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February  2015, 9(1): 55-77. doi: 10.3934/ipi.2015.9.55

## High-order total variation regularization approach for axially symmetric object tomography from a single radiograph

 1 Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong, China 2 Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University, No. 111 Ren'ai Road, Suzhou Industrial Park, Jiangsu Province, China 3 Institute of Applied Physics and Computational Mathematics, Beijing, China 4 Centre de Mathematiques et de Leurs Applications, CNRS, ENS de Cachan, PRES UniverSud, 61 av. du President Wilson, 94235 Cachan Cedex, France 5 Department of Mathematics, University of Bergen, P. O. Box 7800, N-5020, Bergen, Norway

Received  June 2012 Revised  March 2014 Published  January 2015

In this paper, we consider tomographic reconstruction for axially symmetric objects from a single radiograph formed by fan-beam X-rays. All contemporary methods are based on the assumption that the density is piecewise constant or linear. From a practical viewpoint, this is quite a restrictive approximation. The method we propose is based on high-order total variation regularization. Its main advantage is to reduce the staircase effect while keeping sharp edges and enable the recovery of smoothly varying regions. The optimization problem is solved using the augmented Lagrangian method which has been recently applied in image processing. Furthermore, we use a one-dimensional (1D) technique for fan-beam X-rays to approximate 2D tomographic reconstruction for cone-beam X-rays. For the 2D problem, we treat the cone beam as fan beam located at parallel planes perpendicular to the symmetric axis. Then the density of the whole object is recovered layer by layer. Numerical results in 1D show that the proposed method has improved the preservation of edge location and the accuracy of the density level when compared with several other contemporary methods. The 2D numerical tests show that cylindrical symmetric objects can be recovered rather accurately by our high-order regularization model.
Citation: Raymond H. Chan, Haixia Liang, Suhua Wei, Mila Nikolova, Xue-Cheng Tai. High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Problems and Imaging, 2015, 9 (1) : 55-77. doi: 10.3934/ipi.2015.9.55
##### References:
 [1] R. Abraham, M. Bergounioux and E. Trelat, A penalization approach for tomographic reconstruction of binary axially symmetric objects, Applied Mathematics and Optimization, 58 (2008), 345-371. doi: 10.1007/s00245-008-9039-8. [2] T. J. Asaki, Quantitative Abel tomography robust to noisy, corrupted and missing data, Optimization and Engineering, 11 (2010), 381-393. doi: 10.1007/s11081-009-9097-z. [3] T. J. Asaki, R. Chartrand, K. R. Vixie and B. Wohlberg, Abel inversion using total variation regularization, Inverse Problem, 21 (2005), 1895-1903. doi: 10.1088/0266-5611/21/6/006. [4] T. Asaki, P. R. Campbell, R. Chartrand, C. E. Powell, K. R. Vixie and B. E. Wohlberg, Abel inversion using total variation regularization: Applications, Inverse Problem in Science and Engineering, 14 (2006), 873-885. doi: 10.1080/17415970600882549. [5] R. H. T. Bates, K. L. Garden and T. M. Peters, Overview of computerized tomography with emphasis on future developments, Proc. IEEE, 71 (1983), 356-372. doi: 10.1109/PROC.1983.12594. [6] M. Benning, C. Brune, M. Burger and J. Mueller, High-order TV methods - Enhancement via Bregman iteration, J. Sci. Comp., 54 (2013), 269-310. doi: 10.1007/s10915-012-9650-3. [7] K. Bredies, K. Kunisch and T. Pock, Total generalized variation, SIAM Journal on Image Sciences, 3 (2010), 492-526. doi: 10.1137/090769521. [8] K. Bredies and T. Valkonen, Inverse Problems with Second-order Total Generalized Variation Constraints, Proceedings of SampTA 2011 - 9th International Conference on Sampling Theory and Applications, Singapore, 2011. http://sampta2011.ntu.edu.sg/SampTA2011Proceedings/papers/Tu2S03.1-P0182.pdf [9] K. Bredies, K. Kunisch and T. Valkonen, Properties of L1-TGV2: The One-Dimensional Case, SFB-Report 2011-006, Karl-Franzens University of Graz, 2011. http://math.uni-graz.at/mobis/publications/SFB-Report-2011-006.pdf [10] A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011321.19549.88. [11] T. F. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516. doi: 10.1137/S1064827598344169. [12] T. Chen and H. R. Wu, Space variant median filters for the restoration of impulse noise corrupted images, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., 48 (2001), 784-789. [13] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. and Simul., 4 (2005), 1168-1200. doi: 10.1137/050626090. [14] D. Donoho, De-noising by soft-thresholding, IEEE Transactions on Information Theory, 41 (1995), 613-627. doi: 10.1109/18.382009. [15] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, 1999. [16] H. L. Eng and K. K. Ma, Noise adaptive soft-switching median filter, IEEE Trans. Image Process., 10 (2001), 242-251. [17] R. Glowinski and A. Marrocco, Sur l'approximation, Rev. Francćaise Automat. Informat. Recherche Opérationnelle RAIRO Analyse Numérique, 9 (1975), 41-76. [18] T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [19] K. M. Hanson, A Bayesian Approach to Nonlinear Inversion: Abel Inversion from X-ray Attenuation Data, Maximum Entropy and Bayesian Methods in Applied Statistics, edited by J. H. Justice, Cambridge University Press, Cambridge, England, 1986. [20] J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer-Verlag Berlin, 1993. doi: 10.1007/978-3-662-02796-7. [21] H. Hwang and R. A. Haddad, Adaptive median filters: New algorithms and results, IEEE Trans. Image Process., 4 (1995), 499-502. doi: 10.1109/83.370679. [22] F. Knoll, K. Bredies, T. Pock and R. Stollberger, Second order total generalized variation (TGV) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491. doi: 10.1002/mrm.22595. [23] S. Kontogiorgis and R. R. Meyer, A variable-penalty alternating directions method for convex optimizations, Mathematical Programming, 83 (1998), 29-53. doi: 10.1007/BF02680549. [24] M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 1579-1590. doi: 10.1109/TIP.2003.819229. [25] M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, International Journal of Computer Vision, 66 (2006), 5-18. doi: 10.1007/s11263-005-3219-7. [26] M. Nikolova, Local strong homogeneity of a regularized estimator, SIAM J. Appl. Math., 61 (2000), 633-658. doi: 10.1137/S0036139997327794. [27] P. E. Ng and K. K. Ma, A switching median filter with boundary discriminative noise detection for extremely corrupted images, IEEE Trans. Image Process., 15 (2006), 1506-1516. [28] R. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal of Control and Optimization, 14 (1976), 877-898. doi: 10.1137/0314056. [29] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D., 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [30] A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-posed Problems, New York: Wiley, 1977. [31] C. L. Wu and X. C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Science, 3 (2010), 300-339. doi: 10.1137/090767558. [32] C. L. Wu, J. Y. Zhang and X. C. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems and Imaging, 5 (2010), 237-261. doi: 10.3934/ipi.2011.5.237. [33] X. C. Tai and C. L. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, Scale Space and Variational Methods in Computer Vision, Second International Conference, SSVM 2009, Voss, Norway, June 1-5, 2009. Proceedings. Lecture Notes in Computer Science, Springer, 5567 (2009), 502-513.

