Journal of Scientific Computing

, Volume 60, Issue 1, pp 79–100 | Cite as

Domain Decomposition Methods for Nonlocal Total Variation Image Restoration

  • Huibin Chang
  • Xiaoqun Zhang
  • Xue-Cheng Tai
  • Danping Yang


Nonlocal total variation (TV) regularization (Gilboa and Osher in Multiscale Model Simulat 7(3): 1005–1028, 2008; Zhou and Schölkopf in Pattern recognition, proceedings of the 27th DAGM symposium. Springer, Berlin, pp 361–368, 2005) has been widely used for the natural image processing, since it is able to preserve repetitive textures and details of images. However, its applications have been limited in practice, due to the high computational cost for large scale problems. In this paper, we apply domain decomposition methods (DDMs) (Xu et al. in Inverse Probl Imag 4(3):523–545, 2010) to the nonlocal TV image restoration. By DDMs, the original problem is decomposed into much smaller subproblems defined on subdomains. Each subproblem can be efficiently solved by the split Bregman algorithm and Bregmanized operator splitting algorithm in Zhang et al. (SIAM J Imaging Sci 3(3):253–276, 2010). Furthermore, by using coloring technique on the domain decomposition, all subproblems defined on subdomains with same colors can be computed in parallel. Our numerical examples demonstrate that the proposed methods can efficiently solve the nonlocal TV based image restoration problems, such as denoising, deblurring and inpainting. It can be computed in parallel with a considerable speedup ratio and speedup efficiency.


Overlapping domain decomposition Subspace correction  Nonlocal total variation regularization Image restoration  Parallel computing 



The first two authors are grateful for the visit in MAS, SPMS, Nanyang Technological University, Singapore invited by Prof. Xue-Cheng Tai and Prof. Li-Lian Wang, which initiates this project. We also thank the reviewers for the comments, which have greatly improved the paper.


