Advertisement

An Approach to the 2D Hilbert Transform for Image Processing Applications

  • Juan Valentín Lorenzo-Ginori
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4633)

Abstract

The Hilbert Transform (HT) and the analytic signal (AS) are widely used in their one-dimensional version for various applications. However, in the bi-dimensional (2D) case as occur for images, the definition of the 2D-HT is not unique and several approaches to it have been developed, having as one of the main goals to obtain a meaningful 2D-AS or analytic image, which can be used for various practical applications. In this work, one particular approach to the 2D-HT is introduced that allowed the calculation of analytic images which satisfy the basic properties that these functions have in the 1D case, and that produces a 2D spectrum equal to zero in one quadrant. The methods for calculation of the discrete version of the 2D-HT and the associated AS are presented and analyzed, as well as two applications, for edge detection and for envelope detection in a 2D AM modulated radial chirp.

Keywords

Edge Detection Support Region Hilbert Transform Image Processing Application Instantaneous Amplitude 
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.
    Gabor, D.: Theory of Communication. Journal of the IEE 93, 429–457 (1946)Google Scholar
  2. 2.
    Cohen, L.: Time Frequency Analysis. Prentice-Hall, Englewood Cliffs (1995)Google Scholar
  3. 3.
    Hahn, S.L.: Multidimensional complex signals with single-orthant spectra. Proc. IEEE 80(8), 1287–1300 (1992)CrossRefGoogle Scholar
  4. 4.
    Bülow, T., Sommer, G.: Multi-dimensional signal processing using an algebraically extended signal representation. In: Sommer, G. (ed.) AFPAC 1997. LNCS, vol. 1315, pp. 148–163. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  5. 5.
    Kohlmann, K.: Corner detection in natural images based on the 2D Hilbert Transform. Signal Processing 48, 225–234 (1996)MATHCrossRefGoogle Scholar
  6. 6.
    Havlicek, J.P., Havlicek, J.W., Mamuya, N.D., Bovik, A.C.: Skewed 2D Hilbert Transforms and AM-FM models. In: ICIP 1998. Proc. International conference on Image Processing, October 4-7, 1998, vol. 1, pp. 602–606 (1998)Google Scholar
  7. 7.
    Havlicek, J.P., Havlicek, J.W., Bovik, A.C.: The analytic image. In: ICIP 1997. Proc. International conference on Image Processing, October 26-29, 1997, vol. 2, pp. 446–449 (1997)Google Scholar
  8. 8.
    Kovesi, P.: Image Features from Phase Congruency. Videre Journal of Computer Vision Research, vol. 1(3). MIT Press, Cambridge (1999)Google Scholar
  9. 9.
    Pei, S.-C., Ding, J.-J.: The generalized radial Hilbert Transform and its applications to 2-D edge detection (any direction or specified directions). In: ICASSP ’03. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, April 6-10, 2003, vol. 3, pp. 357–360 (2003)Google Scholar
  10. 10.
    Felsberg, M., Sommer, G.: The Monogenic Signal. IEEE Transactions on Signal Processing 49(12), 3136–3144 (2001)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Larkin, K.G., Bone, D.J., Oldfield, M.A.: Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform. Journal of the Optical Society of America 18(8), 1862–1870 (2001)CrossRefGoogle Scholar
  12. 12.
    Lawrence Marple Jr., S.: Computing the discrete-time analytic signal via FFT. IEEE Transactions on Signal Processing 47(9), 2600–2603 (1999)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Juan Valentín Lorenzo-Ginori
    • 1
  1. 1.Center for Studies on Electronics and Information Technologies, Universidad Central de Las Villas, Carretera a Camajuaní, km 5 1/2, Santa Clara, VC, CP 54830Cuba

Personalised recommendations