# American Institute of Mathematical Sciences

June 2018, 12(3): 545-572. doi: 10.3934/ipi.2018024

## Existence and convergence analysis of $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT reconstruction

 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 4 School of Biomedical Engineering, Hubei University of Science and Technology, Xianning 437100, China

* Corresponding author: drlizeng@cqu.edu.cn

Received  April 2017 Revised  December 2017 Published  March 2018

Fund Project: Li Zeng was supported by the National Natural Science Foundation of China (No.61771003) and the National Instrumentation Program of China (2013YQ030629). Liwei Xu was supported in part by a Key Project of the Major Research Plan of NSFC (No.91630205) and by the National Natural Science Foundation of China (No.11771068). Wei Yu was supported by the National Natural Science Foundation of China (No.61701174), the Hubei Provincial Natural Science Foundation of China (No.2017CFB168) and the Ph.D. start-up Fund of HBUST (No. BK1527)

In some practical applications of computed tomography (CT) imaging, the projections of an object are obtained within a limited-angle range due to the restriction of the scanning environment. In this situation, conventional analytic algorithms, such as filtered backprojection (FBP), will not work because the projections are incomplete. An image reconstruction algorithm based on total variation minimization (TVM) can significantly reduce streak artifacts in sparse-view reconstruction, but it will not effectively suppress slope artifacts when dealing with limited-angle reconstruction problems. To solve this problem, we consider a family of image reconstruction model based on $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT and prove the existence of a solution for two CT reconstruction models. The Alternating Direction Method of Multipliers (ADMM)-like method is utilized to solve our model. Furthermore, we prove the convergence of our algorithm under certain conditions. Some numerical experiments are used to evaluate the performance of our algorithm and the results indicate that our algorithm has advantage in suppressing slope artifacts.

Citation: Chengxiang Wang, Li Zeng, Wei Yu, Liwei Xu. Existence and convergence analysis of $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT reconstruction. Inverse Problems & Imaging, 2018, 12 (3) : 545-572. doi: 10.3934/ipi.2018024
##### References:
 [1] M. A. Anastasio, E. Y. Sidky, X. Pan and C. Y. Chou, Boundary reconstruction in limited-angle x-ray phase-contrast tomography, Medical Imaging 2009: Physics of Medical Imaging, 7258 (2009), 725827. doi: 10.1117/12.811918. [2] A. H. Andersen and A. C. Kak, Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm, Ultrasonic Imaging, 6 (1984), 81-94. doi: 10.1016/0161-7346(84)90008-7. [3] A. Auslender and M. Teboulle, Asymptotic cones and functions in optimization and variational inequalities, Springer, Berlin, 2006. [4] 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, Medical Physics, 34 (2007), 4664-4677. doi: 10.1118/1.2799492. [5] T. Blumensath, Accelerated iterative hard thresholding, Signal Processing, 92 (2012), 752-756. doi: 10.1016/j.sigpro.2011.09.017. [6] T. Blumensath and M. E. Davies, Iterative thresholding for sparse approximations, Journal of Fourier Analysis and Applications, 14 (2008), 629-654. doi: 10.1007/s00041-008-9035-z. [7] T. Blumensath and M. E. Davies, Iterative hard thresholding for compressed sensing, Applied and Computational Harmonic Analysis, 27 (2009), 265-274. doi: 10.1016/j.acha.2009.04.002. [8] 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. [9] K. Bredies, D. A. Lorenz and S. Reiterer, Minimization of non-smooth, non-convex functionals by iterative thresholding, Journal of Optimization Theory and Applications, 165 (2015), 78-112. doi: 10.1007/s10957-014-0614-7. [10] M. Burger, J. Müller, E. Papoutsellis and C. B. Schonlieb, Total variation regularization in measurement and image space for