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Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data
1. | School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China |
2. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
3. | Engineering Research Center of Industrial Computed Tomography, Nondestructive Testing of the Education Ministry of China, Chongqing University, Chongqing 400044, China |
The limited-angle projection data of an object, in some practical applications of computed tomography (CT), are obtained due to the restriction of scanning condition. In these situations, since the projection data are incomplete, some limited-angle artifacts will be presented near the edges of reconstructed image using some classical reconstruction algorithms, such as filtered backprojection (FBP). The reconstructed image can be fine approximated by sparse coefficients under a proper wavelet tight frame, and the quality of reconstructed image can be improved by an available prior image. To deal with limited-angle CT reconstruction problem, we propose a minimization model that is based on wavelet tight frame and a prior image, and perform this minimization problem efficiently by iteratively minimizing separately. Moreover, we show that each bounded sequence, which is generated by our method, converges to a critical or a stationary point. The experimental results indicate that our algorithm can efficiently suppress artifacts and noise and preserve the edges of reconstructed image, what's more, the introduced prior image will not miss the important information that is not included in the prior image.
References:
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Image quality and localization accuracy in C-arm tomosynthesis-guided head and neck surgery, Med. Phys., 34 (2007), 4664-4677.
doi: 10.1118/1.2799492. |
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C. Bao, H. Ji and Z. Shen,
Convergence analysis for iterative data-driven tight frame construction scheme, Appl. Comput. Harmon. Anal., 38 (2015), 510-523.
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[3] |
J. G. Bian, J. H. Siewerdsen, X. Han, E. Y. Sidky, J. L. Prince, C. A. Pelizzari and X. C. Pan,
Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT, Physics in Medicine and Biology, 55 (2010), 6575-6599.
doi: 10.1088/0031-9155/55/22/001. |
[4] |
J. Bolte, S. Sabach and M. Teboulle,
Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Mathematical Programming, 146 (2014), 459-494.
doi: 10.1007/s10107-013-0701-9. |
[5] |
T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-beam CT, 1nd edition, Springer-Verlag, Berlin Heidelberg, 2008.
![]() |
[6] |
J. F. Cai, H. Ji, Z. W. Shen and G. B. Ye,
Data-driven tight frame construction and image denoising, Appl. Comput. Harmon. Anal., 37 (2014), 89-105.
doi: 10.1016/j.acha.2013.10.001. |
[7] |
J. F. Cai, S. Osher and Z. Shen,
Split Bregman methods and frame based image restoration, Multiscale Modeling and Simulation, 8 (2009), 337-369.
doi: 10.1137/090753504. |
[8] |
G. H. Chen, J. Tang and S. Leng,
Prior image constrained compressed sensing (PICCS): A method to accurately reconstruct dynamic CT images from highly undersampled projection data sets, Medical Physics, 35 (2008), 660-663.
doi: 10.1118/1.2836423. |
[9] |
G. H. Chen,
Time-resolved interventional cardiac C-arm cone-beam CT: An application of the PICCS algorithm, IEEE Transactions on Medical Imaging, 31 (2012), 907-923.
doi: 10.1109/TMI.2011.2172951. |
[10] |
Z. Q. Chen, X. Jin, L. Li and G. Wang,
A limited-angle CT reconstruction method based on anisotropic TV minimization, Phys. Med. Biol., 58 (2013), 2119-2141.
doi: 10.1088/0031-9155/58/7/2119. |
[11] |
B. Dong and Z. Shen, MRA-based wavelet frames and applications: Image segmentation and surface reconstruction In SPIE Defense, Security, and Sensing, International Society for Optics and Photonics, (2012), 840102.
doi: 10.1117/12.923203. |
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M. M. Eger and P. E. Danielsson,
Scanning of logs with linear cone-beam tomography, Computers and Electronics in Agriculture, 41 (2003), 45-62.
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J. M. Fadili and G. Peyré,
Total variation projection with first order schemes, IEEE Transactions on Image Processing, 20 (2011), 657-669.
