Advertisement

Denoising 3D Medical Images Using a Second Order Variational Model and Wavelet Shrinkage

  • Minh-Phuong Tran
  • Renaud Péteri
  • Maitine Bergounioux
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7325)

Abstract

The aim of this paper is to construct a model which decomposes a 3D image into two components: the first one containing the geometrical structure of the image, the second one containing the noise. The proposed method is based on a second order variational model and an undecimated wavelet thresholding operator. The numerical implementation is described, and some experiments for denoising a 3D MRI image are successfully performed. Future prospects are finally exposed.

Keywords

Image Decomposition Image Denoising Undecimated wavelet Shrinkage Second order variational model 3D medical image 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bergounioux, M.: On Poincare-Wirtinger inequalities in spaces of functions of bounded variation. [hal-00515451] version 2 (June 10, 2011)Google Scholar
  2. 2.
    Chambolle, A.: An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision 20, 89–97 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Holschneider, M., Kronland-Martinet, R., Morlet, J., Tchamitchian, P.: A real time algorithm for signal analysis with the help of the wavelet transform. In: Wavelets, Time-Frequency Methods and Phase Space, pp. 286–297. Springer, Berlin (1989)Google Scholar
  4. 4.
    Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems Numerische Mathematik. Journal of Mathematical Imaging and Vision 77, 167–188 (1997)MathSciNetGoogle Scholar
  5. 5.
    Chan, T., Esedoglu, S., Park, F., Yip, A.: Recent Developments in total Variation Image Restoration. CAM Report 05-01, Department of Mathematics, UCLA (2004)Google Scholar
  6. 6.
    Louchet, C.: Variational and Bayesian models for image denoising: from total variation towards non-local means. Universite Paris Descartes, Ecole Doctoranle Mathematiques Paris-Centre (December 10, 2008)Google Scholar
  7. 7.
    Ekeland, I., Remam, R.: Analyse convex et problemes variationnels. Etudes Mathematiques. Dunod (1974)Google Scholar
  8. 8.
    Aujol, J.-F., Chambolle, A.: Dual norms and image decomposition models. IJCV 63(1), 85–104 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Piffet, L.: Décomposition d’image par modèles variationnels - Débruitage et extraction de texture. Université d’Orléans, Pôle Universités Centre Val de Loire (Novembre 23, 2010)Google Scholar
  10. 10.
    Bergounioux, M., Tran, M.P.: A second order model for 3D texture extraction. In: Mathematical Image Processing. Springer Proceedings in Mathematics, vol. 5. Universite d’Orleans (2011)Google Scholar
  11. 11.
    Starck, J.-L., Candes, E.J., Donoho, D.L.: The Curvelet transform for Image Denoising. IEEE Transactions on Image Processing 11(6) (June 2002)Google Scholar
  12. 12.
    Mallat, S.: A wavelet tour of signal processing. Academic Press Inc. (1998)Google Scholar
  13. 13.
    Jin, Y., Angelini, E., Laine, A.: Wavelets in medical image processing: denoising, sementation and registration. Department of Biomedical Engineering, Columbia University, New York, USAGoogle Scholar
  14. 14.
    Meyer, Y.: Oscillating patterns in image processing and in some nonlinear evolution equations. The 15th Dean Jacquelines B. Lewis Memorial Lectures (March 2001)Google Scholar
  15. 15.
    Steidl, G., Weickert, J., Brox, T., Mrzek, P., Welk, M.: On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and sides, Tech. Rep. 26, Department of Mathematics, University of Bremen, Germany (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Minh-Phuong Tran
    • 1
  • Renaud Péteri
    • 2
  • Maitine Bergounioux
    • 1
  1. 1.Laboratoire MAPMO, UMR 6628, Fédération Denis-PoissonUniversité d’OrléansOrléans Cedex 2France
  2. 2.Laboratoire Mathématiques, Image et ApplicationsUniversité de La RochelleLa RochelleFrance

Personalised recommendations