Skip to main content
SpringerLink
Log in

A dual algorithm for minimization of the LLT model

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We apply the dual algorithm of Chambolle for the minimization of the LLT model. A convergence theorem is given for the proposed algorithm. The algorithm overcomes the numerical difficulties related to the non-differentiability of the LLT model. The dual algorithm is faster than the original gradient descent algorithm. Numerical experiments are supplied to demonstrate the efficiency of the algorithm.

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

Similar content being viewed by others

References

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

    Article  MATH  Google Scholar 

  2. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: partial differential equations and the calculus of variations. Applied Mathematical Science, vol. 147, 2nd edn. Springer (2006)

  3. Andreu, F., Ballester, C., Caselles, V., Mazon, J.M.: Minimizing total variation flow. CRAS I-Mathématique 331(11), 867–872 (2000)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Vese, L.: A study in the BV space of a denosing–deblurring variationl problem. Appl. Math. Optim. 44, 131–161 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Blomgern, P., Chan, T.F., Mulet, P., Wong, C.: Total variation image restoration: Numerical methods and extensions. In: Proceedings, IEEE International Conference on Image Processing, III, pp. 384–387 (1997)

  9. Nikolova, M.: Weakly constrained minimization: application to the estimation of images and signals involving constant regions. J. Math. Imaging Vis. 21, 155–175 (2004)

    Article  MathSciNet  Google Scholar 

  10. Chan, T.F., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. You, Y.-L., Kaveh, M.: Fourth-order partial differential equation for noise removal. IEEE Trans. Image Process. 9(10), 1723–1730 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lysaker, M., Lundervold, A., Tai, X.-C.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)

    Article  Google Scholar 

  13. Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H 1 norm. Multiscale Model. Simul. 1(3), 349–370 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Didas, S., Burgeth, B., Imiya, A., Weickert, J.: Regularity and scale- space properties of fractional high order linear filtering. In: Kimmel, R., Sochen, N., Weickert, J. (eds.) Scale-space and PDE Methods in Computer Vision. Lecture Notes in Computer Science, vol. 3459. Springer, Berlin (2005)

    Google Scholar 

  15. Didas, S., Weickert, J., Burgeth, B.: Stability and local feature enhancement of higher order nonlinear diffusion filtering. In: Kropatsch, W., Sablatnig, R., Hanbury, A. (eds.) Pattern Recognition. Lecture Notes in Computer Science, vol. 3663, pp. 451–458. Springer, Berlin (2005)

    Google Scholar 

  16. Osher, S., Scherzer, O.: G-norm properties of bounded variation regularization. Comm. Math. Sci. 2(2), 237–254 (2004)

    MATH  MathSciNet  Google Scholar 

  17. Obereder, A., Osher, S., Scherzer, O.: On the use of dual norms in bounded variation type regularization. UCLA cam report, cam-04-35. Available from: http://www.math.ucla.edu/applied/cam/ (2004)

  18. Fang, L., Shen, C., Fan, J., Shen, C.: Image restoration combining a total variational filter and a fourth-order filter. J. Vis. Commun. Image Represent. 18(4), 322–330 (2007)

    Article  Google Scholar 

  19. Bertsekas, D.P.: Convex Analysis and Optimization. Tsinghua University Press (2006)

  20. Yuan, Y.X.: Numerical Method For Nonlinear Programming. Shanghai Scientific and Technical Publishers (1992)

  21. Liu, Q., Yao, Z.G., Ke, Y.: Entropy solutions for a fourth-order nonlinear degenerate problem for noise removal. Nonlinear Anal. Theory Methods Appl. 67(6), 1908–1918 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Liu, Q., Yao, Z.G., Ke, Y.: Solutions of fourth-order partial differential equations in a noise removal model. Electron. J. Diff. Equ. 2007(120), 11 (2007)

    MathSciNet  Google Scholar 

  23. Chan, T.F., Esedoglu, S., Park, F.: Image decomposition combining staircase reduction and texture extraction. J. Vis. Commun. Image Represent. 18(6), 464–486 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Henan University, Kaifeng, 475004, China

    Hua-zhu Chen & Jin-ping Song

  2. College of Mathematics and Information Science, Henan University, Kaifeng, 475004, China

    Hua-zhu Chen, Jin-ping Song & Xue-Cheng Tai

  3. Department of Mathematics, University of Bergen, 5009, Bergen, Norway

    Xue-Cheng Tai

Authors
  1. Hua-zhu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jin-ping Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xue-Cheng Tai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xue-Cheng Tai.

Additional information

Communicated by Lixin Shen and Yuesheng Xu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, Hz., Song, Jp. & Tai, XC. A dual algorithm for minimization of the LLT model. Adv Comput Math 31, 115–130 (2009). https://doi.org/10.1007/s10444-008-9097-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-008-9097-0

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation