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The "exterior approach" to solve the inverse obstacle problem for the Stokes system
The Moreau envelope approach for the L1/TV image denoising model
1. | Department of Mathematics, Syracuse University, Syracuse, NY 13244, United States, United States |
2. | Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150 |
3. | School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China |
References:
[1] |
J. E. Aujol, G. Gilboa, T. Chan and S. Osher, Structure-texture image decomposition-modeling, algorithms, and parameter selection, International Journal of Computer Vision, 67 (2006), 111-136.
doi: 10.1007/s11263-006-4331-z. |
[2] |
H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Space, Springer Press, New York, 2011.
doi: 10.1007/978-1-4419-9467-7. |
[3] |
A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Science, 2 (2009), 183-202.
doi: 10.1137/080716542. |
[4] |
P. Bloomfield and W. Steiger, Least Absolute Deviations: Theory, Applications, and Algorithms, Birkhäuser Boston, Boston, 1983. |
[5] |
A. Bovik, Handbook of Image and Video Processing, Academic Press, San Diego, 2000. |
[6] |
A. Chambolle, An algorithm for total variation minimization and applications, Special issue on mathematics and image analysis, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.
doi: 10.1023/B:JMIV.0000011321.19549.88. |
[7] |
A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[8] |
T. Chan and S. Esedoglu, Aspects of total variation regularized l1 function approximation, SIAM Journal on Applied Mathematics, 65 (2005), 1817-1837.
doi: 10.1137/040604297. |
[9] |
R. Chartrand and V. Staneva, Total variation regularization of images corrupted by non-gaussian noise using a quasi-newton method, IET Image Processing, 2 (2008), 295-303.
doi: 10.1049/iet-ipr:20080017. |
[10] |
T. Chen, W. Yin, X. Zhou, D. Comaniciu and T. Huang, Total variation models for variable lighting face recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 28 (2006), 1519-1524.
doi: 10.1109/TPAMI.2006.195. |
[11] |
C. Clason, B. Jin and K. Kunisch, A duality-based splitting method for l1-tv image restoration with automatic regularization parameter choice, SIAM Journal on Scientific Computing, 32 (2010), 1484-1505.
doi: 10.1137/090768217. |
[12] |
P. Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model and Simulation, 4 (2005), 1168-1200.
doi: 10.1137/050626090. |
[13] |
Y. Dong, M. Hintermuller and M. Neri, An Efficient Primal-Dual Method for $l^1$-TV image Restoration, SIAM Journal on Imaging Sciences, 2 (2009), 1168-1189.
doi: 10.1137/090758490. |
[14] |
T. Goldstein and S. Osher, The split bregman method for l1-regularized problems, SIAM Journal on Imaging Science, 2 (2009), 323-343.
doi: 10.1137/080725891. |
[15] |
Y. Gousseau and J.-M. Morel, Are natural images of bounded variation?, SIAM Journal on Mathematical Analysis, 33 (2001), 634-648.
doi: 10.1137/S0036141000371150. |
[16] |
X. Guo, F. Li and M. Ng, A fast $l^1$-TV algorithm for image restoration, SIAM Journal of Scientific Computing, 31 (2009), 2322-2341.
doi: 10.1137/080724435. |
[17] |
B. He and X. Yuan, Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective, SIAM Journal on Imaging Sciences, 5 (2012), 119-149.
doi: 10.1137/100814494. |
[18] |
Q. Li, C. Micchelli, L. Shen and Y. Xu, A proximity algorithm accelerated by Gauss-Seidel iterations for L1/TV denoising models, Inverse Problems, 28 (2012), 20 pp.
doi: 10.1088/0266-5611/28/9/095003. |
[19] |
C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denosing, Inverse Problems, 27 (2011), 30pp.
doi: 10.1088/0266-5611/27/4/045009. |
[20] |
C. A. Micchelli, L. Shen, Y. Xu and X. Zeng, Proximity algorithms for the $l_1$$/$$tv$ image denosing models, Advances in Computational Mathematics, 38 (2013), 401-426.
doi: 10.1007/s10444-011-9243-y. |
[21] |
J.-J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien, C. R. Acad. Sci. Paris Sér. A Math., 255 (1962), 1897-2899. |
[22] |
M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120.
doi: 10.1023/B:JMIV.0000011920.58935.9c. |
[23] |
R. Rockafellar, Augmented lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research, 8 (1976), 421-439.
doi: 10.1287/moor.1.2.97. |
[24] |
R. T. Rockafellar and R. Wets, Variational Analysis, Springer, 1998.
doi: 10.1007/978-3-642-02431-3. |
[25] |
C. Wu, J. Zhang and X. Tai, Augmented lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems and Imaging, 5 (2011), 237-261.
doi: 10.3934/ipi.2011.5.237. |
[26] |
J. Yang, Y. Zhang and W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM Journal on Scientific Computing, 31 (2009), 2842-2865.
doi: 10.1137/080732894. |
[27] |
W. Yin, T. Chen, S. Zhou and A. Chakraborty, Background correction for cdna microarray images using the tv+l1 model, Bioinformatics, 21 (2005), 2410-2416.
doi: 10.1093/bioinformatics/bti341. |
[28] |
W. Yin, D. Goldfard and S. Osher, The total variation l1 model for multiscale decomposition, Multiscale Modelling and Simulation, 6 (2007), 190-211.
doi: 10.1137/060663027. |
[29] |
C. Zach, T. Pock and H. Bischof, A duality based approach for realtime tv-l1 optical flow, Pattern Recognition, 4713 (2007), 214-223. |
show all references
References:
[1] |
J. E. Aujol, G. Gilboa, T. Chan and S. Osher, Structure-texture image decomposition-modeling, algorithms, and parameter selection, International Journal of Computer Vision, 67 (2006), 111-136.
doi: 10.1007/s11263-006-4331-z. |
[2] |
H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Space, Springer Press, New York, 2011.
doi: 10.1007/978-1-4419-9467-7. |
[3] |
A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Science, 2 (2009), 183-202.
doi: 10.1137/080716542. |
[4] |
P. Bloomfield and W. Steiger, Least Absolute Deviations: Theory, Applications, and Algorithms, Birkhäuser Boston, Boston, 1983. |
[5] |
A. Bovik, Handbook of Image and Video Processing, Academic Press, San Diego, 2000. |
[6] |
A. Chambolle, An algorithm for total variation minimization and applications, Special issue on mathematics and image analysis, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.
doi: 10.1023/B:JMIV.0000011321.19549.88. |
[7] |
A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[8] |
T. Chan and S. Esedoglu, Aspects of total variation regularized l1 function approximation, SIAM Journal on Applied Mathematics, 65 (2005), 1817-1837.
doi: 10.1137/040604297. |
[9] |
R. Chartrand and V. Staneva, Total variation regularization of images corrupted by non-gaussian noise using a quasi-newton method, IET Image Processing, 2 (2008), 295-303.
doi: 10.1049/iet-ipr:20080017. |
[10] |
T. Chen, W. Yin, X. Zhou, D. Comaniciu and T. Huang, Total variation models for variable lighting face recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 28 (2006), 1519-1524.
doi: 10.1109/TPAMI.2006.195. |
[11] |
C. Clason, B. Jin and K. Kunisch, A duality-based splitting method for l1-tv image restoration with automatic regularization parameter choice, SIAM Journal on Scientific Computing, 32 (2010), 1484-1505.
doi: 10.1137/090768217. |
[12] |
P. Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model and Simulation, 4 (2005), 1168-1200.
doi: 10.1137/050626090. |
[13] |
Y. Dong, M. Hintermuller and M. Neri, An Efficient Primal-Dual Method for $l^1$-TV image Restoration, SIAM Journal on Imaging Sciences, 2 (2009), 1168-1189.
doi: 10.1137/090758490. |
[14] |
T. Goldstein and S. Osher, The split bregman method for l1-regularized problems, SIAM Journal on Imaging Science, 2 (2009), 323-343.
doi: 10.1137/080725891. |
[15] |
Y. Gousseau and J.-M. Morel, Are natural images of bounded variation?, SIAM Journal on Mathematical Analysis, 33 (2001), 634-648.
doi: 10.1137/S0036141000371150. |
[16] |
X. Guo, F. Li and M. Ng, A fast $l^1$-TV algorithm for image restoration, SIAM Journal of Scientific Computing, 31 (2009), 2322-2341.
doi: 10.1137/080724435. |
[17] |
B. He and X. Yuan, Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective, SIAM Journal on Imaging Sciences, 5 (2012), 119-149.
doi: 10.1137/100814494. |
[18] |
Q. Li, C. Micchelli, L. Shen and Y. Xu, A proximity algorithm accelerated by Gauss-Seidel iterations for L1/TV denoising models, Inverse Problems, 28 (2012), 20 pp.
doi: 10.1088/0266-5611/28/9/095003. |
[19] |
C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denosing, Inverse Problems, 27 (2011), 30pp.
doi: 10.1088/0266-5611/27/4/045009. |
[20] |
C. A. Micchelli, L. Shen, Y. Xu and X. Zeng, Proximity algorithms for the $l_1$$/$$tv$ image denosing models, Advances in Computational Mathematics, 38 (2013), 401-426.
doi: 10.1007/s10444-011-9243-y. |
[21] |
J.-J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien, C. R. Acad. Sci. Paris Sér. A Math., 255 (1962), 1897-2899. |
[22] |
M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120.
doi: 10.1023/B:JMIV.0000011920.58935.9c. |
[23] |
R. Rockafellar, Augmented lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research, 8 (1976), 421-439.
doi: 10.1287/moor.1.2.97. |
[24] |
R. T. Rockafellar and R. Wets, Variational Analysis, Springer, 1998.
doi: 10.1007/978-3-642-02431-3. |
[25] |
C. Wu, J. Zhang and X. Tai, Augmented lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems and Imaging, 5 (2011), 237-261.
doi: 10.3934/ipi.2011.5.237. |
[26] |
J. Yang, Y. Zhang and W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM Journal on Scientific Computing, 31 (2009), 2842-2865.
doi: 10.1137/080732894. |
[27] |
W. Yin, T. Chen, S. Zhou and A. Chakraborty, Background correction for cdna microarray images using the tv+l1 model, Bioinformatics, 21 (2005), 2410-2416.
doi: 10.1093/bioinformatics/bti341. |
[28] |
W. Yin, D. Goldfard and S. Osher, The total variation l1 model for multiscale decomposition, Multiscale Modelling and Simulation, 6 (2007), 190-211.
doi: 10.1137/060663027. |
[29] |
C. Zach, T. Pock and H. Bischof, A duality based approach for realtime tv-l1 optical flow, Pattern Recognition, 4713 (2007), 214-223. |
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