Skip to main content
SpringerLink
Log in

Total Variation with Overlapping Group Sparsity for Removing Mixed Noise

  • Conference paper
  • First Online:
The Proceedings of the International Conference on Sensing and Imaging (ICSI 2017)

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 506))

Included in the following conference series:

  • 650 Accesses

Abstract

The total variation (TV) model has been used for removing the mixed additive and multiplicative noise. However, the restored images inevitably suffer from the staircase artifacts. In order to overcome this disadvantage, we propose two new variational models by combining the TV with overlapping group sparsity. Then the alternating direction method of multiplier (ADMM) is applied to solve the proposed models. Numerical experiments demonstrate that our methods are competitive with the state-of-the-art methods in visual and quantitative measures.

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 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover 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

Notes

  1. 1.

    A function Q(t, t′) is a majorizor of the function P(t), if Q(t, t′) ≥ P(t) for all t, t′ and Q(t, t) = P(t).

References

  1. Rudin LI, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Phys D Nonlinear Phenom 60:259–268. https://doi.org/10.1016/0167-2789(92)90242-F

    Article  MathSciNet  Google Scholar 

  2. Chambolle A (2004) An algorithm for total variation minimization and applications. J Math Imaging Vis 20:89–97. https://doi.org/10.1023/B:JMIV.0000011325.36760.1e

    Article  MathSciNet  Google Scholar 

  3. Chan RH, Tao M, Yuan XM (2013) Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers. SIAM J Imaging Sci 6:680–697. https://doi.org/10.1137/110860185

    Article  MathSciNet  Google Scholar 

  4. Aubert G, Aujo JF (2008) A variational approach to removing multiplicative noise. SIAM J Appl Math 28:925–946. https://doi.org/10.1137/060671814

    Article  MathSciNet  Google Scholar 

  5. Beck A, Teboulle M (2009) Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans Image Process 18:2419–2434. https://doi.org/10.1109/TIP.2009.2028250

    Article  MathSciNet  Google Scholar 

  6. Bioucas-Dias JM, Figueiredo MAT (2010) Multiplicative noise removal using variable splitting and constrained optimization. IEEE Trans Image Process 19:1720–1730. https://doi.org/10.1109/TIP.2010.2045029

    Article  MathSciNet  Google Scholar 

  7. Steidl G, Teuber T (2010) Removing multiplicative noise by Douglas-Rachford splitting methods. J Math Imaging Vis 36:168–184. https://doi.org/10.1007/s10851-009-0179-5

    Article  MathSciNet  Google Scholar 

  8. Zhao XL, Wang F, Ng MK (2014) A new convex optimization model for multiplicative noise and blur removal. SIAM J Imaging Sci 7:456–475. https://doi.org/10.1137/13092472X

    Article  MathSciNet  Google Scholar 

  9. Dong YQ, Zeng TY (2013) A convex variational model for restoring blurred images with multiplicative noise. SIAM J Imaging Sci 6:1598–1625. https://doi.org/10.1137/120870621

    Article  MathSciNet  Google Scholar 

  10. Liu J, Huang TZ, Selesnick IW, Lv XG, Chen PY (2015) Image restoration using total variation with overlapping group sparsity. Inform Sci 295:232–246. https://doi.org/10.1016/j.ins.2014.10.041

    Article  MathSciNet  Google Scholar 

  11. Mei JJ, Huang TZ (2016) Primal-dual splitting method for high-order model with application to image restoration. Appl Math Model 40:2322–2332. https://doi.org/10.1016/j.apm.2015.09.068

    Article  MathSciNet  Google Scholar 

  12. Mei JJ, Dong YQ, Huang TZ, Yin WT (2017) Cauchy noise removal by nonconvex ADMM with convergence guarantees. J Sci Comput 1–24. https://doi.org/10.1007/s10915-017-0460-5

    Article  MathSciNet  Google Scholar 

  13. Lukin VV, Fevralev DV, Ponomarenko NN, Abramov SK, Pogrebnyak O, Egiazarian KO, Astola JT (2010) Discrete cosine transform-based local adaptive filtering of images corrupted by nonstationary noise. J Electron Imaging 19:023007–023007. https://doi.org/10.1117/1.3421973

