Journal of Scientific Computing

, Volume 57, Issue 2, pp 349–371 | Cite as

Image Restoration via Tight Frame Regularization and Local Constraints

  • Fang Li
  • Tieyong Zeng


In this paper, we propose two variational image denosing/deblurring models which combine tight frame regularization with two types of existing local constraints. Additive white Gaussian noise is assumed in the models. By Lagrangian multiplier method, the local constraints correspond to the fidelity term with spatial adaptive parameters. As the fidelity parameter is bigger in the image regions with textures than in the cartoon region, our models can recover more texture while denoising/deblurring. Fast numerical schemes are designed for the two models based on split Bregman (SB) technique and doubly augmented Lagrangian (DAL) method with acceleration. In the experiments, we show that the proposed models have better performance compared with the existing total variation based image restoration models with global or local constraints and the frame based model with global constraint.


Image restoration Tight frame Local constraint  Split Bregman  Doubly augmented Lagrangian 


  1. 1.
    Afonso, M.V., Bioucas-Dias, J.M., Figueiredo, M.A.T.: Fast image recovery using variable splitting and constrained optimization. IEEE Trans. Image Process. 19(9), 2345–2356 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Almansa, A., Ballester, C., Caselles, V., Haro, G.: A TV based restoration model with local constraints. J. Sci. Comput. 34(3), 209–236 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, vol. 147. Springer, New York (2006)Google Scholar
  4. 4.
    Bertalmio, M., Caselles, V., Rougé, B., Solé, A.: TV based image restoration with local constraints. J. Sci. Comput. 19(1), 95–122 (2003)Google Scholar
  5. 5.
    Cai. J.-F., Dong, B., Osher, S., Shen, Z.: Image restoration: total variation, wavelet frames, and beyond. J. Am. Math. Soc. (to appear) (2012)Google Scholar
  6. 6.
    Cai, J.-F., Osher, S., Shen, Z.: Split bregman methods and frame based image restoration. Multiscale Model. Simul. 8(2), 337–369 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cai, J.-F., Shen, Z.: Framelet based deconvolution. J. Comput. Math. 28(3), 289–308 (2010)MathSciNetMATHGoogle Scholar
  8. 8.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1), 89–97 (2004)MathSciNetGoogle Scholar
  9. 9.
    Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dong, B., Shen, Z.: MRA-Based Wavelet Frames and Applications. UCLA CAM reports 10–69 (2010)Google Scholar
  12. 12.
    Dong, B., Zhang, Y.: An efficient algorithm for \(l_0\) minimization in wavelet frame based image restoration. Technical report, UCLA CAM reports, cam11-66 (2011)Google Scholar
  13. 13.
    Dong, Y., Hintermíźller, M., Camacho, M.R.: Automated regularization parameter selection in multi-scale total variation models for image restoration. J. Math. Imaging Vis. 40(1), 82–104 (2011)CrossRefMATHGoogle Scholar
  14. 14.
    Eckstein, J., Bertsekas, D.P.: On the douglasałrachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1), 293–318 (1992)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Faurre, P.: Analyse numéique. Notes d’optimisation. École Polytechnique. Ed. Ellipses (1988)Google Scholar
  16. 16.
    Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Studies in mathematics and its applications, chapter 6, pp. 299–331. North-Holland, Amsterdam (1983)Google Scholar
  17. 17.
    Gilboa, G., Sochen, N., Zeevi, Y.Y.: Variational denoising of partly textured images by spatially varying constraints. IEEE Trans. Image Process. 15(8), 2281–2289 (2006)CrossRefGoogle Scholar
  18. 18.
    Goldstein, T., Osher, S.: The split bregman method for l1 regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hahn, J., Wu, C., Tai, X.C.: Augmented lagrangian method for generalized tv-stokes model. J. Sci. Comput. 50(2), 235–264 (2012)Google Scholar
  20. 20.
    Hintermüller, M., Rincon-Camacho, M.M.: Expected absolute value estimators for a spatially adapted regularization parameter choice rule in l1-tv-based image restoration. Inverse Probl. 26(8), 85005–85034 (2010)CrossRefGoogle Scholar
  21. 21.
    Iusem, A.N.: Augmented lagrangian methods and proximal point methods for convex optimization. Investigación Operativa 8, 11–49 (1999)Google Scholar
  22. 22.
    Li, F., Ng, M.K., Shen, C.: Multiplicative noise removal with spatially varying regularization parameters. SIAM J. Imaging Sci. 3(1), 1–22 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lintner, S., Malgouyres, F.: Solving a variational image restoration model which involves \(l^\infty \) constraints. Inverse Probl. 20, 815–831 (2004)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ma, L., Mosian, L., Yu, J., Zeng, T.: Stable method in solving total variation dictionay model with \(l^\infty \) constraints. HongKong Baptist University (HKBU), Technical report (2011) Google Scholar
  25. 25.
    Malgouyres, F.: Mathematical analysis of a model which combines total variation and wavelet for image restoration. J. Inf. process. 2(1), 1–10 (2002)Google Scholar
  26. 26.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Rudin, L.I., Osher, S.: Total variation based image restoration with free local constraints. In: Image Processing, 1994. Proceedings. ICIP-94., IEEE International Conference, 1, pp. 31–35. IEEE (1994)Google Scholar
  28. 28.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena 60, 259–268 (1992)CrossRefMATHGoogle Scholar
  29. 29.
    Setzer, S.: Operator splittings, bregman methods and frame shrinkage in image processing. Int. J. Comput. Vis. 92(3), 265–280 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Setzer, S., Steidl, G., Teuber, T.: Deblurring poissonian images by split bregman techniques. J. Vis. Commun. Image Represent. 21(3), 193–199 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Strong, D., Chan, T.: Edge-preserving and scale-dependent properties of total variation regularization. Inverse Probl. 19(6), S165–S187 (2003)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Tai, X.C., Hahn, J., Chung, G.J.: A fast algorithm for euler’s elastica model using augmented lagrangian method. SIAM J. Imaging Sci. 4, 313–344 (2011)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wu, C., Tai, X.C.: Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Zeng, T., Ng, K.: On the total variation dictionary model. IEEE Trans. Image Process. 19(3), 821–825 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsEast China Normal UniversityShanghaiChina
  2. 2.Department of Mathematics, Centre for Mathematical Imaging and VisionHong Kong Baptist UniversityKowloon TongHong Kong

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