Local Structure Analysis by Isotropic Hilbert Transforms

  • Lennart Wietzke
  • Oliver Fleischmann
  • Anne Sedlazeck
  • Gerald Sommer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6376)


This work presents the isotropic and orthogonal decomposition of 2D signals into local geometrical and structural components. We will present the solution for 2D image signals in four steps: signal modeling in scale space, signal extension by higher order generalized Hilbert transforms, signal representation in classical matrix form, followed by the most important step, in which the matrix-valued signal will be mapped to a so called multivector. We will show that this novel multivector-valued signal representation is an interesting space for complete geometrical and structural signal analysis. In practical computer vision applications lines, edges, corners, and junctions as well as local texture patterns can be analyzed in one unified algebraic framework. Our novel approach will be applied to parameter-free multilayer decomposition.


Scale Space Geometric Algebra Convolution Kernel Signal Extension Inverse Radon 
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.


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  1. 1.
    Oppenheim, A.V., Lim, J.S.: The importance of phase in signals. Proceedings of the IEEE 69(5), 529–541 (1981)CrossRefGoogle Scholar
  2. 2.
    Huang, T., Burnett, J., Deczky, A.: The importance of phase in image processing filters. IEEE Trans. on Acoustics, Speech and Signal Processing 23(6), 529–542 (1975)CrossRefGoogle Scholar
  3. 3.
    Felsberg, M., Sommer, G.: The monogenic scale-space: A unifying approach to phase-based image processing in scale-space. Journal of Mathematical Imaging and Vision 21, 5–26 (2004)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Delanghe, R.: On some properties of the Hilbert transform in Euclidean space. Bull. Belg. Math. Soc. Simon Stevin 11(2), 163–180 (2004)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Köthe, U., Felsberg, M.: Riesz-transforms vs. derivatives: On the relationship between the boundary tensor and the energy tensor. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds.) Scale-Space 2005. LNCS, vol. 3459, pp. 179–191. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Lowe, D.G.: Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision 60, 91–110 (2004)CrossRefGoogle Scholar
  7. 7.
    Wietzke, L., Sommer, G., Fleischmann, O.: The geometry of 2D image signals. In: IEEE Computer Society on Computer Vision and Pattern Recognition, CVPR 2009, pp. 1690–1697 (2009)Google Scholar
  8. 8.
    Felsberg, M.: Low-level image processing with the structure multivector. Technical Report 2016, Kiel University, Department of Computer Science (2002)Google Scholar
  9. 9.
    Hahn, S.L.: Hilbert Transforms in Signal Processing. Artech House Inc., Boston (1996)zbMATHGoogle Scholar
  10. 10.
    Pan, W., Bui, T.D., Suen, C.Y.: Rotation invariant texture classification by ridgelet transform and frequency-orientation space decomposition. Signal Process. 88(1), 189–199 (2008)zbMATHCrossRefGoogle Scholar
  11. 11.
    Perwass, C.: Geometric Algebra with Applications in Engineering. Geometry and Computing, vol. 4. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  12. 12.
    Sobczyk, G., Erlebacher, G.: Hybrid matrix geometric algebra. In: Li, H., Olver, P.J., Sommer, G. (eds.) IWMM-GIAE 2004. LNCS, vol. 3519, pp. 191–206. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Danielsson, P.E., Lin, Q., Ye, Q.Z.: Efficient detection of second-degree variations in 2D and 3D images. Journal of Visual Communication and Image Representation 12(3), 255–305 (2001)CrossRefGoogle Scholar
  14. 14.
    Gabor, D.: Theory of communication. Journal IEE, London 93(26), 429–457 (1946)Google Scholar
  15. 15.
    Stuke, I., Aach, T., Barth, E., Mota, C.: Analysing superimposed oriented patterns. In: 6th IEEE Southwest Symposium on Image Analysis and Interpretation, pp. 133–137. IEEE Computer Society, Los Alamitos (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Lennart Wietzke
    • 1
  • Oliver Fleischmann
    • 2
  • Anne Sedlazeck
    • 2
  • Gerald Sommer
    • 2
  1. 1.Raytrix GmbHGermany
  2. 2.Cognitive Systems Group, Department of Computer ScienceKiel University 

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