Skip to main content
SpringerLink
Log in

A Variational Model for Multiplicative Structured Noise Removal

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

We consider the problem of restoring images impaired by noise that is simultaneously structured and multiplicative. Our primary motivation for this setting is the selective plane illumination microscope which often suffers from severe inhomogeneities due to light absorption and scattering. This type of degradation arises in other imaging devices such as ultrasonic imaging. We model the multiplicative noise as a stationary process with known distribution. This leads to a novel convex image restoration model based on a maximum a posteriori estimator. After establishing some analytical properties of the minimizers, we finally propose a fast optimization method on GPU. Numerical experiments on 2D fluorescence microscopy images demonstrate the usefulness of the proposed models in practical applications.

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. A sufficient condition for existence of such a \(\bar{\pmb {\lambda }}\) is that the Fourier transform \(\hat{\pmb {\psi }}\) does not vanish.

References

  1. Aizenberg, I., Butakoff, C.: A windowed Gaussian notch filter for quasi-periodic noise removal. Image Vis. Comput. 26(10), 1347–1353 (2008)

    Article  Google Scholar 

  2. Anas, E.M.A., Lee, S.Y., Kamrul Hasan, M.: Classification of ring artifacts for their effective removal using type adaptive correction schemes. Comput. Biol. Med. 41(6), 390–401 (2011)

    Article  Google Scholar 

  3. Aubert, G., Aujol, J.-F.: A variational approach to removing multiplicative noise. SIAM J. Appl. Math. 68(4), 925–946 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Boas, F.E., Fleischmann, D.: CT artifacts: causes and reduction techniques. Imaging Med. 4(2), 229–240 (2012)

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  6. 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chang, Y., Fang, H., Yan, L., Liu, H.: Robust destriping method with unidirectional total variation and framelet regularization. Opt. Express 21(20), 23307–23323 (2013)

    Article  Google Scholar 

  8. Chen, S.-W., Pellequer, J.-L.: DeStripe: frequency-based algorithm for removing stripe noises from AFM images. BMC Struct. Biol. 11(1), 7 (2011)

    Article  Google Scholar 

  9. Cornelis, B., Dooms, A., Cornelis, J., Schelkens, P.: Digital canvas removal in paintings. Signal Process. 92(4), 1166–1171 (2012)

    Article  Google Scholar 

  10. Deledalle, C., Denis, L., Tupin, F.: Iterative weighted maximum likelihood denoising with probabilistic patch-based weights. IEEE Trans. Image Process. 18(12), 2661–2672 (2009)

    Article  MathSciNet  Google Scholar 

  11. Dong, Y., Zeng, T.: A convex variational model for restoring blurred images with multiplicative noise. SIAM J. Imaging Sci. 6(3), 1598–1625 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Durand, S., Fadili, J., Nikolova, M.: Multiplicative noise removal using l1 fidelity on frame coefficients. J. Math. Imaging Vis. 36(3), 201–226 (2010)

    Article  Google Scholar 

  13. Fehrenbach, J., Weiss, P.: Processing stationary noise: model and parameter selection in variational methods. SIAM J. Imaging Sci. 7(2), 613–640 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fehrenbach, J., Weiss, P., Lorenzo, C.: Variational algorithms to remove stationary noise: applications to microscopy imaging. Image Process., IEEE Trans. 21(10), 4420–4430 (2012)

    Article  MathSciNet  Google Scholar 

  15. Fitschen, J.H., Ma, J., Schuff, S.: Removal of curtaining effects by a variational model with directional first and second order differences. arXiv preprint arXiv:1507.00112 (2015)

  16. Gómez-Chova, L., Alonso, L., Guanter, L., Camps-Valls, G., Calpe, J., Moreno, J.: Correction of systematic spatial noise in push-broom hyperspectral sensors: application to CHRIS/PROBA images. Appl. Opt. 47(28), F46–F60 (2008)

    Article  Google Scholar 

  17. Hsieh, J.: Computed Tomography: principles, Design, Artifacts, and Recent Advances. SPIE, Bellingham (2009)

    Google Scholar 

  18. Huang, Y.-M., Moisan, L., Ng, M.K., Zeng, T.: Multiplicative noise removal via a learned dictionary. Image Process., IEEE Trans. 21(11), 4534–4543 (2012)

    Article  MathSciNet  Google Scholar 

  19. Huisken, J., Swoger, J., Del Bene, F., Wittbrodt, J., Stelzer, E.H.: Optical sectioning deep inside live embryos by selective plane illumination microscopy. Science 305(5686), 1007–1009 (2004)

    Article  Google Scholar 

  20. Kim, Y., Baek, J., Hwang, D.: Ring artifact correction using detector line-ratios in computed tomography. Opt. Express 22(11), 13380–13392 (2014)

    Article  Google Scholar 

  21. Kryvanos, A., Hesser, J., Steidl, G.: Nonlinear image restoration methods for marker extraction in 3D fluorescent microscopy. In Electronic Imaging 2005, pp. 432–443. International Society for Optics and Photonics (2005)

  22. Levin, A., Nadler, B.: Natural image denoising: Optimality and inherent bounds. In Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, pp. 2833–2840. IEEE (2011)

  23. Moschopoulos, P.G.: The distribution of the sum of independent gamma random variables. Ann. Inst. Stat. Math. 37(1), 541–544 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  24. Münch, B., Trtik, P., Marone, F., Stampanoni, M.: Stripe and ring artifact removal with combined wavelet Fourier filtering. Opt. Express 17(10), 8567–8591 (2009)

    Article  Google Scholar 

  25. Panin, V., Zeng, G., Gullberg, G.: Total variation regulated EM algorithm. Nucl. Sci., IEEE Trans. 46(6), 2202–2210 (1999)

    Article  Google Scholar 

  26. Pizurica, A., Philips, W., Lemahieu, I., Acheroy, M.: A versatile wavelet domain noise filtration technique for medical imaging. Med. Imaging, IEEE Trans. 22(3), 323–331 (2003)

    Article  Google Scholar 

  27. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (2015)

    MATH  Google Scholar 

  28. Rudin, L., Lions, P.-L., Osher, S.: Multiplicative denoising and deblurring: theory and algorithms. In: Geometric Level Set Methods in Imaging, Vision, and Graphics, pp. 103–119. Springer, New York (2003)

  29. Schindelin, J., Arganda-Carreras, I., Frise, E., Kaynig, V., Longair, M., Pietzsch, T., Preibisch, S., Rueden, C., Saalfeld, S., Schmid, B., et al.: Fiji: an open-source platform for biological-image analysis. Nat. Methods 9(7), 676–682 (2012)

    Article  Google Scholar 

  30. Shi, J., Osher, S.: A nonlinear inverse scale space method for a convex multiplicative noise model. SIAM J. Imaging Sci. 1(3), 294–321 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Steidl, G., Teuber, T.: Removing multiplicative noise by Douglas-Rachford splitting methods. J. Math. Imaging Vis. 36(2), 168–184 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sur, F., Grédiac, M.: Automated removal of quasiperiodic noise using frequency domain statistics. J. Electron. Imaging 24(1), 013003–013003 (2015)

    Article  Google Scholar 

  33. Tsai, F., Chen, W.W.: Striping noise detection and correction of remote sensing images. Geosci. Remote Sens., IEEE Trans. 46(12), 4122–4131 (2008)

    Article  Google Scholar 

  34. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. Image Process., IEEE Trans. 13(4), 600–612 (2004)

    Article  Google Scholar 

  35. Zhao, X.-L., Wang, F., Ng, M.K.: A new convex optimization model for multiplicative noise and blur removal. SIAM J. Imaging Sci. 7(1), 456–475 (2014)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Jérôme Fehrenbach for fruitful discussions and support. They thank the anonymous reviewers for their careful reading which helped in improving the paper. W. Zhang was supported by the ANR SPH-IM-3D (ANR-12-BSV5-0008) and support by the NNSFC Grant 11301055. P. Escande is pursuing a Ph.D. degree supported by the MODIM project funded by the PRES of Toulouse University and Midi-Pyrénées region. P. Weiss was supported by the OPTIMUS Project (fondation RITC, France).

Author information

Authors and Affiliations

  1. DISC, ISAE, Université de Toulouse, Toulouse, France

    Paul Escande

  2. ITAV and IMT, Université de Toulouse, CNRS, 31106, Toulouse, France

    Pierre Weiss

  3. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Wenxing Zhang

Authors
  1. Paul Escande
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre Weiss
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenxing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre Weiss.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Escande, P., Weiss, P. & Zhang, W. A Variational Model for Multiplicative Structured Noise Removal. J Math Imaging Vis 57, 43–55 (2017). https://doi.org/10.1007/s10851-016-0667-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-016-0667-3

Keywords

Mathematics Subject Classification

Search

Navigation