Advertisement

A Generic Neighbourhood Filtering Framework for Matrix Fields

  • Luis Pizarro
  • Bernhard Burgeth
  • Stephan Didas
  • Joachim Weickert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5304)

Abstract

The Nonlocal Data and Smoothness (NDS) filtering framework for greyvalue images has been recently proposed by Mrázek et al. This model for image denoising unifies M-smoothing and bilateral filtering, and several well-known nonlinear filters from the literature become particular cases. In this article we extend this model to so-called matrix fields. These data appear, for example, in diffusion tensor magnetic resonance imaging (DT-MRI). Our matrix-valued NDS framework includes earlier filters developped for DT-MRI data, for instance, the affine-invariant and the log-Euclidean regularisation of matrix fields. Experiments performed with synthetic matrix fields and real DT-MRI data showed excellent performance with respect to restoration quality as well as speed of convergence.

Keywords

Image Denoising Complexity Order Smoothness Term Matrix Field Restoration Quality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Lee, J.S.: Digital image smoothing and the sigma filter. Computer Vision, Graphics, and Image Processing 24, 255–269 (1983)CrossRefGoogle Scholar
  2. 2.
    Chu, C.K., Glad, I.K., Godtliebsen, F., Marron, J.S.: Edge-preserving smoothers for image processing. Journal of the American Statistical Association 93, 526–541 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Polzehl, J., Spokoiny, V.: Adaptive weights smoothing with applications to image restoration. Journal of the Royal Statistical Society, Series B 62, 335–354 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Winkler, G., Aurich, V., Hahn, K., Martin, A.: Noise reduction in images: Some recent edge-preserving methods. Pattern Recognition and Image Analysis 9, 749–766 (1999)Google Scholar
  5. 5.
    Griffin, L.D.: Mean, median and mode filtering of images. Proceedings of the Royal Society of London A 456, 2995–3004 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Tomasi, C., Manduchi, R.: Bilateral filtering for gray and colour images. In: Proc. of the 1998 IEEE International Conference on Computer Vision, Bombay, India, pp. 839–846. Narosa Publishing House (1998)Google Scholar
  7. 7.
    Mrázek, P., Weickert, J., Bruhn, A.: On robust estimation and smoothing with spatial and tonal kernels. In: Klette, R., Kozera, R., Noakes, L., Weickert, J. (eds.) Geometric Properties for Incomplete Data, pp. 335–352. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Didas, S., Mrázek, P., Weickert, J.: Energy-based image simplification with nonlocal data and smoothness terms. In: Iske, A., Levesley, J. (eds.) Algorithms for Approximation, pp. 51–60. Springer, Heidelberg (2006)Google Scholar
  9. 9.
    Pizarro, L., Didas, S., Bauer, F., Weickert, J.: Evaluating a general class of filters for image denoising. In: Ersbøll, B.K., Pedersen, K.S. (eds.) SCIA 2007. LNCS, vol. 4522, pp. 601–610. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Tschumperlé, D., Deriche, R.: Diffusion tensor regularization with constraints preservation. In: Proc. of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2001), vol. 1, pp. 948–953 (2001)Google Scholar
  11. 11.
    Weickert, J., Brox, T.: Diffusion and regularization of vector- and matrix-valued images. In: Nashed, M.Z., Scherzer, O. (eds.) Inverse Problems, Image Analysis, and Medical Imaging. Contemporary Mathematics, vol. 313, AMS, Providence (2002)Google Scholar
  12. 12.
    Chefd’hotel, C., Tschumperlé, D., Deriche, R., Faugeras, O.: Regularizing flows for constrained matrix-valued images. Journal of Mathematical Imaging and Vision 20, 147–162 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Batchelor, P.G., Moakher, M., Atkinson, D., Calamante, F., Connelly, A.: A rigorous framework for diffusion tensor calculus. Magnetic Resonance in Medicine 53, 221–225 (2005)CrossRefGoogle Scholar
  14. 14.
    Moakher, M.: A differential geometry approach to the geometric mean of symmetric positive-definite matrices. SIAM Journal on Matrix Analysis and Applications, 735–747 (2005)Google Scholar
  15. 15.
    Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. International Journal of Computer Vision 66, 41–66 (2006)CrossRefzbMATHGoogle Scholar
  16. 16.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine 56, 411–421 (2006)CrossRefGoogle Scholar
  17. 17.
    Fletcher, P.T., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Processing 87, 250–262 (2007)CrossRefzbMATHGoogle Scholar
  18. 18.
    Gur, Y., Sochen, N.: Coordinate-free diffusion over compact Lie-groups. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 580–591. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Burgeth, B., Didas, S., Florack, L., Weickert, J.: A generic approach to diffusion filtering of matrix-fields. Computing 81, 179–197 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Steidl, G., Setzer, S., Popilka, B., Burgeth, B.: Restoration of matrix fields by second order cone programming. Computing 81, 161–178 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fillard, P., Arsigny, V., Pennec, X., Ayache, N.: Joint estimation and smoothing of clinical DT-MRI with a log-Euclidean metric. Research Report RR-5607, INRIA, Sophia-Antipolis, France (2005)Google Scholar
  22. 22.
    Hamarneh, G., Hradsky, J.: Bilateral filtering of diffusion tensor magnetic resonance images. IEEE Transactions on Image Processing 16, 2463–2475 (2007)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nashed, M.Z., Scherzer, O.: Least squares and bounded variation regularization with nondifferentiable functionals. Numerical Functional Analysis and Optimization 19, 873–901 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 629–639 (1990)CrossRefGoogle Scholar
  25. 25.
    Gudbjartsson, H., Patz, S.: The Rician distribution of noisy MRI data. Magnetic Resonance in Medicine 34, 910–914 (1995) [published erratum appears in Magnetic Resonance in Medicine 36, 332 (1996)]CrossRefGoogle Scholar
  26. 26.
    Pajevic, S., Basser, P.J.: Parametric and non-parametric statistical analysis of DT-MRI data. Journal of Magnetic Resonance 161, 1–14 (2003)CrossRefGoogle Scholar
  27. 27.
    Mori, S., van Zijl, P.C.: Fiber tracking: principles and strategies - a technical review. NRM in Biomedicine 15, 468–480 (2002)Google Scholar
  28. 28.
    Nucifora, P.G., Verma, R., Lee, S.K., Melhem, E.R.: Diffusion-tensor MR imaging and tractography: Exploring brain microstructure and connectivity. Radiology 245, 367–384 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Luis Pizarro
    • 1
  • Bernhard Burgeth
    • 1
  • Stephan Didas
    • 1
  • Joachim Weickert
    • 1
  1. 1.Mathematical Image Analysis Group, Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany

Personalised recommendations