PET reconstruction, Inverse Problems, 30 (2014), 105003. doi: 10.1088/0266-5611/30/10/105003. [11] T. M. Buzug, Computed tomography: From photon statistics to modern cone-beam CT, Springer Handbook of Medical Technology, (2008), 311-342. doi: 10.1007/978-3-540-74658-4_16. [12] J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling & Simulation, 8 (2009), 337-369. [13] Y. Censor and A. Segal, Iterative projection methods in biomedical inverse problems, Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), 10 (2008), 65-96. [14] C. Chen, R. H. Chan, S. Ma and J. Yang, Inertial proximal ADMM for linearly constrained separable convex optimization, SIAM Journal on Imaging Sciences, 8 (2015), 2239-2267. [15] Z. Q. Chen, X. Jin, L. Li and G. Wang, A limited-angle CT reconstruction method based on anisotropic TV minimization, Physics in Medicine and Biology, 58 (2013), 2119-2141. doi: 10.1088/0031-9155/58/7/2119. [16] M. K. Cho, H. K. Kim, H. Youn and S. S. Kim, A feasibility study of digital tomosynthesis for volumetric dental imaging, J. Instrum., 7 (2012), 1-6. doi: 10.1088/1748-0221/7/03/P03007. [17] I. Daubechies, M. Defrise and M. C. De, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Communications on Pure and Applied Mathematics, 57 (2004), 1413-1457. doi: 10.1002/cpa.20042. [18] I. Daubechies, Ten Lectures on Wavelets, 1nd edition, Society for industrial and applied mathematics, Philadelphia, 1992. doi: 10.1137/1.9781611970104.fm. [19] I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Applied and Computational Harmonic Analysis, 14 (2003), 1-46. doi: 10.1016/S1063-5203(02)00511-0. [20] B. Dong and Y. Zhang, An efficient algorithm for $\ell_{0}$ minimization in wavelet frame based image restoration, Journal of Scientific Computing, 54 (2013), 350-368. doi: 10.1007/s10915-012-9597-4. [21] M. Elad, J. L. Starck, P. Querre and D. L. Donoho, Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA), Applied and Computational Harmonic Analysis, 19 (2005), 340-358. doi: 10.1016/j.acha.2005.03.005. [22] M. Filipović and A. Jukić, Restoration of images corrupted by mixed Gaussian-impulse noise by iterative soft-hard thresholding, In Signal Processing Conference (EUSIPCO), 2014 Proceedings of the 22nd European, (2014), 1637-1641. [23] J. Frikel and E. T. Quinto, Characterization and reduction of artifacts in limited angle tomography, Inverse Problems, 29(2013), 125007. doi: 10.1088/0266-5611/29/12/125007. [24] H. Gao, J. F. Cai, Z. W. Shen and H. Zhao, Robust principal component analysis-based four-dimensional computed tomography, Physics in Medicine and Biology, 56 (2011), 3781-3798. doi: 10.1088/0031-9155/56/11/002. [25] 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. [26] H. Gao, H. Y. Yu, S. Osher and G. Wang, Multi-energy CT based on a prior rank, intensity and sparsity model (PRISM), Inverse Problems, 27 (2011), 1-22. doi: 10.1088/0266-5611/27/11/115012. [27] 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), 057003-057003. [28] T. Goldstein and S. Osher, The split Bregman method for $\ell_1$ regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [29] R. Gordon, A tutorial on ART (algebraic reconstruction techniques), IEEE Transactions on Nuclear Science, 21 (1974), 78-93. doi: 10.1109/TNS.1974.6499238. [30] B. Han, On dual wavelet tight frames, Applied and Computational Harmonic Analysis, 4 (1997), 380-413. doi: 10.1006/acha.1997.0217. [31] X. Han, J. Bian, E. L. Ritman, E. Y. Sidky and X. Pan, Optimization-based reconstruRit-manction of sparse images from few-view projections, Physics in Medicine and Biology, 57 (2012), p5245. doi: 10.1088/0031-9155/57/16/5245. [32] B. S. He, A class of projection and contraction methods for monotone variational inequalities, Applied Mathematics and Optimization, 35 (1997), 69-76. doi: 10.1007/BF02683320. [33] B. S. He and M. H. Xu, A general framework of contraction methods for monotone variational inequalities, Pacific Journal of Optimization, 4 (2008), 195-212. [34] K. Ito and