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A first-order image denoising model for staircase reduction

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Abstract

In this paper, we consider a total variation–based image denoising model that is able to alleviate the well-known staircasing phenomenon possessed by the Rudin-Osher-Fatemi model (Rudin et al., Phys. D 60, 259–268, 30). To minimize this variational model, we employ augmented Lagrangian method (ALM). Convergence analysis is established for the proposed algorithm. Numerical experiments are presented to demonstrate the features of the proposed model and also show the efficiency of the proposed numerical method.

Keywords

Image denoising Augmented Lagrangian method Variational model 

Mathematics Subject Classification (2010)

94A08 65K10 65M32 

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Notes

Acknowledgments

The author would like to thank the anonymous referees for their valuable comments and suggestions, which have helped very much to improve the presentation of this paper.

References

  1. 1.
    Ambrosio, L.: A compactness theorem for a new class of functions of bounded variation. Bollettino della Unione Matematica Italiana VII(4), 857–881 (1989)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aubert, G., Kornprobst, P.: Mathematical problems in image processing: partial differential equations and the calculus of variations. Springer Science and Business Media (2006)Google Scholar
  3. 3.
    Aubert, G., Vese, L.: A variational method in image recovery. SIAM J. Numer. Anal. 34(5), 1948–1979 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Brito-Loeza, C., Chen, K.: Multigrid algorithm for high order denoising. SIAM J. Imaging. Sciences 3(3), 363–389 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bae, E., Tai, X.C., Zhu, W.: Augmented Lagrangian method for an Euler’s elastica based segmentation model that promotes convex contours. Inverse Probl. Imag. 11(1), 1–23 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Beck, A.: First-Order Methods in Optimization, vol. 25, SIAM (2017)Google Scholar
  7. 7.
    Bellettini, G., Caselles, V., Novaga, M.: The total variation flow in \(\mathbb {R}^{n}\). J. Differ. Equations 184, 475–525 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bertalmio, M., Vese, L., Sapiro, G., Osher, S.: Simultaneous structure and texture image inpainting. IEEE Trans. on Image Process. 12(8), 882–889 (2003)CrossRefGoogle Scholar
  9. 9.
    Bredies, K, Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends\(^{{\circledR }}{\circledR }\) in Machine learning 3(1), 1–122 (2011)zbMATHGoogle Scholar
  11. 11.
    Buttazzo, G.: Semicontinuity, relaxation and integral representation in the calculus of variations, Pitman Research Notes in Mathematics 207, Longman Scientific and Technical (1989)Google Scholar
  12. 12.
    Chang, Q.S., Che, Z.Y.: An adaptive algorithm for TV-based model of three norms \(L_{q}, (q=\frac {1}{2},1,2)\) in image restoration. Applied Math and Comput. 329, 251–265 (2018)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math Imaging Vis. 40, 120–145 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Chan, T., Esedoglu, S.: Aspects of total variation regularized l 1 function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Chan, T., Esedoglu, S., Park, F., Yip, M.H.: Recent developments in total variation image restoration. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Handbook of Mathematical Models in Computer Vision. Springer, Berlin (2005)Google Scholar
  16. 16.
    Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Chan, T., Shen, J.: Mathematical models for local nontexture inpaintings. SIAM J. Appl. Math. 62(3), 1019–1043 (2001)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Chan, T., Wong, C.K.: Total variation blind deconvolution. IEEE Trans. Image Process. 7(3), 370–375 (1998)CrossRefGoogle Scholar
  20. 20.
    Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2, 323–343 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)zbMATHCrossRefGoogle Scholar
  22. 22.
    Huber, P.J.: Robust estimation of a location parameter. Ann. Stat. 53(1), 73–101 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Lysaker, M., Lundervold, A., Tai, X.C.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE. Trans. Image Process. 12, 1579–1590 (2003)zbMATHCrossRefGoogle Scholar
  24. 24.
    Lysaker, M., Osher, S., Tai, X.C.: Noise removal using smoothed normals and surface fitting. IEEE. Trans. Image Process. 13(10), 1345–1457 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations, University Lecture Series, Vol 22, Amer. Math. Soc.Google Scholar
  26. 26.
    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577–685 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Osher, S., Burger, M., Goldfarb, D., Xu, J.J., Yin, W.T.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4, 460–489 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the h − 1norm. SIAM Multiscale Model. Simul. 1, 349–370 (2003)zbMATHCrossRefGoogle Scholar
  29. 29.
    Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1(2), 97–116 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithm. Phys. D 60, 259–268 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Tai, X.C., Hahn, J., Chung, G.J.: A fast algorithm for Euler’s Elastica model using augmented Lagrangian method. SIAM J. Imaging Sciences 4(1), 313–344 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Vese, L.: A study in the BV space of a denoising-deblurring variational problem. Appl. Math Optim. 44, 131–161 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE. Trans. Image Process. 13(4), 600–612 (2004)CrossRefGoogle Scholar
  34. 34.
    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 Sciences 3(3), 300–339 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Zhu, W., Chan, T.: Image denoising using mean curvature of image surface. SIAM J. Imaging Sciences 5(1), 1–32 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Zhu, W., Tai, X.C., Chan, T.: Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Probl. Imag. 7(4), 1409–1432 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Zhu, W., Tai, X.C., Chan, T.: Image segmentation using Euler’s elastica as the regularization. J. Sci. Comput. 57(2), 414–438 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Zhu, W.: A numerical study of a mean curvature denoising model using a novel augmented Lagrangian method. Inverse Probl. Imag. 11(6), 975–996 (2017)MathSciNetzbMATHCrossRefGoogle Scholar

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

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AlabamaTuscaloosaUSA

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