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

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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

Raymond H. Chan ^1, , Haixia Liang ^2, , Suhua Wei ^3, , Mila Nikolova ^4, and Xue-Cheng Tai ^5,

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

Abstract
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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.

Keywords: high-order total variation, augmented Lagrangian method., radiograph, Tomography, Abel inversion.

Mathematics Subject Classification: Primary: 65R32, 65K10; Secondary: 49M25, 93B4.

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 & 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. doi: 10.1007/s00245-008-9039-8. Google Scholar
[2]	T. J. Asaki, Quantitative Abel tomography robust to noisy, corrupted and missing data,, Optimization and Engineering, 11 (2010), 381. doi: 10.1007/s11081-009-9097-z. Google Scholar
[3]	T. J. Asaki, R. Chartrand, K. R. Vixie and B. Wohlberg, Abel inversion using total variation regularization,, Inverse Problem, 21 (2005), 1895. doi: 10.1088/0266-5611/21/6/006. Google Scholar
[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. doi: 10.1080/17415970600882549. Google Scholar
[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. doi: 10.1109/PROC.1983.12594. Google Scholar
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[7]	K. Bredies, K. Kunisch and T. Pock, Total generalized variation,, SIAM Journal on Image Sciences, 3 (2010), 492. doi: 10.1137/090769521. Google Scholar
[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, (2011). Google Scholar
[9]	K. Bredies, K. Kunisch and T. Valkonen, Properties of L1-TGV2: The One-Dimensional Case,, SFB-Report 2011-006, (2011), 2011. Google Scholar
[10]	A. Chambolle, An algorithm for total variation minimization and applications,, Journal of Mathematical Imaging and Vision, 20 (2004), 89. doi: 10.1023/B:JMIV.0000011321.19549.88. Google Scholar
[11]	T. F. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration,, SIAM J. Sci. Comput., 22 (2000), 503. doi: 10.1137/S1064827598344169. Google Scholar
[12]	T. Chen and H. R. Wu, Space variant median filters for the restoration of impulse noise corrupted images,, IEEE Trans. Circuits Syst. II, 48 (2001), 784. Google Scholar
[13]	P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting,, Multiscale Model. and Simul., 4 (2005), 1168. doi: 10.1137/050626090. Google Scholar
[14]	D. Donoho, De-noising by soft-thresholding,, IEEE Transactions on Information Theory, 41 (1995), 613. doi: 10.1109/18.382009. Google Scholar
[15]	I. Ekeland and R. Témam, Convex Analysis and Variational Problems,, 1999., (). Google Scholar
[16]	H. L. Eng and K. K. Ma, Noise adaptive soft-switching median filter,, IEEE Trans. Image Process., 10 (2001), 242. Google Scholar
[17]	R. Glowinski and A. Marrocco, Sur l'approximation,, Rev. Francćaise Automat. Informat. Recherche Opérationnelle RAIRO Analyse Numérique, 9 (1975), 41. Google Scholar
[18]	T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323. doi: 10.1137/080725891. Google Scholar
[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, (1986). Google Scholar
[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. Google Scholar
[21]	H. Hwang and R. A. Haddad, Adaptive median filters: New algorithms and results,, IEEE Trans. Image Process., 4 (1995), 499. doi: 10.1109/83.370679. Google Scholar
[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. doi: 10.1002/mrm.22595. Google Scholar
[23]	S. Kontogiorgis and R. R. Meyer, A variable-penalty alternating directions method for convex optimizations,, Mathematical Programming, 83 (1998), 29. doi: 10.1007/BF02680549. Google Scholar
[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. doi: 10.1109/TIP.2003.819229. Google Scholar
[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. doi: 10.1007/s11263-005-3219-7. Google Scholar
[26]	M. Nikolova, Local strong homogeneity of a regularized estimator,, SIAM J. Appl. Math., 61 (2000), 633. doi: 10.1137/S0036139997327794. Google Scholar
[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. Google Scholar
[28]	R. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM Journal of Control and Optimization, 14 (1976), 877. doi: 10.1137/0314056. Google Scholar
[29]	L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D., 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar
[30]	A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-posed Problems,, New York: Wiley, (1977). Google Scholar
[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. doi: 10.1137/090767558. Google Scholar
[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. doi: 10.3934/ipi.2011.5.237. Google Scholar
[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, 5567 (2009), 1. Google Scholar

show all references

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. doi: 10.1007/s00245-008-9039-8. Google Scholar
[2]	T. J. Asaki, Quantitative Abel tomography robust to noisy, corrupted and missing data,, Optimization and Engineering, 11 (2010), 381. doi: 10.1007/s11081-009-9097-z. Google Scholar
[3]	T. J. Asaki, R. Chartrand, K. R. Vixie and B. Wohlberg, Abel inversion using total variation regularization,, Inverse Problem, 21 (2005), 1895. doi: 10.1088/0266-5611/21/6/006. Google Scholar
[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. doi: 10.1080/17415970600882549. Google Scholar
[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. doi: 10.1109/PROC.1983.12594. Google Scholar
[6]	M. Benning, C. Brune, M. Burger and J. Mueller, High-order TV methods - Enhancement via Bregman iteration,, J. Sci. Comp., 54 (2013), 269. doi: 10.1007/s10915-012-9650-3. Google Scholar
[7]	K. Bredies, K. Kunisch and T. Pock, Total generalized variation,, SIAM Journal on Image Sciences, 3 (2010), 492. doi: 10.1137/090769521. Google Scholar
[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, (2011). Google Scholar
[9]	K. Bredies, K. Kunisch and T. Valkonen, Properties of L1-TGV2: The One-Dimensional Case,, SFB-Report 2011-006, (2011), 2011. Google Scholar
[10]	A. Chambolle, An algorithm for total variation minimization and applications,, Journal of Mathematical Imaging and Vision, 20 (2004), 89. doi: 10.1023/B:JMIV.0000011321.19549.88. Google Scholar
[11]	T. F. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration,, SIAM J. Sci. Comput., 22 (2000), 503. doi: 10.1137/S1064827598344169. Google Scholar
[12]	T. Chen and H. R. Wu, Space variant median filters for the restoration of impulse noise corrupted images,, IEEE Trans. Circuits Syst. II, 48 (2001), 784. Google Scholar
[13]	P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting,, Multiscale Model. and Simul., 4 (2005), 1168. doi: 10.1137/050626090. Google Scholar
[14]	D. Donoho, De-noising by soft-thresholding,, IEEE Transactions on Information Theory, 41 (1995), 613. doi: 10.1109/18.382009. Google Scholar
[15]	I. Ekeland and R. Témam, Convex Analysis and Variational Problems,, 1999., (). Google Scholar
[16]	H. L. Eng and K. K. Ma, Noise adaptive soft-switching median filter,, IEEE Trans. Image Process., 10 (2001), 242. Google Scholar
[17]	R. Glowinski and A. Marrocco, Sur l'approximation,, Rev. Francćaise Automat. Informat. Recherche Opérationnelle RAIRO Analyse Numérique, 9 (1975), 41. Google Scholar
[18]	T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323. doi: 10.1137/080725891. Google Scholar
[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, (1986). Google Scholar
[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. Google Scholar
[21]	H. Hwang and R. A. Haddad, Adaptive median filters: New algorithms and results,, IEEE Trans. Image Process., 4 (1995), 499. doi: 10.1109/83.370679. Google Scholar
[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. doi: 10.1002/mrm.22595. Google Scholar
[23]	S. Kontogiorgis and R. R. Meyer, A variable-penalty alternating directions method for convex optimizations,, Mathematical Programming, 83 (1998), 29. doi: 10.1007/BF02680549. Google Scholar
[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. doi: 10.1109/TIP.2003.819229. Google Scholar
[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. doi: 10.1007/s11263-005-3219-7. Google Scholar
[26]	M. Nikolova, Local strong homogeneity of a regularized estimator,, SIAM J. Appl. Math., 61 (2000), 633. doi: 10.1137/S0036139997327794. Google Scholar
[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. Google Scholar
[28]	R. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM Journal of Control and Optimization, 14 (1976), 877. doi: 10.1137/0314056. Google Scholar
[29]	L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D., 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar
[30]	A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-posed Problems,, New York: Wiley, (1977). Google Scholar
[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. doi: 10.1137/090767558. Google Scholar
[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. doi: 10.3934/ipi.2011.5.237. Google Scholar
[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, 5567 (2009), 1. Google Scholar

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