K. Kunisch, A note on the existence of nonsmooth nonconvex optimization problems, Journal of Optimization Theory and Applications, 163 (2014), 697-706. doi: 10.1007/s10957-014-0552-4. [35] X. Jia, B. Dong, Y. Lou and S. B. jiang, GPU-based iterative cone-beam CT reconstruction using tight frame regularization, Physics in Medicine and Biology, 56 (2010), 3787-3806. doi: 10.1088/0031-9155/56/13/004. [36] M. Jiang and G. Wang, Development of iterative algorithms for image reconstruction, Journal of X-ray Science and Technology, 10 (2001), 77-86. [37] M. Jiang and G. Wang, Convergence of the simultaneous algebraic reconstruction technique (SART), IEEE Transactions on Image Processing, 12 (2003), 957-961. doi: 10.1109/TIP.2003.815295. [38] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, Medical physics, IEEE Press, New York, 1988. [39] V. Kolehmainen, S. Siltanen, S. Järvenpää, J. P. Kaipio, P. Koistinenand, M. Lassas, J. Pirttilä and E. Somersalo, Statistical inversion for medical x-ray tomography with few radiographs: Ⅱ. Application to dental radiology, Physics in Medicine and Biology, 48 (2003), 1465-1490. doi: 10.1088/0031-9155/48/10/315. [40] H. Kudo, F. Noo, M. Defrise and R. Clackdoyle, New super-short-scan algorithms for fan-beam and cone-beam reconstruction, Nuclear Science Symposium Conference Record, 2002 IEEE, 2 (2002), 902-906. doi: 10.1109/NSSMIC.2002.1239470. [41] S. J. LaRoque, E. Y. Sidky and X. 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. [42] 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. [43] X. Lu, Y. Sun and Y. Yuan, Optimization for limited angle tomography in medical image processing, Pattern Recognition, 44 (2011), 2427-2435. doi: 10.1016/j.patcog.2010.12.016. [44] F. Noo and D. J. Heuscher, Image reconstruction from cone-beam data on a circular short-scan, Medical Imaging 2002: Image Processing, 4684 (2002), 50-59. doi: 10.1117/12.467199. [45] F. Noo, M. Defrise, R. Clackdoyle and H. Kudo, Image reconstruction from fan-beam projections on less than a short scan, Physics in Medicine and Biology, 47 (2002), 2525-2546. doi: 10.1088/0031-9155/47/14/311. [46] X. Pan, E. Y. Sidky and M. Vannier, Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?, Inverse Problems, 25 (2009), 123009. doi: 10.1088/0266-5611/25/12/123009. [47] C. Ravazzi, S. M. Fosson and E. Magli, Distributed iterative thresholding for L0/L1-regularized linear inverse problems, IEEE Transactions on Information Theory, 61 (2015), 2081-2100. doi: 10.1109/TIT.2015.2403263. [48] R. T. Rockafellarr and R. J. B. Wets, Variational Analysis, 1nd edition, Springer, Berlin, 2009. doi: 10.1007/978-3-642-02431-3. [49] A. Ron and Z. Shen, Affine systems in $L_{2}(R^{d})$ Ⅱ: Dual systems, Journal of Fourier Analysis and Applications, 3 (1997), 617-637. doi: 10.1007/BF02648888. [50] W. P. Segars, D. S. Lalush and B. M. W. Tsui, A realistic spline-based dynamic heart phantom, IEEE Transactions on Nuclear Science, 46 (1999), 503-506. doi: 10.1109/NSSMIC.1998.774369. [51] M. M. Seger 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. [52] R. L. Siddon, Fast calculation of the exact radiological path for a three-dimensional CT array, Medical Physics, 12 (1985), 252-255. doi: 10.1118/1.595715. [53] E. Y. Sidky and X. Pan, Accurate image reconstruction in circular cone-beam computed tomography by total variation minimization: a preliminary investigation, In Nuclear Science Symposium Conference Record, 5 (2006), 2904-2907. doi: 10.1109/NSSMIC.2006.356484. [54] 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), 4777. doi: 10.1088/0031-9155/53/17/021. [55] E. Soubies, L. Blanc-Féraud and G. Aubert, A Continuous Exact $\ell_{0}$ Penalty (CEL0) for Least Squares Regularized Problem, SIAM Journal on Imaging Sciences, 8 (2015), 1607-1639. doi: 10.1137/151003714. [56] C. Soussen, J. Idier, J. Duan and D. Brie, Homotopy Based Algorithms for-Regularized Least-Squares, IEEE Transactions on Signal Processing, 63 (2015), 3301-3316. doi: 