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May  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

* Correspondence: Email: z.wang@cqu.edu.au; Tel: +61-7-49232815

Academic Editor: Mingfeng Wang

Received  April 2022 Accepted  May 2022 Published  June 2022

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.

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.

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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.

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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.

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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.

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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.

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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.

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[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 1.  Parallel beam geometry: a detector measures an integral of attenuation along the line at angle θ and distance r to the iso-center
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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