Nonlinear Diffusion on the 2D Euclidean Motion Group

  • Erik Franken
  • Remco Duits
  • Bart ter Haar Romeny
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4485)

Abstract

Linear and nonlinear diffusion equations are usually considered on an image, which is in fact a function on the translation group. In this paper we study diffusion on orientation scores, i.e. on functions on the Euclidean motion group SE(2). An orientation score is obtained from an image by a linear invertible transformation. The goal is to enhance elongated structures by applying nonlinear left-invariant diffusion on the orientation score of the image. For this purpose we describe how we can use Gaussian derivatives to obtain regularized left-invariant derivatives that obey the non-commutative structure of the Lie algebra of SE(2). The Hessian constructed with these derivatives is used to estimate local curvature and orientation strength and the diffusion is made nonlinearly dependent on these measures. We propose an explicit finite difference scheme to apply the nonlinear diffusion on orientation scores. The experiments show that preservation of crossing structures is the main advantage compared to approaches such as coherence enhancing diffusion.

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References

  1. 1.
    Iijima, T.: Basic theory of pattern observation (in Japanese). Papers of Technical Group on Automata and Automatic Control, IECE, Japan (1959)Google Scholar
  2. 2.
    Witkin, A.P.: Scale-space filtering. In: 8th Int. Joint Conf. Artificial Intelligence, pp. 1019–1022 (1983)Google Scholar
  3. 3.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Analysis and Machine Intelligence 12, 629–639 (1990)CrossRefGoogle Scholar
  4. 4.
    Weickert, J.: Coherence-enhancing diffusion filtering. Int. J. Comp. Vision 31(2-3), 111–127 (1999)CrossRefGoogle Scholar
  5. 5.
    Kalitzin, S.N., ter Haar Romeny, B.M., Viergever, M.A.: Invertible orientation bundles on 2D scalar images. In: ter Haar Romeny, B.M., et al. (eds.) Scale Space Theory in Computer Vision, pp. 77–88 (1997)Google Scholar
  6. 6.
    Duits, R., et al.: Image analysis and reconstruction using a wavelet transform constructed from a reducible representation of the euclidean motion group. Int. J. Comp. Vision 72(1), 79–102 (2007)CrossRefGoogle Scholar
  7. 7.
    Brox, T., et al.: Nonlinear structure tensors. Image and Vision Computing 24(1), 41–55 (2006)CrossRefGoogle Scholar
  8. 8.
    Duits, R., Burgeth, B.: Scale Spaces on Lie Groups. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 300–312. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Antoine, J.-P., Murenzi, R.: Two-dimensional directional wavelets and the scale-angle representation. Signal Processing 52(3), 241–272 (1996)CrossRefGoogle Scholar
  10. 10.
    Granlund, G.H., Knutsson, H.: Signal Processing for Computer Vision. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  11. 11.
    Duits, R., van Almsick, M.: The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D-euclidean motion group. To appear in: Quarterly of Applied Mathematics, American Mathetical Society (2007)Google Scholar
  12. 12.
    van Ginkel, M.: Image Analysis using Orientation Space based on Steerable Filters. PhD thesis, Technische Universiteit Delft, The Netherlands (2002)Google Scholar
  13. 13.
    Weickert, J., Steidl, G., Welk, M.: From Tensor-Driven Diffusion to Anisotropic Wavelet Shrinkage. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3951, pp. 391–403. Springer, Heidelberg (2006)Google Scholar
  14. 14.
    Weickert, J., Scharr, H.: A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance. Journal of Visual Communication and Image Representation, 103–118 (2002)Google Scholar
  15. 15.
    Unser, M.: Splines: A perfect fit for signal and image processing. IEEE Signal Processing Magazine 16(6), 22–38 (1999)CrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Erik Franken
    • 1
  • Remco Duits
    • 1
  • Bart ter Haar Romeny
    • 1
  1. 1.Eindhoven University of Technology, Dept. of Biomedical EngineeringThe Netherlands

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