Skip to main content

Texture analysis with Hermite basic elementary functions

  • Chapter
Book cover Wavelets and Their Applications

Part of the book series: NATO ASI Series ((ASIC,volume 442))

  • 279 Accesses

Abstract

Recently the use of Hermite functions in vision and image analysis has gained interest in the image processing community. Hermite functions are related to Gaussian functions through a differential operator. In the search for a suitable spatially localized, but not-redundant, alternative to the Gabor approach, Gaussian derivatives as basis functions were introduced. The literature reports that Hermite functions fit the profiles of many cells in the striate cortex very well. We will present some results on texture analysis using Hermite functions.

This research was supported by the USAF Armstrong Laboratory, Williams AFB, Arizona and USAF Contract F33615-90-C-0005 to the University of Dayton Research Institute. All the authors would like to thank Professor Norman Zabusky, head of the Visualization and Quantification Laboratory at Rutgers University, for supporting this project. We also thank Mr. Y. Tan and A. Feher for software support.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. L. Auslander and I. Gertner. Wide-band ambiguity function and ax + b group. In IMA Proceedings on Signal Processing, Minneapolis, 1990.

    Google Scholar 

  2. L. Auslander, I. Gertner, and R. Tolimieri. The discrete Zak transform application to time-frequency analysis and synthesis of non-stationary signals. IEEE Trans. Signal Processing, 39(4), April 1991.

    Google Scholar 

  3. J.W. Cooley and J.W. Tukey. An algorithm for the machine computation of complex Fourier series. Math. Comp., 19:297–301, April 1965.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.G. Daugman. Two-dimensional spectral analysis of cortical receptive field profiles. Vision Res., 20:847–856, 1980.

    Article  Google Scholar 

  5. J.G. Daugman. Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression. IEEE Trans. ASSP, 36:1169–1179, 1988.

    Article  MATH  Google Scholar 

  6. D. Gabor. Theory of communication. J. of IEEE, 93:429–457, November 1946.

    Google Scholar 

  7. I. Gertner. An alternative approach to nonlinear filtering. Stochastic Processes and Their Applications, 1976.

    Google Scholar 

  8. I. Gertner. A new efficient algorithm to compute the two dimensional discrete Fourier transform. IEEE Trans. ASSP, 36(7):1036–1050, 1988.

    Article  MATH  Google Scholar 

  9. I. Gertner and R. Tolimieri. The group theoretic approach to image representation. J. Visual Communication and Image Representation, 1(1):67–82, September 1990.

    Article  Google Scholar 

  10. I. Gertner and Y.Y. Zeevi. On the Zak-Gabor representation of images. In Proceedings Visual Communications and Image Processing’90, volume 1360, pages 1738–1748. SPIE, 1990.

    Google Scholar 

  11. I. Gertner and Y.Y. Zeevi. Image representation with position frequency localization. Proc. ICASSP, 1991.

    Google Scholar 

  12. A.J.E.M. Janssen. The Zak transform: A signal transform for sampled time-continuous signals. Philips J. of Research, 43:23–69, 1988.

    MATH  Google Scholar 

  13. J.P. Jones and L.A. Palmer. An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex. J. Neurophysiology, 58:1233–1258, 1987.

    Google Scholar 

  14. S.A. Klein and B. Beutter. Minimizing and maximizing the joint space-spatial frequency uncertainty of Gabor like functions. JOSA Communications, 9(2):337–340, February 1992.

    Article  Google Scholar 

  15. J.J. Koenderink. The structure of images. Biological Cybernetics, 50:363–370, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  16. J.J. Koenderink and A.J. van Doom. Receptive field families. Biological Cybernetics, 63:291–297, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  17. J.J. Kulikowski, S. Marcelja, and P.O. Bishop. Theory of spatial position and spatial frequency relations in the receptive fields of simple cells in the visual cortex. Biological Cybernetics, 43:187–198, 1982.

    Article  MATH  Google Scholar 

  18. S. Marcelja. Mathematical description of the responses of simple cortical cells. J. Opt. Soc. Amer., 70:1297–1300, 1980.

    Article  MathSciNet  Google Scholar 

  19. D. Marr. Vision. W.H. Freeman, San Francisco, New York, 1982.

    Google Scholar 

  20. D. Marr and E. Hildreth. Theory of edge detection. Proc. Roy. Soc. London, B207:187–217, 1980.

    Google Scholar 

  21. J.-B. Martens. The Hermite transform—theory. IEEE Trans. ASSP, 38:1595–1606, 1990.

    Article  MATH  Google Scholar 

  22. M. Porat and Y.Y. Zeevi. The generalized Gabor scheme of image representation. IEEE Trans. on Pattern Anal and Machine Intel., 10:452–468, 1988.

    Article  MATH  Google Scholar 

  23. M. Porat and Y.Y. Zeevi. The generalized Gabor scheme of image representation in biological and machine vision. IEEE Trans. on Pattern Anal. and Machine Intel., 10:452–468, 1989.

    Article  Google Scholar 

  24. R.W. Rodieck and J.S. Stone. Analysis of receptive fields of cat retinal ganglion cells. J. Neurophysiology, 28:965–980, 1965.

    Google Scholar 

  25. D.G. Stork and H.R. Wilson. Do Gabor functions provide appropriate descriptions of visual cortical receptive fields? J. Opt. Soc. Amer., 7:1362–1373, 1990.

    Article  Google Scholar 

  26. A.B. Watson. The cortex transform: Rapid computation of simulated neural images. Computer Vision, Graphics, and Image Processing, 39:311–327, 1987.

    Article  Google Scholar 

  27. G. Westheimer. Spatial frequency and light-spread descriptions of visual acuity and hyperacuity. J. Opt. Soc. Amer., 67:207–212, 1977.

    Article  Google Scholar 

  28. J. Yang. Do Gabor functions provide appropriate descriptions of visual cortical receptive fields?: comment. J. Opt. Soc. Amer., 9:334–336, 1992.

    Article  Google Scholar 

  29. R.A. Young. The Gaussian derivative model for spatial vision: I. Retinal mechanisms. Spatial Vision, 2:273–293, 1987.

    Article  Google Scholar 

  30. J. Zak. Finite translations in solid physics. Phys. Rev. Lett., 19:1385–1397, 1967.

    Article  Google Scholar 

  31. Y.Y. Zeevi and I. Gertner. The finite Zak transform: An efficient tool for image representation and analysis. J. Visual Communication and Image Representation, 3(1):13–23, March 1992.

    Article  Google Scholar 

  32. S.W. Zucker and R.A. Hummel. Receptive fields and the representation of visual information. Human Neurobiol., 5:121–128, 1986.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Gertner, I., Geri, G.A., Pierce, B. (1994). Texture analysis with Hermite basic elementary functions. In: Byrnes, J.S., Byrnes, J.L., Hargreaves, K.A., Berry, K. (eds) Wavelets and Their Applications. NATO ASI Series, vol 442. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1028-0_19

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-1028-0_19

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4448-6

  • Online ISBN: 978-94-011-1028-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics