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Image representation using Hermite functions

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Abstract

We present a mathematical technique for analyzing images based on two-dimensional Hermite functions that are translated in both space and spatial frequency. Although the translated functions are not orthogonal, they do constitute a frame and hence can be used for image expansion. The technique has the practical advantage that fast algorithms based on the Zak transform (ZT) can be used to compute expansion coefficients. We describe properties of the ZT that are relevant to image representation and which allow us to use it both to compute expansion coefficients efficiently and to reconstruct images from them. Finally, we use a Hermite function frame to decompose and reconstruct a texture image.

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References

  • Assaleh K, Zeevi YY, Gertner I (1991) On the realization of the Zak-Gabor representation of images. SPIE Proc Visual Comm Image Process, Boston

    Google Scholar 

  • Braddick O, Campbell FW, Atkinson J (1978) Channels in vision: basic aspects. (Handbook of sensory physiology, vol 8). Springer, Berlin New York pp 3–38

    Google Scholar 

  • Daubechies I (1990) The wavelet transform, time-frequency localization and signal analysis. IEEE Trans Inform Theory 36:961–1005

    Article  Google Scholar 

  • Daubechies I (1992) Ten lectures on wavelets. SIAM, Philadelphia, pp 53–105

    Google Scholar 

  • Daugman JG (1980) Two-dimensional spectral analysis of cortical receptive field profiles. Vision Res 20:847–856

    Article  PubMed  Google Scholar 

  • Daugman JG (1988) Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression. IEEE Trans Acoust Speech Signal Proc 36:1169–1179

    Article  Google Scholar 

  • Gabor D (1946) Theory of communication J IEE 93:429–459

    Google Scholar 

  • Gertner I, Zeevi YY (1990) On the Zak-Gabor representation of images. SPIE Proc Visual Comm Image Process 1360:1738–1748

    Google Scholar 

  • Jacobson LD, Wechsler H (1988) Joint spatial/spatial-frequency representation. Signal Proc 14:37–68

    Article  Google Scholar 

  • Janssen AJEM (1994) Signal analytic proofs of two basic results on lattice expansions. Appl Comput Harmonic Anal (in press)

  • Jones JP, Palmer LA (1987) An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex. J Neurophysiol 58:1233–1258

    PubMed  Google Scholar 

  • Klein SA, Beutter B (1992) Minimizing and maximizing the joint space-spatial frequency uncertainty of Gabor-like functions. J Opt Soc Am [A] 9:337–340

    Google Scholar 

  • Koenderink JJ (1984) The structure of images. Biol Cybern 50:363–370

    Article  PubMed  Google Scholar 

  • Koenderink JJ, van Doom AJ (1990) Receptive field families. Biol Cybern 63:291–297

    Article  Google Scholar 

  • Kulikowski JJ, Marcelja S, Bishop PO (1982) Theory of spatial position and spatial frequency relations in the receptive fields of simple cells in the visual cortex. Biol Cybern 43:187–198

    Article  PubMed  Google Scholar 

  • Lebedev NN (1972) Special functions and their applications. Dover, New York

    Google Scholar 

  • Marr D (1982) Vision Freeman, New York

  • Marr D, Hildreth E (1980) Theory of edge detection. Proc R Soc Lond [Biol] 207:187–217

    Google Scholar 

  • Marcelja S (1980) Mathematical description of the responses of simple cortical cells. J Opt Soc Am 70:1297–1300

    PubMed  Google Scholar 

  • Martens J-B (1990) The Hermite transform — theory. IEEE Trans Acoust Speech Signal Proc 38:1595–1606

    Article  Google Scholar 

  • Rodieck RW, Stone JS (1965) Analysis of receptive fields of cat retinal ganglion cells. J Neurophysiol 28:965–980

    Google Scholar 

  • Stork DG, Wilson HR (1990) Do Gabor functions provide appropriate descriptions of visual cortical receptive fields? J Opt Soc Am [A] 7:1362–1373

    Google Scholar 

  • Yang J (1992) Do Gabor functions provide appropriate descriptions of visual cortical receptive fields? Comment. J Opt Soc Am [A] 9:334–336

    Google Scholar 

  • Young RA (1987) The Gaussian derivative model for spatial vision. I. Retinal mechanisms. Spatial Vis 2:273–293

    PubMed  Google Scholar 

  • Zak J (1987) Finite translations in solid-state physics. Phys Rev Lett 19:1385–1397

    Article  Google Scholar 

  • Zeevi YY, Gertner I (1992) The finite Zak transform: an efficient tool for image representation and analysis. J Visual Comm Image Representation 3:13–23

    Google Scholar 

  • Zucker SW, Hummel RA (1986) Receptive fields and the representation of visual information. Hum Neurobiol 5:121–128

    PubMed  Google Scholar 

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Authors and Affiliations

  1. Computer Science Department, City College of CUNY, Convent Avenue and 138th Street, 10031, New York, NY, USA

    Izidor Gertner

  2. University of Dayton Research Institute, 85236-2020, Higley, AZ, USA

    George A. Geri

Authors
  1. Izidor Gertner
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  2. George A. Geri
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Gertner, I., Geri, G.A. Image representation using Hermite functions. Biol. Cybern. 71, 147–151 (1994). https://doi.org/10.1007/BF00197317

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  • DOI: https://doi.org/10.1007/BF00197317

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