Skip to main content
SpringerLink
Log in

A Unified Framework for the Restoration of Images Corrupted by Additive White Noise

  • Conference paper
  • First Online:
Scale Space and Variational Methods in Computer Vision (SSVM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10302))

  • 1380 Accesses

Abstract

We propose a robust variational model for the restoration of images corrupted by blur and the general class of additive white noises. The solution of the non-trivial optimization problem, due to the non-smooth non-convex proposed model, is efficiently obtained by an Alternating Directions Method of Multipliers (ADMM), which in particular reduces the solution to a sequence of convex optimization sub-problems. Numerical results show the potentiality of the proposed model for restoring blurred images corrupted by several kinds of additive white noises.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ben-Israel, A.: A concentrated cauchy distribution with finite moments. Ann. Oper. Res. 208(1), 147–153 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bouman, C., Sauer, K.: A generalized Gaussian image model for edge-preserving MAP estimation. IEEE Trans. Image Process. 2(3), 296–310 (1993)

    Article  Google Scholar 

  3. Bovik, A.C.: Handbook of Image and Video Processing. Academic Press, Cambridge (2010)

    MATH  Google Scholar 

  4. Lanza, A., Morigi, S., Sgallari, F., Wen, Y.W.: Image restoration with Poisson-Gaussian mixed noise. Comput. Methods Biomech. Biomed. Eng.: Imaging Vis. 2(1), 12–24 (2013)

    Google Scholar 

  5. Clason, C.: \(L^\infty \) fitting for inverse problems with uniform noise. Inverse Prob. 28(10), 104007 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Keren, D., Werman, M.: Probabilistic analysis of regularization. IEEE Trans. Pattern Anal. Mach. Intell. 15(10), 982–995 (1993)

    Article  Google Scholar 

  8. Lanza, A., Morigi, S., Sgallari, F., Yezzi, A.J.: Variational image denoising based on autocorrelation whiteness. SIAM J. Imaging Sci. 6(4), 1931–1955 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lanza, A., Morigi, S., Sgallari, F.: Variational image restoration with constraints on noise whiteness. J. Math. Imaging Vis. 53(1), 61–77 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chan, R.H., Lanza, A., Morigi, S., Sgallari, F.: An adaptive strategy for the restoration of textured images using fractional order regularization. Numer. Math.: Theory Methods Appl. 6(1), 276–296 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(1), 99–120 (2004)

    Article  MathSciNet  Google Scholar 

  12. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sciacchitano, F., Dong, Y., Zeng, T.: Variational approach for restoring blurred images with cauchy noise. SIAM J. Imaging Sc. 8(3), 1894–1922 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Widrow, B., Kollár, I.: Quantization Noise: Roundoff Error in Digital Computation, Signal Processing, Control, and Communications. Cambridge University Press, New York (2008)

    Book  Google Scholar 

  15. Lanza, A., Morigi, S., Sgallari, F.: Constrained TV\(_p\)-\(\ell _2\) model for image restoration. J. Sci. Comput. 68(1), 64–91 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)

    Article  MATH  Google Scholar 

  17. Wu, C., Zhang, J., Tai, X.C.: Augmented lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Probl. Imaging 5(1), 237–261 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the “National Group for Scientific Computation (GNCS-INDAM)” and by ex60% project by the University of Bologna “Funds for selected research topics”.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bologna, Bologna, Italy

    Alessandro Lanza, Serena Morigi & Fiorella Sgallari

  2. Department of Mathematics, University of Genova, Genova, Italy

    Federica Sciacchitano

Authors
  1. Alessandro Lanza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Federica Sciacchitano
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Serena Morigi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fiorella Sgallari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serena Morigi .

Editor information

Editors and Affiliations

  1. University of Copenhagen , Copenhagen, Denmark

    François Lauze

  2. Technical University of Denmark , Kongens Lyngby, Denmark

    Yiqiu Dong

  3. Technical University of Denmark , Kongens Lyngby, Denmark

    Anders Bjorholm Dahl

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Lanza, A., Sciacchitano, F., Morigi, S., Sgallari, F. (2017). A Unified Framework for the Restoration of Images Corrupted by Additive White Noise. In: Lauze, F., Dong, Y., Dahl, A. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2017. Lecture Notes in Computer Science(), vol 10302. Springer, Cham. https://doi.org/10.1007/978-3-319-58771-4_40

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-58771-4_40

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-58770-7

  • Online ISBN: 978-3-319-58771-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation