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The Conformal Monogenic Signal of Image Sequences

  • Lennart Wietzke
  • Gerald Sommer
  • Oliver Fleischmann
  • Christian Schmaltz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5604)

Abstract

Based on the research results of the Kiel University Cognitive Systems Group in the field of multidimensional signal processing and Computer Vision, this book chapter presents new ideas in 2D/3D and multidimensional signal theory. The novel approach, called the conformal monogenic signal, is a rotationally invariant quadrature filter for extracting i(ntrinsic)1D and i2D local features of any curved 2D signal - such as lines, edges, corners and circles - without the use of any heuristics or steering techniques. The conformal monogenic signal contains the monogenic signal as a special case for i1D signals - such as lines and edges - and combines monogenic scale space, local energy, direction/orientation, both i1D and i2D phase 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 of the conformal monogenic signal is to lift up 2D signals by stereographic projection to a higher dimensional conformal space where the local signal features can be analyzed with more degrees of freedom compared to the flat two-dimensional space of the original signal domain. The philosophy of the conformal monogenic signal is based on the idea to make use of the direct relation of the original two-dimensional signal and abstract geometric entities such as lines, circles, planes and spheres. Furthermore, the conformal monogenic signal can not only be extended to 3D signals (image sequences) but also to signals of any dimension.

The main advantages of the conformal monogenic signal in practical applications are the completeness with respect to the intrinsic dimension of the signal, the rotational invariance, the low computational time complexity, the easy implementation into existing Computer Vision software packages and the numerical robustness of calculating exact local curvature of signals without the need of any derivatives.

Keywords

Conformal Space Stereographic Projection Local Phase Phase Congruency Monogenic Signal 
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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References

  1. 1.
    Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory (Graduate Texts in Mathematics), vol. 137. Springer, Heidelberg (2002)Google Scholar
  2. 2.
    Bernstein, S.: Inverse Probleme. Technical report, TU Bergakademie Freiberg (2007)Google Scholar
  3. 3.
    Brackx, F., De Knock, B., De Schepper, H.: Generalized multidimensional Hilbert transforms in Clifford analysis. International Journal of Mathematics and Mathematical Sciences (2006)Google Scholar
  4. 4.
    Delanghe, R.: Clifford analysis: History and perspective. Computational Methods and Function Theory 1(1), 107–153 (2001)MathSciNetzbMATHGoogle Scholar
  5. 5.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)Google Scholar
  6. 6.
    Felsberg, M.: Low-level image processing with the structure multivector. Technical Report 2016, Kiel University, Department of Computer Science (2002)Google Scholar
  7. 7.
    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
  8. 8.
    Grau, V., Becher, H., Alison Noble, J.: Phase-based registration of multi-view real-time three-dimensional echocardiographic sequences. In: Larsen, R., Nielsen, M., Sporring, J. (eds.) MICCAI 2006. LNCS, vol. 4190, pp. 612–619. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Gürlebeck, K., Habetha, K., Sprössig, W.: Funktionentheorie in der Ebene und im Raum (Grundstudium Mathematik). Birkhäuser, Basel (2006)Google Scholar
  10. 10.
    Hahn, S.L.: Hilbert Transforms in Signal Processing. Artech House Inc., Boston (1996)Google Scholar
  11. 11.
    Kovesi, P.: Phase congruency detects corners and edges. In: The Australian Pattern Recognition Society Conference, pp. 309–318 (2003)Google Scholar
  12. 12.
    Kovesi, P., Videre, A.: Image features from phase congruency. Journal of Computer Vision Research 1(3) (1999)Google Scholar
  13. 13.
    Krause, M., Sommer, G.: A 3D isotropic quadrature filter for motion estimation problems. In: Proc. Visual Communications and Image Processing, Beijing, China, vol. 5960, pp. 1295–1306. The International Society for Optical Engineering, Bellingham (2005)Google Scholar
  14. 14.
    Lichtenauer, J., Hendriks, E.A., Reinders, M.J.T.: Isophote properties as features for object detection. In: CVPR, vol. (2), pp. 649–654 (2005)Google Scholar
  15. 15.
    Needham, T.: Visual Complex Analysis. Oxford University Press, Oxford (1997)zbMATHGoogle Scholar
  16. 16.
    Reisfeld, D.: The constrained phase congruency feature detector: Simultaneous localization, classification and scale determination  17(11), 1161–1169 (1996)Google Scholar
  17. 17.
    Toft, P.: The Radon Transform - Theory and Implementation. PhD thesis, Technical University of Denmark (1996)Google Scholar
  18. 18.
    Wietzke, L., Fleischmann, O., Sommer, G.: 2D image analysis by generalized Hilbert transforms in conformal space. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part II. LNCS, vol. 5303, pp. 638–649. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    Wietzke, L., Sommer, G., Schmaltz, C., Weickert, J.: Differential geometry of monogenic signal representations. In: Sommer, G., Klette, R. (eds.) RobVis 2008. LNCS, vol. 4931, pp. 454–465. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  21. 21.
    Zang, D., Sommer, G.: Detecting intrinsically two-dimensional image structures using local phase. In: Franke, K., Müller, K.-R., Nickolay, B., Schäfer, R. (eds.) DAGM 2006. LNCS, vol. 4174, pp. 222–231. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  22. 22.
    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 2009

Authors and Affiliations

  • Lennart Wietzke
    • 1
  • Gerald Sommer
    • 1
  • Oliver Fleischmann
    • 1
  • Christian Schmaltz
    • 2
  1. 1.Institute of Computer Science, Chair of Cognitive SystemsChristian-Albrechts-UniversityKielGermany
  2. 2.Mathematical Image Analysis Group, Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany

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