imaging technology. Annual review of biomedical engineering, 2014, 16: 431-453. https://doi.org/10.1146/annurev-bioeng-121813-113601
doi: 10.1146/annurev-bioeng-121813-113601. |
| [2] |
Cnudde, V. and Boone, M.N., High-resolution X-ray computed tomography in geosciences: A review of the current technology and applications. Earth-Science Reviews, 2013, 123: 1-17. https://doi.org/10.1016/j.earscirev.2013.04.003
doi: 10.1016/j.earscirev.2013.04.003. |
| [3] |
Bonse, U. and Busch, F., X-ray computed microtomography (μCT) using synchrotron radiation (SR). Progress in biophysics and molecular biology, 1996, 65(1): 133-169. https://doi.org/10.1016/S0079-6107(96)00011-9
doi: 10.1016/S0079-6107(96)00011-9. |
| [4] |
Murrie, R.P., Morgan, K.S., Maksimenko, A., Fouras, A., Paganin, D.M., Hall, C., et al., Live small-animal X-ray lung velocimetry and lung micro-tomography at the Australian Synchrotron Imaging and Medical Beamline. Journal of Synchrotron Radiation, 2015, 22(4): 1049-1055. https://doi.org/10.1107/S1600577515006001
doi: 10.1107/S1600577515006001. |
| [5] |
Liu, Y., Nelson, J., Holzner, C., Andrews, J.C. and Pianetta, P., Recent advances in synchrotron-based hard x-ray phase contrast imaging. Journal of Physics D: Applied Physics, 2013, 46(49): 494001. https://doi.org/10.1088/0022-3727/46/49/494001
doi: 10.1088/0022-3727/46/49/494001. |
| [6] |
Bian, J., Siewerdsen, J.H., Han, X., Sidky, E.Y., Prince, J.L., Pelizzari, C.A., et al., Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT. Physics in medicine and biology, 2010, 55(22): 6575. https://doi.org/10.1088/0031-9155/55/22/001
doi: 10.1088/0031-9155/55/22/001. |
| [7] |
Zhu, Z., Wahid, K., Babyn, P., Cooper, D., Pratt, I. and Carter, Y., Improved Compressed Sensing-Based Algorithm for Sparse-View CT Image Reconstruction. Computational and Mathematical Methods in Medicine, 2013, 2013: 15. https://doi.org/10.1155/2013/185750
doi: 10.1155/2013/185750. |
| [8] |
Hashemi, S., Beheshti, S., Gill, P.R., Paul, N.S. and Cobbold, R.S., Fast fan/parallel beam CS-based low-dose CT reconstruction. Acoustics, Speech and Signal Processing (ICASSP), 2013 IEEE International Conference on. 2013. https://doi.org/10.1109/ICASSP.2013.6637820
doi: 10.1109/ICASSP.2013.6637820. |
| [9] |
Pan, X., Sidky, E.Y. and Vannier, M., Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction? Inverse problems, 2009, 25(12): 1230009-1230009. https://doi.org/10.1088/0266-5611/25/12/123009
doi: 10.1088/0266-5611/25/12/123009. |
| [10] |
Gordon, R., Bender, R. and Herman, G.T., Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography. Journal of Theoretical Biology, 1970, 29(3): 471-481. https://doi.org/10.1016/0022-5193(70)90109-8
doi: 10.1016/0022-5193(70)90109-8. |
| [11] |
Min, J. and Ge, W., Convergence of the simultaneous algebraic reconstruction technique (SART). Image Processing, IEEE Transactions on, 2003, 12(8): 957-961. https://doi.org/10.1109/TIP.2003.815295
doi: 10.1109/TIP.2003.815295. |
| [12] |
Andersen, A.H. and Kak, A.C., Simultaneous Algebraic Reconstruction Technique (SART): A superior implementation of the ART algorithm. Ultrasonic Imaging, 1984, 6(1): 81-94. https://doi.org/10.1177/016173468400600107
doi: 10.1177/016173468400600107. |
| [13] |
Donoho, D.L., Tsaig, Y., Drori, I. and Starck, J.L., Sparse Solution of Underdetermined Systems of Linear Equations by Stagewise Orthogonal Matching Pursuit. IEEE Transactions on Information Theory, 2012, 58(2): 1094-1121. https://doi.org/10.1109/TIT.2011.2173241
doi: 10.1109/TIT.2011.2173241. |
| [14] |
Candès, E.J., Romberg, J. and Tao, T., Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 2006, 52(2): 489-509. https://doi.org/10.1109/TIT.2005.862083