10.1109/TSP.2015.2421476. [57] J. L. Starck, M. Elad and D. L. Donoho, Image decomposition via the combination of sparse representations and a variational approach, IEEE Transactions on Image Processing, 14 (2005), 1570-1582. doi: 10.1109/TIP.2005.852206. [58] M. Storath, A. Weinmann, J. Frikel 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. [59] 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. [60] C. Wang and L. Zeng, Error bounds and stability in the $\ell_{0}$ regularized for CT reconstruction from small projections, Inverse Problems and Imaging, 10 (2016), 829-853. doi: 10.3934/ipi.2016023. [61] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861. [62] Y. Xiao, T. Zeng, J. Yu and M. K. Ng, Restoration of images corrupted by mixed Gaussian-impulse noise via l1-l0 minimization, Pattern Recognition, 44 (2011), 1708-1720. [63] G. L. Zeng, Medical Image Reconstruction, 1nd edition, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-05368-9. [64] 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. [65] 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. [66] 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. [67] 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.

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##### References:
 [1] M. A. Anastasio, E. Y. Sidky, X. Pan and C. Y. Chou, Boundary reconstruction in limited-angle x-ray phase-contrast tomography, Medical Imaging 2009: Physics of Medical Imaging, 7258 (2009), 725827. doi: 10.1117/12.811918. [2] A. H. Andersen and A. C. Kak, Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm, Ultrasonic Imaging, 6 (1984), 81-94. doi: 10.1016/0161-7346(84)90008-7. [3] A. Auslender and M. Teboulle, Asymptotic cones and functions in optimization and variational inequalities, Springer, Berlin, 2006. [4] 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, Medical Physics, 34 (2007), 4664-4677. doi: 10.1118/1.2799492. [5] T. Blumensath, Accelerated iterative hard thresholding, Signal Processing, 92 (2012), 752-756. doi: 10.1016/j.sigpro.2011.09.017. [6] T. Blumensath and M. E. Davies, Iterative thresholding for sparse approximations, Journal of Fourier Analysis and Applications, 14 (2008), 629-654. doi: 10.1007/s00041-008-9035-z. [7] T. Blumensath and M. E. Davies, Iterative hard thresholding for compressed sensing, Applied and Computational Harmonic Analysis, 27 (2009), 265-274. doi: 10.1016/j.acha.2009.04.002. [8] 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. [9] K. Bredies, D. A. Lorenz and S. Reiterer, Minimization of non-smooth, non-convex functionals by iterative thresholding, Journal of Optimization Theory and Applications, 165 (2015), 78-112. doi: 10.1007/s10957-014-0614-7. [10] M. Burger, J. Müller, E. Papoutsellis and C. B. Schonlieb, Total variation regularization in measurement and image space for PET reconstruction, Inverse Problems, 30 (2014), 105003. doi: 10.1088/0266-5611/30/10/105003. [11] T. M. Buzug, Computed tomography: From photon statistics to modern cone-beam CT, Springer Handbook of Medical Technology, (2008), 311-342. doi: 10.1007/978-3-540-74658-4_16. [12] J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling & Simulation, 8 (2009), 337-369. [13] Y. Censor and A. Segal, Iterative projection methods in biomedical inverse problems, Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), 10 (2008), 65-96. [14] C. Chen, R. H. Chan, S. Ma and J. Yang, Inertial proximal ADMM for linearly constrained separable convex optimization, SIAM Journal on Imaging Sciences, 8 (2015), 2239-2267. [15] Z. Q. Chen, X. Jin, L. Li and G. Wang, A limited-angle CT reconstruction method based on anisotropic TV minimization, Physics in Medicine and Biology, 58 (2013), 2119-2141. doi: 10.1088/0031-9155/58/7/2119. [16] M. K. Cho, H. K. Kim, H. Youn and S. S. Kim, A feasibility study of digital tomosynthesis for volumetric dental imaging, J. Instrum., 7 (2012), 1-6. doi: 10.1088/1748-0221/7/03/P03007. [17] I. Daubechies, M. Defrise and M. C. De, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Communications