Generalized Analytic Signals in Image Processing: Comparison, Theory and Applications

  • Swanhild Bernstein
  • Jean-Luc Bouchot
  • Martin Reinhardt
  • Bettina Heise
Part of the Trends in Mathematics book series (TM)


This article is intended as a mathematical overview of the generalizations of analytic signals to higher-dimensional problems, as well as of their applications to and of their comparison on artificial and real-world image samples.

We first start by reviewing the basic concepts behind analytic signal theory and derive its mathematical background based on boundary value problems of one-dimensional analytic functions. Following that, two generalizations are motivated by means of higher-dimensional complex analysis or Clifford analysis. Both approaches are proven to be valid generalizations of the known analytic signal concept.

In the last part we experimentally motivate the choice of such higherdimensional analytic or monogenic signal representations in the context of image analysis. We see how one can take advantage of one or the other representation depending on the application.


Monogenic signal monogenic functional theory image processing texture Riesz transform. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Baker, D. Scharstein, J. Lewis, S. Roth, M. Black, and R. Szeliski. A database and evaluation methodology for optical flow. International Journal of Computer Vision, 92:1–31, 2011. See also website: Scholar
  2. [2]
    T. Batard and M. Berthier. The spinor representation of images. In K. Gürlebeck, editor, 9th International Conference on Clifford Algebras and their Applications, Weimar, Germany, 15–20 July 2011.Google Scholar
  3. [3]
    T. Batard, M. Berthier, and C. Saint-Jean. Clifford Fourier transform for color image processing. In E.J. Bayro-Corrochano and G. Scheuermann, editors, Geometric Algebra Computing in Engineering and Computer Science, pages 135–162. Springer, London, 2010.Google Scholar
  4. [4]
    F. Brackx, R. Delanghe, and F. Sommen. Clifford Analysis, volume 76. Pitman, Boston, 1982.Google Scholar
  5. [5]
    T. Bülow, D. Pallek, and G. Sommer. Riesz transform for the isotropic estimation of the local phase of Moiré interferograms. In G. Sommer, N. Krüger, and C. Perwass, editors, DAGM-Symposium, Informatik Aktuell, pages 333–340. Springer, 2000.Google Scholar
  6. [6]
    T. Bülow and G. Sommer. Hypercomplex signals – a novel extension of the analytic signal to the multidimensional case. IEEE Transactions on Signal Processing, 49(11):2844–2852, Nov. 2001.MathSciNetCrossRefGoogle Scholar
  7. [7]
    V. Chandrasekhar, D.M. Chen, A. Lin, G. Takacs, S.S. Tsai, N.M. Cheung, Y. Reznik, R. Grzeszczuk, and B. Girod. Comparison of local feature descriptors for mobile visual search. In Image Processing (ICIP), 2010 17th IEEE International Conference on, pages 3885–3888. IEEE, 2010.Google Scholar
  8. [8]
    A. Dzhuraev. On Riemann–Hilbert boundary problem in several complex variables. Complex Variables and Elliptic Equations, 29(4):287–303, 1996.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    M. Felsberg and G. Sommer. A new extension of linear signal processing for estimating local properties and detecting features. Proceedings of the DAGM 2000, pages 195–202, 2000.Google Scholar
  10. [10]
    M. Felsberg and G. Sommer. The monogenic signal. IEEE Transactions on Signal Processing, 49(12):3136–3144, Dec. 2001.MathSciNetCrossRefGoogle Scholar
  11. [11]
    D. Gabor. Theory of communication. Journal of the Institution of Electrical Engineers, 93(26):429–457, 1946. Part III.Google Scholar
  12. [12]
    K. Gürlebeck, K. Habetha, and W. Sprössig. Holomorphic Functions in the Plane and n-dimensional Space. Birkh¨auser, 2008.Google Scholar
  13. [13]
    S.L. Hahn. Multidimensional complex signals with single-orthant spectra. Proceedings of the IEEE, 80(8):1287–1300, Aug. 1992.CrossRefGoogle Scholar
  14. [14]
    R.M. Haralick, K. Shanmugam, and I.H. Dinstein. Textural features for image classification. IEEE Transactions on Systems, Man and Cybernetics, 3(6):610–621, 1973.CrossRefGoogle Scholar
  15. [15]
    B. Heise, S.E. Schausberger, C. Maurer, M. Ritsch-Marte, S. Bernet, and D. Stifter. Enhancing of structures in coherence probe microscopy imaging. In Proceedings of SPIE, pages 83350G–83350G–7, 2012.Google Scholar
  16. [16]
    E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations. Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    U. Köthe and M. Felsberg. Riesz-transforms versus derivatives: On the relationship between the boundary tensor and the energy tensor. Scale Space and PDE Methods in Computer Vision, pages 179–191, 2005.Google Scholar
  18. [18]
    K.G. Larkin, D.J. Bone, and M.A. Oldfield. Natural demodulation of two-dimensional fringe patterns. I. general background of the spiral phase quadrature transform. Journal of the Optical Society of America A, 18(8):1862–1870, 2001.Google Scholar
  19. [19]
    P. Lounesto. Clifford Algebras and Spinors, volume 286 of London Mathematical Society Lecture Notes. Cambridge University Press, 1997.Google Scholar
  20. [20]
    K. Mikolajczyk and C. Schmid. A performance evaluation of local descriptors. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(10):1615–1630, 2005.CrossRefGoogle Scholar
  21. [21]
    W. Rudin. Function Theory in the Unit Ball ofn. Springer, 1980.Google Scholar
  22. [22]
    V. Schlager, S. Schausberger, D. Stifter, and B. Heise. Coherence probe microscopy imaging and analysis for fiber-reinforced polymers. Image Analysis, pages 424–434, 2011.Google Scholar
  23. [23]
    E.M. Stein. Singular Integrals and Differentiability Properties of Functions, volume 30 of Princeton Mathematical Series. Princeton University Press, 1970.Google Scholar
  24. [24]
    The Mathworks, Inc. MATLAB® R2012b documentation: colormap. Software documentation available at: html, 1994–2012.
  25. [25]
    J. Ville. Théorie et applications de la notion de signal analytique. Cables et Transmission, 2A:61–74, 1948.Google Scholar
  26. [26]
    D. Zang and G. Sommer. Signal modeling for two-dimensional image structures. Journal of Visual Communication and Image Representation, 18(1):81–99, 2007.CrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Swanhild Bernstein
    • 1
  • Jean-Luc Bouchot
    • 2
  • Martin Reinhardt
    • 1
  • Bettina Heise
    • 3
    • 4
  1. 1.Fakultät für Mathematik und Informatik Institut für Angewandte AnalysisTechnische Universität Bergakademie FreibergFreibergGermany
  2. 2.Department of Knowledge-Based Mathematical Systems, FLLLJohannes Kepler UniversityLinzAustria
  3. 3.Department of Knowledge-Based Mathematical Systems, FLLLJohannes Kepler UniversityLinzAustria
  4. 4.Christian Doppler Laboratory MS-MACH Center for Surface- and Nanoanalytics, ZONALinzAustria

Personalised recommendations