American Institute of Mathematical Sciences

Advanced
December  2017, 11(6): 917-948. doi: 10.3934/ipi.2017043

Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data

Chengxiang Wang 1,2, , Li Zeng 2,3,, , Yumeng Guo 2, and  Lingli Zhang 2,

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

1The corresponding author: drlizeng@cqu.edu.cn

Received  December 2015 Revised  August 2017 Published  September 2017

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.

Keywords: Inverse problems, image reconstruction, wavelet transform, $\ell_0$ regularized, prior information.
Mathematics Subject Classification: Primary: 90C90, 15A29, 44A12; Secondary: 94A08.
Citation: Chengxiang Wang, Li Zeng, Yumeng Guo, Lingli Zhang. Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data. Inverse Problems & Imaging, 2017, 11 (6) : 917-948. doi: 10.3934/ipi.2017043
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5] T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-beam CT, 1nd edition, Springer-Verlag, Berlin Heidelberg, 2008.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  doi: 10.1016/S0168-1699(03)00041-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

P. C. Hansen, E. Y. Sidky and X. C. Pan, Accelerated gradient methods for total-variation-based CT image reconstruction, arXiv: 1105.4002. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

F. Natterer, The Mathmetics of Computed Tomography, 1nd edition, B. G. Teubner, Stuttgart., 1986. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

Z. Shen, Wavelet frames and image restorations, Proceedings of the International congress of Mathematicians, 4 (2010), 2834-2863.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

H. K. Tuy, An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math., 43 (1983), 546-552.  doi: 10.1137/0143035.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

chest phantom website, http://lgdv.cs.fau.de/External/vollib/. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5] T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-beam CT, 1nd edition, Springer-Verlag, Berlin Heidelberg, 2008.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

P. C. Hansen, E. Y. Sidky and X. C. Pan, Accelerated gradient methods for total-variation-based CT image reconstruction, arXiv: 1105.4002. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

F. Natterer, The Mathmetics of Computed Tomography, 1nd edition, B. G. Teubner, Stuttgart., 1986. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

Z. Shen, Wavelet frames and image restorations, Proceedings of the International congress of Mathematicians, 4 (2010), 2834-2863.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

H. K. Tuy, An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math., 43 (1983), 546-552.  doi: 10.1137/0143035.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

