A Nonlocal Version of the Osher-Solé-Vese Model | Journal of Mathematical Imaging and Vision
Skip to main content

Advertisement

Springer Nature Link
Log in

A Nonlocal Version of the Osher-Solé-Vese Model

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

Abstract

In this paper, we present a nonlocal version of the OSV model by using the new results of Bartholdi-Schick-Smale-Smale. We compare this model with other nonlocal models, both theoretically and in computer experiments.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubert, G., Aujol, J.-F.: Modeling very oscillating signals, application to image processing. Appl. Math. Optim. 51 (2005)

  2. Aujol, J.F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63, 85–104 (2005)

    Article  Google Scholar 

  3. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations, 2nd edn. Applied Mathematical Sciences, vol. 147. Springer, Berlin (2006)

    MATH  Google Scholar 

  4. Aujol, J.F., Gilboa, G.: Implementation and parameter selection for BV-Hilbert space regularizations. UCLA CAM-Reports 04-66 (2004)

  5. Aujol, J.-F., Gilboa, G.: Constrained and SNR-based solutions for TV-Hilbert space image denoising. J. Math. Imaging Vis. 26, 217–237 (2006)

    Article  MathSciNet  Google Scholar 

  6. Aujol, J.-F., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition–modeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67(1), 111–136

  7. Bartholdi, L., Schick, T., Smale, N., Smale, S.: Hodge theory on metric spaces. (2009). arXiv:0912.0284v1

  8. Bresson, X., Chan, T.F.: Nonlocal unsupervised variational image segmentation models. UCLA CAM Report 08-67 (2008)

  9. Buades, A., Coll, B., Morel, J.-M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Chan, T.F., Shen, J.: Image Processing and Analysis. Variational, PDE, Wavelet, and Stochastic Methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2005)

    MATH  Google Scholar 

  14. Garnett, J.B., Le, T.M., Meyer, Y., Vese, L.A.: Image decompositions using bounded variation and generalized homogeneous Besov spaces. Appl. Comput. Harmon. Anal. 23, 25–56 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gilboa, G., Darbon, J., Osher, S., Chan, T.: Nonlocal convex functionals for image regularization. Technical Report 06-57, UCLA CAM Report (2006)

  16. Gilboa, G., Osher, S.: Nonlocal linear image regularization and supervised segmentation. Multiscale Model. Simul. 6(2), 595–630 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gilboa, G., Osher, S.: Nonlocal linear image regularization and supervised segmentation. UCLA CAM Report 06-47 (2006)

  19. Jin, Y., Jost, J., Wang, G.: A new nonlocal variational setting for image processing. Preprint (2010)

  20. Jost, J.: Equilibrium maps between metric spaces. Calc. Var. 2, 17305 (1994)

    Article  MathSciNet  Google Scholar 

  21. Jung, M., Vese, L.A.: Nonlocal Variational Image Deblurring Models in the Presence of Gaussian or Impulse Noise. Lecture Notes in Computer Science, vol. 5567, pp. 401–412. Springer, Berlin (2009)

    Google Scholar 

  22. Kimmel, R., Malladi, R., Sochen, N.: Images as embedding maps and minimal surfaces: movies, color, texture, and volumetric medical images. Int. J. Comput. Vis. 39(2), 111–129 (2000)

    Article  MATH  Google Scholar 

  23. Kimmel, R., Sochen, N., Malladi, R.: From high energy physics to low level vision. In: First International Conference on Scal-Space Theory in Computer Vision, pp. 236–247 (1997)

    Chapter  Google Scholar 

  24. Kindermann, S., Osher, S., Jones, P.W.: Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul. 4(4), 1091–1115 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Le, Triet M., Vese, Luminita A.: Image decomposition and restoration using total variation minimization and the H −1 norm. Multiscale Model. Simul. 4(2), 390–423 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lou, Y., Zhang, X., Osher, S., Bertozzi, A.: Image recovery via nonlocal operators. J. Sci. Comput. 42, 185–197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series, vol. 22. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Peyré, G.: Image processing with nonlocal spectral bases. Multiscale Model. Simul. 7(2), 703–730 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Peyré, G., Bougleux, S., Cohen, L.: Nonlocal regularization of inverse problems. In: ECCV 8: European Conference on Computer Vision, p. 578. Springer, Berlin (2008)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  32. Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19, 553–572 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Schönlieb, C.B.: Total variation minimization with an H −1 constraint. Preprint

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

    MATH  Google Scholar 

  35. Yuan, J., Schnörr, C., Steidl, G.: Convex Hodge decomposition and regularization of image flows. J. Math. Imaging Vis. 33(2), 16977 (2009)

    Article  Google Scholar 

  36. Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. UCLA CAM Report (2009)

  37. Zhang, X., Chan, T.: Wavelet inpainting by nonlocal total variation. Inverse Probl. Imaging 4, 191–210 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhou, D., Schölkopf, B.: A regularization framework for learning from graph data. In: ICML Workshop on Stat. Relational Learning and Its Connections to Other Fields (2004)

    Google Scholar 

  39. Zhou, D., Schölkopf, B.: Regularization on discrete spaces. In: Pattern Recognition, Proceedings of the 27th DAGM Symposium, Berlin, Germany, pp. 361–369 (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Information Engineering, Zhejiang University of Technology, Hangzhou, 310023, China

    Yan Jin

  2. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103, Leipzig, Germany

    Yan Jin & Jürgen Jost

  3. Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, 79104, Freiburg, Germany

    Guofang Wang

Authors
  1. Yan Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jürgen Jost
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guofang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jürgen Jost.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jin, Y., Jost, J. & Wang, G. A Nonlocal Version of the Osher-Solé-Vese Model. J Math Imaging Vis 44, 99–113 (2012). https://doi.org/10.1007/s10851-011-0313-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-011-0313-z

Keywords

Search

Navigation