on Pure and Applied Mathematics, 57 (2004), 1413-1457. doi: 10.1002/cpa.20042. [18] I. Daubechies, Ten Lectures on Wavelets, 1nd edition, Society for industrial and applied mathematics, Philadelphia, 1992. doi: 10.1137/1.9781611970104.fm. [19] I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Applied and Computational Harmonic Analysis, 14 (2003), 1-46. doi: 10.1016/S1063-5203(02)00511-0. [20] B. Dong and Y. Zhang, An efficient algorithm for $\ell_{0}$ minimization in wavelet frame based image restoration, Journal of Scientific Computing, 54 (2013), 350-368. doi: 10.1007/s10915-012-9597-4. [21] M. Elad, J. L. Starck, P. Querre and D. L. Donoho, Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA), Applied and Computational Harmonic Analysis, 19 (2005), 340-358. doi: 10.1016/j.acha.2005.03.005. [22] M. Filipović and A. Jukić, Restoration of images corrupted by mixed Gaussian-impulse noise by iterative soft-hard thresholding, In Signal Processing Conference (EUSIPCO), 2014 Proceedings of the 22nd European, (2014), 1637-1641. [23] J. Frikel and E. T. Quinto, Characterization and reduction of artifacts in limited angle tomography, Inverse Problems, 29(2013), 125007. doi: 10.1088/0266-5611/29/12/125007. [24] H. Gao, J. F. Cai, Z. W. Shen and H. Zhao, Robust principal component analysis-based four-dimensional computed tomography, Physics in Medicine and Biology, 56 (2011), 3781-3798. doi: 10.1088/0031-9155/56/11/002. [25] 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. [26] H. Gao, H. Y. Yu, S. Osher and G. Wang, Multi-energy CT based on a prior rank, intensity and sparsity model (PRISM), Inverse Problems, 27 (2011), 1-22. doi: 10.1088/0266-5611/27/11/115012. [27] 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), 057003-057003. [28] T. Goldstein and S. Osher, The split Bregman method for $\ell_1$ regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [29] R. Gordon, A tutorial on ART (algebraic reconstruction techniques), IEEE Transactions on Nuclear Science, 21 (1974), 78-93. doi: 10.1109/TNS.1974.6499238. [30] B. Han, On dual wavelet tight frames, Applied and Computational Harmonic Analysis, 4 (1997), 380-413. doi: 10.1006/acha.1997.0217. [31] X. Han, J. Bian, E. L. Ritman, E. Y. Sidky and X. Pan, Optimization-based reconstruRit-manction of sparse images from few-view projections, Physics in Medicine and Biology, 57 (2012), p5245. doi: 10.1088/0031-9155/57/16/5245. [32] B. S. He, A class of projection and contraction methods for monotone variational inequalities, Applied Mathematics and Optimization, 35 (1997), 69-76. doi: 10.1007/BF02683320. [33] B. S. He and M. H. Xu, A general framework of contraction methods for monotone variational inequalities, Pacific Journal of Optimization, 4 (2008), 195-212. [34] K. Ito and K. Kunisch, A note on the existence of nonsmooth nonconvex optimization problems, Journal of Optimization Theory and Applications, 163 (2014), 697-706. doi: 10.1007/s10957-014-0552-4. [35] X. Jia, B. Dong, Y. Lou and S. B. jiang, GPU-based iterative cone-beam CT reconstruction using tight frame regularization, Physics in Medicine and Biology, 56 (2010), 3787-3806. doi: 10.1088/0031-9155/56/13/004. [36] M. Jiang and G. Wang, Development of iterative algorithms for image reconstruction, Journal of X-ray Science and Technology, 10 (2001), 77-86. [37] M. Jiang and G. Wang, Convergence of the simultaneous algebraic reconstruction technique (SART), IEEE Transactions on Image Processing, 12 (2003), 957-961. doi: 10.1109/TIP.2003.815295. [38] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, Medical physics, IEEE Press, New York, 1988. [39] V. Kolehmainen, S. Siltanen, S. Järvenpää, J. P. Kaipio, P. Koistinenand, M. Lassas, J. Pirttilä and E. Somersalo, Statistical inversion for medical x-ray tomography with few radiographs: Ⅱ. Application to dental radiology, Physics in Medicine and Biology, 48 (2003), 1465-1490. doi: 10.1088/0031-9155/48/10/315. [40] H. Kudo, F. Noo, M. Defrise and R. Clackdoyle, New super-short-scan algorithms for fan-beam and cone-beam reconstruction, Nuclear Science Symposium Conference Record, 2002 IEEE, 2 (2002), 902-906. doi: 10.1109/NSSMIC.2002.1239470. [41] S. J. LaRoque, E. Y. Sidky and X. 