chest phantom website, http://lgdv.cs.fau.de/External/vollib/. Google Scholar

Figure 1.  Scanning geometry of limited-angle CT. $\textrm{S}$ denotes the X-ray source, $\textrm{o}$ denotes the rotation center of object, $\textrm{D}$ denotes the detector, and $\theta$ denotes the rotation angle which is less than $180^{0}$ plus a fan-angle
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Figure 2.  Reconstructed result of NCAT phantom using FBP algorithm for the scanning angle $[0,120^{0}]$. The limited-angle artifacts are labelled by the red rectangles
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Figure 3.  (a) and (b) are the reconstructed results using FBP algorithm for scanning range $[0,360^{0}]$ and $[0,120^{0}]$, respectively
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Figure 3 under B-spline frame">Figure 4.  The wavelet transform of (a) of Figure 3 under B-spline frame
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Figure 3 under B-spline frame">Figure 5.  The wavelet transform of (b) of Figure 3 under B-spline frame
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Figure 6.  NCAT phantom and prior image. The first column is NCAT phantom, the second column is the prior image and the third column is the absolute value of the difference between phantom and prior image. The prior image is the same with NCAT phantom
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Figure 7.  The reconstructed results for different scanning ranges using FBP algorithm, PICCS algorithm and our algorithm. The display window is $[0,255]$. The noise levels are $0.5\%\|g\|_{\infty}$
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Figure 8.  The reconstructed results for different scanning ranges using FBP algorithm, PICCS algorithm and our algorithm. The display window is $[0,255]$. The noise levels are $1\%\|g\|_{\infty}$
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Figure 9.  Phantom, prior image, and the absolute value of the difference between phantom and prior image. Three circles are labelled by red rectangles
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Figure 10.  The reconstructed results for the different scanning ranges using FBP algorithm, PICCS algorithm and our algorithm. Three circles are labelled by red rectangles. The display window is $[0,255]$
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Figure 11.  Phantom and prior image. The first column is the phantom, the subsequent columns are the prior image and the absolute value of the difference between phantom and prior image
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Figure 12.  The reconstructed results using PICCS algorithm and our algorithm. The display window for the first row is $[0,1]$ and for the second row is $[0.8, 0.9]$
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Figure 13.  The reconstructed results for the different scanning ranges using $\ell_{1}-\ell_{0}$ algorithm and $\ell_{2}-\ell_{0}$ algorithm. Three circles are labelled by red rectangles. The display window is $[0,255]$
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Figure 14.  The reconstructed results of simulation phantom for the different scanning ranges using $\ell_{1}-\ell_{0}$ algorithm and $\ell_{2}-\ell_{0}$ algorithm. The display window for the first row is $[0,1]$ and for the second row is $[0.8, 0.9]$
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Figure 15.  The reconstructed results of gear using FBP algorithm and our algorithm. ROI is labelled by red rectangle. The display window is $[0, 0.0096]mm^{-1}$
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Figure 15">Figure 16.  The zoom-in view of ROI of Figure 15
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Table 1.  Geometrical scanning parameters of simulated CT system
The distance between source and object center$981mm$
The angle interval of two adjacent projection views $1^{0}$
The angle interval of two adjacent rays $0.00329^{0}$
The diameter of field of view $143.6222mm$
Detector numbers $256$
Pixel size $0.5632\times0.5632mm^{2}$
Image size $256\times256$
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The distance between source and object center$981mm$
The angle interval of two adjacent projection views $1^{0}$
The angle interval of two adjacent rays $0.00329^{0}$
The diameter of field of view $143.6222mm$
Detector numbers $256$
Pixel size $0.5632\times0.5632mm^{2}$
Image size $256\times256$
Table 2.  Quantitatively characterize the reconstruction quality
Scanning rangesVariancesAlgorithmRMSEPSNRMSSIM
$0\sim 360^{0}$ $0.5\% \|g\|_{\infty}$FBP8.40329.640.9921
$1\% \|g\|_{\infty}$FBP12.8825.930.9800
$0\sim 100^{0}$ $0.5\% \|g\|_{\infty}$our algorithm2.11641.620.9995
PICCS3.74336.670.9985
$1\% \|g\|_{\infty}$our algorithm4.74734.600.9973
PICCS5.95332.640.9953
$0\sim 80^{0}$ $0.5\% \|g\|_{\infty}$our algorithm2.24041.120.9994
