Multidimensional Systems and Signal Processing

, Volume 2, Issue 4, pp 421–436

Gaussian wavelet transform: Two alternative fast implementations for images

  • Rafael Navarro
  • Antonio Tabernero


A series of schemes for pyramid multiresolution image coding has been proposed, all of them based on sets of orthogonal functions. Several of them are implementable in the spatial domain (such as wavelets), whereas others are more suitable for Fourier domain implementation (as for instance the cortex transform). Gabor functions have many important advantages, allowing easy and fast implementations in either domain, but are usually discarded by their lack of orthogonality which causes incomplete transforms. In this paper we quantify such effect, showing a Gaussian Wavelet Transform, GWT, withquasiorthogonal Gabor functions, which allows robust and efficient coding. Our particular GWT is based on a human visual model. Its incompleteness causes small amounts of reconstruction errors (due to small indentations in the MTF), which, however, are irrelevant under criteria based on visual perception.


Gaussian wavelets Gabor functions image coding completeness space and frequency domains implementations 


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  1. M.J. Bastiaans, “A sampling theorem for the complex spectrogram, and Gabor's expansion of a signal in Gaussian elementary signals,”Opt. Engineer. 20, pp. 594–598, 1981.Google Scholar
  2. P.L. Burt and E.H. Adelson, “The laplacian pyramid as a compact image code,”IEEE Transactions on Communications COM-31, pp. 532–540, 1983.Google Scholar
  3. F.W. Campbell and J.J. Kulikowski, “Orientation selectivity of the human visual system”J. of Physiology 187, pp. 437–445, 1966.Google Scholar
  4. J.G. Daugman, “Spatial visual channels in the Fourier plane,”Vision Research 24, pp. 891–910, 1984.Google Scholar
  5. J.G. Daugman, “Complete discrete 2-D Gabor transform by neural networks for image analysis and compression,”IEEE Transactions on Acoustic Speech and Signal Processing ASSP-36, pp. 1169–1169, 1988.Google Scholar
  6. R.L. De Valois, D.G. Albrecht, and L.G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,”Vision Research 22, pp. 545–559, 1982.Google Scholar
  7. D.J. Field, “Relation between the statistics of natural images and the response properties of cortical cells,”J. of the Optical Society of America A 4, pp. 2379–2394, 1987.Google Scholar
  8. E. Freysz, B. Pouligny, F. Argoul, and A. Arneodo, “Optical wavelet transform of fractal aggregates,”Phys. Rev. Lett. 64, pp. 745–748, 1990.Google Scholar
  9. D. Gabor, “Theory of communication,”J. Inst. Elect. Eng. 93, pp. 429–457, 1946.Google Scholar
  10. R.C. Gonzalez and P. Wintz,Digital Image Processing. Addison-Wesley: London, 1977.Google Scholar
  11. L.D. Jacobson and H. Wechsler, “Joint spatial/spatial-frequency representations,”Signal Processing 14, pp. 37–68, 1988.Google Scholar
  12. J. Jones and L. Palmer, “An evaluation of the two-dimensional Gabor filters model of simple receptive fields in cat striate cortex,”J. of Neurophysiology 58, pp. 538–539, 1987.Google Scholar
  13. S.G. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,”IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-11, pp. 674–693, 1989.Google Scholar
  14. S. Marcelja, “Mathematical description of the response of simple cortical cells,”J. of the Optical Society of America 70, pp. 1297–1300, 1980.Google Scholar
  15. J. Morlet, G. Arens, I. Forgeau, and D. Giard, “Wave propagation and sampling theory,”Geophysics 47, pp. 203–236, 1982.Google Scholar
  16. R. Navarro, J. Santamaria, and R. Gómez, “Automatic log spectrum restoration of atmospheric seeing,”Astronomy and Astrophysics 174, pp. 334–351, 1987.Google Scholar
  17. D.A. Pollen and S.F. Ronnen, “Visual cortical neurons as localized spatial filters,”IEEE Transactions on Systems, Man and Cybernetics SMC-13, pp. 284–302, 1983.Google Scholar
  18. M. Porat and Y.Y. Zeevi, “The generalized Gabor scheme for image representation in biological and machine vision,”IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-10, pp. 452–468, 1988.Google Scholar
  19. A. Rosenfeld,Multiresolution Image Processing and Analysis. Springer-Verlag: New York/Berlin, 1984.Google Scholar
  20. E.P. Simoncelli and E.H. Adelson, “Nonseparable extensions of quadrature mirror filters to multiple dimensions,”Proc. of the IEEE 78, pp. 652–664, 1990.Google Scholar
  21. A. Tabernero and R. Navarro, “Performance of Gabor functions for texture analysis,”IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI (submitted).Google Scholar
  22. A.B. Watson, “The cortex transform: rapid computation of simulated neural images,”Computer Vision, Graphics and Image Processing 39, pp. 311–327, 1987.Google Scholar
  23. A.B. Watson, “Perceptual-components architecture for digital video,”J. of the Optical Society of America A 7, pp. 1943–1954, 1990.Google Scholar
  24. J.W. Woods and S.D. O'Neal, “Subband coding of images,”IEEE Trans. on Acoustic Speach and Signal Procesing ASSP-34, pp. 1278–1288, 1986.Google Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Rafael Navarro
    • 1
  • Antonio Tabernero
    • 1
  1. 1.Instituto de Optica “Daza de Valdés” (CSIC)MadridSpain

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