doi: 10.1109/TIT.2005.862083. |
| [15] |
Lustig, M., Donoho, D. and Pauly, J.M., Sparse MRI: The application of compressed sensing for rapid MR imaging. Magnetic Resonance in Medicine, 2007, 58(6): 1182-1195. https://doi.org/10.1002/mrm.21391
doi: 10.1002/mrm.21391. |
| [16] |
Lauzier, P.T., Tang, J. and Chen, G.-H., Prior image constrained compressed sensing: Implementation and performance evaluation. Medical Physics, 2012, 39(1): 66-80. https://doi.org/10.1118/1.3666946
doi: 10.1118/1.3666946. |
| [17] |
Li, X. and Luo, S., A compressed sensing-based iterative algorithm for CT reconstruction and its possible application to phase contrast imaging. Biomedical Engineering Online, 2011, 10(1): 1-14. https://doi.org/10.1186/1475-925X-10-73
doi: 10.1186/1475-925X-10-73. |
| [18] |
Sidky, E.Y. and Pan, X., Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization. Physics in medicine and biology, 2008, 53(17): 4777. https://doi.org/10.1088/0031-9155/53/17/021
doi: 10.1088/0031-9155/53/17/021. |
| [19] |
Emil, Y.S., Jakob, H.J.r. and Xiaochuan, P., Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle Pock algorithm. Physics in Medicine and Biology, 2012, 57(10): 3065. https://doi.org/10.1088/0031-9155/57/10/3065
doi: 10.1088/0031-9155/57/10/3065. |
| [20] |
Hsieh, J., Computed tomography: principles, design, artifacts, and recent advances. 2009, SPIE Press: Bellingham, WA. |
| [21] |
Gottleib, D., Gustafsson, B. and Forssen, P., On the direct Fourier method for computer tomography. Medical Imaging, IEEE Transactions on, 2000, 19(3): 223-232. https://doi.org/10.1109/42.845180
doi: 10.1109/42.845180. |
| [22] |
Averbuch, A., Coifman, R.R., Donoho, D.L., Elad, M. and Israeli, M., Fast and accurate Polar Fourier transform. Applied and Computational Harmonic Analysis, 2006, 21(2): 145-167. https://doi.org/10.1016/j.acha.2005.11.003
doi: 10.1016/j.acha.2005.11.003. |
| [23] |
Duijndam, A.J.W. and Schonewille, M.A., Nonuniform fast Fourier transform. GEOPHYSICS, 1999, 64(2): 539-551. https://doi.org/10.1190/1.1444560
doi: 10.1190/1.1444560. |
| [24] |
Fessler, J.A. and Sutton, B.P., Nonuniform fast Fourier transforms using min-max interpolation. Signal Processing, IEEE Transactions on, 2003, 51(2): 560-574. https://doi.org/10.1109/TSP.2002.807005
doi: 10.1109/TSP.2002.807005. |
| [25] |
Jacob, M., Optimized Least-Square Nonuniform Fast Fourier Transform. Signal Processing, IEEE Transactions on, 2009, 57(6): 2165-2177. https://doi.org/10.1109/TSP.2009.2014809
doi: 10.1109/TSP.2009.2014809. |
| [26] |
Yang, Z. and Jacob, M., Mean square optimal NUFFT approximation for efficient non-Cartesian MRI reconstruction. Journal of Magnetic Resonance, 2014, 242: 126-135. https://doi.org/10.1016/j.jmr.2014.01.016
doi: 10.1016/j.jmr.2014.01.016. |
| [27] |
Keiner, J., Kunis, S. and Potts, D., Using NFFT 3---A Software Library for Various Nonequispaced Fast Fourier Transforms. ACM Transactions on Mathematical Software, 2009, 36(4): 1-30. https://doi.org/10.1145/1555386.1555388
doi: 10.1145/1555386.1555388. |
| [28] |
Fahimian, B.P., Zhao, Y., Huang, Z., Fung, R., Mao, Y., Zhu, C., et al., Radiation dose reduction in medical x-ray CT via Fourier-based iterative reconstruction. Medical Physics, 2013, 40(3): 031914. https://doi.org/10.1118/1.4791644
doi: 10.1118/1.4791644. |
| [29] |
Matej, S., Fessler, J.A. and Kazantsev, I.G., Iterative tomographic image reconstruction using Fourier-based forward and back-projectors. Medical Imaging, IEEE Transactions on, 2004, 23(4): 401-412. https://doi.org/10.1109/TMI.2004.824233
doi: 10.1109/TMI.2004.824233. |
| [30] |