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Sparse regularization in limited angle tomography, Appl. Comput. Harmon. Anal., 34 (2013), 117-141.
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Robust principal component analysis-based four-dimensional computed tomography, Phys. Med. Biol., 56 (2011), 3781-3798.
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H. Gao, R. Li, Y. Lin and L. Xing,
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H. Gao, L. Zhang, Z. Chen, Y. Xing, J. Cheng and Z. Qi,
Direct filtered-backprojection-type reconstruction from a straight-line trajectory, Optical Engineering, 46 (2007), 057003.
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[18] |
B. Han, G. Kutyniok and Z. Shen,
Adaptive multiresolution analysis structures and shearlet systems, SIAM. J. Numer. Anal., 49 (2011), 1921-1946.
doi: 10.1137/090780912. |
[19] |
P. C. Hansen, E. Y. Sidky and X. C. Pan, Accelerated gradient methods for total-variation-based CT image reconstruction, arXiv: 1105.4002. |
[20] |
G. T. Herman and R. Davidi,
Image reconstruction from a small number of projections, Inverse Problems, 24 (2008), 045011.
doi: 10.1088/0266-5611/24/4/045011. |
[21] |
X. Jia, B. Dong, Y. Lou and S. B. jiang,
GPU-based iterative cone-beam CT reconstruction using tight frame regularization, Phys. Med. Biol., 56 (2010), 3787-3806.
doi: 10.1088/0031-9155/56/13/004. |
[22] |
V. Kolehmainen, S. Siltanen, S. Jarvenpaa, J. P. Kaipio, P. Koistinenand, M. Lassas, J. Pirttila and E. Somersalo,
Statistical inversion for medical x-ray tomography with few radiographs: Ⅱ. Application to dental radiology, Phys. Med. Biol., 48 (2003), 1465-1490.
doi: 10.1088/0031-9155/48/10/315. |
[23] |
S. J. LaRoque, E. Y. Sidky and X. C. Pan,
Accurate image reconstruction from few-view and limited-angle data in diffraction tomography, JOSA A, 25 (2008), 1772-1782.
doi: 10.1364/JOSAA.25.001772. |
[24] |
X. Lu, Y. Sun and Y. Yuan,
Image reconstruction by an alternating minimisation, Neurocomputing, 74 (2011), 661-670.
doi: 10.1016/j.neucom.2010.08.003. |
[25] |
X. Lu, Y. Sun and Y. Yuan,
Optimization for limited angle tomography in medical image processing, Phys. Med. Biol., 44 (2011), 2427-2435.
doi: 10.1016/j.patcog.2010.12.016. |
[26] |
M. G. Lubner,
Prospective evaluation of prior image constrained compressed sensing (PICCS) algorithm in abdominal CT: A comparison of reduced dose with standard dose imaging, Abdominal Imaging, 40 (2015), 207-221.
doi: 10.1007/s00261-014-0178-x. |
[27] |
F. Natterer, The Mathmetics of Computed Tomography, 1nd edition, B. G. Teubner, Stuttgart., 1986. |
[28] |
B. Nett, J. Tang, S. Leng and G. H. Chen,
Tomosynthesis via total variation minimization reconstruction and prior image constrained compressed sensing (PICCS) on a C-arm system, Medical Imaging. International Society for Optics and Photonics, 6913 (2008), 1-10.
doi: 10.1117/12.771294. |
[29] |
E. T. Quinto,
Exterior and limited-angle tomography in non-destructive evaluation, Inverse Problems, 14 (1998), 1339-1353.
doi: 10.1088/0266-5611/14/2/009. |
[30] |
A. Ron and Z. Shen,
Affine systems in $L_{2}(R^{d})$: The analysis of the analysis operator, Journal of Functional Analysis, 148 (1997), 408-447.
doi: 10.1006/jfan.1996.3079. |
[31] |
L. Shen, Y. Xu and X. Zeng,
Wavelet inpainting with the $\ell_{0}$ sparse regularization, Appl. Comput. Harmon. Anal., 41 (2016), 26-53.