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##### References:
 [1] R. Abraham, M. Bergounioux and E. Trelat, A penalization approach for tomographic reconstruction of binary axially symmetric objects, Applied Mathematics and Optimization, 58 (2008), 345-371. doi: 10.1007/s00245-008-9039-8. [2] T. J. Asaki, Quantitative Abel tomography robust to noisy, corrupted and missing data, Optimization and Engineering, 11 (2010), 381-393. doi: 10.1007/s11081-009-9097-z. [3] T. J. Asaki, R. Chartrand, K. R. Vixie and B. Wohlberg, Abel inversion using total variation regularization, Inverse Problem, 21 (2005), 1895-1903. doi: 10.1088/0266-5611/21/6/006. [4] T. Asaki, P. R. Campbell, R. Chartrand, C. E. Powell, K. R. Vixie and B. E. Wohlberg, Abel inversion using total variation regularization: Applications, Inverse Problem in Science and Engineering, 14 (2006), 873-885. doi: 10.1080/17415970600882549. [5] R. H. T. Bates, K. L. Garden and T. M. Peters, Overview of computerized tomography with emphasis on future developments, Proc. IEEE, 71 (1983), 356-372. doi: 10.1109/PROC.1983.12594. [6] M. Benning, C. Brune, M. Burger and J. Mueller, High-order TV methods - Enhancement via Bregman iteration, J. Sci. Comp., 54 (2013), 269-310. doi: 10.1007/s10915-012-9650-3. [7] K. Bredies, K. Kunisch and T. Pock, Total generalized variation, SIAM Journal on Image Sciences, 3 (2010), 492-526. doi: 10.1137/090769521. [8] K. Bredies and T. Valkonen, Inverse Problems with Second-order Total Generalized Variation Constraints, Proceedings of SampTA 2011 - 9th International Conference on Sampling Theory and Applications, Singapore, 2011. http://sampta2011.ntu.edu.sg/SampTA2011Proceedings/papers/Tu2S03.1-P0182.pdf [9] K. Bredies, K. Kunisch and T. Valkonen, Properties of L1-TGV2: The One-Dimensional Case, SFB-Report 2011-006, Karl-Franzens University of Graz, 2011. http://math.uni-graz.at/mobis/publications/SFB-Report-2011-006.pdf [10] A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011321.19549.88. [11] T. F. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516. doi: 10.1137/S1064827598344169. [12] T. Chen and H. R. Wu, Space variant median filters for the restoration of impulse noise corrupted images, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., 48 (2001), 784-789. [13] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. and Simul., 4 (2005), 1168-1200. doi: 10.1137/050626090. [14] D. Donoho, De-noising by soft-thresholding, IEEE Transactions on Information Theory, 41 (1995), 613-627. doi: 10.1109/18.382009. [15] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, 1999. [16] H. L. Eng and K. K. Ma, Noise adaptive soft-switching median filter, IEEE Trans. Image Process., 10 (2001), 242-251. [17] R. Glowinski and A. Marrocco, Sur l'approximation, Rev. Francćaise Automat. Informat. Recherche Opérationnelle RAIRO Analyse Numérique, 9 (1975), 41-76. [18] T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [19] K. M. Hanson, A Bayesian Approach to Nonlinear Inversion: Abel Inversion from X-ray Attenuation Data, Maximum Entropy and Bayesian Methods in Applied Statistics, edited by J. H. Justice, Cambridge University Press, Cambridge, England, 1986. [20] J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer-Verlag Berlin, 1993. doi: 10.1007/978-3-662-02796-7. [21] H. Hwang and R. A. Haddad, Adaptive median filters: New algorithms and results, IEEE Trans. Image Process., 4 (1995), 499-502. doi: 10.1109/83.370679. [22] F. Knoll, K. Bredies, T. Pock and R. Stollberger, Second order total generalized variation (TGV) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491. doi: 10.1002/mrm.22595. [23] S. Kontogiorgis and R. R. Meyer, A variable-penalty alternating directions method for convex optimizations, Mathematical Programming, 83 (1998), 29-53. doi: 10.1007/BF02680549. [24] M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 1579-1590. doi: 10.1109/TIP.2003.819229. [25] M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, International Journal of Computer Vision, 66 (2006), 5-18. doi: 10.1007/s11263-005-3219-7. [26] M. Nikolova, Local strong homogeneity of a regularized estimator, SIAM J. Appl. Math., 61 (2000), 633-658. doi: 10.1137/S0036139997327794. [27] P. E. Ng and K. K. Ma, A switching median filter with boundary discriminative noise detection for extremely corrupted images, IEEE Trans. Image Process., 15 (2006), 1506-1516. [28] R. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal of Control and Optimization, 14 (1976), 877-898. doi: 10.1137/0314056. [29] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D., 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [30] A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-posed Problems, New York: Wiley, 1977. [31] C. L. Wu and X. C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Science, 3 (2010), 300-339. doi: 10.1137/090767558. [32] C. L. Wu, J. Y. Zhang and X. C. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems and Imaging, 5 (2010), 237-261. doi: 10.3934/ipi.2011.5.237. [33] X. C. Tai and C. L. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, Scale Space and Variational Methods in Computer Vision, Second International Conference, SSVM 2009, Voss, Norway, June 1-5, 2009. Proceedings. Lecture Notes in Computer Science, Springer, 5567 (2009), 502-513.
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