Journal of Mathematical Imaging and Vision

, Volume 40, Issue 3, pp 305–325 | Cite as

Image Analysis by Conformal Embedding

  • Oliver FleischmannEmail author
  • Lennart Wietzke
  • Gerald Sommer


This work presents new ideas in isotropic multi-dimensional phase based signal theory. The novel approach, called the conformal monogenic signal, is a rotational invariant quadrature filter for extracting local features of any curved signal without the use of any heuristics or steering techniques. The conformal monogenic signal contains the recently introduced monogenic signal as a special case and combines Poisson scale space, local amplitude, direction, phase and curvature in one unified algebraic framework. The conformal monogenic signal will be theoretically illustrated and motivated in detail by the relation between the Radon transform and the generalized Hilbert transform. The main idea of the conformal monogenic signal is to lift up n-dimensional signals by inverse stereographic projections to a n-dimensional sphere in ℝn+1 where the local signal features can be analyzed with more degrees of freedom compared to the flat n-dimensional space of the original signal domain. As result, it delivers a novel way of computing the isophote curvature of signals without partial derivatives. The philosophy of the conformal monogenic signal is based on the idea to use the direct relation between the original signal and geometric entities such as lines, circles, hyperplanes and hyperspheres. Furthermore, the 2D conformal monogenic signal can be extended to signals of any dimension. The main advantages of the conformal monogenic signal in practical applications are its compatibility with intrinsically one dimensional and special intrinsically two dimensional signals, the rotational invariance, the low computational time complexity, the easy implementation into existing software packages and the numerical robustness of calculating exact local curvature of signals without the need of any derivatives.


Unit sphere Signal processing Generalized Hilbert transform Riesz transform Radon transform Isotropic Local phase based signal analysis Clifford analysis Monogenic signal Analytic signal Isophote curvature Poisson scale space Stereographic projection Conformal space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Carneiro, G., Jepson, A.D.: Phase-based local features. In: 7th European Conference on Computer Vision-Part I. LNCS, vol. 2350, pp. 282–296. Springer, Berlin, Heidelberg, New York (2002) Google Scholar
  2. 2.
    Coope, I.D.: Circle fitting by linear and nonlinear least squares. J. Optim. Theory Appl. 76(2), 381–388 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, New York (1976) zbMATHGoogle Scholar
  4. 4.
    Felsberg, M., Sommer, G.: The monogenic signal. IEEE Trans. Signal Process. 49(12), 3136–3144 (2001) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Felsberg, M., Sommer, G.: The monogenic scale-space: a unifying approach to phase-based image processing in scale-space. J. Math. Imaging Vis. 21, 5–26 (2004) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fleet, D.J., Jepson, A.D.: Stability of phase information. IEEE Trans. Pattern Anal. Mach. Intell. 15(12), 1253–1268 (1993) CrossRefGoogle Scholar
  7. 7.
    Fleet, D.J., Jepson, A.D., Jenkin, M.R.M.: Phase-based disparity measurement. CVGIP, Image Underst. 53, 198–210 (1991) zbMATHCrossRefGoogle Scholar
  8. 8.
    Gabor, D.: Theory of communication. J. IEE (Lond.) 93, 429–457 (1946) Google Scholar
  9. 9.
    Gander, W., Golub, G.H., Strebel, R.: Least-squares fitting of circles and ellipses. BIT 34(4), 558–578 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Huang, T., Burnett, J., Deczky, A.: The importance of phase in image processing filters. IEEE Trans. Acoust. Speech Signal Process. 23(6), 529–542 (1975) CrossRefGoogle Scholar
  11. 11.
    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
  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.
    Luo, Y., Al-Dossary, S., Marhoon, M., Alfaraj, M.: Generalized Hilbert transform and its applications in geophysics. Lead. Edge 22(3), 198–202 (2003) CrossRefGoogle Scholar
  14. 14.
    Oppenheim, A.V., Lim, J.S.: The importance of phase in signals. Proc. IEEE 69(5), 529–541 (1981) CrossRefGoogle Scholar
  15. 15.
    Romeny, B.M. (ed.): Geometry-Driven Diffusion in Computer Vision. Kluwer Academic, Dordrecht (1994) zbMATHGoogle Scholar
  16. 16.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions (PMS-30). Princeton University Press, Princeton (1971) Google Scholar
  17. 17.
    van de Weijer, J., van Vliet, L.J., Verbeek, P.W., van Ginkel, M.: Curvature estimation in oriented patterns using curvilinear models applied to gradient vector fields. In: IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, pp. 1035–1042 (2001) Google Scholar
  18. 18.
    van Ginkel, M., van de Weijer, J., van Vliet, L.J., Verbeek, P.W.: Curvature estimation from orientation fields. In: Ersboll, B.K. (ed.) 11th Scandinavian Conference on Image Analysis, pp. 545–551. Pattern Recognition Society of Denmark (1999) Google Scholar
  19. 19.
    Wietzke, L., Sommer, G.: The signal multi-vector. J. Math. Imaging Vis. 37, 132–150 (2010) CrossRefGoogle Scholar
  20. 20.
    Xiaoxun, Z., Yunde, J.: Local Steerable Phase (LSP) feature for face representation and recognition. In: CVPR’06: Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 1363–1368. IEEE Comput. Soc., Los Alamitos (2006) Google Scholar
  21. 21.
    Zang, D., Wietzke, L., Schmaltz, C., Sommer, G.: Dense optical flow estimation from the monogenic curvature tensor. In: Scale Space and Variational Methods. LNCS, vol. 4485, pp. 239–250. Springer, Berlin, Heidelberg, New York (2007) CrossRefGoogle Scholar
  22. 22.
    Zetzsche, C., Barth, E.: Fundamental limits of linear filters in the visual processing of two-dimensional signals. Vis. Res. 30, 1111–1117 (1990) CrossRefGoogle Scholar
  23. 23.
    Zhang, L., Qian, T., Zeng, Q.: Radon measure formulation for edge detection using rotational wavelets. Commun. Pure Appl. Anal. 6(3), 899–915 (2007) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Oliver Fleischmann
    • 1
    Email author
  • Lennart Wietzke
    • 1
  • Gerald Sommer
    • 1
  1. 1.Cognitive Systems Group, Department of Computer ScienceKiel UniversityKielGermany

Personalised recommendations