doi: 10.1016/j.acha.2015.03.001. |
[32] |
Z. Shen,
Wavelet frames and image restorations, Proceedings of the International congress of Mathematicians, 4 (2010), 2834-2863.
|
[33] |
E. Y. Sidky, C. M. Kao and X. C. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT, arXiv: 0904.4495. |
[34] |
E. Y. Sidky and X. C. Pan,
Accurate image reconstruction in circular cone-beam computed tomography by total variation minimization: A preliminary investigation, IEEE Nuclear Science Symposium Conference Record, 5 (2006), 2904-2907.
doi: 10.1109/NSSMIC.2006.356484. |
[35] |
E. Y. Sidky and X. C. Pan,
Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Physics in Medicine and Biology, 53 (2008), 3572277-3572284.
doi: 10.1088/0031-9155/53/17/021. |
[36] |
W. P. Segars, D. S. Lalush and B. M. Tsui,
A realistic splinebased dynamic heart phantom, IEEE Trans. Nucl. Sci., 46 (1999), 503-506.
doi: 10.1109/23.775570. |
[37] |
M. Storath, A. Weinmann, J. Frikeland and M. Unser,
Joint image reconstruction and segmentation using the Potts model, Inverse Problems, 31 (2015), 025003.
doi: 10.1088/0266-5611/31/2/025003. |
[38] |
J. Tang, J. Hsieh and G. H. Chen,
Temporal resolution improvement in cardiac CT using PICCS (TRI-PICCS): Performance studies, Medical Physics, 37 (2010), 4377-4388.
doi: 10.1118/1.3460318. |
[39] |
A. Tingberg,
X-ray tomosynthesis: A review of its use for breast and chest imaging, Radiation Protection Dosimetry, 139 (2010), 100-107.
doi: 10.1093/rpd/ncq099. |
[40] |
H. K. Tuy,
An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math., 43 (1983), 546-552.
doi: 10.1137/0143035. |
[41] |
Z. Wang, A. Bovik, H. Sheikhand and E. Simoncelli,
Image quality assessment: From error visibility to structural similarity, IEEE Trans Image Process, 13 (2004), 600-612.
doi: 10.1109/TIP.2003.819861. |
[42] |
Z. Wang, Z. Huang, Z. Chen, L. Zhang, X. Jiang, K. Kang, H. Yin, Z. Wang and M. Stampanoni, Low-dose multiple-information retrieval algorithm for x-ray grating-based imaging, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 635 (2011), 103-107.
doi: 10.1016/j.nima.2011.01.079. |
[43] |
F. Yang, Y. Shen and Z. S. Liu,
The proximal alternating iterative hard thresholding method for $\ell_{0}$ minimization, with complexity $O(\frac{1}{\sqrt{k}})$, Journal of Computational and Applied Mathematics, 311 (2017), 115-129.
doi: 10.1016/j.cam.2016.07.013. |
[44] |
W. Yu and L. Zeng,
$\ell_{0}$ gradient minimization based image reconstruction for limited-angle computed tomography, PLoS ONE, 10 (2015), e0130793.
doi: 10.1371/journal.pone.0130793. |
[45] |
L. Zeng, J. Q. Guo and B. D. Liu,
Limited-angle cone-beam computed tomography image reconstruction by total variation minimization and piecewise-constant modification, Journal of Inverse and Ill-Posed Problems, 21 (2013), 735-754.
doi: 10.1515/jip-2011-0010. |
[46] |
Y. Zhang, B. Dong and Z. S. Lu,
$\ell_{0}$ Minimization for wavelet frame based image restoration, Mathematics of Computation, 82 (2013), 995-1015.
doi: 10.1090/S0025-5718-2012-02631-7. |
[47] |
B. Zhao, H. Gao, H. Ding and S. Molloi,
Tight-frame based iterative image reconstruction for spectral breast CT, Medical Physics, 40 (2013), 031905.
doi: 10.1118/1.4790468. |
[48] |
W. Zhou, J. F. Cai and H. Gao,
Adaptive tight frame based medical image reconstruction: a proof-of-concept study for computed tomography, Inverse problems, 29 (2013), 1-18.
doi: 10.1088/0266-5611/29/12/125006. |
[49] |
chest phantom website,
http://lgdv.cs.fau.de/External/vollib/. |
show all references
References:
[1] |
G. Bachar, J. H. Siewerdsen, M. J. Daly, D. A. Jaffray and J. C. Irish,
Image quality and localization accuracy in C-arm tomosynthesis-guided head and neck surgery, Med. Phys., 34 (2007), 4664-4677.
doi: 10.1118/1.2799492. |
[2] |
C. Bao, H. Ji and Z. Shen,
Convergence analysis for iterative data-driven tight frame construction scheme, Appl. Comput. Harmon. Anal., 38 (2015), 510-523.
doi: 10.1016/j.acha.2014.06.007. |
[3] |
J. G. Bian, J. H. Siewerdsen, X. Han, E. Y. Sidky, J. L. Prince, C. A. Pelizzari and X. C. Pan,
Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT, Physics in Medicine and Biology, 55 (2010), 6575-6599.
doi: 10.1088/0031-9155/55/22/001. |
[4] |
J. Bolte, S. Sabach and M. Teboulle,
Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Mathematical Programming, 146 (2014), 459-494.
doi: 10.1007/s10107-013-0701-9. |
[5] |
T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-beam CT, 1nd edition, Springer-Verlag, Berlin Heidelberg, 2008.
![]() |
[6] |
J. F. Cai, H. Ji, Z. W. Shen and G. B. Ye,
Data-driven tight frame construction and image denoising, Appl. Comput. Harmon. Anal., 37 (2014), 89-105.
doi: 10.1016/j.acha.2013.10.001. |
[7] |
J. F. Cai, S. Osher and Z. Shen,
Split Bregman methods and frame based image restoration, Multiscale Modeling and Simulation, 8 (2009), 337-369.
doi: 10.1137/090753504. |
[8] |
G. H. Chen, J. Tang and S. Leng,
Prior image constrained compressed sensing (PICCS): A method to accurately reconstruct dynamic CT images from highly undersampled projection data sets, Medical Physics, 35 (2008), 660-663.
doi: 10.1118/1.2836423. |
[9] |
G. H. Chen,
Time-resolved interventional cardiac C-arm cone-beam CT: An application of the PICCS algorithm, IEEE Transactions on Medical Imaging, 31 (2012), 907-923.
doi: 10.1109/TMI.2011.2172951. |
[10] |
Z. Q. Chen, X. Jin, L. Li and G. Wang,
A limited-angle CT reconstruction method based on anisotropic TV minimization, Phys. Med. Biol., 58 (2013), 2119-2141.
doi: 10.1088/0031-9155/58/7/2119. |
[11] |
B. Dong and Z. Shen, MRA-based wavelet frames and applications: Image segmentation and surface reconstruction In SPIE Defense, Security, and Sensing, International Society for Optics and Photonics, (2012), 840102.
doi: 10.1117/12.923203. |
[12] |
M. M. Eger and P. E. Danielsson,
Scanning of logs with linear cone-beam tomography, Computers and Electronics in Agriculture, 41 (2003), 45-62.
doi: 10.1016/S0168-1699(03)00041-3. |
[13] |
J. M. Fadili and G. Peyré,
Total variation projection with first order schemes, IEEE Transactions on Image Processing, 20 (2011), 657-669.
doi: 10.1109/TIP.2010.2072512. |
[14] |
J. Frikel,
Sparse regularization in limited angle tomography, Appl. Comput. Harmon. Anal., 34 (2013), 117-141.
doi: 10.1016/j.acha.2012.03.005. |
[15] |
H. Gao, J. F. Cai, Z. W. Shen and H. Zhao,
Robust principal component analysis-based four-dimensional computed tomography, Phys. Med. Biol., 56 (2011), 3781-3798.
doi: 10.1088/0031-9155/56/11/002. |
[16] |
H. Gao, R. Li, Y. Lin and L. Xing,
4D cone beam CT via spatiotemporal tensor framelet, Medical Physics, 39 (2012), 6943-6946.
doi: 10.1118/1.4762288. |
[17] |
H. Gao, L. Zhang, Z. Chen, Y. Xing, J. Cheng and Z. Qi,
Direct filtered-backprojection-type reconstruction from a straight-line trajectory, Optical Engineering, 46 (2007), 057003.
doi: 10.1117/1.2739624. |
[18] |
B. Han, G. Kutyniok and Z. Shen,
Adaptive multiresolution analysis structures and shearlet systems, SIAM. J. Numer. Anal., 49 (2011), 1921-1946.
doi: 10.1137/090780912. |
[19] |
P. C. Hansen, E. Y. Sidky and X. C. Pan, Accelerated gradient methods for total-variation-based CT image reconstruction, arXiv: 1105.4002. |
[20] |
G. T. Herman and R. Davidi,
Image reconstruction from a small number of projections, Inverse Problems, 24 (2008), 045011.
doi: 10.1088/0266-5611/24/4/045011. |
[21] |
X. Jia, B. Dong, Y. Lou and S. B. jiang,
GPU-based iterative cone-beam CT reconstruction using tight frame regularization, Phys. Med. Biol., 56 (2010), 3787-3806.
doi: 10.1088/0031-9155/56/13/004. |
[22] |
V. Kolehmainen, S. Siltanen, S. Jarvenpaa, J. P. Kaipio, P. Koistinenand, M. Lassas, J. Pirttila and E. Somersalo,
Statistical inversion for medical x-ray tomography with few radiographs: Ⅱ. Application to dental radiology, Phys. Med. Biol., 48 (2003), 1465-1490.
doi: 10.1088/0031-9155/48/10/315. |
[23] |
S. J. LaRoque, E. Y. Sidky and X. C. Pan,
Accurate image reconstruction from few-view and limited-angle data in diffraction tomography, JOSA A, 25 (2008), 1772-1782.
doi: 10.1364/JOSAA.25.001772. |
[24] |
X. Lu, Y. Sun and Y. Yuan,
Image reconstruction by an alternating minimisation, Neurocomputing, 74 (2011), 661-670.
doi: 10.1016/j.neucom.2010.08.003. |
[25] |
X. Lu, Y. Sun and Y. Yuan,
Optimization for limited angle tomography in medical image processing, Phys. Med. Biol., 44 (2011), 2427-2435.
doi: 10.1016/j.patcog.2010.12.016. |
[26] |
M. G. Lubner,
Prospective evaluation of prior image constrained compressed sensing (PICCS) algorithm in abdominal CT: A comparison of reduced dose with standard dose imaging, Abdominal Imaging, 40 (2015), 207-221.
doi: 10.1007/s00261-014-0178-x. |
[27] |
F. Natterer, The Mathmetics of Computed Tomography, 1nd edition, B. G. Teubner, Stuttgart., 1986. |
[28] |
B. Nett, J. Tang, S. Leng and G. H. Chen,
Tomosynthesis via total variation minimization reconstruction and prior image constrained compressed sensing (PICCS) on a C-arm system, Medical Imaging. International Society for Optics and Photonics, 6913 (2008), 1-10.
doi: 10.1117/12.771294. |
[29] |
E. T. Quinto,
Exterior and limited-angle tomography in non-destructive evaluation, Inverse Problems, 14 (1998), 1339-1353.
doi: 10.1088/0266-5611/14/2/009. |
[30] |
A. Ron and Z. Shen,
Affine systems in $L_{2}(R^{d})$: The analysis of the analysis operator, Journal of Functional Analysis, 148 (1997), 408-447.
doi: 10.1006/jfan.1996.3079. |
[31] |
L. Shen, Y. Xu and X. Zeng,
Wavelet inpainting with the $\ell_{0}$ sparse regularization, Appl. Comput. Harmon. Anal., 41 (2016), 26-53.