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. [42] 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. [43] X. Lu, Y. Sun and Y. Yuan, Optimization for limited angle tomography in medical image processing, Pattern Recognition, 44 (2011), 2427-2435. doi: 10.1016/j.patcog.2010.12.016. [44] F. Noo and D. J. Heuscher, Image reconstruction from cone-beam data on a circular short-scan, Medical Imaging 2002: Image Processing, 4684 (2002), 50-59. doi: 10.1117/12.467199. [45] F. Noo, M. Defrise, R. Clackdoyle and H. Kudo, Image reconstruction from fan-beam projections on less than a short scan, Physics in Medicine and Biology, 47 (2002), 2525-2546. doi: 10.1088/0031-9155/47/14/311. [46] X. Pan, E. Y. Sidky and M. Vannier, Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?, Inverse Problems, 25 (2009), 123009. doi: 10.1088/0266-5611/25/12/123009. [47] C. Ravazzi, S. M. Fosson and E. Magli, Distributed iterative thresholding for L0/L1-regularized linear inverse problems, IEEE Transactions on Information Theory, 61 (2015), 2081-2100. doi: 10.1109/TIT.2015.2403263. [48] R. T. Rockafellarr and R. J. B. Wets, Variational Analysis, 1nd edition, Springer, Berlin, 2009. doi: 10.1007/978-3-642-02431-3. [49] A. Ron and Z. Shen, Affine systems in $L_{2}(R^{d})$ Ⅱ: Dual systems, Journal of Fourier Analysis and Applications, 3 (1997), 617-637. doi: 10.1007/BF02648888. [50] W. P. Segars, D. S. Lalush and B. M. W. Tsui, A realistic spline-based dynamic heart phantom, IEEE Transactions on Nuclear Science, 46 (1999), 503-506. doi: 10.1109/NSSMIC.1998.774369. [51] M. M. Seger 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. [52] R. L. Siddon, Fast calculation of the exact radiological path for a three-dimensional CT array, Medical Physics, 12 (1985), 252-255. doi: 10.1118/1.595715. [53] E. Y. Sidky and X. Pan, Accurate image reconstruction in circular cone-beam computed tomography by total variation minimization: a preliminary investigation, In Nuclear Science Symposium Conference Record, 5 (2006), 2904-2907. doi: 10.1109/NSSMIC.2006.356484. [54] 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), 4777. doi: 10.1088/0031-9155/53/17/021. [55] E. Soubies, L. Blanc-Féraud and G. Aubert, A Continuous Exact $\ell_{0}$ Penalty (CEL0) for Least Squares Regularized Problem, SIAM Journal on Imaging Sciences, 8 (2015), 1607-1639. doi: 10.1137/151003714. [56] C. Soussen, J. Idier, J. Duan and D. Brie, Homotopy Based Algorithms for-Regularized Least-Squares, IEEE Transactions on Signal Processing, 63 (2015), 3301-3316. doi: 10.1109/TSP.2015.2421476. [57] J. L. Starck, M. Elad and D. L. Donoho, Image decomposition via the combination of sparse representations and a variational approach, IEEE Transactions on Image Processing, 14 (2005), 1570-1582. doi: 10.1109/TIP.2005.852206. [58] M. Storath, A. Weinmann, J. Frikel 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. [59] 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. [60] C. Wang and L. Zeng, Error bounds and stability in the $\ell_{0}$ regularized for CT reconstruction from small projections, Inverse Problems and Imaging, 10 (2016), 829-853. doi: 10.3934/ipi.2016023. [61] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861. [62] Y. Xiao, T. Zeng, J. Yu and M. K. Ng, Restoration of images corrupted by mixed Gaussian-impulse noise via l1-l0 minimization, Pattern Recognition, 44 (2011), 1708-1720. [63] G. L. Zeng, Medical Image Reconstruction, 1nd edition, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-05368-9. [64] 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. [65] 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. [66] 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. [67] 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.
