Abstract

Two-dimensional spatial linear filters are constrained by general uncertainty relations that limit their attainable information resolution for orientation, spatial frequency, and two-dimensional (2D) spatial position. The theoretical lower limit for the joint entropy, or uncertainty, of these variables is achieved by an optimal 2D filter family whose spatial weighting functions are generated by exponentiated bivariate second-order polynomials with complex coefficients, the elliptic generalization of the one-dimensional elementary functions proposed in Gabor’s famous theory of communication [ J. Inst. Electr. Eng. 93, 429 ( 1946)]. The set includes filters with various orientation bandwidths, spatial-frequency bandwidths, and spatial dimensions, favoring the extraction of various kinds of information from an image. Each such filter occupies an irreducible quantal volume (corresponding to an independent datum) in a four-dimensional information hyperspace whose axes are interpretable as 2D visual space, orientation, and spatial frequency, and thus such a filter set could subserve an optimally efficient sampling of these variables. Evidence is presented that the 2D receptive-field profiles of simple cells in mammalian visual cortex are well described by members of this optimal 2D filter family, and thus such visual neurons could be said to optimize the general uncertainty relations for joint 2D-spatial–2D-spectral information resolution. The variety of their receptive-field dimensions and orientation and spatial-frequency bandwidths, and the correlations among these, reveal several underlying constraints, particularly in width/length aspect ratio and principal axis organization, suggesting a polar division of labor in occupying the quantal volumes of information hyperspace. Such an ensemble of 2D neural receptive fields in visual cortex could locally embed coarse polar mappings of the orientation–frequency plane piecewise within the global retinotopic mapping of visual space, thus efficiently representing 2D spatial visual information by localized 2D spectral signatures.

© 1985 Optical Society of America

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    [CrossRef] [PubMed]
  3. U. Neisser, Cognitive Psychology (Prentice-Hall, Englewood Cliffs, N.J., 1967).
  4. F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).
  5. D. A. Pollen, J. R. Lee, J. H. Taylor, “How does the striate cortex begin the reconstruction of the visual world?” Science 173, 74–77 (1971).
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  6. L. Maffei, A. Fiorentini, “The visual cortex as a spatial frequency analyzer,” Vision Res. 13, 1255–1267 (1973).
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  8. R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Cortical cells: bar and edge detectors, or spatial frequency filters,” in Frontiers of Visual Science, S. J. Cool, E. L. Smith, eds. (Springer-Verlag, New York, 1978).
  9. K. K. DeValois, R. L. DeValois, E. W. Yund, “Responses of striate cortical cells to grating and checkerboard patterns,” J. Physiol. (London) 291, 483–505 (1979).
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  12. D. G. Albrecht, R. L. DeValois, L. G. Thorell, “Visual cortical neurons: are bars or gratings the optimal stimuli?” Science 207, 88–90 (1981).
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    [CrossRef] [PubMed]
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  17. D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
    [CrossRef] [PubMed]
  18. B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybern. 43, 97–108 (1982).
    [CrossRef] [PubMed]
  19. J. J. Kulikowski, S. Marčelja, P. O. Bishop, “Theory of spatial position and spatial frequency relations in the receptive fields of simple cells in the visual cortex,” Biol. Cybern. 43, 187–198 (1982).
    [CrossRef] [PubMed]
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  22. J. A. Movshon, D. J. Tolhurst, “On the response linearity of neurons in cat visual cortex,” J. Physiol. (London) 249, 56P–57P (1975).
  23. B. W. Andrews, D. A. Pollen, “Relationship between spatial frequency selectivity and receptive field profile of simple cells,” J. Physiol. (London) 287, 163–176 (1979).
  24. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
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    [CrossRef] [PubMed]
  26. J. Wilson, S. Sherman, “Receptive field characteristics of neurones in cat striate cortex: changes with visual field eccentricity,” J. Neurophysiol. 39, 512–533 (1976).
    [PubMed]
  27. W. H. Mullikin, J. P. Jones, L. A. Palmer, “Periodic simple cells in cat area 17,” J. Neurophysiol. 52, 372–387 (1984).
    [PubMed]
  28. J. P. Jones, L. A. Palmer, J. G. Daugman, “Information management in the visual cortex,” Science (to be published).
  29. R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
    [CrossRef]
  30. G. H. Henry, B. Dreher, P. O. Bishop, “Orientation selectivity of cells in cat striate cortex,” J. Neurophysiol. 37, 1394–1409 (1974).
    [PubMed]
  31. D. Rose, C. Blakemore, “An analysis of orientation selectivity in the cat’s visual cortex,” Exp. Brain Res. 20, 1–17 (1974).
    [CrossRef] [PubMed]
  32. D. W. Watkins, M. A. Berkley, “The orientation selectivity of single neurons in cat striate cortex,” Exp. Brain Res. 19, 433–446 (1974).
    [CrossRef] [PubMed]
  33. P. Heggelund, K. Albus, “Orientation selectivity of single cells in the striate cortex of cat: the shape of orientation tuning curves,” Vision Res. 18, 1067–1071 (1978).
    [CrossRef]
  34. D. G. Albrecht, L. G. Thorell, R. L. DeValois, “Spatial and temporal properties of receptive fields in monkey and cat visual cortex,” Society for Neuroscience, Abstracts (9th Annual Meeting) (Society for Neuroscience, Atlanta, Ga.1979), p. 775.
  35. R. L. DeValois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
    [CrossRef]
  36. J. A. Movshon, “Two-dimensional spatial frequency tuning of cat striate cortical neurons,” Society for Neuroscience, Abstracts (9th Annual Meeting) (Society for Neuroscience, Atlanta, Ga., 1979), p. 799.
  37. R. Tootell, M. Silverman, R. L. DeValois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
    [CrossRef] [PubMed]
  38. J. L. Flanagan, Speech Analysis, Synthesis, and Perception, 2nd ed. (Springer-Verlag, Berlin, 1972).
    [CrossRef]
  39. H. Stark, “Sampling theorems in polar coordinates,” J. Opt. Soc Am. 69, 1519–1525 (1979).
    [CrossRef]

