Skip to main content
SpringerLink
Log in

Denoising images corrupted by impulsive noise using projections onto the epigraph set of the total variation function (PES-TV)

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

Abstract

In this article, a novel algorithm for denoising images corrupted by impulsive noise is presented. Impulsive noise generates pixels whose gray level values are not consistent with the neighboring pixels. The proposed denoising algorithm is a two-step procedure. In the first step, image denoising is formulated as a convex optimization problem, whose constraints are defined as limitations on local variations between neighboring pixels. We use Projections onto the Epigraph Set of the TV function (PES-TV) to solve this problem. Unlike other approaches in the literature, the PES-TV method does not require any prior information about the noise variance. It is only capable of utilizing local relations among pixels and does not fully take advantage of correlations between spatially distant areas of an image with similar appearance. In the second step, a Wiener filtering approach is cascaded to the PES-TV-based method to take advantage of global correlations in an image. In this step, the image is first divided into blocks and those with similar content are jointly denoised using a 3D Wiener filter. The denoising performance of the proposed two-step method was compared against three state-of-the-art denoising methods under various impulsive noise models.

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

Similar content being viewed by others

References

  1. Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-d transform-domain collaborative filtering. Image Process. IEEE Trans. 16(8), 2080–2095 (2007)

    Article  MathSciNet  Google Scholar 

  2. Cetin, A.E., Bozkurt, A., Gunay, O., Habiboglu, Y.H., Kose, K., Onaran, I., Sevimli, R.A., Tofighi, M.: Projections onto convex sets (pocs) based optimization by lifting. IEEE GlobalSIP, Austin, Texas, USA (2013)

  3. Tofighi, M., Kose, K., Cetin, A.E.: Denoising using projections onto the epigraph set of convex cost functions. In: Image Processing (ICIP), 2014 IEEE International Conference on, October 2014. pp. 2709–2713, (2014)

  4. Cetin, A.E., Tofighi, M.: Projection-based wavelet denoising [lecture notes]. Signal Process. Mag. IEEE, 32(5), 120–124 (2015)

  5. Censor, Y., Chen, W., Combettes, P.L., Davidi, R., Herman, G.: On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints. Comput. Optim. Appl. 51(3), 1065–1088 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Youla, D., Webb, H.: Image restoration by the method of convex projections: part 1 theory. Med. Imaging IEEE Trans. 1(2), 81–94 (1982)

    Article  Google Scholar 

  7. Trussell, H., Civanlar, M.R.: The Landweber iteration and projection onto convex set. IEEE Trans. Acoust. Speech Signal Process. 33(6), 1632–1634 (1985)

    Article  Google Scholar 

  8. Combettes, P.L., Pesquet, J.: Image restoration subject to a total variation constraint. IEEE Trans. Image Process. 13, 1213–1222 (2004)

    Article  Google Scholar 

  9. Yamada, I., Yukawa, M., Yamagishi, M.: Minimizing the moreau envelope of nonsmooth convex functions over the fixed point set of certain quasi-nonexpansive mappings. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering. pp. 345–390. Springer, (2011)

  10. Censor, Y., Herman, G.T.: On some optimization techniques in image reconstruction from projections. Appl. Numer. Math. 3(5), 365–391 (1987)

  11. Sezan, I., Stark, H.: Image restoration by the method of convex projections: part 2-applications and numerical results. IEEE Trans. Med. Imaging 1(2), 95–101 (1982)

  12. Chierchia, G., Pustelnik, N., Pesquet, J.-C., Pesquet-Popescu, B.: An epigraphical convex optimization approach for multicomponent image restoration using non-local structure tensor. In: Acoustics, Speech and Signal Processing (ICASSP), 2013 IEEE International Conference on, 2013. pp. 1359–1363

  13. Gubin, L., Polyak, B., Raik, E.: The method of projections for finding the common point of convex sets. USSR Comput. Math. Math. Phys. 7(6), 1–24 (1967)

