# American Institute of Mathematical Sciences

August  2016, 10(3): 829-853. doi: 10.3934/ipi.2016023

## Error bounds and stability in the $l_{0}$ regularized for CT reconstruction from small projections

 1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China, China

Received  April 2015 Revised  May 2016 Published  August 2016

Due to the restriction of the scanning environment and the energy of X-ray, few projections of an object can be obtained in some practical applications of computed tomography (CT). In these situations, the projection data are incomplete and inconsistent, and the conventional analytic algorithm such as filtered backprojection (FBP) algorithm will not work. The streak artifacts can be significantly reduced in few-view reconstruction if the total variation minimization (TVM) based CT reconstruction algorithm is used. However, in the premise of preserving the resolution of image, it will not effectively suppress slope artifacts and metal artifacts when dealing with some few-view of the limited-angle reconstruction problems. To solve this problem, we focus on the image reconstruction algorithm base on $\ell_{0}$ regularized of wavelet coefficients. In this paper, the error bound between the reference or desire image and the reconstructed result, and the stability of solution were shown in theoretical and experimental, a reconstruction experiment on metal laths from few-view of the limited-angle projections was given. The experimental results indicate that this algorithm outperforms classical CT reconstruction algorithms in preserving the resolution of reconstructed image and suppressing the metal artifacts.
Citation: Chengxiang Wang, Li Zeng. Error bounds and stability in the $l_{0}$ regularized for CT reconstruction from small projections. Inverse Problems & Imaging, 2016, 10 (3) : 829-853. doi: 10.3934/ipi.2016023
##### References:
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Osher and Z. Shen, Split Bregman methods and frame based image restoration,, Multiscale modeling and simulation, 8 (2009), 337. [13] 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. doi: 10.1088/1748-0221/7/03/P03007. [14] X. Cai, J. H. Fitschen and M. Nikolova, Disparity and optical flow partitioning using extended potts priors,, Information and Inference, 4 (2015), 43. doi: 10.1093/imaiai/iau010. [15] 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. [16] 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. doi: 10.1088/0031-9155/58/7/2119. [17] B. Dong and Y. Zhang, An efficient algorithm for $l_{0}$ minimization in wavelet frame based image restoration,, Journal of Scientific Computing, 54 (2013), 350. doi: 10.1007/s10915-012-9597-4. [18] 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 (2000), 1413. doi: 10.1002/cpa.20042. [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. doi: 10.1016/S1063-5203(02)00511-0. [20] I. Daubechies, Ten Lectures on Wavelets,, 1nd edition, (1992). doi: 10.1137/1.9781611970104.fm. [21] M. B. De, J. Nuyts, P. dupont, G. marcchal and P. suetens, Reduction of metal streak artifacts in x-ray computed tomography using a transmission maximum a posteriori algorithm,, IEEE transactions on nuclear science, 47 (2000), 977. doi: 10.1109/23.856534. [22] 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. doi: 10.1016/j.acha.2005.03.005. [23] M. M. Eger and P. E. Danielsson, Scanning of logs with linear cone-beam tomography,, Computers and electronics in agriculture, 41 (2003), 45. doi: 10.1016/S0168-1699(03)00041-3. [24] 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. 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. 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. 