1984 (1)

W. H. Mullikin, J. P. Jones, L. A. Palmer, “Periodic simple cells in cat area 17,” J. Neurophysiol. 52, 372–387 (1984).
[PubMed]

1982 (4)

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

R. L. DeValois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
[CrossRef]

B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybern. 43, 97–108 (1982).
[CrossRef] [PubMed]

J. J. Kulikowski, S. Marčelja, P. O. Bishop, “Theory of spatial position and spatial frequency relations in the receptive fields of simple cells in the visual cortex,” Biol. Cybern. 43, 187–198 (1982).
[CrossRef] [PubMed]

1981 (5)

D. G. Albrecht, R. L. DeValois, L. G. Thorell, “Visual cortical neurons: are bars or gratings the optimal stimuli?” Science 207, 88–90 (1981).
[CrossRef]

D. M. MacKay, “Strife over visual cortical function,” Nature 289, 117–118 (1981).
[CrossRef] [PubMed]

J. J. Kulikowski, P. O. Bishop, “Fourier analysis and spatial representation in the visual cortex,” Experientia 37, 160–163 (1981).
[CrossRef] [PubMed]

R. Tootell, M. Silverman, R. L. DeValois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
[CrossRef] [PubMed]

D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
[CrossRef] [PubMed]

1980 (2)

S. Marčelja, “Mathematical description of the responses of simple cortical cells,” J. Opt. Soc. Am. 70, 1297–1300 (1980).
[CrossRef]

J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 847–856 (1980).
[CrossRef] [PubMed]

1979 (3)

K. K. DeValois, R. L. DeValois, E. W. Yund, “Responses of striate cortical cells to grating and checkerboard patterns,” J. Physiol. (London) 291, 483–505 (1979).

B. W. Andrews, D. A. Pollen, “Relationship between spatial frequency selectivity and receptive field profile of simple cells,” J. Physiol. (London) 287, 163–176 (1979).

H. Stark, “Sampling theorems in polar coordinates,” J. Opt. Soc Am. 69, 1519–1525 (1979).
[CrossRef]

1978 (3)

P. Heggelund, K. Albus, “Orientation selectivity of single cells in the striate cortex of cat: the shape of orientation tuning curves,” Vision Res. 18, 1067–1071 (1978).
[CrossRef]

C. W. Tyler, “Selectivity for spatial frequency and bar width in cat visual cortex,” Vision Res. 18, 121–122 (1978).
[CrossRef] [PubMed]

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Spatial summation in the receptive fields of simple cells in the cat’s striate cortex,” J. Physiol. (London) 283, 53–77 (1978).

1976 (1)

J. Wilson, S. Sherman, “Receptive field characteristics of neurones in cat striate cortex: changes with visual field eccentricity,” J. Neurophysiol. 39, 512–533 (1976).
[PubMed]

1975 (1)

J. A. Movshon, D. J. Tolhurst, “On the response linearity of neurons in cat visual cortex,” J. Physiol. (London) 249, 56P–57P (1975).

