Skip to main content
SpringerLink
Log in

Domain Decomposition Methods for Nonlocal Total Variation Image Restoration

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Aujol, J.-F., Ladjal, S., Masnou, S.: Exemplar-based inpainting from a variational point of view UCLA CAM, Report, 09-04, (2009)

  2. Bresson, X., Chan, T.: Non-local unsupervised variational image segmentation models, pp. 08–67. UCLA CAM, Report (2008)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bertalmio, M., Vese, L., Sapiro, G., Osher, S.: Simultaneous structure and texture image inpainting. IEEE Trans. Image Process. 12(8), 882–889 (2003)

    Article  Google Scholar 

  5. Bertalm, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of SIGGRAPH 2000. pp. 417–424 (2000)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chan, T.F., Shen, J.: Mathematical models for local non-texture inpainting. SIAM J. Appl. Math. 62(3), 1019–1043 (2001)

    MathSciNet  Google Scholar 

  9. Chen, K., Tai, X.C.: On semismooth Newton’s methods for total variation minimization. J. Sci. Comput. 33, 115–138 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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. Duan, Y., Tai, X.C.: Domain decomposition methods with graph cuts algorithms for total variation minimization. Adv. Comput. Math. 36(2), 175–199 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

    Article  MathSciNet  Google Scholar 

  15. Esedoglu, S., Shen, J.: Digital inpainting based on the mumford-shah-euler image model. Eur. J. Appl. Math. 13(4), 353–370 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fornasier, M., Langer, A., Schönlieb, C.B.: Domain decomposition methods for compressed sensing. (2009, in print)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Griebel, M., Oswald, P.: On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math. 70(2), 163–180 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Goldstein, T., Osher, S.: The split bregman method for \(l^1\) regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  25. Langer, A., Osher, S., Schonlieb, C.-B.: Bregmanized Domain Decomposition for Image restoration. UCLA CAM Report, CAM 11–30 (2011)

  26. Masnou, S., Morel, J.-M.: Level lines based disocclusion. In: International Conference on Image Processing vol. 3, pp. 259–263 (1998)

  27. Lou, Y., Zhang, X., Osher, S., Bertozzi, A.: Image recovery via nonlocal operators. J. Sci. Comput. 42(2), 185–197 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  28. Firsov, D., Lui, S.H.: Domain decomposition methods in image denoising using Gaussian curvature. J. Comput. Appl. Math. 193(2), 460–473 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  30. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

    Article  MATH  Google Scholar 

  31. Scherzer, O., et al.: Handbook of Mathematical Methods in Imaging. Springer, New York (2011)

    Book  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  34. Tai, X.C.: Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities. Numer. Math. 93(4), 755–786 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Xu, J.C.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  44. Zhang, X.Q., Chan, T.F.: Wavelet inpainting by nonlocal total variation. Inverse Probl. Imag. 4(1), 191–210 (2010)

    Google Scholar 

  45. Zhou, D., Schölkopf, B.: Regularization on discrete spaces. In: Pattern Recognition, Proceedings of the 27th DAGM Symposium, pp. 361–368. Springer, Berlin (2005)

Download references

Acknowledgments

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.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tianjin Normal University, Tianjin , 300387, People’s Republic of China

    Huibin Chang

  2. Department of Mathematics, MOE-LSC and Institute of Natural Science, Shanghai Jiao Tong University, Shanghai , 200240, People’s Republic of China

    Xiaoqun Zhang

  3. Department of Mathematics, University of Bergen, 125007 , Bergen, Norway

    Xue-Cheng Tai

  4. Department of Mathematics, East China Normal University, Shanghai , 200241, People’s Republic of China

    Danping Yang

Authors
  1. Huibin Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoqun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xue-Cheng Tai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Danping Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huibin Chang.

Additional information

The authors’ research was supported by MOE IDM project NRF2007IDM-IDM002-010, Singapore. The second author was partially supported by NSFC11101277, NSFC11161130004, and by the Shanghai Pujiang Talent Program under Grant Number 11PJ1405900. The last author were partially supported by NSFC11071080.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chang, H., Zhang, X., Tai, XC. et al. Domain Decomposition Methods for Nonlocal Total Variation Image Restoration. J Sci Comput 60, 79–100 (2014). https://doi.org/10.1007/s10915-013-9786-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-013-9786-9

Keywords

Search

Navigation