    Article  Google Scholar 

  14. Bregman, L.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7(3), 200–217 (1967)

    Article  Google Scholar 

  15. Kose, K., Cevher, V., Cetin, A.E.: Filtered variation method for denoising and sparse signal processing. IEEE international conference on acoustics, speech and signal processing (ICASSP), pp. 3329–3332. (2012)

  16. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vision 20(1–2), 89–97 (2004)

    MathSciNet  Google Scholar 

  17. Combettes, P.L., Pesquet, J.-C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H.: (eds.) Fixed-pointalgorithms for inverse problems in science and engineering, ser. Springer optimization and its applications, pp. 185–212. Springer, New York. (2011)

  18. Bovik, A., Huang, T., Munson, J.D.C.: A generalization of median filtering using linear combinations of order statistics. Acoust. Speech Signal Process. IEEE Trans. 31(6), 1342–1350 (1983)

    Article  MATH  Google Scholar 

  19. Luisier, F., Blu, T., Unser, M.: A new sure approach to image denoising: interscale orthonormal wavelet thresholding. Image Process. IEEE Trans. 16(3), 593–606 (2007)

    Article  MathSciNet  Google Scholar 

  20. Micchelli, C.A., Shen, L., Xu, Y.: Proximity algorithms for image models: denoising. Inverse Probl. 27(4), 045009 (2011)

    Article  MathSciNet  Google Scholar 

  21. Cetin, A.E., Tekalp, A.: Robust reduced update kalman filtering. Circuits Syst. IEEE Trans. 37(1), 155–156 (1990)

    Article  MathSciNet  Google Scholar 

  22. Theodoridis, S., Slavakis, K., Yamada, I.: Adaptive learning in a world of projections. Signal Process. Mag. IEEE 28(1), 97–123 (2011)

    Article  Google Scholar 

  23. Chierchia, G., Pustelnik, N., Pesquet, J.-C., Pesquet-Popescu, B.: Epigraphical projection and proximal tools for solving constrained convex optimization problems. Signal Image Video Process. 9(8), 1737–1749 (2015)

    Article  Google Scholar 

  24. Tofighi, M., Kose, K., Cetin, A.E.: Signal reconstruction framework based on projections onto epigraph set of a convex cost function (PESC). ArXiv e-prints Feberuary 2014

  25. Efron, I.J.B., Hastie, T., Tibshirani, R.: Least angle regression. Ann. Stat. 32(2), 407–499 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. PES-TV software. http://signal.ee.bilkent.edu.tr/PES-TV.html

  27. Windyga, P.: Fast impulsive noise removal. Image Process. IEEE Trans. 10(1), 173–179 (2001)

    Article  Google Scholar 

  28. Lee, Y.-H., Kassam, S.: Generalized median filtering and related nonlinear filtering techniques. Acousti. Speech Signal Process. IEEE Trans. 33(3), 672–683 (1985)

    Article  Google Scholar 

  29. Kodak lossless true color image suite. http://r0k.us/graphics/kodak/

  30. Chan, R., Ho, C.-W., Nikolova, M.: Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization. Image Process. IEEE Trans. 14(10), 1479–1485 (2005)

    Article  Google Scholar 

Download references

Acknowledgments

This work is supported by the Scientific and Technological Research Council of Turkey (TUBITAK), under project 113E069.

Author information

Authors and Affiliations

  1. Department of Electrical and Electronics Engineering, Bilkent University, Ankara, Turkey

    Mohammad Tofighi & A. Enis Cetin

  2. Dermatology Service, Memorial Sloan-Kettering Cancer Center, New York, NY, 10022, USA

    Kivanc Kose

Authors
  1. Mohammad Tofighi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kivanc Kose
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Enis Cetin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad Tofighi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tofighi, M., Kose, K. & Cetin, A.E. Denoising images corrupted by impulsive noise using projections onto the epigraph set of the total variation function (PES-TV). SIViP 9 (Suppl 1), 41–48 (2015). https://doi.org/10.1007/s11760-015-0827-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-015-0827-8

Keywords

Search

Navigation