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. doi: 10.1117/1.2739624. [28] T. Goldstein and S. Osher, The split Bregman method for $l_1$ regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323. doi: 10.1137/080725891. [29] B. Han, On dual wavelet tight frames,, Applied and Computational Harmonic Analysis, 4 (1997), 380. doi: 10.1006/acha.1997.0217. [30] B. S. He, A class of projection and contraction methods for monotone variational inequalities,, Applied Mathematics and Optimization, 35 (1997), 69. doi: 10.1007/BF02683320. [31] B. S. He and M. H. Xu, A general framework of contraction methods for monotone variational inequalities,, Pacific Journal of Optimization, 4 (2008), 195. [32] T. J. Hubertus, M. Klaus and T. Eberhard, Optimization Theory,, 1nd edition, (2004). doi: 10.1007/b130886. [33] K. Ito and K. Kunisch, A note on the existence of nonsmooth nonconvex optimization problems,, Journal of Optimization Theory and Applications, 163 (2014), 697. doi: 10.1007/s10957-014-0552-4. [34] M. Jiang and G. Wang, Development of iterative algorithms for image reconstruction,, Journal of X-ray Science and Technology, 10 (2001), 77. [35] M. Jiang and G. Wang, Convergence of the simultaneous algebraic reconstruction technique (SART),, IEEE Transactions on Image Processing, 12 (2003), 957. doi: 10.1109/TIP.2003.815295. [36] 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. doi: 10.1088/0031-9155/56/13/004. [37] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging,, Med. Phys, (1988). [38] 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. doi: 10.1088/0031-9155/48/10/315. [39] X. Lu, Y. Sun and Y. Yuan, Image reconstruction by an alternating minimisation,, Neurocomputing, 74 (2011), 661. doi: 10.1016/j.neucom.2010.08.003. [40] E. T. Quinto, Exterior and limited-angle tomography in non-destructive evaluation,, Inverse Problems, 14 (1998), 339. doi: 10.1088/0266-5611/14/2/009. [41] A. Ron and Z. Shen, Affine systems in $L_{2}(R^d)$ II: Dual systems,, Journal of Fourier Analysis and Applications, 3 (1997), 617. doi: 10.1007/BF02648888. [42] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. [43] R. T. Rockafellarr and R. J. B. Wets, Variational Analysis,, 1nd edition, (1998). doi: 10.1007/978-3-642-02431-3. [44] 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. doi: 10.1088/0031-9155/53/17/021. [45] E. Y. Sidky, C. M. Kao and X. C. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,, preprint, (). [46] J. 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. doi: 10.1109/TIP.2005.852206. [47] W. P. Segars, D. S. Lalush and B. M. W.Tsui, A realistic spline-based dynamic heart phantom,, IEEE Transactions on Nuclear Science, 2 (1998), 1175. doi: 10.1109/NSSMIC.1998.774369. [48] A. Tingberg, X-ray tomosynthesis: A review of its use for breast and chest imaging,, Radiation Protection Dosimetry, 139 (2010), 100. doi: 10.1093/rpd/ncq099. [49] H. K. Tuy, An inversion formula for cone-beam reconstruction,, SIAM J. Appl. Math., 43 (1983), 546. doi: 10.1137/0143035. [50] L. Xu, C. Lu, Y. Xu and J. Jia, Image smoothing via $l_{0}$ gradient minimization,, ACM Trans. Graph, 30 (2011), 1. doi: 10.1109/TIP.2005.852206. [51] Y. H. Xiao, J. F. Yang and X. M. Yuan, Alternating algorithms for total variation image reconstruction from random projections,, Inverse Problems and Imaging, 6 (2012), 547. doi: 10.3934/ipi.2012.6.547. [52] B. Zhao, H. Gao, H. Ding and S. Molloi, Tight-frame based iterative image reconstruction for spectral breast CT,, Medical Physics, 40 (2013). doi: 10.1118/1.4790468. [53] G. L. Zeng, Medical Image Reconstruction,, 1nd edition, (2010). doi: 10.1007/978-3-642-05368-9. [54] 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. doi: 10.1515/jip-2011-0010. [55] 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. doi: 10.1088/0266-5611/29/12/125006. [56] Y. Zhang, B. Dong and Z. S. Lu, $l_{0}$ Minimization for wavelet frame based image restoration,, Mathematics of Computation, 82 (2013), 995. doi: 10.1090/S0025-5718-2012-02631-7. [57] Abdomen phantom website., Available from:, , ().