  1. 1.
    Aujol, J.-F., Ladjal, S., Masnou, S.: Exemplar-based inpainting from a variational point of view UCLA CAM, Report, 09-04, (2009)Google Scholar
  2. 2.
    Bresson, X., Chan, T.: Non-local unsupervised variational image segmentation models, pp. 08–67. UCLA CAM, Report (2008)Google Scholar
  3. 3.
    Buades, A., Coll, B., Morel, J.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bertalmio, M., Vese, L., Sapiro, G., Osher, S.: Simultaneous structure and texture image inpainting. IEEE Trans. Image Process. 12(8), 882–889 (2003)CrossRefGoogle Scholar
  5. 5.
    Bertalm, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of SIGGRAPH 2000. pp. 417–424 (2000)Google Scholar
  6. 6.
    Burger, M., He, L., Schoenlieb, C.: Cahn-Hilliard inpainting and a generalization for grayvalue images. SIAM J. Imaging Sci. 2(4), 1129–1167 (2009)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Chan, T.F., Shen, J.: Mathematical models for local non-texture inpainting. SIAM J. Appl. Math. 62(3), 1019–1043 (2001)MathSciNetGoogle Scholar
  9. 9.
    Chen, K., Tai, X.C.: On semismooth Newton’s methods for total variation minimization. J. Sci. Comput. 33, 115–138 (2007)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Darbon, J., Cunha, A., Chan, T.F., Osher, S., Jensen, G.J.: Fast nonlocal filtering applied to electron cryomicroscopy. In: Proceedings of ISBI. pp. 1331–1334 (2008)Google Scholar
  11. 11.
    Dryja, M., Widlund, O.B.: Towards a unified theory of domain decomposition algorithms for elliptic problems. In: Chan, T., et al. (eds.) Third International Symposiumon Domain Decomposition Methods for Partial Differential Equations. Houston, Texas (1989)Google Scholar
  12. 12.
    Duan, Y., Tai, X.C.: Domain decomposition methods with graph cuts algorithms for total variation minimization. Adv. Comput. Math. 36(2), 175–199 (2012)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Efros, A., Leung, T.: Texture synthesis by non-parametric sampling. In: Proceedings of the IEEE international conference on computer vision, vol. 2, pp. 1033–1038. Corfu, Greece (1999)Google Scholar
  14. 14.
    Elmoataz, A., Lezoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing. IEEE Trans. Image Process. 17(7), 1047–1060 (2008)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Esedoglu, S., Shen, J.: Digital inpainting based on the mumford-shah-euler image model. Eur. J. Appl. Math. 13(4), 353–370 (2002)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Fornasier, M., Langer, A., Schönlieb, C.B.: Domain decomposition methods for compressed sensing. (2009, in print)Google Scholar
  17. 17.
    Fornasier, M., Langer, A., Schönlieb, C.B.: A convergent overlapping domain decomposition method for total variation minimization. Numer. Math. 116(4), 645–685 (2010)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Fornasier, M., Schönlieb, C.-B.: Subspace correction methods for total variation and \(L_1\) minimization. SIAM J. Numer. Anal. 47(5), 3397–3428 (2009)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Fornasier, M., Kim, Y., Langer, A., Schönlieb, C.B.: Wavelet decomposition method for L2/TV-image deblurring. SIAM J. Imaging Sci. 5(3), 857–885 (2012)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Griebel, M., Oswald, P.: On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math. 70(2), 163–180 (1995)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Goldstein, T., Osher, S.: The split bregman method for \(l^1\) regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Hale, E.T., Yin, W., Zhang, Y.: Fixed-point continuation for \(L1\)-regularized minimization: methodology and convergence. SIAM J. Optim. 19(3), 1107–1130 (2008)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Hintermüller, M., Langer, A.: Subspace correction methods for a class of non-smooth and non-additive convex variational problems in image processing. Oct (2012, submitted)Google Scholar
  25. 25.
    Langer, A., Osher, S., Schonlieb, C.-B.: Bregmanized Domain Decomposition for Image restoration. UCLA CAM Report, CAM 11–30 (2011)Google Scholar
  26. 26.
    Masnou, S., Morel, J.-M.: Level lines based disocclusion. In: International Conference on Image Processing vol. 3, pp. 259–263 (1998)Google Scholar
  27. 27.
    Lou, Y., Zhang, X., Osher, S., Bertozzi, A.: Image recovery via nonlocal operators. J. Sci. Comput. 42(2), 185–197 (2010)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Firsov, D., Lui, S.H.: Domain decomposition methods in image denoising using Gaussian curvature. J. Comput. Appl. Math. 193(2), 460–473 (2006)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Ng, M., Qi, L., Yang, Y., Huang, Y.: On semismooth Newton’s methods for total variation minimization. J. Math. Imaging Vis. 27, 265–276 (2007)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)CrossRefMATHGoogle Scholar
  31. 31.
    Scherzer, O., et al.: Handbook of Mathematical Methods in Imaging. Springer, New York (2011)CrossRefMATHGoogle Scholar
  32. 32.
    Smith, B.F., Bjørstad, P.E., Gropp, W.D.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996)MATHGoogle Scholar
  33. 33.
    Tai, X.C., Duan, Y.P.: domain decomposition methods with graph cuts algorithms for image segmentation. Int. J. Numer. Anal. Model. 8(1), 137–155 (2011)MATHMathSciNetGoogle Scholar
  34. 34.
    Tai, X.C.: Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities. Numer. Math. 93(4), 755–786 (2003)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Tai, X.C., Espedal, M.: Applications of a space decomposition method to linear and nonlinear elliptic problems. Numer. Methods Partial Differ. Equ. 14(6), 717–737 (1998)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Peyré, G., Bougleux, S., Cohen L.: Non-local regularization of inverse problems. In: ECCV 2008, Part III, Lecture Notes in Computer Science 5304, pp. 57–68. Springer, Berlin, Heidelberg (2008)Google Scholar
  37. 37.
    Tai, X.C., Espedal, M.: Rate of convergence of some space decomposition methods for linear and nonlinear problems. SIAM J. Numer. Anal. 35(4), 1558–1570 (1998)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Tai, X.C., Tseng, P.: Convergence rate analysis of an asynchronous space decomposition method for convex minimization. Math. Comput. 71(239), 1105–1136 (2002)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Tai, X.C., Xu, J.: Global and uniform convergence of subspace correction methods for some convex optimization problems. Math. Comput. 71(237), 105–124 (2002)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Wu, C.L., Tai, X.C.: Augmented lagrangian method, dual methods and split-bregman iterations for ROF, vectorial TV and higher order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Xu, J., Tai, X.C., Wang, L.L.: A two-level domain decomposition method for image restoration. Inverse Probl. Imag. 4(3), 523–545 (2010)Google Scholar
  42. 42.
    Xu, J.C.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Zhang, X.Q., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3(3), 253–276 (2010)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Zhang, X.Q., Chan, T.F.: Wavelet inpainting by nonlocal total variation. Inverse Probl. Imag. 4(1), 191–210 (2010)Google Scholar
  45. 45.
    Zhou, D., Schölkopf, B.: Regularization on discrete spaces. In: Pattern Recognition, Proceedings of the 27th DAGM Symposium, pp. 361–368. Springer, Berlin (2005)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Huibin Chang
    • 1
  • Xiaoqun Zhang
    • 2
  • Xue-Cheng Tai
    • 3
  • Danping Yang
    • 4
  1. 1.School of Mathematical SciencesTianjin Normal UniversityTianjin People’s Republic of China
  2. 2.Department of Mathematics, MOE-LSC and Institute of Natural ScienceShanghai Jiao Tong UniversityShanghai People’s Republic of China
  3. 3.Department of MathematicsUniversity of BergenBergenNorway
  4. 4.Department of MathematicsEast China Normal UniversityShanghai People’s Republic of China

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