1974 (5)

I. D. G. Macleod, A. Rosenfeld, “The visibility of gratings: spatial frequency channels or bar-detecting units?” Vision Res. 14, 909–915 (1974).
[CrossRef] [PubMed]

D. G. Hubel, T. N. Wiesel, “Sequence regularity and geometry of orientation columns in the monkey striate cortex,” J. Comp. Neurol. 158, 267–293 (1974).
[CrossRef] [PubMed]

G. H. Henry, B. Dreher, P. O. Bishop, “Orientation selectivity of cells in cat striate cortex,” J. Neurophysiol. 37, 1394–1409 (1974).
[PubMed]

D. Rose, C. Blakemore, “An analysis of orientation selectivity in the cat’s visual cortex,” Exp. Brain Res. 20, 1–17 (1974).
[CrossRef] [PubMed]

D. W. Watkins, M. A. Berkley, “The orientation selectivity of single neurons in cat striate cortex,” Exp. Brain Res. 19, 433–446 (1974).
[CrossRef] [PubMed]

1973 (1)

L. Maffei, A. Fiorentini, “The visual cortex as a spatial frequency analyzer,” Vision Res. 13, 1255–1267 (1973).
[CrossRef] [PubMed]

1971 (1)

D. A. Pollen, J. R. Lee, J. H. Taylor, “How does the striate cortex begin the reconstruction of the visual world?” Science 173, 74–77 (1971).
[CrossRef] [PubMed]

1968 (1)

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

1965 (1)

R. W. Rodieck, “Quantitative analysis of cat retinal ganglion cell response to visual stimuli,” Vision Res. 5, 583–601 (1965).
[CrossRef] [PubMed]

1962 (1)

D. G. Hubel, T. N. Wiesel, “Receptive fields, binocular interaction, and functional architecture in the cat’s visual cortex,” J. Physiol. (London) 160, 106–154 (1962).

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Albrecht, D. G.

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

D. G. Albrecht, R. L. DeValois, L. G. Thorell, “Visual cortical neurons: are bars or gratings the optimal stimuli?” Science 207, 88–90 (1981).
[CrossRef]

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Cortical cells: bar and edge detectors, or spatial frequency filters,” in Frontiers of Visual Science, S. J. Cool, E. L. Smith, eds. (Springer-Verlag, New York, 1978).

D. G. Albrecht, L. G. Thorell, R. L. DeValois, “Spatial and temporal properties of receptive fields in monkey and cat visual cortex,” Society for Neuroscience, Abstracts (9th Annual Meeting) (Society for Neuroscience, Atlanta, Ga.1979), p. 775.

Albus, K.

P. Heggelund, K. Albus, “Orientation selectivity of single cells in the striate cortex of cat: the shape of orientation tuning curves,” Vision Res. 18, 1067–1071 (1978).
[CrossRef]

Andrews, B. W.

B. W. Andrews, D. A. Pollen, “Relationship between spatial frequency selectivity and receptive field profile of simple cells,” J. Physiol. (London) 287, 163–176 (1979).

Barlow, H. B.

B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybern. 43, 97–108 (1982).
[CrossRef] [PubMed]

Berkley, M. A.

D. W. Watkins, M. A. Berkley, “The orientation selectivity of single neurons in cat striate cortex,” Exp. Brain Res. 19, 433–446 (1974).
[CrossRef] [PubMed]

Bishop, P. O.

J. J. Kulikowski, S. Marčelja, P. O. Bishop, “Theory of spatial position and spatial frequency relations in the receptive fields of simple cells in the visual cortex,” Biol. Cybern. 43, 187–198 (1982).
[CrossRef] [PubMed]

J. J. Kulikowski, P. O. Bishop, “Fourier analysis and spatial representation in the visual cortex,” Experientia 37, 160–163 (1981).
[CrossRef] [PubMed]

G. H. Henry, B. Dreher, P. O. Bishop, “Orientation selectivity of cells in cat striate cortex,” J. Neurophysiol. 37, 1394–1409 (1974).
[PubMed]

Blakemore, C.

D. Rose, C. Blakemore, “An analysis of orientation selectivity in the cat’s visual cortex,” Exp. Brain Res. 20, 1–17 (1974).
[CrossRef] [PubMed]

Campbell, F. W.

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Daugman, J. G.

J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 847–856 (1980).
[CrossRef] [PubMed]

J. P. Jones, L. A. Palmer, J. G. Daugman, “Information management in the visual cortex,” Science (to be published).

DeValois, K. K.