    Article  Google Scholar 

  14. Hirakawa K, Parks TW (2006) Image denoising using total least squares. IEEE Trans Image Process 15:2730–2742. https://doi.org/10.1109/TIP.2006.877352

    Article  Google Scholar 

  15. Chumchob N, Chen K, Brito-Loeza C (2013) A new variational model for removal of combined additive and multiplicative noise and a fast algorithm for its numerical approximation. Int J Comput Math 90:140–161. https://doi.org/10.1080/00207160.2012.709625

    Article  MathSciNet  Google Scholar 

  16. Almgren F (1987) Review: Enrico Giusti, minimal surfaces and functions of bounded variation. Bull Am Math Soc (NS) 16:167–171

    Article  Google Scholar 

  17. Ambrosio L, Fusco N, Pallara D (2000) Functions of bounded variation and free discontinuity problems. Oxford mathematical monographs. The Clarendon Press/Oxford University Press, New York

    MATH  Google Scholar 

  18. Selesnick IW, Chen PY (2013) Total variation denoising with overlapping group sparsity. In: 2013 IEEE international conference on acoustics, speech and signal processing (ICASSP), pp 5696–5700. https://doi.org/10.1109/ICASSP.2013.6638755

  19. Liu G, Huang TZ, Liu J, Lv XG (2015) Total variation with overlapping group sparsity for image deblurring under impulse noise. PLOS ONE 10:1–23. https://doi.org/10.1371/journal.pone.0122562

    Google Scholar 

  20. Wang Z, Bovik AC, Sheikh HR, Simoncelli EP (2004) Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process 13:600–612. https://doi.org/10.1109/TIP.2003.819861

    Article  Google Scholar 

  21. Yang JF, Zhang Y, Yin WT (2010) A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data. IEEE J Sel Top Signal Process 4:288–297. https://doi.org/10.1109/JSTSP.2010.2042333

    Article  Google Scholar 

  22. He BS, Yang H (1998) Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities. Oper Res Lett 23:151–161. https://doi.org/10.1016/S0167-6377(98)00044-3

    Article  MathSciNet  Google Scholar 

  23. Chen C, Ng MK, Zhao XL (2015) Alternating direction method of multipliers for nonlinear image restoration problems. IEEE Trans Image Process 24:33–43. https://doi.org/10.1109/TIP.2014.2369953

    Article  MathSciNet  Google Scholar 

  24. Glowinski R (1984) Numerical methods for nonlinear variational problems. Springer, Berlin/Heidelberg. https://doi.org/10.1007/978-3-662-12613-4

    Book  Google Scholar 

Download references

Acknowledgements

This research is supported by NSFC (61772003, 61402082, 11401081) and the Fundamental Research Funds for the Central Universities (ZYGX2016J129).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, China

    Jin-Jin Mei & Ting-Zhu Huang

  2. School of Mathematics and Statistics, Fuyang Normal University, Fuyang, Anhui, China

    Jin-Jin Mei & Ting-Zhu Huang

Authors
  1. Jin-Jin Mei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ting-Zhu Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jin-Jin Mei .

Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, China

    Ming Jiang

  2. Department of Electrical and Computer Engineering, The University of Akron, Akron, OH, USA

    Nathan Ida

  3. Institute of Applied Mathematics, Saarland University, Saarbrücken, Germany

    Alfred K. Louis

  4. Department of Mathematics, Tufts University, Medford, MA, USA

    Eric Todd Quinto

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Mei, JJ., Huang, TZ. (2019). Total Variation with Overlapping Group Sparsity for Removing Mixed Noise. In: Jiang, M., Ida, N., Louis, A., Quinto, E. (eds) The Proceedings of the International Conference on Sensing and Imaging. ICSI 2017. Lecture Notes in Electrical Engineering, vol 506. Springer, Cham. https://doi.org/10.1007/978-3-319-91659-0_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-91659-0_16

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-91658-3

  • Online ISBN: 978-3-319-91659-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation