Journal of Scientific Computing

, Volume 75, Issue 2, pp 713–732 | Cite as

A Semi-smooth Newton Method for Inverse Problem with Uniform Noise

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Abstract

In this paper we study inverse problems where observations are corrupted by uniform noise. By using maximum a posteriori approach, an \(L_\infty \)-norm constrained minimization problem can be formulated for uniform noise removal. The main difficulty of solving such minimization problem is how to deal with non-differentiability of the \(L_\infty \)-norm constraint and how to estimate the level of uniform noise. The main contribution of this paper is to develop an efficient semi-smooth Newton method for solving this minimization problem. Here the \(L_\infty \)-norm constraint can be handled by active set constraints arising from the optimality conditions. In the proposed method, linear systems based on active set constraints are required to solve in each Newton step. We also employ the method of moments (MoM) to estimate the level of uniform noise for the minimization problem. The combination of the proposed method and MoM is quite effective for solving inverse problems with uniform noise. Numerical examples are given to demonstrate that our proposed method outperforms the other testing methods.

Keywords

Inverse problem Uniform noise Semi-smooth Newton method \(L_\infty \)-norm constraint Linear systems 

Notes

Acknowledgements

The authors would like to thank the two anonymous referees and the editor for their helpful comments and suggestions.

References

  1. 1.
    Alter, F., Durand, S., Froment, J.: Adapted total variation for artifact free decompression of JPEG images. J. Math. Imaging Vis. 23(2), 199–211 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrew, H., Hunt, B.: Digital Image Restoration. Prentice-Hall, Englewood Cliffs (1977)MATHGoogle Scholar
  3. 3.
    Avriel, M.: Nonlinear Programming: Analysis and Methods. Dover Publications, Mineola (2003)MATHGoogle Scholar
  4. 4.
    Bar, L., Brook, A., Schen, N., Kiryati, N.: Deblurring of color images corrupted by salt-and-pepper noise. IEEE Trans. Image Process. 16, 1101–1111 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bar, L., Schen, N., Kiryati, N.: Image deblurring in the presence of impulsive noise. Int. J. Comput. Vis. 70, 279–298 (2006)CrossRefGoogle Scholar
  6. 6.
    Bertero, M., De Mol, C., Pike, E.R.: Linear inverse problems with discrete data. I. General formulation and singular system analysis. Inverse Prob. 1(4), 301 (1985)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bjorck, A.: Numerical methods for least squares problems. Number 51, Society for Industrial Mathematics (1996)Google Scholar
  8. 8.
    Bovik, A.: Handbook of Image and Video Processing. Academic Press, Cambridge (2010)MATHGoogle Scholar
  9. 9.
    Bredies, K., Holler, M.: A total variation-based JPEG decompression model. SIAM J. Imaging Sci. 5(1), 366–393 (2012)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.
    Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38(4), 1200–1216 (2000)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Clarke, F.: Optimization and Nonsmooth Analysis. Wiley, Hoboken (1983)MATHGoogle Scholar
  13. 13.
    Clason, C.: \(l^\infty \) fitting for inverse problems with uniform noise. Inverse Prob. 28(10), 104007 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Clason, C., Ito, K., Kunisch, K.: Minimal invasion: an optimal \(l_\infty \) state constraint problem. ESAIM Math. Model. Numer. Anal. 45(03), 505–522 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fletcher, R.: Practical Methods of Optimization. Wiley, Hoboken (2013)MATHGoogle Scholar
  16. 16.
    Griesse, R., Lorenz, D.: A semismooth newton method for tikhonov functionals with sparsity constraints. Inverse Prob. 24(3), 035007 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hans, E., Raasch, T.: Global convergence of damped semismooth newton methods for \(l_1\) Tikhonov regularization. Inverse Prob. 31(2), 025005 (2015)CrossRefMATHGoogle Scholar
  18. 18.
    Ito, K., Kunisch, K.: Semi-smooth newton methods for variational inequalities of the first kind. ESAIM. Math. Model. Numer. Anal. 37(01), 41–62 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jung, C., Jiao, L., Qi, H., Sun, T.: Image deblocking via sparse representation. Signal Process. Image Commun. 27(6), 663–677 (2012)CrossRefGoogle Scholar
  20. 20.
    Kotz, S., Van Dorp, J.: Other Continuous Families of Distributions with Bounded Support and Applications. World Scientific, Singapore (2004)CrossRefMATHGoogle Scholar
  21. 21.
    Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15(6), 959–972 (1977)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(1–2), 99–120 (2004)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Pang, J., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3(3), 443–465 (1993)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58(1–3), 353–367 (1993)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Schay, G.: Introduction to Probability with Statistical Applications. Springer, Berlin (2007)MATHGoogle Scholar
  27. 27.
    Sun, D., Cham, W.: Postprocessing of low bit-rate block dct coded images based on a fields of experts prior. IEEE Trans. Image Process. 16(11), 2743–2751 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Ulbrich, M.: Nonsmooth Newton-like methods for variational inequalities and constrained optimization problems in function spaces. PhD thesis, Habilitation thesis, Fakultät für Mathematik, Technische Universität München (2002)Google Scholar
  29. 29.
    Williams, J., Kalogiratou, Z.: Least squares and chebyshev fitting for parameter estimation in ODEs. Adv. Comput. Math. 1(3), 357–366 (1993)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Wu, X., Bao, P.: \(l_\infty \); constrained high-fidelity image compression via adaptive context modeling. IEEE Trans. Image Process. 9(4), 536–542 (2000)CrossRefMATHGoogle Scholar
  31. 31.
    Yang, Y., Galatsanos, N., Katsaggelos, A.: Regularized reconstruction to reduce blocking artifacts of block discrete cosine transform compressed images. IEEE Trans. Circuits Syst. Video Technol. 3(6), 421–432 (1993)CrossRefGoogle Scholar
  32. 32.
    Yang, Y., Galatsanos, N.P., Katsaggelos, A.K.: Projection-based spatially adaptive reconstruction of block-transform compressed images. IEEE Trans. Image Process. 4(7), 896–908 (1995)CrossRefGoogle Scholar
  33. 33.
    Zhang, Z., Wen, Y.: Primal-dual approach for uniform noise removal. In: First International Conference on Information Science and Electronic Technology (ISET 2015), pp. 103–106 (2015)Google Scholar
  34. 34.
    Zhen, L., Delp, E.: Block artifact reduction using a transform-domain Markov random field model. IEEE Trans. Circuits Syst. Video Technol. 15(12), 1583–1593 (2005)CrossRefGoogle Scholar
  35. 35.
    Zhou, J., Wu, X., Zhang, L.: \(l_2\) restoration of \(l_\infty \)-decoded images via soft-decision estimation. IEEE Trans. Image Process. 21(12), 4797–4807 (2012)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP), College of Mathematics and Computer ScienceHunan Normal UniversityChangshaPeople’s Republic of China
  2. 2.Advanced Modeling and Applied Computing Laboratory, Department of MathematicsThe University of Hong KongHong KongPeople’s Republic of China
  3. 3.Department of Mathematics Hong KongBaptist UniversityHong KongPeople’s Republic of China

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