Skip to main content
Log in

An Algorithm for Total Variation Minimization and Applications

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

Abstract

We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. Braides, Gamma-Convergence for Beginners, No. 22 in Oxford Lecture Series in Mathematics and Its Applications.Oxford University Press, 2002.

  2. K.A. Brakke, The Motion of a Surface by its Mean Curva-ture, Vol. 20 of Mathematical Notes. Princeton University Press: Princeton, NJ, 1978.

  3. J.L. Carter, "Dual methods for total variation-Based image restoration," Ph.D. thesis, U.C.L.A. (Advisor: T. F. Chan), 2001.

  4. A. Chambolle, "An algorithm for mean curvature motion," to appear in Interfaces Free Bound.

  5. A. Chambolle and P.-L. Lions, "Image recovery via total vari-ation minimization and related problems," Numer. Math.,Vol. 76, No. 2, pp. 167–188, 1997.

    Google Scholar 

  6. T.F. Chan, G.H. Golub, and P. Mulet, "A nonlinear primal-dual method for total variation-based image restoration," SIAM J. Sci. Comput.,Vol. 20, No.6, pp. 1964–1977, 1999 (electronic).

    Google Scholar 

  7. P.G. Ciarlet, EsIntroduction à l'analyse numèrique matricielle et à l'optimisation, Collection Mathèmatiques Appliquèes pour la Mâýtrise. [Collection of Applied Mathematics for the Master's Degree]. Masson: Paris, 1982.

    Google Scholar 

  8. P.L. Combettes and J. Luo, "An adaptive level set method for nondifferentiable constrained image recovery," IEEE Trans.Image Process.,Vol. 11, 2002.

  9. F. Dibos and G. Koepfler, "Global total variation minimiza-tion," SIAM J. Numer. Anal.,Vol. 37, No. 2, pp. 646–664, 2000 (electronic).

    Google Scholar 

  10. D.C. Dobson and C.R. Vogel, "Convergence of an iterative method for total variation denoising," SIAM J. Numer. Anal., Vol. 34, No. 5, pp. 1779–1791, 1997.

    Google Scholar 

  11. I. Ekeland and R. Temam, Convex Analysis and Variational Problems. Amsterdam: North Holland, 1976.

    Google Scholar 

  12. E. Giusti, Minimal Surfaces and Functions of Bounded Varia-tion. Birkhäuser Verlag: Basel, 1984.

    Google Scholar 

  13. F. Guichard and F. Malgouyres, "Total variation based inter-polation," in Proceedings of the European Signal Processing Conference,Vol. 3, pp. 1741–1744, 1998.

    Google Scholar 

  14. J.-B. Hiriart-Urruty and C. Lemarèchal, Convex Analysis and Minimization Algorithms. I, II,Vol. 305–306 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag: Berlin, 1993 (two volumes).

    Google Scholar 

  15. Y. Li and F. Santosa, "Acomputational algorithm for minimizing total variation in image restoration," IEEE Trans. Image Process-ing, Vol. 5, pp. 987–995, 1996.

    Google Scholar 

  16. F. Malgouyres and F. Guichard, "Edge direction preserving im-age zooming: A mathematical and numerical analysis," SIAM J. Numer. Anal.,Vol. 39, No. 1, pp. 1–37, 2001 (electronic).

    Google Scholar 

  17. L.I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D,Vol. 60, pp. 259–268, 1992.

    Google Scholar 

  18. J.A. Sethian, "Fast marching methods," SIAM Rev., Vol. 41, No. 2, pp. 199–235, 1999 (electronic).

    Google Scholar 

  19. C.R. Vogel and M.E. Oman, "Iterative methods for total varia-tion denoising," SIAM J. Sci. Comput.,Vol. 17, No. 1, pp. 227–238, 1996. Special issue on iterative methods in numerical linear algebra (Breckenridge, CO, 1994).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chambolle, A. An Algorithm for Total Variation Minimization and Applications. Journal of Mathematical Imaging and Vision 20, 89–97 (2004). https://doi.org/10.1023/B:JMIV.0000011325.36760.1e

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JMIV.0000011325.36760.1e

Navigation