doi: 10.1016/j.acha.2015.03.001. |
[32] |
Z. Shen,
Wavelet frames and image restorations, Proceedings of the International congress of Mathematicians, 4 (2010), 2834-2863.
|
[33] |
E. Y. Sidky, C. M. Kao and X. C. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT, arXiv: 0904.4495. |
[34] |
E. Y. Sidky and X. C. Pan,
Accurate image reconstruction in circular cone-beam computed tomography by total variation minimization: A preliminary investigation, IEEE Nuclear Science Symposium Conference Record, 5 (2006), 2904-2907.
doi: 10.1109/NSSMIC.2006.356484. |
[35] |
E. Y. Sidky and X. C. Pan,
Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Physics in Medicine and Biology, 53 (2008), 3572277-3572284.
doi: 10.1088/0031-9155/53/17/021. |
[36] |
W. P. Segars, D. S. Lalush and B. M. Tsui,
A realistic splinebased dynamic heart phantom, IEEE Trans. Nucl. Sci., 46 (1999), 503-506.
doi: 10.1109/23.775570. |
[37] |
M. Storath, A. Weinmann, J. Frikeland and M. Unser,
Joint image reconstruction and segmentation using the Potts model, Inverse Problems, 31 (2015), 025003.
doi: 10.1088/0266-5611/31/2/025003. |
[38] |
J. Tang, J. Hsieh and G. H. Chen,
Temporal resolution improvement in cardiac CT using PICCS (TRI-PICCS): Performance studies, Medical Physics, 37 (2010), 4377-4388.
doi: 10.1118/1.3460318. |
[39] |
A. Tingberg,
X-ray tomosynthesis: A review of its use for breast and chest imaging, Radiation Protection Dosimetry, 139 (2010), 100-107.
doi: 10.1093/rpd/ncq099. |
[40] |
H. K. Tuy,
An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math., 43 (1983), 546-552.
doi: 10.1137/0143035. |
[41] |
Z. Wang, A. Bovik, H. Sheikhand and E. Simoncelli,
Image quality assessment: From error visibility to structural similarity, IEEE Trans Image Process, 13 (2004), 600-612.
doi: 10.1109/TIP.2003.819861. |
[42] |
Z. Wang, Z. Huang, Z. Chen, L. Zhang, X. Jiang, K. Kang, H. Yin, Z. Wang and M. Stampanoni, Low-dose multiple-information retrieval algorithm for x-ray grating-based imaging, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 635 (2011), 103-107.
doi: 10.1016/j.nima.2011.01.079. |
[43] |
F. Yang, Y. Shen and Z. S. Liu,
The proximal alternating iterative hard thresholding method for $\ell_{0}$ minimization, with complexity $O(\frac{1}{\sqrt{k}})$, Journal of Computational and Applied Mathematics, 311 (2017), 115-129.
doi: 10.1016/j.cam.2016.07.013. |
[44] |
W. Yu and L. Zeng,
$\ell_{0}$ gradient minimization based image reconstruction for limited-angle computed tomography, PLoS ONE, 10 (2015), e0130793.
doi: 10.1371/journal.pone.0130793. |
[45] |
L. Zeng, J. Q. Guo and B. D. Liu,
Limited-angle cone-beam computed tomography image reconstruction by total variation minimization and piecewise-constant modification, Journal of Inverse and Ill-Posed Problems, 21 (2013), 735-754.
doi: 10.1515/jip-2011-0010. |
[46] |
Y. Zhang, B. Dong and Z. S. Lu,
$\ell_{0}$ Minimization for wavelet frame based image restoration, Mathematics of Computation, 82 (2013), 995-1015.
doi: 10.1090/S0025-5718-2012-02631-7. |
[47] |
B. Zhao, H. Gao, H. Ding and S. Molloi,
Tight-frame based iterative image reconstruction for spectral breast CT, Medical Physics, 40 (2013), 031905.
doi: 10.1118/1.4790468. |
[48] |
W. Zhou, J. F. Cai and H. Gao,
Adaptive tight frame based medical image reconstruction: a proof-of-concept study for computed tomography, Inverse problems, 29 (2013), 1-18.
