# American Institute of Mathematical Sciences

December  2017, 10(6): 1207-1232. doi: 10.3934/dcdss.2017066

## 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

* Corresponding author: Sören Bartels

Dedicated to Professor T. Roubíčcek on the occasion of his 60th birthday.

Received  May 2016 Revised  November 2016 Published  June 2017

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.

Citation: Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066
##### 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

##### 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.
$L^2$-error between approximate minimizer $\tilde{u}_h$ of $E$ (generated by the split-split method) and iterates $u_h^j$, $s_h^j$ of the Heron-penalty method, $h=\sqrt{2}2^{-6}$, $\varepsilon=h$, for Example 1. Note that the iterates $u_h^j$ serve as good approximations of $\tilde{u}_h$ when $\delta=h/\alpha$
$L^2$-error between approximate minimizer $\tilde{u}_h$ of $E$ (generated by the split-split method) and iterates $u_h^j$, $s_h^j$ of the Heron-penalty method, $h=\sqrt{2}2^{-6}$, $\varepsilon=h$, for Example 2. Again, the iterates $u_h^j$ approximate $\tilde{u}_h$ properly when $\delta=h/\alpha$
Original image and outputs of Algorithm 5 using the split-split, Heron-penalty and Heron-split method in step (2), respectively (horizontal white lines are due to image conversion)
Iteration numbers with (5), (6), (7) for Example 1 and $\sigma_1=h^{-3/2}$ for the split-split method and $\tau=1$ for the Heron-penalty and the Heron-split method
 Split-split Heron-penalty Heron-split $\sigma_2$ $\delta$ $\sigma$ $1$ $\alpha$ $1/h$ $h/\alpha$ $h$ $1$ $\alpha$ $1/h$ $\sqrt{2}/2^5$ $344$ $39$ $37$ $89$ $42$ $892$ $52$ $57$ $\sqrt{2}/2^6$ $289$ $79$ $79$ $143$ $55$ $576$ $77$ $77$ $\sqrt{2}/2^7$ $283$ $77$ $78$ $211$ $69$ $473$ $94$ $94$ $\sqrt{2}/2^8$ $287$ $115$ $120$ $426$ $101$ $373$ $148$ $151$
 Split-split Heron-penalty Heron-split $\sigma_2$ $\delta$ $\sigma$ $1$ $\alpha$ $1/h$ $h/\alpha$ $h$ $1$ $\alpha$ $1/h$ $\sqrt{2}/2^5$ $344$ $39$ $37$ $89$ $42$ $892$ $52$ $57$ $\sqrt{2}/2^6$ $289$ $79$ $79$ $143$ $55$ $576$ $77$ $77$ $\sqrt{2}/2^7$ $283$ $77$ $78$ $211$ $69$ $473$ $94$ $94$ $\sqrt{2}/2^8$ $287$ $115$ $120$ $426$ $101$ $373$ $148$ $151$
Iteration numbers with (5), (6), (7) for Example 2 and $\sigma_1=h^{-3/2}$ for the split-split method and $\tau=1$ for the Heron-penalty and the Heron-split method
 Split-split Heron-penalty Heron-split $\sigma_2$ $\delta$ $\sigma$ $1$ $\alpha$ $1/h$ $h/\alpha$ $h$ $1$ $\alpha$ $1/h$ $\sqrt{2}/2^5$ $2745$ $10$ $124$ $74$ $24$ $4592$ $15$ $189$ $\sqrt{2}/2^6$ $2808$ $15$ $65$ $156$ $27$ $3586$ $21$ $72$ $\sqrt{2}/2^7$ $2847$ $23$ $35$ $298$ $35$ $3133$ $25$ $42$ $\sqrt{2}/2^8$ $-$ $29$ $30$ $572$ $43$ $-$ $33$ $37$
 Split-split Heron-penalty Heron-split $\sigma_2$ $\delta$ $\sigma$ $1$ $\alpha$ $1/h$ $h/\alpha$ $h$ $1$ $\alpha$ $1/h$ $\sqrt{2}/2^5$ $2745$ $10$ $124$ $74$ $24$ $4592$ $15$ $189$ $\sqrt{2}/2^6$ $2808$ $15$ $65$ $156$ $27$ $3586$ $21$ $72$ $\sqrt{2}/2^7$ $2847$ $23$ $35$ $298$ $35$ $3133$ $25$ $42$ $\sqrt{2}/2^8$ $-$ $29$ $30$ $572$ $43$ $-$ $33$ $37$
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