Skip to main content
SpringerLink
Log in

A New Nonlocal H 1 Model for Image Denoising

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

Abstract

Following ideas of Kindermann et al. (Multiscale Model. Simul. 4(4):1091–1115, 2005) and Gilboa and Osher (Multiscale Model. Simul. 7:1005–1028, 2008) we introduce new nonlocal operators to interpret the nonlocal means filter (NLM) as a regularization of the corresponding Dirichlet functional. Then we use these nonlocal operators to propose a new nonlocal H 1 model, which is (slightly) different from the nonlocal H 1 model of Gilboa and Osher (Multiscale Model. Simul. 6(2):595–630, 2007; Proc. SPIE 6498:64980U, 2007). The key point is that both the fidelity and the smoothing term are derived from the same geometric principle. We compare this model with the nonlocal H 1 model of Gilboa and Osher and the nonlocal means filter, both theoretically and in computer experiments. The experiments show that this new nonlocal H 1 model also provides good results in image denoising and closer to the nonlocal means filter than the H 1 model of Gilboa and Osher. This means that the new nonlocal operators yield a better interpretation of the nonlocal means filter than the nonlocal operators given in Gilboa and Osher (Multiscale Model. Simul. 7:1005–1028, 2008).

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Aharon, M., Elad, M., Bruckstein, A.: K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54, 4311–4322 (2006)

    Article  Google Scholar 

  2. 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 

  3. Bresson, X., Chan, T.F.: Nonlocal unsupervised variational image segmentation models. Ucla c.a.m. report 08-67, (2008)

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Buades, A., Coll, B., Morel, J.-M.: Image enhancement by nonlocal reverse heat equation. Technical report 22, Centre de Mathématiques et Leurs Applications, Cachan, France (2006)

  6. Chan, T.F., Shen, J., Image Processing and Analysis. Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  7. Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3D transform-domain collaborative ltering. IEEE Trans. Image Process. 16, 2080–2095 (2007)

    Article  MathSciNet  Google Scholar 

  8. Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image restoration by sparse 3d transform-domain collaborative filtering. SPIE Electron. E Imaging 08, 6812 (2008)

  9. Deledalle, C.-A., Duval, V., Salmon, J.: Non-local methods with shape-adaptive patches (NLM-SAP). J. Math. Imaging Vis. (2011). doi:10.1007/s10851-011-0294-y

    Google Scholar 

  10. Dong, B., Ye, J., Osher, S., Dinov, I.: Level set based nonlocal surfaces restoration. Multiscale Model. Simul. 7, 589–598 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Foi, A., Katkovnik, V., Egiazarian, K.: Pointwise shape-adaptive DCT for high-quality denoising and deblocking of grayscale and color images. IEEE Trans. Image Process. 16(5), 1395–1411 (2007)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Gilboa, G., Osher, S.: Nonlocal evolutions for image regularization. Proc. SPIE 6498, 64980U (2007). doi:10.1117/12.714701

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

  16. Jin, Y., Jost, J., Wang, G.: Nonlocal version of Osher-Solé-Vese model. J. Math. Imaging Vis. 44, 99–113 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jin, Y., Jost, J., Wang, G.: A new nonlocal variational setting for image processing. Preprint

  18. Jung, M., Vese, L.A.: Nonlocal variational image deblurring models in the presence of Gaussian or impulse noise. In: Lecture Notes in Computer Science, vol. 5567, pp. 401–412. Springer, Berlin (2009)

    Google Scholar 

  19. Katkovnik, V., Foi, A., Egiazarian, K., Astola, J.: From local kernel to nonlocal multiple-model image denoising. Int. J. Comput. Vis. 86(1), 1–32 (2010). doi:10.1007/s11263-009-0272-7

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  23. Osher, S., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153. Springer, New York (2003)

    MATH  Google Scholar 

  24. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MATH  Google Scholar 

  28. Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, New York (2006)

    MATH  Google Scholar 

  29. Sochen, N., Kimmel, R., Malladi, R.: A general framework for low level vision. IEEE Trans. Image Process. 7, 310–318 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Proceedings of the 6th IEEE International Conference on Computer Vision (ICCV’98), pp. 839–846 (1998)

    Google Scholar 

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

    MATH  Google Scholar 

  32. Yaroslavsky, L.P.: Digital Picture Processing: An Introduction. Springer Series in Information Sciences, vol. 9. Springer, Berlin (1979)

    Google Scholar 

  33. Zhang, X., Chan, T.: Wavelet Inpainting by Nonlocal Total Variation Inverse Problems and Imaging, vol. 4, pp. 191–210 (2010)

    Google Scholar 

  34. Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3, 253–276 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  35. 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 

  36. 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

Acknowledgements

A part of this work was carried out, while the first named author was visiting Max Planck Institute for Mathematics in the Sciences, Leipzig. She would like to thank the institute for the warm hospitality and a good working condition. The authors would like to thank both anonymous referees for their careful reading and helpful suggestions.

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

    Jürgen Jost

  3. Fakultät für Mathematik und Informatik, Universität Leipzig, 04091, Leipzig, Germany

    Jürgen Jost

  4. 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 Guofang Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jin, Y., Jost, J. & Wang, G. A New Nonlocal H 1 Model for Image Denoising. J Math Imaging Vis 48, 93–105 (2014). https://doi.org/10.1007/s10851-012-0395-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-012-0395-2

Keywords

Search

Navigation