Advertisement

2D Image Analysis by Generalized Hilbert Transforms in Conformal Space

  • Lennart Wietzke
  • Oliver Fleischmann
  • Gerald Sommer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5303)

Abstract

This work presents a novel rotational invariant quadrature filter approach - called the conformal monogenic signal - for analyzing i(ntrinsic)1D and i2D local features of any curved 2D signal such as lines, edges, corners and junctions without the use of steering. The conformal monogenic signal contains the monogenic signal as a special case for i1D signals and combines monogenic scale space, phase, direction/orientation, energy and curvature in one unified algebraic framework. The conformal monogenic signal will be theoretically illustrated and motivated in detail by the relation of the 3D Radon transform and the generalized Hilbert transform on the sphere. The main idea is to lift up 2D signals to the higher dimensional conformal space where the signal features can be analyzed with more degrees of freedom. Results of this work are the low computational time complexity, the easy implementation into existing Computer Vision applications and the numerical robustness of determining curvature without the need of any derivatives.

Keywords

Conformal Space Poisson Kernel Conformal Curvature Monogenic Signal Radon Space 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hahn, S.L.: Hilbert Transforms in Signal Processing. Artech House Inc., Boston (1996)zbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Felsberg, M.: Low-Level Image Processing with the Structure Multivector, Technical Report No. 2016. Ph.D thesis, Kiel University, Department of Computer Science (2002)Google Scholar
  4. 4.
    Brackx, F., Knock, B.D., Schepper, H.D.: Generalized multidimensional Hilbert transforms in Clifford analysis. International Journal of Mathematics and Mathematical Sciences (2006)Google Scholar
  5. 5.
    Wietzke, L., Sommer, G., Schmaltz, C., Weickert, J.: Differential geometry of monogenic signal representations. In: Sommer, G. (ed.) RobVis 2008. LNCS, vol. 4931, pp. 454–465. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Needham, T.: Visual Complex Analysis. Oxford University Press, Oxford (1997)zbMATHGoogle Scholar
  7. 7.
    Toft, P.: The Radon Transform - Theory and Implementation. Ph.D thesis, Technical University of Denmark (1996)Google Scholar
  8. 8.
    Delanghe, R.: Clifford analysis: History and perspective. Computational Methods and Function Theory 1(1), 107–153 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)zbMATHGoogle Scholar
  10. 10.
    Bernstein, S.: Inverse Probleme. Technical report, TU Bergakdemie Freiberg (2007)Google Scholar
  11. 11.
    Wietzke, L., Sommer, G.: The Conformal Monogenic Signal. In: Rigoll, G. (ed.) DAGM 2008. LNCS, vol. 5096, pp. 527–536. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Lichtenauer, J., Hendriks, E.A., Reinders, M.J.T.: Isophote properties as features for object detection. CVPR (2), 649–654 (2005)Google Scholar
  13. 13.
    Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory Graduate Texts in Mathematics, vol. 137. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  14. 14.
    Zang, D., Wietzke, L., Schmaltz, C., Sommer, G.: Dense optical flow estimation from the monogenic curvature tensor. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 239–250. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Lennart Wietzke
    • 1
  • Oliver Fleischmann
    • 1
  • Gerald Sommer
    • 1
  1. 1.Department of Computer ScienceKiel UniversityKielGermany

Personalised recommendations