K. K. DeValois, R. L. DeValois, E. W. Yund, “Responses of striate cortical cells to grating and checkerboard patterns,” J. Physiol. (London) 291, 483–505 (1979).

DeValois, R. L.

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

R. L. DeValois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
[CrossRef]

D. G. Albrecht, R. L. DeValois, L. G. Thorell, “Visual cortical neurons: are bars or gratings the optimal stimuli?” Science 207, 88–90 (1981).
[CrossRef]

R. Tootell, M. Silverman, R. L. DeValois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
[CrossRef] [PubMed]

K. K. DeValois, R. L. DeValois, E. W. Yund, “Responses of striate cortical cells to grating and checkerboard patterns,” J. Physiol. (London) 291, 483–505 (1979).

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Cortical cells: bar and edge detectors, or spatial frequency filters,” in Frontiers of Visual Science, S. J. Cool, E. L. Smith, eds. (Springer-Verlag, New York, 1978).

D. G. Albrecht, L. G. Thorell, R. L. DeValois, “Spatial and temporal properties of receptive fields in monkey and cat visual cortex,” Society for Neuroscience, Abstracts (9th Annual Meeting) (Society for Neuroscience, Atlanta, Ga.1979), p. 775.

Dreher, B.

G. H. Henry, B. Dreher, P. O. Bishop, “Orientation selectivity of cells in cat striate cortex,” J. Neurophysiol. 37, 1394–1409 (1974).
[PubMed]

Fiorentini, A.

L. Maffei, A. Fiorentini, “The visual cortex as a spatial frequency analyzer,” Vision Res. 13, 1255–1267 (1973).
[CrossRef] [PubMed]

Flanagan, J. L.

J. L. Flanagan, Speech Analysis, Synthesis, and Perception, 2nd ed. (Springer-Verlag, Berlin, 1972).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Heggelund, P.

P. Heggelund, K. Albus, “Orientation selectivity of single cells in the striate cortex of cat: the shape of orientation tuning curves,” Vision Res. 18, 1067–1071 (1978).
[CrossRef]

Henry, G. H.

G. H. Henry, B. Dreher, P. O. Bishop, “Orientation selectivity of cells in cat striate cortex,” J. Neurophysiol. 37, 1394–1409 (1974).
[PubMed]

Hepler, N.

R. L. DeValois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
[CrossRef]

Hubel, D. G.

D. G. Hubel, T. N. Wiesel, “Sequence regularity and geometry of orientation columns in the monkey striate cortex,” J. Comp. Neurol. 158, 267–293 (1974).
[CrossRef] [PubMed]

D. G. Hubel, T. N. Wiesel, “Receptive fields, binocular interaction, and functional architecture in the cat’s visual cortex,” J. Physiol. (London) 160, 106–154 (1962).

Jones, J. P.

W. H. Mullikin, J. P. Jones, L. A. Palmer, “Periodic simple cells in cat area 17,” J. Neurophysiol. 52, 372–387 (1984).
[PubMed]

J. P. Jones, L. A. Palmer, J. G. Daugman, “Information management in the visual cortex,” Science (to be published).

Kulikowski, J. J.

J. J. Kulikowski, S. Marčelja, P. O. Bishop, “Theory of spatial position and spatial frequency relations in the receptive fields of simple cells in the visual cortex,” Biol. Cybern. 43, 187–198 (1982).
[CrossRef] [PubMed]

J. J. Kulikowski, P. O. Bishop, “Fourier analysis and spatial representation in the visual cortex,” Experientia 37, 160–163 (1981).
[CrossRef] [PubMed]

Lee, J. R.

D. A. Pollen, J. R. Lee, J. H. Taylor, “How does the striate cortex begin the reconstruction of the visual world?” Science 173, 74–77 (1971).
[CrossRef] [PubMed]

MacKay, D. M.

D. M. MacKay, “Strife over visual cortical function,” Nature 289, 117–118 (1981).
[CrossRef] [PubMed]

Macleod, I. D. G.

I. D. G. Macleod, A. Rosenfeld, “The visibility of gratings: spatial frequency channels or bar-detecting units?” Vision Res. 14, 909–915 (1974).
[CrossRef] [PubMed]

Maffei, L.

L. Maffei, A. Fiorentini, “The visual cortex as a spatial frequency analyzer,” Vision Res. 13, 1255–1267 (1973).
[CrossRef] [PubMed]

Marcelja, S.