The scanning geometry of limited-angle CT. $T$ denotes the objective table, $\textrm{O}^{'}$ denotes the scanned object, $\textrm{D}$ denotes the detector array, $\textrm{o}$ denotes the rotation center of X-ray source, $\textrm{S}$ denotes X-ray source that rotates anticlockwise around $\textrm{o}$ with $\textrm{D}$ synchronously and $\theta$ denotes the rotation angle which is less than $180^{0}$ plus a fan-angle
The reconstructed results of the NCAT phantom. The image on the first row is the original image (or reference image). The subsequent rows are the results reconstructed for the scan ranges $[0,120^{0}]$ and $[0,140^{0}]$, respectively. The images from left to right in each column are the results reconstructed using the SART algorithm, the ASD-POCS algorithm and our algorithm. The location of red arrows present slope artifacts. The grey-scale display window is $[0.2, 0.85]$
The reconstructed results of the practical gear data. The images on the left column are the results reconstructed using the SART algorithm. The subsequent columns are the ASD-POCS algorithm and our algorithm, respectively. The images from top to bottom in each row present the results reconstructed from scanning ranges $[0,100^{0}]$, $[0,120^{0}]$, $[0,140^{0}]$, and $[0,160^{0}]$. The location of red arrows present slope artifacts. The display window is $[0.005, 0.0068]$ $mm^{-1}$
The transform results of image under the one level piecewise-constant linear B-spline framelets transform
Sketch map of slope artifacts correction for the limited-angle CT image reconstruction if some proper parameters are chose. The wavelet tight framlet transform is piecewise-constant linear B-spline framelets transform
The images from left to right present the Shepp-Logan phantom, the reconstructed result using the $\ell_{0}$ and the reconstructed result $\ell_{0}-\ell_{2}$ regularization. The display window is $[0, 1]$
The reconstructed image from the projection data of the scanning angular ranges $[0,360^{0}]$ using the FBP algorithm. Metal artifacts are labelled by red rectangles
The reconstructed image from the projection data of the scanning angular ranges $[0,180^{0}]$ using the ASD-POCS algorithm and our algorithm. ROIs are labelled by red rectangles. The display window is $[0.0051, 0.0068]$ $mm^{-1}$
The reconstructed image from the projection data of the scanning angular ranges $[0,160^{0}]$ using the ASD-POCS algorithm and our algorithm. ROIs are labelled by red rectangles. The display window is $[0.0051, 0.0062]$ $mm^{-1}$
The zoomed-in view of the image ROIs of Figure 8 and Figure 9. The upper plane of Figure are the reconstructed results of ROIs for the scan ranges $[0,180^{0}]$, and the bottom plane of Figure are the reconstructed results of ROIs for the scan ranges $[0,160^{0}]$. The Figure on the first and three rows are the results using the ASD-POCS algorithm, and the second and four rows are the results using the our algorithm. The display window is $[0.0051, 0.0068]$ $mm^{-1}$ for the last column, and $[0.005, 0.005007]$ $mm^{-1}$ for the rest