PICCS4.08735.900.9984
$1\% \|g\|_{\infty}$our algorithm4.25835.550.9978
PICCS4.91532.690.9953
$0\sim 60^{0}$ $0.5\% \|g\|_{\infty}$our algorithm1.84342.820.9996
PICCS5.90532.710.9956
$1\% \|g\|_{\infty}$our algorithm3.56637.090.9985
PICCS6.25032.210.9952
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Scanning rangesVariancesAlgorithmRMSEPSNRMSSIM
$0\sim 360^{0}$ $0.5\% \|g\|_{\infty}$FBP8.40329.640.9921
$1\% \|g\|_{\infty}$FBP12.8825.930.9800
$0\sim 100^{0}$ $0.5\% \|g\|_{\infty}$our algorithm2.11641.620.9995
PICCS3.74336.670.9985
$1\% \|g\|_{\infty}$our algorithm4.74734.600.9973
PICCS5.95332.640.9953
$0\sim 80^{0}$ $0.5\% \|g\|_{\infty}$our algorithm2.24041.120.9994
PICCS4.08735.900.9984
$1\% \|g\|_{\infty}$our algorithm4.25835.550.9978
PICCS4.91532.690.9953
$0\sim 60^{0}$ $0.5\% \|g\|_{\infty}$our algorithm1.84342.820.9996
PICCS5.90532.710.9956
$1\% \|g\|_{\infty}$our algorithm3.56637.090.9985
PICCS6.25032.210.9952
Table 3.  Geometrical scanning parameters of simulated CT system
The distance between source and object center$981mm$
The angle interval of two adjacent projection views $0.703^{0}$
The angle interval of two adjacent rays $0.0005^{0}$
The diameter of field of view $279.5mm$
Detector numbers $560$
Pixel size $0.5\times0.5mm^{2}$
Image size $512\times512$
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The distance between source and object center$981mm$
The angle interval of two adjacent projection views $0.703^{0}$
The angle interval of two adjacent rays $0.0005^{0}$
The diameter of field of view $279.5mm$
Detector numbers $560$
Pixel size $0.5\times0.5mm^{2}$
Image size $512\times512$
Table 4.  Quantitatively characterize the reconstruction quality
Scanning rangesAlgorithmRMSEPSNRMSSIM
$0\sim 360^{0}$FBP4.97834.190.9961
$0\sim 120^{0}$our algorithm3.60736.990.9979
PICCS4.20835.650.9972
$0\sim 100^{0}$our algorithm3.90136.310.9976
PICCS4.19035.690.9972
$0\sim 80^{0}$our algorithm3.69336.780.9979
PICCS4.17735.710.9972
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Scanning rangesAlgorithmRMSEPSNRMSSIM
$0\sim 360^{0}$FBP4.97834.190.9961
$0\sim 120^{0}$our algorithm3.60736.990.9979
PICCS4.20835.650.9972
$0\sim 100^{0}$our algorithm3.90136.310.9976
PICCS4.19035.690.9972
$0\sim 80^{0}$our algorithm3.69336.780.9979
PICCS4.17735.710.9972
Table 5.  The parameters of simulated phantom
$I$$h$ $v$ $x_{0}$ $y_{0}$ $r$
10.740.74000
-10.50.5000
-10.10.10.430.430
-10.10.1-0.43-0.430
-10.10.1-0.430.430
-10.10.10.43-0.430
-10.120.0060.250.55-18
-10.080.0060.25-0.55-240
-10.080.006-0.550.320
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$I$$h$ $v$ $x_{0}$ $y_{0}$ $r$
10.740.74000
-10.50.5000
-10.10.10.430.430
-10.10.1-0.43-0.430
-10.10.1-0.430.430
-10.10.10.43-0.430
-10.120.0060.250.55-18
-10.080.0060.25-0.55-240
-10.080.006-0.550.320
Table 6.  Quantitatively characterize the reconstruction quality
Scanning rangesAlgorithmRMSEPSNRMSSIM
$0\sim 120^{0}$our algorithm0.04027.970.9949
PICCS0.05728.260.9897
$0\sim 100^{0}$our algorithm0.037828.450.9954
PICCS0.054628.650.9906
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Scanning rangesAlgorithmRMSEPSNRMSSIM
$0\sim 120^{0}$our algorithm0.04027.970.9949
PICCS0.05728.260.9897
$0\sim 100^{0}$our algorithm0.037828.450.9954
PICCS0.054628.650.9906
Table 7.  Quantitatively characterize the reconstruction quality
Scanning rangesAlgorithmRMSEPSNRMSSIM
$0\sim 120^{0}$$\ell_{1}-\ell_{0}$3.72536.710.9978
$0\sim 120^{0}$$\ell_{2}-\ell_{0}$3.60736.990.9979
$0\sim 100^{0}$$\ell_{1}-\ell_{0}$3.85636.410.9977
$0\sim 100^{0}$$\ell_{2}-\ell_{0}$3.90136.310.9976
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Scanning rangesAlgorithmRMSEPSNRMSSIM
$0\sim 120^{0}$$\ell_{1}-\ell_{0}$3.72536.710.9978
$0\sim 120^{0}$$\ell_{2}-\ell_{0}$3.60736.990.9979
$0\sim 100^{0}$$\ell_{1}-\ell_{0}$3.85636.410.9977
$0\sim 100^{0}$$\ell_{2}-\ell_{0}$3.90136.310.9976
Table 8.  Quantitatively characterize the reconstruction quality
Scanning rangesAlgorithmRMSEPSNRMSSIM
$0\sim 120^{0}$$\ell_{1}-\ell_{0}$0.040427.860.9948
$0\sim 120^{0}$$\ell_{2}-\ell_{0}$0.040027.970.9949
$0\sim 100^{0}$$\ell_{1}-\ell_{0}$0.038328.340.9954
$0\sim 100^{0}$$\ell_{2}-\ell_{0}$0.037828.450.9954
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Scanning rangesAlgorithmRMSEPSNRMSSIM
$0\sim 120^{0}$$\ell_{1}-\ell_{0}$0.040427.860.9948
$0\sim 120^{0}$$\ell_{2}-\ell_{0}$0.040027.970.9949
$0\sim 100^{0}$$\ell_{1}-\ell_{0}$0.038328.340.9954
$0\sim 100^{0}$$\ell_{2}-\ell_{0}$0.037828.450.9954
Table 9.  Quantitatively characterize the reconstruction quality
Scanning rangesAlgorithmRMSEPSNRMSSIM
$0\sim 360^{0}$FBP6.55931.790.9965
$0\sim 80^{0}$our algorithm4.54334.980.9983
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Scanning rangesAlgorithmRMSEPSNRMSSIM
$0\sim 360^{0}$FBP6.55931.790.9965
$0\sim 80^{0}$our algorithm4.54334.980.9983
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