J. J. Kulikowski, S. Marčelja, P. O. Bishop, “Theory of spatial position and spatial frequency relations in the receptive fields of simple cells in the visual cortex,” Biol. Cybern. 43, 187–198 (1982).
[CrossRef] [PubMed]

S. Marčelja, “Mathematical description of the responses of simple cortical cells,” J. Opt. Soc. Am. 70, 1297–1300 (1980).
[CrossRef]

Movshon, J. A.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Spatial summation in the receptive fields of simple cells in the cat’s striate cortex,” J. Physiol. (London) 283, 53–77 (1978).

J. A. Movshon, D. J. Tolhurst, “On the response linearity of neurons in cat visual cortex,” J. Physiol. (London) 249, 56P–57P (1975).

J. A. Movshon, “Two-dimensional spatial frequency tuning of cat striate cortical neurons,” Society for Neuroscience, Abstracts (9th Annual Meeting) (Society for Neuroscience, Atlanta, Ga., 1979), p. 799.

Mullikin, W. H.

W. H. Mullikin, J. P. Jones, L. A. Palmer, “Periodic simple cells in cat area 17,” J. Neurophysiol. 52, 372–387 (1984).
[PubMed]

Neisser, U.

U. Neisser, Cognitive Psychology (Prentice-Hall, Englewood Cliffs, N.J., 1967).

Palmer, L. A.

W. H. Mullikin, J. P. Jones, L. A. Palmer, “Periodic simple cells in cat area 17,” J. Neurophysiol. 52, 372–387 (1984).
[PubMed]

J. P. Jones, L. A. Palmer, J. G. Daugman, “Information management in the visual cortex,” Science (to be published).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Pollen, D. A.

D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
[CrossRef] [PubMed]

B. W. Andrews, D. A. Pollen, “Relationship between spatial frequency selectivity and receptive field profile of simple cells,” J. Physiol. (London) 287, 163–176 (1979).

D. A. Pollen, J. R. Lee, J. H. Taylor, “How does the striate cortex begin the reconstruction of the visual world?” Science 173, 74–77 (1971).
[CrossRef] [PubMed]

D. A. Pollen, J. H. Taylor, “The striate cortex and the spatial analysis of visual space,” in The Neurosciences, Third Study Program, F. O. Schmitt, F. G. Worden, eds. (MIT, Cambridge, Mass.1974), pp. 239–247.

Robson, J. G.

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Rodieck, R. W.

R. W. Rodieck, “Quantitative analysis of cat retinal ganglion cell response to visual stimuli,” Vision Res. 5, 583–601 (1965).
[CrossRef] [PubMed]

Ronner, S. F.

D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
[CrossRef] [PubMed]

Rose, D.

D. Rose, C. Blakemore, “An analysis of orientation selectivity in the cat’s visual cortex,” Exp. Brain Res. 20, 1–17 (1974).
[CrossRef] [PubMed]

Rosenfeld, A.

I. D. G. Macleod, A. Rosenfeld, “The visibility of gratings: spatial frequency channels or bar-detecting units?” Vision Res. 14, 909–915 (1974).
[CrossRef] [PubMed]

Sakitt, B.

B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybern. 43, 97–108 (1982).
[CrossRef] [PubMed]

Sherman, S.

J. Wilson, S. Sherman, “Receptive field characteristics of neurones in cat striate cortex: changes with visual field eccentricity,” J. Neurophysiol. 39, 512–533 (1976).
[PubMed]

Silverman, M.

R. Tootell, M. Silverman, R. L. DeValois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
[CrossRef] [PubMed]

Stark, H.

H. Stark, “Sampling theorems in polar coordinates,” J. Opt. Soc Am. 69, 1519–1525 (1979).
[CrossRef]

Taylor, J. H.

D. A. Pollen, J. R. Lee, J. H. Taylor, “How does the striate cortex begin the reconstruction of the visual world?” Science 173, 74–77 (1971).
[CrossRef] [PubMed]

D. A. Pollen, J. H. Taylor, “The striate cortex and the spatial analysis of visual space,” in The Neurosciences, Third Study Program, F. O. Schmitt, F. G. Worden, eds. (MIT, Cambridge, Mass.1974), pp. 239–247.

Thompson, I. D.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Spatial summation in the receptive fields of simple cells in the cat’s striate cortex,” J. Physiol. (London) 283, 53–77 (1978).

Thorell, L. G.

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

D. G. Albrecht, R. L. DeValois, L. G. Thorell, “Visual cortical neurons: are bars or gratings the optimal stimuli?” Science 207, 88–90 (1981).
[CrossRef]

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Cortical cells: bar and edge detectors, or spatial frequency filters,” in Frontiers of Visual Science, S. J. Cool, E. L. Smith, eds. (Springer-Verlag, New York, 1978).

