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Spectral gradient method for impulse noise removal

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Abstract

In this paper, we propose a new spectral gradient method for removing impulse noise in the second phase of the two-phase method. An attractive property of the proposed method is that the search direction satisfies the sufficient descent property at each iteration, which is independent of any line search. Under Armijo-type line search, the global convergence of the proposed method is simplify established for general smooth functions. The preliminary numerical experiments are given to indicate the efficiency of the proposed method for impulse noise removal.

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References

  1. Hwang, H., Haddad, R.A.: Adaptive median filters: new algorithms and results. IEEE Trans. Image Process. 4, 499–502 (1995)

    Article  Google Scholar 

  2. Huang, T.S., Yang, G.J., Tang, G.Y.: Fast two-dimensional median filtering algorithm. IEEE Trans. Acoust. Speech Signal Process. 1, 13–18 (1979)

    Article  Google Scholar 

  3. Nodes, T.A., Gallagher Jr, N.C.: The output distribution of median type filters. IEEE Trans. Commun. 32, 532–541 (1984)

    Article  Google Scholar 

  4. Astola, J., Kuosmanen, P.: Fundamentals of Nonlinear Digital Filtering. CRC, Boca Raton (1997)

    Google Scholar 

  5. Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20, 99–120 (2004)

    Article  MathSciNet  Google Scholar 

  6. Chan, R.H., Ho, C.W., Nikolova, M.: Salt-and-pepper noise removal by median-type noise detector and edge-preserving regularization. IEEE Trans. Image Process. 14, 1479–1485 (2005)

    Article  Google Scholar 

  7. Chan, R.H., Hu, C., Nikolova, M.: An iterative procedure for removing random-valued impulsenoise. IEEE Signal Process. Lett. 11, 921–924 (2004)

    Article  Google Scholar 

  8. Chan, R.H., Ho, C.W., Leung, C.Y., Nikolova, M.: Minimization of detail-preserving regularization functional by Newton’s method with continuation. In: Proceedings of IEEE International Conference on Image Processing, pp. 125–128. Genova, Italy (2005)

  9. Cai, J.F., Chan, R.H., Morini, B.: Minimization of an edge-preserving regularization functional by conjugate gradient types methods. In: Image Processing Based on Partial Differential Equations: Prceedings of the International Conference on PDE-Based Image Processsing and Related Inverse Problems, CMA, Oslo, August 8–12, 2005, pp.109–122. Springer, Berlin (2007)

  10. Dong, Y., Chan, R.H.: A detecation statisitic for random-valued impulse noise. IEEE Trans. Image Process 16, 1112–1120 (2007)

    Article  MathSciNet  Google Scholar 

  11. Barizilai, J.M., Borwein, M.: Two point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)

    Article  MathSciNet  Google Scholar 

  12. Grippo, L., Lampariello, F., Lucidi, S.: A non-monotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23, 707–716 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  13. Raydan, M.: The Barzilain and Borwein gradient method for the large unconstrained minimization problem. SIAM J. Optim. 7, 26–33 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dai, Y.H., Zhang, H.C.: Adaptive two-point stepsize gradient algorithm. Numer. Algor. 27, 377–385 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Yu, G., Qi, L., Dai, Y.: On nonmonotone Chambolle gradient projection algorithms for total variation image restoration. J. Math. Imaging Vis. 35, 143–154 (2009)

    Article  MathSciNet  Google Scholar 

  16. Yu, G., Qi, L., Sun, Y., Zhou, Y.: Impulse noise removal by a nonmonotone adaptive gradient method. Signal Process. 90, 2891–2897 (2010)

    Article  MATH  Google Scholar 

  17. Yu, G., Xue, W., Zhou, Y.: A nonmonotone adaptive projected gradient method for primal-dual total variation image restoration. Signal Process. 103, 242–249 (2014)

    Article  Google Scholar 

  18. Yu, G., Huang, J.H., Zhou, Y.: A descent spectral conjugate gradient method for impulse noise removal. Appl. Math. Lett. 23, 555–560 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. De Leone, R., Gaudioso, M., Grippo, L.: Stopping criteria for line search methods without derivatives. Math. Program. 30, 285–300 (1984)

    Article  MATH  Google Scholar 

  20. Huber, P.J.: Robust regression: asymptotics, conjectures, and Monte Carlo. Ann. Stat. 1, 799–821 (1973)

    Article  MATH  Google Scholar 

  21. Cai, J.F., Chan, R.H., Fiore, C.D.: Minimization of a detail-preserving regularization functional for impulse noise removal. J. Math. Imaging Vis. 29, 79–91 (2007)

    Article  Google Scholar 

  22. Bovik, A.: Handbook of Image and Video Processing. Academic, New York (2000)

    MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to express their thanks to Professor J.F. Cai for his kind offer of the source codes for PRCG method compared in this paper. This research was supported by the National Natural Science Foundation of China (Grant Number: 11171362) and Specialized Research Fund for the Doctoral Program of Higher Education (Grant Number: 20120191110031).

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Correspondence to Shengjie Li.

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Liu, J., Li, S. Spectral gradient method for impulse noise removal. Optim Lett 9, 1341–1351 (2015). https://doi.org/10.1007/s11590-014-0845-4

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  • DOI: https://doi.org/10.1007/s11590-014-0845-4

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