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##### References:
 [1] A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities,, 1nd edition, (2003). doi: 10.1007/b97594. [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. doi: 10.1177/016173468400600107. [3] M. A. Anastasio, E. Y. Sidky, X. C. Pan and C. Y. Chou, Boundary reconstruction in limited-angle x-ray phase-contrast tomography,, In SPIE Medical Imaging, 7258 (2009), 725827. doi: 10.1117/12.811918. [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,, Med. Phys., 34 (2007), 4664. doi: 10.1118/1.2799492. [5] J. Bolte, S. Sabach and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems,, Mathematical Programming, 146 (2014), 459. doi: 10.1007/s10107-013-0701-9. [6] K. Bredies and D. A. Lorenz, Minimization of non-smooth, non-convex functionals by iterative thresholding,, Journal of Optimization Theory and Applications, 165 (2015), 78. doi: 10.1007/s10957-014-0614-7. [7] M. Burger, J. Müller, E. Papoutsellis and C. B. Schönlieb, Total Variation Regularisation in Measurement and Image space for PET reconstruction,, preprint, (). [8] T. Blumensath and M. E. Davies, Iterative thresholding for sparse approximations,, Journal of Fourier Analysis and Applications, 14 (2008), 629. doi: 10.1007/s00041-008-9035-z. [9] T. Blumensath and M. E. Davies, Iterative hard thresholding for compressed sensing,, Applied and Computational Harmonic Analysis, 27 (2009), 265. doi: 10.1016/j.acha.2009.04.002. [10] T. Blumensath, Accelerated iterative hard thresholding,, Signal Processing, 92 (2012), 752. doi: 10.1016/j.sigpro.2011.09.017. [11] T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-beam CT,, 1nd edition, (2008). [12] J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration,, Multiscale modeling and simulation, 8 (2009), 337. [13] 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. doi: 10.1088/1748-0221/7/03/P03007. [14] X. Cai, J. H. Fitschen and M. Nikolova, Disparity and optical flow partitioning using extended potts priors,, Information and Inference, 4 (2015), 43. doi: 10.1093/imaiai/iau010. [15] 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. [16] 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. doi: 10.1088/0031-9155/58/7/2119. [17] B. Dong and Y. Zhang, An efficient algorithm for $l_{0}$ minimization in wavelet frame based image restoration,, Journal of Scientific Computing, 54 (2013), 350. doi: 10.1007/s10915-012-9597-4. [18] 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 (2000), 1413. doi: 10.1002/cpa.20042. [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. doi: 10.1016/S1063-5203(02)00511-0. [20] I. Daubechies, Ten Lectures on Wavelets,, 1nd edition, (1992). doi: 10.1137/1.9781611970104.fm. [21] M. B. De, J. Nuyts, P. dupont, G. marcchal and P. suetens, Reduction of metal streak artifacts in x-ray computed tomography using a transmission maximum a posteriori algorithm,, IEEE transactions on nuclear science, 47 (2000), 977. doi: 10.1109/23.856534. [22] 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. doi: 10.1016/j.acha.2005.03.005. [23] M. M. Eger and P. E. Danielsson, Scanning of logs with linear cone-beam tomography,, Computers and electronics in agriculture, 41 (2003), 45. doi: 10.1016/S0168-1699(03)00041-3. [24] 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. 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. 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. 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. doi: 10.1117/1.2739624. [28] T. Goldstein and S. Osher, The split Bregman method for $l_1$ regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323. doi: 10.1137/080725891. [29] B. Han, On dual wavelet tight frames,, Applied and Computational Harmonic Analysis, 4 (1997), 380. doi: 10.1006/acha.1997.0217. [30] B. S. He, A class of projection and contraction methods for monotone variational inequalities,, Applied Mathematics and Optimization, 35 (1997), 69. doi: 10.1007/BF02683320. [31] B. S. He and M. H. Xu, A general framework of contraction methods for monotone variational inequalities,, Pacific Journal of Optimization, 4 (2008), 195. [32] T. J. Hubertus, M. Klaus and T. Eberhard, Optimization Theory,, 1nd edition, (2004). doi: 10.1007/b130886. [33] K. Ito and K. Kunisch, A note on the existence of nonsmooth nonconvex optimization problems,, Journal of Optimization Theory and Applications, 163 (2014), 697. doi: 10.1007/s10957-014-0552-4. [34] M. Jiang and G. Wang, Development of iterative algorithms for image reconstruction,, Journal of X-ray Science and Technology, 10 (2001), 77. [35] M. Jiang and G. Wang, Convergence of the simultaneous algebraic reconstruction technique (SART),, IEEE Transactions on Image Processing, 12 (2003), 957. doi: 10.1109/TIP.2003.815295. [36] 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. doi: 10.1088/0031-9155/56/13/004. [37] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging,, Med. Phys, (1988). [38] 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. doi: 10.1088/0031-9155/48/10/315. [39] X. Lu, Y. Sun and Y. Yuan, Image reconstruction by an alternating minimisation,, Neurocomputing, 74 (2011), 661. doi: 10.1016/j.neucom.2010.08.003. [40] E. T. Quinto, Exterior and limited-angle tomography in non-destructive evaluation,, Inverse Problems, 14 (1998), 339. doi: 10.1088/0266-5611/14/2/009. [41] A. Ron and Z. Shen, Affine systems in $L_{2}(R^d)$ II: Dual systems,, Journal of Fourier Analysis and Applications, 3 (1997), 617. doi: 10.1007/BF02648888. [42] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. [43] R. T. Rockafellarr and R. J. B. Wets, Variational Analysis,, 1nd edition, (1998). doi: 10.1007/978-3-642-02431-3. [44] 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. doi: 10.1088/0031-9155/53/17/021. [45] E. Y. Sidky, C. M. Kao and X. C. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,, preprint, (). [46] J. 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. doi: 10.1109/TIP.2005.852206. [47] W. P. Segars, D. S. Lalush and B. M. W.Tsui, A realistic spline-based dynamic heart phantom,, IEEE Transactions on Nuclear Science, 2 (1998), 1175. doi: 10.1109/NSSMIC.1998.774369. [48] A. Tingberg, X-ray tomosynthesis: A review of its use for breast and chest imaging,, Radiation Protection Dosimetry, 139 (2010), 100. doi: 10.1093/rpd/ncq099. [49] H. K. Tuy, An inversion formula for cone-beam reconstruction,, SIAM J. Appl. Math., 43 (1983), 546. doi: 10.1137/0143035. [50] L. Xu, C. Lu, Y. Xu and J. Jia, Image smoothing via $l_{0}$ gradient minimization,, ACM Trans. Graph, 30 (2011), 1. doi: 10.1109/TIP.2005.852206. [51] Y. H. Xiao, J. F. Yang and X. M. Yuan, Alternating algorithms for total variation image reconstruction from random projections,, Inverse Problems and Imaging, 6 (2012), 547. doi: 10.3934/ipi.2012.6.547. [52] B. Zhao, H. Gao, H. Ding and S. Molloi, Tight-frame based iterative image reconstruction for spectral breast CT,, Medical Physics, 40 (2013). doi: 10.1118/1.4790468. [53] G. L. Zeng, Medical Image Reconstruction,, 1nd edition, (2010). doi: 10.1007/978-3-642-05368-9. [54] 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. doi: 10.1515/jip-2011-0010. [55] W. Zhou, J. F. Cai and H. 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