D. G. Albrecht, L. G. Thorell, R. L. DeValois, “Spatial and temporal properties of receptive fields in monkey and cat visual cortex,” Society for Neuroscience, Abstracts (9th Annual Meeting) (Society for Neuroscience, Atlanta, Ga.1979), p. 775.

Tolhurst, D. J.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Spatial summation in the receptive fields of simple cells in the cat’s striate cortex,” J. Physiol. (London) 283, 53–77 (1978).

J. A. Movshon, D. J. Tolhurst, “On the response linearity of neurons in cat visual cortex,” J. Physiol. (London) 249, 56P–57P (1975).

Tootell, R.

R. Tootell, M. Silverman, R. L. DeValois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
[CrossRef] [PubMed]

Tyler, C. W.

C. W. Tyler, “Selectivity for spatial frequency and bar width in cat visual cortex,” Vision Res. 18, 121–122 (1978).
[CrossRef] [PubMed]

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D. W. Watkins, M. A. Berkley, “The orientation selectivity of single neurons in cat striate cortex,” Exp. Brain Res. 19, 433–446 (1974).
[CrossRef] [PubMed]

Wiesel, T. N.

D. G. Hubel, T. N. Wiesel, “Sequence regularity and geometry of orientation columns in the monkey striate cortex,” J. Comp. Neurol. 158, 267–293 (1974).
[CrossRef] [PubMed]

D. G. Hubel, T. N. Wiesel, “Receptive fields, binocular interaction, and functional architecture in the cat’s visual cortex,” J. Physiol. (London) 160, 106–154 (1962).

Wilson, J.

J. Wilson, S. Sherman, “Receptive field characteristics of neurones in cat striate cortex: changes with visual field eccentricity,” J. Neurophysiol. 39, 512–533 (1976).
[PubMed]

Yund, E. W.

R. L. DeValois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
[CrossRef]

K. K. DeValois, R. L. DeValois, E. W. Yund, “Responses of striate cortical cells to grating and checkerboard patterns,” J. Physiol. (London) 291, 483–505 (1979).

Biol. Cybern. (2)

B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybern. 43, 97–108 (1982).
[CrossRef] [PubMed]

J. J. Kulikowski, S. Marčelja, P. O. Bishop, “Theory of spatial position and spatial frequency relations in the receptive fields of simple cells in the visual cortex,” Biol. Cybern. 43, 187–198 (1982).
[CrossRef] [PubMed]

Exp. Brain Res. (2)

D. Rose, C. Blakemore, “An analysis of orientation selectivity in the cat’s visual cortex,” Exp. Brain Res. 20, 1–17 (1974).
[CrossRef] [PubMed]

D. W. Watkins, M. A. Berkley, “The orientation selectivity of single neurons in cat striate cortex,” Exp. Brain Res. 19, 433–446 (1974).
[CrossRef] [PubMed]

Experientia (1)

J. J. Kulikowski, P. O. Bishop, “Fourier analysis and spatial representation in the visual cortex,” Experientia 37, 160–163 (1981).
[CrossRef] [PubMed]

J. Comp. Neurol. (1)

D. G. Hubel, T. N. Wiesel, “Sequence regularity and geometry of orientation columns in the monkey striate cortex,” J. Comp. Neurol. 158, 267–293 (1974).
[CrossRef] [PubMed]

J. Inst. Electr. Eng. (1)

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J. Wilson, S. Sherman, “Receptive field characteristics of neurones in cat striate cortex: changes with visual field eccentricity,” J. Neurophysiol. 39, 512–533 (1976).
[PubMed]

W. H. Mullikin, J. P. Jones, L. A. Palmer, “Periodic simple cells in cat area 17,” J. Neurophysiol. 52, 372–387 (1984).
[PubMed]

G. H. Henry, B. Dreher, P. O. Bishop, “Orientation selectivity of cells in cat striate cortex,” J. Neurophysiol. 37, 1394–1409 (1974).
[PubMed]

J. Opt. Soc Am. (1)

H. Stark, “Sampling theorems in polar coordinates,” J. Opt. Soc Am. 69, 1519–1525 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Physiol. (London) (6)

D. G. Hubel, T. N. Wiesel, “Receptive fields, binocular interaction, and functional architecture in the cat’s visual cortex,” J. Physiol. (London) 160, 106–154 (1962).