Yan, B., Jin, Z., Zhang, H., Li, L. and Cai, A., NUFFT-based iterative image reconstruction via alternating direction total variation minimization for sparse-view CT. Computational and Mathematical Methods in Medicine, 2015, 2015: 1-9. https://doi.org/10.1155/2015/691021
doi: 10.1155/2015/691021. |
| [31] |
Liu, D.C. and Nocedal, J., On the limited memory BFGS method for large scale optimization. Mathematical Programming, 1989, 45: 503-528. https://doi.org/10.1007/BF01589116
doi: 10.1007/BF01589116. |
| [32] |
Besson, G., CT image reconstruction from fan-parallel data. Medical Physics, 1999, 26(3): 415-426. https://doi.org/10.1118/1.598533
doi: 10.1118/1.598533. |
| [33] |
Swan, P., Discrete Fourier transforms of nonuniformly spaced data. The Astronomical Journal, 1982, 87: 1608. https://doi.org/10.1086/113252
doi: 10.1086/113252. |
| [34] |
Candès, E.J. and Romberg, J., 11-magic: Recovery of sparse signals via convex programming. 2005, Applied & Computational Mathematics, California Institute of Technology, Pasadena. |
| [35] |
Chen, G.-H., Tang, J. and Leng, S., Prior image constrained compressed sensing (PICCS): A method to accurately reconstruct dynamic CT images from highly undersampled projection data sets. Medical Physics, 2008, 35(2): 660-663. https://doi.org/10.1118/1.2836423
doi: 10.1118/1.2836423. |
| [36] |
Wang, Z., Bovik, A.C., Sheikh, H.R. and Simoncelli, E.P., Image quality assessment: from error visibility to structural similarity. Image Processing, IEEE Transactions on, 2004, 13(4): 600-612. https://doi.org/10.1109/TIP.2003.819861
doi: 10.1109/TIP.2003.819861. |
Figure 2.
Sampling distribution in Fourier space based on the Fourier slice theorem
Figure 3.
An example of the Fourier slice theorem for CT imaging
Figure 4.
Pseudo code implementation for proposed algorithm
Figure 5.
Performance comparison with different number of views
Figure 6.
Visual comparison for different algorithms with 60 views, 90 views, 180 views
Figure 7.
Reconstruction with 180 projections: FBP (left) vs Proposed method (right)
Figure 8.
Reconstruction with 1800 projections: FBP (left) vs Proposed method (right)
Table 1.
Reconstruction with 60 projections
| Methods | SNR | RMSE | SSIM | Time (s) |
| FBP | 9.76 | 0.080 | 0.851 | 0.25 |
| Zhu et al. | 9.83 | 0.079 | 0.922 | 7.62 |
| Hashemi et al. | 2.31 | 0.188 | 0.686 | 10.33 |
| Proposed | 10.95 | 0.069 | 0.911 | 5.07 |
| Methods | SNR | RMSE | SSIM | Time (s) |
| FBP | 9.76 | 0.080 | 0.851 | 0.25 |
| Zhu et al. | 9.83 | 0.079 | 0.922 | 7.62 |
| Hashemi et al. | 2.31 | 0.188 | 0.686 | 10.33 |
| Proposed | 10.95 | 0.069 | 0.911 | 5.07 |
Table 2.
Reconstruction with 90 projections
| Methods | SNR | RMSE | SSIM | Time (s) |
| FBP | 10.80 | 0.071 | 0.916 | 0.38 |
| Zhu et al. | 10.41 | 0.074 | 0.946 | 9.16 |
| Hashemi et al. | 3.60 | 0.162 | 0.760 | 9.70 |
| Proposed | 12.23 | 0.060 | 0.936 | 5.74 |
| Methods | SNR | RMSE | SSIM | Time (s) |
| FBP | 10.80 | 0.071 | 0.916 | 0.38 |
| Zhu et al. | 10.41 | 0.074 | 0.946 | 9.16 |
| Hashemi et al. | 3.60 | 0.162 | 0.760 | 9.70 |
| Proposed | 12.23 | 0.060 | 0.936 | 5.74 |
Table 3.
Reconstruction with 180 projections
| Methods | SNR | RMSE | SSIM | Time (s) |
| FBP | 11.22 | 0.067 | 0.946 | 0.74 |
| Zhu et al. | 10.47 | 0.073 | 0.949 | 15.71 |
| Hashemi et al. | 10.23 | 0.075 | 0.934 | 9.67 |
| Proposed | 13.43 | 0.052 | 0.950 | 7.79 |
| Methods | SNR | RMSE | SSIM | Time (s) |
| FBP | 11.22 | 0.067 | 0.946 | 0.74 |
| Zhu et al. | 10.47 | 0.073 | 0.949 | 15.71 |
| Hashemi et al. | 10.23 | 0.075 | 0.934 | 9.67 |
| Proposed | 13.43 | 0.052 | 0.950 | 7.79 |
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