Geometrical scanning parameters for simulated CT imaging system
 The distance between source and rotation center $981mm$ The angle interval of projection views $1^{0}$ The distance between source and detector $1200mm$ The diameter of field of view $143.6222mm$ The detector bin numbers $256$ The angle interval of rays $0.00329^{0}$ Pixel size $0.5632\times0.5632mm^{2}$ Image size $256\times256$
 The distance between source and rotation center $981mm$ The angle interval of projection views $1^{0}$ The distance between source and detector $1200mm$ The diameter of field of view $143.6222mm$ The detector bin numbers $256$ The angle interval of rays $0.00329^{0}$ Pixel size $0.5632\times0.5632mm^{2}$ Image size $256\times256$
Characterize quantitatively the reconstruction quality for NCAT
 Scanning ranges Algorithm RMSE PSNR MSSIM SART 0.0514 25.78 0.9997 $[0,120^{0}]$ ASD-POCS 0.0287 30.83 0.9999 our method 0.0250 32.05 0.9999 SART 0.0429 27.35 0.9998 $[0,140^{0}]$ ASD-POCS 0.0209 33.60 1.0000 our method $0.0215$ $33.33$ 1.0000
 Scanning ranges Algorithm RMSE PSNR MSSIM SART 0.0514 25.78 0.9997 $[0,120^{0}]$ ASD-POCS 0.0287 30.83 0.9999 our method 0.0250 32.05 0.9999 SART 0.0429 27.35 0.9998 $[0,140^{0}]$ ASD-POCS 0.0209 33.60 1.0000 our method $0.0215$ $33.33$ 1.0000
Characterize quantitatively the reconstruction quality for Shepp-Logan phantom
 Scanning ranges Algorithm RMSE PSNR MSSIM $[0,150^{0}]$ $\ell_{0}$ regularization 0.0628 24.04 0.9676 $\ell_{0}-\ell_{2}$ regularization 0.0618 24.17 0.9681
 Scanning ranges Algorithm RMSE PSNR MSSIM $[0,150^{0}]$ $\ell_{0}$ regularization 0.0628 24.04 0.9676 $\ell_{0}-\ell_{2}$ regularization 0.0618 24.17 0.9681
Quantitatively characterize the reconstruction quality of gear
 Scan ranges Algorithm RMSE PSNR MSSIM SART 40.157 16.056 0.752 $0^{0}\sim100^{0}$ ASD-POCS 17.296 23.372 0.784 our method 8.000 30.069 0.805 SART 37.156 16.730 0.776 $0^{0}\sim120^{0}$ ASD-POCS 10.728 27.521 0.787 our method 7.691 30.412 0.815 SART 30.425 18.466 0.790 $0^{0}\sim140^{0}$ ASD-POCS 9.096 28.954 0.788 our method 6.984 31.248 0.814 SART 19.766 22.212 0.805 $0^{0}\sim160^{0}$ ASD-POCS 9.680 28.413 0.807 our method 6.570 31.779 0.807
 Scan ranges Algorithm RMSE PSNR MSSIM SART 40.157 16.056 0.752 $0^{0}\sim100^{0}$ ASD-POCS 17.296 23.372 0.784 our method 8.000 30.069 0.805 SART 37.156 16.730 0.776 $0^{0}\sim120^{0}$ ASD-POCS 10.728 27.521 0.787 our method 7.691 30.412 0.815 SART 30.425 18.466 0.790 $0^{0}\sim140^{0}$ ASD-POCS 9.096 28.954 0.788 our method 6.984 31.248 0.814 SART 19.766 22.212 0.805 $0^{0}\sim160^{0}$ ASD-POCS 9.680 28.413 0.807 our method 6.570 31.779 0.807
Characterize quantitatively the reconstruction quality for metal lath
 Algorithm RMSE PSNR MSSIM $0\sim 180^{0}$ ASD-POCS 105.7 7.6481 0.8050 our method 107.4 7.5139 0.8060 $0\sim 160^{0}$ ASD-POCS 108.5 7.4209 0.8014 our method 107.9 7.4704 0.8014
 Algorithm RMSE PSNR MSSIM $0\sim 180^{0}$ ASD-POCS 105.7 7.6481 0.8050 our method 107.4 7.5139 0.8060 $0\sim 160^{0}$ ASD-POCS 108.5 7.4209 0.8014 our method 107.9 7.4704 0.8014
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