K. K. DeValois, R. L. DeValois, E. W. Yund, “Responses of striate cortical cells to grating and checkerboard patterns,” J. Physiol. (London) 291, 483–505 (1979).

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Spatial summation in the receptive fields of simple cells in the cat’s striate cortex,” J. Physiol. (London) 283, 53–77 (1978).

J. A. Movshon, D. J. Tolhurst, “On the response linearity of neurons in cat visual cortex,” J. Physiol. (London) 249, 56P–57P (1975).

B. W. Andrews, D. A. Pollen, “Relationship between spatial frequency selectivity and receptive field profile of simple cells,” J. Physiol. (London) 287, 163–176 (1979).

Nature (1)

D. M. MacKay, “Strife over visual cortical function,” Nature 289, 117–118 (1981).
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Science (4)

D. G. Albrecht, R. L. DeValois, L. G. Thorell, “Visual cortical neurons: are bars or gratings the optimal stimuli?” Science 207, 88–90 (1981).
[CrossRef]

D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
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D. A. Pollen, J. R. Lee, J. H. Taylor, “How does the striate cortex begin the reconstruction of the visual world?” Science 173, 74–77 (1971).
[CrossRef] [PubMed]

R. Tootell, M. Silverman, R. L. DeValois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
[CrossRef] [PubMed]

Vision Res. (8)

R. L. DeValois, E. W. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
[CrossRef]

P. Heggelund, K. Albus, “Orientation selectivity of single cells in the striate cortex of cat: the shape of orientation tuning curves,” Vision Res. 18, 1067–1071 (1978).
[CrossRef]

R. W. Rodieck, “Quantitative analysis of cat retinal ganglion cell response to visual stimuli,” Vision Res. 5, 583–601 (1965).
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R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
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L. Maffei, A. Fiorentini, “The visual cortex as a spatial frequency analyzer,” Vision Res. 13, 1255–1267 (1973).
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I. D. G. Macleod, A. Rosenfeld, “The visibility of gratings: spatial frequency channels or bar-detecting units?” Vision Res. 14, 909–915 (1974).
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C. W. Tyler, “Selectivity for spatial frequency and bar width in cat visual cortex,” Vision Res. 18, 121–122 (1978).
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J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 847–856 (1980).
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D. A. Pollen, J. H. Taylor, “The striate cortex and the spatial analysis of visual space,” in The Neurosciences, Third Study Program, F. O. Schmitt, F. G. Worden, eds. (MIT, Cambridge, Mass.1974), pp. 239–247.

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Cortical cells: bar and edge detectors, or spatial frequency filters,” in Frontiers of Visual Science, S. J. Cool, E. L. Smith, eds. (Springer-Verlag, New York, 1978).

J. P. Jones, L. A. Palmer, J. G. Daugman, “Information management in the visual cortex,” Science (to be published).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

D. G. Albrecht, L. G. Thorell, R. L. DeValois, “Spatial and temporal properties of receptive fields in monkey and cat visual cortex,” Society for Neuroscience, Abstracts (9th Annual Meeting) (Society for Neuroscience, Atlanta, Ga.1979), p. 775.

J. A. Movshon, “Two-dimensional spatial frequency tuning of cat striate cortical neurons,” Society for Neuroscience, Abstracts (9th Annual Meeting) (Society for Neuroscience, Atlanta, Ga., 1979), p. 799.

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[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

An even-symmetric member of the family of 2D Gabor filters, with unity aspect ratio, and its 2D Fourier transform. Members of this filter family generated by Eqs. (3) have the sharpest possible joint resolution of information in the two 2D domains. The number of significant sidelobes in the space-domain profile inversely determines the filter’s spatial-frequency bandwidth and orientation bandwidth; the spatial periodicity and orientation of the lobes specifies the filter’s preferred spatial frequency and orientation. Different members of this optimal filter family are an excellent description of the 2D neural receptive fields found in the visual cortex, as illustrated in Fig. 3. CPD, cycles/degree. (From Ref. 20.)

