Skip to main content
SpringerLink
Log in

Locally Adaptive Total Variation Regularization

  • Conference paper
Scale Space and Variational Methods in Computer Vision (SSVM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5567))

Abstract

We introduce a locally adaptive parameter selection method for total variation regularization applied to image denoising. The algorithm iteratively updates the regularization parameter depending on the local smoothness of the outcome of the previous smoothing step. In addition, we propose an anisotropic total variation regularization step for edge enhancement. Test examples demonstrate the capability of our method to deal with varying, unknown noise levels.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10(6), 1217–1229 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aubert, G., Kornprobst, P.: Mathematical problems in image processing. In: Partial differential equations and the calculus of variations, With a foreword by Olivier Faugeras, 2nd edn. Applied Mathematical Sciences, vol. 147. Springer, New York (2006)

    Google Scholar 

  3. Burger, M., Osher, S.: Convergence rates of convex variational regularization. Inverse Probl. 20(5), 1411–1421 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  5. Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Davies, P.L., Kovac, A.: Local extremes, runs, strings and multiresolution. Ann. Statist. 29(1), 1–65 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)

    MATH  Google Scholar 

  8. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of inverse problems. Mathematics and its Applications, vol. 375. Kluwer Academic Publishers Group, Dordrecht (1996)

    Book  MATH  Google Scholar 

  9. Frigaard, I.A., Ngwa, G., Scherzer, O.: On effective stopping time selection for visco-plastic nonlinear BV diffusion filters used in image denoising. SIAM J. Appl. Math. 63(6), 1911–1934 (electronic) (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Frigaard, I.A., Scherzer, O.: Herschel–Bulkley diffusion filtering: non-Newtonian fluid mechanics in image processing. Z. Angew. Math. Mech. 86(6), 474–494 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gilboa, G., Sochen, N., Zeevi, Y.Y.: Variational denoising of partly-textured images by spatially varying constraints. IEEE Trans. Image Process. 15(8), 2281–2289 (2006)

    Article  Google Scholar 

  12. Ito, K., Kunisch, K.: Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. 41A, 591–616 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nashed, M.Z., Scherzer, O.: Least squares and bounded variation regularization with nondifferentiable functional. Numer. Funct. Anal. Optim. 19(7-8), 873–901 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Applied Mathematical Sciences, vol. 167. Springer, New York (2008)

    MATH  Google Scholar 

  16. Strong, D.M.: Adaptive Total Variation Minimizing Image Restoration. CAM Report 97-38, University of California, Los Angeles (1997)

    Google Scholar 

  17. Strong, D.M., Aujol, J.-F., Chan, T.F.: Scale recognition, regularization parameter selection, and Meyer’s G norm in total variation regularization. Multiscale Model. Simul. 5(1), 273–303 (electronic) (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Innsbruck, Technikerstr. 21a, A-6020, Innsbruck, Austria

    Markus Grasmair

Authors
  1. Markus Grasmair
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008, Bergen, Norway

    Xue-Cheng Tai

  2. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Oslo, Norway

    Knut Mørken

  3. Simula Research Laboratory, M. Linges v 17, Fornebu, P.O.Box 134, N-1325, Lysaker, Norway

    Marius Lysaker

  4. SINTEF ICT, Oslo, Norway

    Knut-Andreas Lie

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Grasmair, M. (2009). Locally Adaptive Total Variation Regularization. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02256-2_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02255-5

  • Online ISBN: 978-3-642-02256-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation