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A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
K. Bredies, K. Kunisch and T. Pock, Total generalized variation,, SIAM Journal on Image Sciences, 3 (2010), 492.
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, (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. |
[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. |
[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. |
[14] |
D. Donoho, De-noising by soft-thresholding,, IEEE Transactions on Information Theory, 41 (1995), 613.
doi: 10.1109/18.382009. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
M. Nikolova, Local strong homogeneity of a regularized estimator,, SIAM J. Appl. Math., 61 (2000), 633.
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. 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. |
[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. |
[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.
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.
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, 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
K. Bredies, K. Kunisch and T. Pock, Total generalized variation,, SIAM Journal on Image Sciences, 3 (2010), 492.
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, (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. |
[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. |
[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. |
[14] |
D. Donoho, De-noising by soft-thresholding,, IEEE Transactions on Information Theory, 41 (1995), 613.
doi: 10.1109/18.382009. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
M. Nikolova, Local strong homogeneity of a regularized estimator,, SIAM J. Appl. Math., 61 (2000), 633.
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. 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. |
[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. |
[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.
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.
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, 5567 (2009), 1. Google Scholar |
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