Fig. 2
Fig. 2

Bird’s-eye view of three members of the set of 2D Gabor optimal filters, all having the same preferred spatial frequency and orientation. The three pairs of panels illustrate the dependence of a filter’s spatial-frequency bandwidth and orientation bandwidth on its space-domain envelope dimensions; its preferred frequency and orientation are independent of those dimensions. A, A circular filter envelope in the space domain is supported in the frequency domain by the sum of two circular regions whose centers correspond to the filter’s modulation frequency and whose spatial-frequency bandwidth and orientation bandwidth are inversely related to the space-domain envelope diameter. B, Elongating the filter’s receptive field in the direction parallel to its modulation sharpens its orientation bandwidth Δθ½ but has no effect on its spatial-frequency bandwidth ΔF. C, Elongating the field instead in the perpendicular direction sharpens its spatial-frequency bandwidth ΔF but has no effect on its orientation bandwidth Δθ½. Thus such filters can negotiate the inescapable trade-offs for resolution in different ways, attaining, for example, sharp spatial resolution in the y direction (at the expense of orientation selectivity) or sharp spatial resolution in the x direction (at the expense of spatial-frequency selectivity). Such a division of labor among filters, or visual neurons, permits the extraction of differentially resolved spatial–spectral information from the image. Always, however, for 2D Gabor filters the product of the 2D resolutions in the two 2D domains is the same and equals the theoretically attainable limit.

Fig. 3
Fig. 3

Illustration of experimentally measured 2D receptive-field profiles of three simple cells in cat striate cortex (top row) obtained in the laboratory of L. A. Palmer and J. P. Jones (University of Pennsylvania Medical School). Each plot shows the excitatory or inhibitory effect of a small flashing light or dark spot on the firing rate of the cell, as a function of the (x, y) location of the stimulus, computed by reverse correlation of the 2D stimulus sequence with the neural-response sequence. The second row shows the best-fitting 2D Gabor function for each cell’s receptive-field profile, based on Eqs. (3) with the parameters fitted by least squares. The third row shows the residual error between the measured response profile of each cell and Eqs. (3). In formal statistical tests, the residuals were indistinguishable from random error for 33 of the 36 simple cells tested. (From Ref. 28.)

Tables (4)

Tables Icon

Table 1 Corresponding Filter Properties in Space and Spectral Domains

Tables Icon

Table 2 Orientation Bandwidths of Cat Cortical Simple Cells: Half-Width at Half-Response

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Table 3 Spatial-Frequency Bandwidths of Cat Cortical Simple Cells: Full Width at Half-Response

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Table 4 Predicted Correlation between Orientation Bandwidth and Spatial-Frequency Bandwidth for 2D Gabor Filters with 0.6 Width/Length Spatial Aspect Ratioa

Equations (12)

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( Δ x ) 2 = f f * x 2 d x f f * d x ,
x 2 f ( x , y ) f * ( x , y ) d x d y , y 2 f ( x , y ) f * ( x , y ) d x d y , xyf ( x , y ) f * ( x , y ) d x d y .
( Δ x ) ( Δ u ) = [ ( x x 0 ) 2 f ( x , y ) f * ( x , y ) d x d y f ( x , y ) f * ( x , y ) d x d y ] 1 / 2 × [ ( u u 0 ) 2 F ( u , υ ) F * ( u , υ ) d u d υ F ( u , υ ) F * ( u , υ ) d u d υ ] 1 / 2 1 4 π ,
( Δ y ) ( Δ υ ) = [ ( y y 0 ) 2 f ( x , y ) f * ( x , y ) d x d y f ( x , y ) f * ( x , y ) d x d y ] 1 / 2 × [ ( υ υ 0 ) 2 F ( u , υ ) F * ( u , υ ) d u d υ F ( u , υ ) F * ( u , υ ) d u d υ ] 1 / 2 1 4 π .
( Δ x ) ( Δ y ) ( Δ u ) ( Δ υ ) 1 / 16 π 2 .
f ( x , y ) = exp { π [ ( x x 0 ) 2 a 2 + ( y y 0 ) 2 b 2 ] } × exp { 2 π i [ u 0 ( x x 0 ) + υ 0 ( y y 0 ) ] } , F ( u , υ ) = exp { π [ ( u u 0 ) 2 / a 2 + ( υ υ 0 ) 2 / b 2 ] } × exp { 2 π i [ x 0 ( u u 0 ) + y 0 ( υ υ 0 ) ] } .
( Δ x ) = 1 2 a π , ( Δ y ) = 1 2 b π , ( Δ u ) = a 2 π , ( Δ υ ) = b 2 π .
exp [ ( A x 2 + B x y + C y 2 + D x + E y + F ) ] ,
( Δ υ ) = 2 ω 0 sin ( Δ θ 1 / 2 ) .
Δ ω = log 2 [ ω 0 + ( Δ u ) / 2 ω 0 ( Δ υ ) / 2 ] ,
( Δ u ) = 2 ω 0 [ 2 Δ ω 1 2 Δ ω + 1 ] .
Δ θ 1 / 2 = arcsin [ λ ( 2 Δ ω 1 ) ( 2 Δ ω + 1 ) ] .

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