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Tomáš Roubíček celebrates his sixtieth anniversary
Iterative finite element solution of a constrained total variation regularized model problem
Department of Applied Mathematics, Mathematical Institute, University of Freiburg, Hermann-Herder-Str. 9,79104 Freiburg i. Br., Germany |
The discretization of a bilaterally constrained total variation minimization problem with conforming low order finite elements is analyzed and three iterative schemes are proposed which differ in the treatment of the non-differentiable terms. Unconditional stability and convergence of the algorithms is addressed, an application to piecewise constant image segmentation is presented and numerical experiments are shown.
References:
[1] |
H. Attouch, G. Buttazzo and G. Michaille,
Variational Analysis in Sobolev and BV Spaces, MPS/SIAM Series on Optimization (vol. 6), Philadelphia, 2006. |
[2] |
S. Bartels,
Numerical Methods for Nonlinear Partial Differential Equations, Springer, Heidelberg, 2015.
doi: 10.1007/978-3-319-13797-1. |
[3] |
S. Bartels and M. Milicevic,
Stability and experimental comparison of prototypical iterative schemes for total variation regularized problems, Computational Methods in Applied Mathematics, 16 (2016), 361-388.
doi: 10.1515/cmam-2016-0014. |
[4] |
S. Bartels, R. H. Nochetto and A. J. Salgado,
Discrete total variation flows without regularization, SIAM J. Numer. Anal., 52 (2014), 363-385.
doi: 10.1137/120901544. |
[5] |
S. Bartels, R. H. Nochetto and A. J. Salgado,
A total variation diminishing interpolation operator and applications, Mathematics of Computation, 84 (2015), 2569-2587.
doi: 10.1090/mcom/2942. |
[6] |
A. Beck and M. Teboulle,
Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Transactions on Image Processing, 18 (2009), 2419-2434.
doi: 10.1109/TIP.2009.2028250. |
[7] |
B. Berkels, An unconstrained multiphase thresholding approach for image segmentation, in Scale Space and Variational Methods in Computer Vision (eds. X.-C. Tai, K. Mørken, M. Lysaker, K.-A. Lie), 5567 (2009), 26–37.
doi: 10.1007/978-3-642-02256-2_3. |
[8] |
B. Berkels, A. Effland and M. Rumpf,A posteriori error control for the binary Mumford-Shah model, Math. Comp., 86 (2017), 1769-1791, arXiv: 1505.05284
doi: 10.1090/mcom/3138. |
[9] |
S. C. Brenner and L. R. Scott,
The Mathematical Theory of Finite Element Methods, 3rd edition, Texts in Applied Mathematics, vol. 15, Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
[10] |
E. Casas, K. Kunisch and C. Pola,
Regularization by functions of bounded variation and application to image enhancement, Appl. Math. Optim., 40 (1999), 229-257.
doi: 10.1007/s002459900124. |
[11] |
A. Chambolle,
An algorithm for total variation minmization and applications, J. Math. Imaging Vision, 20 (2004), 89-97.
doi: 10.1023/B:JMIV.0000011321.19549.88. |
[12] |
A. Chambolle and T. Pock,
A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[13] |
T. F. Chan, S. Esedoglu and M. Nikolova,
Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math., 66 (2006), 1632-1648.
doi: 10.1137/040615286. |
[14] |
R. H. Chan, M. Tao and X. Yuan,
Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers, SIAM J. Imaging Sci., 6 (2013), 680-697.
doi: 10.1137/110860185. |
[15] |
M. Fortin and R. Glowinski,
Augmented Lagrangian Methods, 1st edition, North-Holland Publishing Co. , Amsterdam, 1983. |
[16] |
R. Glowinski,
Numerical Methods for Nonlinear Variational Problems, Springer, New York, 1984.
doi: 10.1007/978-3-662-12613-4. |
[17] |
M. Hintermüller, C. N. Rautenberg and J. Hahn, Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction, Inverse Problems, 30 (2014), 055014, 34pp.
doi: 10.1088/0266-5611/30/5/055014. |
[18] |
R. T. Rockafellar,
Convex Analysis, Princeton University Press, New Jersey, 1970. |
[19] |
T. Roubiček and J. Valdman,
Perfect plasticity with damage and healing at small strains, its modelling, analysis, and computer implementation, SIAM J. Appl. Math., 76 (2016), 314-340.
doi: 10.1137/15M1019647. |
[20] |
L. I. 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. |
[21] |
M. Thomas,
Quasistatic damage evolution with spatial BV-regularization, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 235-255.
doi: 10.3934/dcdss.2013.6.235. |
[22] |
C. Wu and X.-C. Tai,
Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and higher order models, SIAM J. Imaging Sci., 3 (2010), 300-339.
doi: 10.1137/090767558. |
[23] |
M. Zhu,
Fast Numerical Algorithms for Total Variation Based Image Restoration, Ph. D thesis, University of California in Los Angeles, 2008. |
show all references
Dedicated to Professor T. Roubíčcek on the occasion of his 60th birthday.
References:
[1] |
H. Attouch, G. Buttazzo and G. Michaille,
Variational Analysis in Sobolev and BV Spaces, MPS/SIAM Series on Optimization (vol. 6), Philadelphia, 2006. |
[2] |
S. Bartels,
Numerical Methods for Nonlinear Partial Differential Equations, Springer, Heidelberg, 2015.
doi: 10.1007/978-3-319-13797-1. |
[3] |
S. Bartels and M. Milicevic,
Stability and experimental comparison of prototypical iterative schemes for total variation regularized problems, Computational Methods in Applied Mathematics, 16 (2016), 361-388.
doi: 10.1515/cmam-2016-0014. |
[4] |
S. Bartels, R. H. Nochetto and A. J. Salgado,
Discrete total variation flows without regularization, SIAM J. Numer. Anal., 52 (2014), 363-385.
doi: 10.1137/120901544. |
[5] |
S. Bartels, R. H. Nochetto and A. J. Salgado,
A total variation diminishing interpolation operator and applications, Mathematics of Computation, 84 (2015), 2569-2587.
doi: 10.1090/mcom/2942. |
[6] |
A. Beck and M. Teboulle,
Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Transactions on Image Processing, 18 (2009), 2419-2434.
doi: 10.1109/TIP.2009.2028250. |
[7] |
B. Berkels, An unconstrained multiphase thresholding approach for image segmentation, in Scale Space and Variational Methods in Computer Vision (eds. X.-C. Tai, K. Mørken, M. Lysaker, K.-A. Lie), 5567 (2009), 26–37.
doi: 10.1007/978-3-642-02256-2_3. |
[8] |
B. Berkels, A. Effland and M. Rumpf,A posteriori error control for the binary Mumford-Shah model, Math. Comp., 86 (2017), 1769-1791, arXiv: 1505.05284
doi: 10.1090/mcom/3138. |
[9] |
S. C. Brenner and L. R. Scott,
The Mathematical Theory of Finite Element Methods, 3rd edition, Texts in Applied Mathematics, vol. 15, Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
[10] |
E. Casas, K. Kunisch and C. Pola,
Regularization by functions of bounded variation and application to image enhancement, Appl. Math. Optim., 40 (1999), 229-257.
doi: 10.1007/s002459900124. |
[11] |
A. Chambolle,
An algorithm for total variation minmization and applications, J. Math. Imaging Vision, 20 (2004), 89-97.
doi: 10.1023/B:JMIV.0000011321.19549.88. |
[12] |
A. Chambolle and T. Pock,
A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[13] |
T. F. Chan, S. Esedoglu and M. Nikolova,
Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math., 66 (2006), 1632-1648.
doi: 10.1137/040615286. |
[14] |
R. H. Chan, M. Tao and X. Yuan,
Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers, SIAM J. Imaging Sci., 6 (2013), 680-697.
doi: 10.1137/110860185. |
[15] |
M. Fortin and R. Glowinski,
Augmented Lagrangian Methods, 1st edition, North-Holland Publishing Co. , Amsterdam, 1983. |
[16] |
R. Glowinski,
Numerical Methods for Nonlinear Variational Problems, Springer, New York, 1984.
doi: 10.1007/978-3-662-12613-4. |
[17] |
M. Hintermüller, C. N. Rautenberg and J. Hahn, Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction, Inverse Problems, 30 (2014), 055014, 34pp.
doi: 10.1088/0266-5611/30/5/055014. |
[18] |
R. T. Rockafellar,
Convex Analysis, Princeton University Press, New Jersey, 1970. |
[19] |
T. Roubiček and J. Valdman,
Perfect plasticity with damage and healing at small strains, its modelling, analysis, and computer implementation, SIAM J. Appl. Math., 76 (2016), 314-340.
doi: 10.1137/15M1019647. |
[20] |
L. I. 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. |
[21] |
M. Thomas,
Quasistatic damage evolution with spatial BV-regularization, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 235-255.
doi: 10.3934/dcdss.2013.6.235. |
[22] |
C. Wu and X.-C. Tai,
Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and higher order models, SIAM J. Imaging Sci., 3 (2010), 300-339.
doi: 10.1137/090767558. |
[23] |
M. Zhu,
Fast Numerical Algorithms for Total Variation Based Image Restoration, Ph. D thesis, University of California in Los Angeles, 2008. |



Split-split | Heron-penalty | Heron-split | |||||||||
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Split-split | Heron-penalty | Heron-split | |||||||||
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Split-split | Heron-penalty | Heron-split | |||||||||
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