doi: 10.1088/0266-5611/29/12/125006. |
[49] |
chest phantom website,
http://lgdv.cs.fau.de/External/vollib/. |














The distance between source and object center | |
The angle interval of two adjacent projection views | |
The angle interval of two adjacent rays | |
The diameter of field of view | |
Detector numbers | |
Pixel size | |
Image size | |
The distance between source and object center | |
The angle interval of two adjacent projection views | |
The angle interval of two adjacent rays | |
The diameter of field of view | |
Detector numbers | |
Pixel size | |
Image size | |
Scanning ranges | Variances | Algorithm | RMSE | PSNR | MSSIM |
| | FBP | 8.403 | 29.64 | 0.9921 |
| FBP | 12.88 | 25.93 | 0.9800 | |
| our algorithm | 2.116 | 41.62 | 0.9995 | |
PICCS | 3.743 | 36.67 | 0.9985 | ||
| our algorithm | 4.747 | 34.60 | 0.9973 | |
PICCS | 5.953 | 32.64 | 0.9953 | ||
| | our algorithm | 2.240 | 41.12 | 0.9994 |
PICCS | 4.087 | 35.90 | 0.9984 | ||
| our algorithm | 4.258 | 35.55 | 0.9978 | |
PICCS | 4.915 | 32.69 | 0.9953 | ||
| | our algorithm | 1.843 | 42.82 | 0.9996 |
PICCS | 5.905 | 32.71 | 0.9956 | ||
| our algorithm | 3.566 | 37.09 | 0.9985 | |
PICCS | 6.250 | 32.21 | 0.9952 |
Scanning ranges | Variances | Algorithm | RMSE | PSNR | MSSIM |
| | FBP | 8.403 | 29.64 | 0.9921 |
| FBP | 12.88 | 25.93 | 0.9800 | |
| our algorithm | 2.116 | 41.62 | 0.9995 | |
PICCS | 3.743 | 36.67 | 0.9985 | ||
| our algorithm | 4.747 | 34.60 | 0.9973 | |
PICCS | 5.953 | 32.64 | 0.9953 | ||
| | our algorithm | 2.240 | 41.12 | 0.9994 |
PICCS | 4.087 | 35.90 | 0.9984 | ||
| our algorithm | 4.258 | 35.55 | 0.9978 | |
PICCS | 4.915 | 32.69 | 0.9953 | ||
| | our algorithm | 1.843 | 42.82 | 0.9996 |
PICCS | 5.905 | 32.71 | 0.9956 | ||
| our algorithm | 3.566 | 37.09 | 0.9985 | |
PICCS | 6.250 | 32.21 | 0.9952 |
The distance between source and object center | |
The angle interval of two adjacent projection views | |
The angle interval of two adjacent rays | |
The diameter of field of view | |
Detector numbers | |
Pixel size | |
Image size | |
The distance between source and object center | |
The angle interval of two adjacent projection views | |
The angle interval of two adjacent rays | |
The diameter of field of view | |
Detector numbers | |
Pixel size | |
Image size | |
Scanning ranges | Algorithm | RMSE | PSNR | MSSIM |
| FBP | 4.978 | 34.19 | 0.9961 |
| our algorithm | 3.607 | 36.99 | 0.9979 |
PICCS | 4.208 | 35.65 | 0.9972 | |
| our algorithm | 3.901 | 36.31 | 0.9976 |
