Skip to main content
SpringerLink
Log in

An Improved LOT Model for Image Restoration

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

Abstract

Some second order PDE-based image restoration models such as total variation (TV) minimization or ROF model of Rudin et al. (Physica D 60, 259–268, 1992) can easily give rise to staircase effect, which may produce undesirable blocky image. LOT model proposed by Laysker, Osher and Tai (IEEE Trans. Image Process. 13(10), 1345–1357, 2004) has alleviated the staircase effect successfully, but the algorithms are complicated due to three nonlinear second-order PDEs to be computed, besides, when we have no information about the noise, the model cannot preserve edges or textures well. In this paper, we propose an improved LOT model for image restoration. First, we smooth the angle θ rather than the unit normal vector n, where n=(cos θ,sin θ). Second, we add an edge indicator function in order to preserve fine structures such as edges and textures well. And then the dual formulation of TV-norm and TV g -norm are used in the numerical algorithms. Finally, some numerical experiments prove our proposed model and algorithms to be effective.

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. Alliney, S.: Digital filters as absolute norm regularizers. IEEE Trans. Signal Process. 40(6), 1548–1562 (1992)

    Article  MATH  Google Scholar 

  2. Alliney, S.: Recursive median filters of increasing order: a variational approach. IEEE Trans. Signal Process. 44(6), 1346–1354 (1996)

    Article  Google Scholar 

  3. Alliney, S.: A property of the minimum vectors of a regularizing functional defined by means of the absolute norm. IEEE Trans. Signal Process. 45(4), 913–917 (1997)

    Article  Google Scholar 

  4. Alliney, S.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(12), 99–120 (2004)

    Google Scholar 

  5. Alliney, S.: Weakly constrained minimization: application to the estimation of images and signals involving constant regions. J. Math. Imaging Vis. 21(2), 155–175 (2004)

    Article  Google Scholar 

  6. Ballaster, C., Bertalmio, M., Caselles, V., Sapiro, G., Verdera, J.: Filling-in by joint interpolation of vector fields and gray levels. IEEE Trans. Image Process. 10, 1200–1211 (2000)

    Article  Google Scholar 

  7. Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J.-P., Osher, S.: Fast global minimization of the active contour/snake model. J. Math. Imaging Vis. 28, 151–167 (2007)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. 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 

  10. Chan, T., Esedoglu, S.: Aspects of total variation regularized L 1 function approximation. UCLA CAM Report, 04-07 (2004)

  11. Chan, T., 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 

  12. Cheon, E., Paranjpye, A.: Noise removal by total variation minimization. UCLA MATH 199 project report (2002)

  13. El-Fallah, A., Ford, G.: The evolution of mean curvature in image filtering. In: Proceedings of ICIP, pp. 298–303 (1994)

  14. Hong, L., Wan, Y.F., Jain, A.K.: Fingerprint image enhancement: algorithm and performance evaluation. IEEE Trans. Pattern Anal. Mach. Intell. 20(8), 777–789 (1998)

    Article  Google Scholar 

  15. Joo, K., Kim, S.: PDE-based image restoration: Anti-staircasing and anti-diffusion. http://www.ms.uky.edu/math/MAreport/PDF/2003-07.pdf (2003)

  16. Kenney, C., Langan, J.: A new image processing primitive: reconstructing images from modified flow fields. Univ. California, Santa Barbara, pp. 1–10 (1999)

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

    Article  Google Scholar 

  18. Lysaker, M., Osher, S., Tai, X.-C.: Noise removal using smoothed normals and surface fitting. IEEE Trans. Image Process. 13(10), 1345–1357 (2004)

    Article  MathSciNet  Google Scholar 

  19. Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. SIAM J. Numer. Anal. 40(3), 965–994 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Perona, P.: Orientation diffusions. IEEE Trans. Image Process. 7(3), 457–467 (1998)

    Article  Google Scholar 

  21. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)

    Article  Google Scholar 

  22. Rahman, T., Tai, X.-C., Osher, S.: A TV-stokes denoising algorithm. In: SSVM, pp. 473–483 (2007)

  23. Ratha, N.K., Chen, S.Y., Jain, A.K.: Adaptive flow orientation-based feature extraction in fingerprint images. Pattern Recognit. 28(11), 1657–1672 (1995)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  25. Tasdizen, T., Whitaker, R., Burchard, P., Osher, S.: Geometric surface processing via normal maps. ACM Trans. Graph. 22(4), 1012–1033 (2003)

    Article  Google Scholar 

  26. Vese, L., Osher, S.: Numerical methods for p-harmonic flows and application to image processing. SIAM J. Numer. Anal. 40(6), 2085–2104 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Fangfang Dong, Zhen Liu & Kefeng Liu

  2. Department of Mathematics, Zhejiang University of Technology, Hangzhou, 310023, China

    Zhen Liu

  3. Department of Mathematics, Zheijiang University, Hangzhou, 310027, China

    Dexing Kong

Authors
  1. Fangfang Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhen Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dexing Kong
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kefeng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhen Liu.

Additional information

This work is supported by National Natural Science Foundation of China (Grant No.10801045), Postdoctoral Foundation of ZheJiang province and Foundation (Grant No.1098129 and No.109001329) from the Zhejiang University of Technology.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dong, F., Liu, Z., Kong, D. et al. An Improved LOT Model for Image Restoration. J Math Imaging Vis 34, 89–97 (2009). https://doi.org/10.1007/s10851-008-0132-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-008-0132-z

Keywords

Search

Navigation