A Finite Element Nonoverlapping Domain Decomposition Method with Lagrange Multipliers for the Dual Total Variation Minimizations

  • Chang-Ock Lee
  • Jongho ParkEmail author


In this paper, we consider a primal-dual domain decomposition method for total variation regularized problems appearing in mathematical image processing. The model problem is transformed into an equivalent constrained minimization problem by tearing-and-interconnecting domain decomposition. Then, the continuity constraints on the subdomain interfaces are treated by introducing Lagrange multipliers. The resulting saddle point problem is solved by the first order primal-dual algorithm. We apply the proposed method to image denoising, inpainting, and segmentation problems with either \(L^2\)-fidelity or \(L^1\)-fidelity. Numerical results show that the proposed method outperforms the existing state-of-the-art methods.


Total variation Lagrange multipliers Domain decomposition Parallel computation Image processing 

Mathematics Subject Classification

65N30 65N55 65Y05 68U10 



  1. 1.
    Bartels, S.: Total variation minimization with finite elements: convergence and iterative solution. SIAM J. Numer. Anal. 50(3), 1162–1180 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chambolle, A., Levine, S.E., Lucier, B.J.: An upwind finite-difference method for total variation-based image smoothing. SIAM J. Imaging Sci. 4(1), 277–299 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chambolle, A., Pock, T.: An introduction to continuous optimization for imaging. Acta Numer. 25, 161–319 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chan, T.F., Esedoglu, S.: Aspects of total variation regularized \({L}^1\) function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chan, T.F., Wong, C.-K.: Total variation blind deconvolution. IEEE Trans. Image Process. 7(3), 370–375 (1998)CrossRefGoogle Scholar
  8. 8.
    Chang, H., Tai, X.-C., Wang, L.-L., Yang, D.: Convergence rate of overlapping domain decomposition methods for the Rudin–Osher–Fatemi model based on a dual formulation. SIAM J. Imaging Sci. 8(1), 564–591 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dong, Y., Hintermüller, M., Neri, M.: An efficient primal-dual method for \({L}^1 {T}{V}\) image restoration. SIAM J. Imaging Sci. 2(4), 1168–1189 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Duan, Y., Chang, H., Tai, X.-C.: Convergent non-overlapping domain decomposition methods for variational image segmentation. J. Sci. Comput. 69(2), 532–555 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Farhat, C., Roux, F.-X.: A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng. 32(6), 1205–1227 (1991)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fornasier, M.: Domain decomposition methods for linear inverse problems with sparsity constraints. Inverse Probl. 23(6), 2505–2526 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fornasier, M., Langer, A., Schönlieb, C.-B.: A convergent overlapping domain decomposition method for total variation minimization. Numer. Math. 116(4), 645–685 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fornasier, M., Schönlieb, C.-B.: Subspace correction methods for total variation and \(l1\)-minimization. SIAM J. Numer. Anal. 47(5), 3397–3428 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hermann, M., Herzog, R., Schmidt, S., Vidal-Núñez, J., Wachsmuth, G.: Discrete total variation with finite elements and applications to imaging. J. Math. Imaging Vis. 61(4), 411–431 (2019)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hintermüller, M., Langer, A.: Subspace correction methods for a class of nonsmooth and nonadditive convex variational problems with mixed \({L}^1\)/\({L}^2\) data-fidelity in image processing. SIAM J. Imaging Sci. 6(4), 2134–2173 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hintermüller, M., Langer, A.: Non-overlapping domain decomposition methods for dual total variation based image denoising. J. Sci. Comput. 62(2), 456–481 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lee, C.-O., Lee, J.H., Woo, H., Yun, S.: Block decomposition methods for total variation by primal-dual stitching. J. Sci. Comput. 68(1), 273–302 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lee, C.-O., Nam, C.: Primal domain decomposition methods for the total variation minimization, based on dual decomposition. SIAM J. Sci. Comput. 39(2), B403–B423 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lee, C.-O., Nam, C., Park, J.: Domain decomposition methods using dual conversion for the total variation minimization with \({L}^1\) fidelity term. J. Sci. Comput. 78(2), 951–970 (2019)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lee, C.-O., Park, E.-H., Park, J.: A finite element approach for the dual Rudin–Osher–Fatemi model and its nonoverlapping domain decomposition methods. SIAM J. Sci. Comput. 41(2), B205–B228 (2019)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(1), 99–120 (2004)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (2015)Google Scholar
  24. 24.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Shen, J., Chan, T.F.: Mathematical models for local nontexture inpaintings. SIAM J. Appl. Math. 62(3), 1019–1043 (2002)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Strong, D., Chan, T.F.: Edge-preserving and scale-dependent properties of total variation regularization. Inverse Probl. 19(6), S165–S187 (2003)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, J., Lucier, B.J.: Error bounds for finite-difference methods for Rudin–Osher–Fatemi image smoothing. SIAM J. Numer. Anal. 49(2), 845–868 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1(3), 248–272 (2008)MathSciNetCrossRefGoogle Scholar
  29. 29.
    You, Y.-L., Kaveh, M.: A regularization approach to joint blur identification and image restoration. IEEE Trans. Image Process. 5(3), 416–428 (1996)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKAISTDaejeonKorea

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