PICCS | 4.190 | 35.69 | 0.9972 | |
| our algorithm | 3.693 | 36.78 | 0.9979 |
PICCS | 4.177 | 35.71 | 0.9972 |
Scanning ranges | Algorithm | RMSE | PSNR | MSSIM |
| FBP | 4.978 | 34.19 | 0.9961 |
| our algorithm | 3.607 | 36.99 | 0.9979 |
PICCS | 4.208 | 35.65 | 0.9972 | |
| our algorithm | 3.901 | 36.31 | 0.9976 |
PICCS | 4.190 | 35.69 | 0.9972 | |
| our algorithm | 3.693 | 36.78 | 0.9979 |
PICCS | 4.177 | 35.71 | 0.9972 |
| | | | | |
1 | 0.74 | 0.74 | 0 | 0 | 0 |
-1 | 0.5 | 0.5 | 0 | 0 | 0 |
-1 | 0.1 | 0.1 | 0.43 | 0.43 | 0 |
-1 | 0.1 | 0.1 | -0.43 | -0.43 | 0 |
-1 | 0.1 | 0.1 | -0.43 | 0.43 | 0 |
-1 | 0.1 | 0.1 | 0.43 | -0.43 | 0 |
-1 | 0.12 | 0.006 | 0.25 | 0.55 | -18 |
-1 | 0.08 | 0.006 | 0.25 | -0.55 | -240 |
-1 | 0.08 | 0.006 | -0.55 | 0.3 | 20 |
| | | | | |
1 | 0.74 | 0.74 | 0 | 0 | 0 |
-1 | 0.5 | 0.5 | 0 | 0 | 0 |
-1 | 0.1 | 0.1 | 0.43 | 0.43 | 0 |
-1 | 0.1 | 0.1 | -0.43 | -0.43 | 0 |
-1 | 0.1 | 0.1 | -0.43 | 0.43 | 0 |
-1 | 0.1 | 0.1 | 0.43 | -0.43 | 0 |
-1 | 0.12 | 0.006 | 0.25 | 0.55 | -18 |
-1 | 0.08 | 0.006 | 0.25 | -0.55 | -240 |
-1 | 0.08 | 0.006 | -0.55 | 0.3 | 20 |
Scanning ranges | Algorithm | RMSE | PSNR | MSSIM |
| our algorithm | 0.040 | 27.97 | 0.9949 |
PICCS | 0.057 | 28.26 | 0.9897 | |
| our algorithm | 0.0378 | 28.45 | 0.9954 |
PICCS | 0.0546 | 28.65 | 0.9906 |
Scanning ranges | Algorithm | RMSE | PSNR | MSSIM |
| our algorithm | 0.040 | 27.97 | 0.9949 |
PICCS | 0.057 | 28.26 | 0.9897 | |
| our algorithm | 0.0378 | 28.45 | 0.9954 |
PICCS | 0.0546 | 28.65 | 0.9906 |
Scanning ranges | Algorithm | RMSE | PSNR | MSSIM |
| 3.725 | 36.71 | 0.9978 | |
| 3.607 | 36.99 | 0.9979 | |
| 3.856 | 36.41 | 0.9977 | |
| 3.901 | 36.31 | 0.9976 |
Scanning ranges | Algorithm | RMSE | PSNR | MSSIM |
| 3.725 | 36.71 | 0.9978 | |
| 3.607 | 36.99 | 0.9979 | |
| 3.856 | 36.41 | 0.9977 | |
| 3.901 | 36.31 | 0.9976 |
Scanning ranges | Algorithm | RMSE | PSNR | MSSIM |
| 0.0404 | 27.86 | 0.9948 | |
| 0.0400 | 27.97 | 0.9949 | |
| 0.0383 | 28.34 | 0.9954 | |
| 0.0378 | 28.45 | 0.9954 |
Scanning ranges | Algorithm | RMSE | PSNR | MSSIM |
| 0.0404 | 27.86 | 0.9948 | |
| 0.0400 | 27.97 | 0.9949 | |
| 0.0383 | 28.34 | 0.9954 | |
| 0.0378 | 28.45 | 0.9954 |
Scanning ranges | Algorithm | RMSE | PSNR | MSSIM |
| FBP | 6.559 | 31.79 | 0.9965 |
| our algorithm | 4.543 | 34.98 | 0.9983 |
Scanning ranges | Algorithm | RMSE | PSNR | MSSIM |
| FBP | 6.559 | 31.79 | 0.9965 |
| our algorithm | 4.543 | 34.98 | 0.9983 |
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