Abstract

On the basis of measured receptive field profiles and spatial frequency tuning characteristics of simple cortical cells, it can be concluded that the representation of an image in the visual cortex must involve both spatial and spatial frequency variables. In a scheme due to Gabor, an image is represented in terms of localized symmetrical and antisymmetrical elementary signals. Both measured receptive fields and measured spatial frequency tuning curves conform closely to the functional form of Gabor elementary signals. It is argued that the visual cortex representation corresponds closely to the Gabor scheme owing to its advantages in treating the subsequent problem of pattern recognition.

© 1980 Optical Society of America

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  1. D. G. Hubel and T. N. Wiesel, "Receptive fields, binocular interaction, and functional architecture in the cat's visual cortex," J. Physiol. London 160, 106–154 (1962).
  2. P. O. Bishop, J. S. Coombs, and G. H. Henry, "Receptive fields of simple cells in the cat striate cortex," J. Physiol. London 231, 31–60 (1973).
  3. A. W. Goodwin, G. H. Henry, and P. O. Bishop, "Direction selectivity of simple striate cells: Properties and mechanism," J. Neurophysiol. 38, 1500–1523 (1975).
  4. G. F. Cooper and J. G. Robson, "Successive transformation of spatial information in the visual system," IEE/NPL Conference on Pattern Recognition, IEE Conf. Publ. London 42, 134–143 (1968).
  5. F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, "The spatial selectivity of the visual cells of the cat," J. Physiol. London 203, 223–235 (1969).
  6. F. W. Campbell and J. G. Robson, "Application of Fourier analysis to the visibility of gratings," J. Physiol. London 197, 551–566 (1968).
  7. L. Maffei and A. Fiorentini, "The visual cortex as a spatial frequency analyser," Vision Res. 13, 1255–1267 (1973).
  8. J. G. Robson, "Receptive fields: Neural representation of the spatial and intensive attributes of the visual image," in Handbook of Perception, edited by E. C. Carterette and M. P. Friedman (Academic, New York, 1975), Vol. V.
  9. D. Gabor, "Theory of communication," J. IEE London 93, 429–457 (1946).
  10. J. A. Movshon, I. D. Thompson, and 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).
  11. J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Receptive field organization of complex cells in the cat's striate cortex," J. Physiol. London 283, 79–99 (1978).
  12. L. Maffei, C. Morrone, M. Pirchio, and G. Sandini, "Responses of visual cortical cells to periodic and non-periodic stimuli," J. Physiol. London 296, 27–47 (1979).
  13. G. A. Orban, H. Kato, and P. O. Bishop, "End-zone region in receptive fields of hypercomplex and other striate neurons in the cat," J. Neurophysiol. 42, 818–832 (1979).
  14. C. W. Helstrom, "An expansion of a signal in Gaussian elementary signals," IEEE Trans. Inf. Theory IT-12, 81–82 (1966).
  15. L. K. Montgomery and I. S. Reed, "A generalization of the Gabor-Helstrom transform," IEEE Trans. Inf. Theory IT-13, 344–345 (1967).
  16. A. W. Rihaczek, "Signal energy distribution in time and frequency," IEEE Trans. Inf. Theory IT-14, 369–374(1968).
  17. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University, Princeton, 1955).
  18. R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766–2788 (1963).
  19. R. M. Lerner, "Representation of signals," in Lectures on Communication System Theory, edited by E. Baghdady (McGraw-Hill, New York, 1961).
  20. V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder, "On the completeness of the coherent states," Rep. Math. Phys. 2, 221–228 (1971).
  21. D. J. Tolhurst, "Symmetry and receptive fields," in Spatial Contrast, edited by H. Spekreijse and L. H. Van der Tweel (North-Holland, Amsterdam, 1977). The requirement of pure symmetric and pure antisymmetric elementary signals is not essential to the scheme. When the phase ø in Eq. (4) differs from zero, one obtains a pair of elementary signals that are a linear combination of the symmetric and antisymmetric Gabor signals. Such elementary signals may correspond to experimentally measured receptive fields that do not show perfect symmetry. I am grateful to J. J. Kulikowski, J. A. Movshon, and J. G. Robson for pointing out the need to accommodate receptive fields without perfect symmetry within the mathematical scheme.
  22. H. Ikeda and M. J. Wright, "Spatial and temporal properties of 'sustained' and 'transient' neurons in area 17 of the cat's visual cortex," Exp. Brain Res. 22, 363–383 (1975).
  23. B. W. Andrews and D. A. Pollen, "Relationship between spatial frequency selectivity and receptive field profile of simple cells," J. Physiol. London 287, 163–176 (1979).
  24. R. L. De Valois, D. G. Albrecht, and L. G. Thorell, "Cortical cells: Bar and edge detectors, or spatial frequency filters," in Frontiers in Visual Science, edited by S. J. Cool and E. L. Smith, III (Springer, New York, 1978).
  25. T. Y. Young and T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974).

1979

L. Maffei, C. Morrone, M. Pirchio, and G. Sandini, "Responses of visual cortical cells to periodic and non-periodic stimuli," J. Physiol. London 296, 27–47 (1979).

G. A. Orban, H. Kato, and P. O. Bishop, "End-zone region in receptive fields of hypercomplex and other striate neurons in the cat," J. Neurophysiol. 42, 818–832 (1979).

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

1978

J. A. Movshon, I. D. Thompson, and 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, I. D. Thompson, and D. J. Tolhurst, "Receptive field organization of complex cells in the cat's striate cortex," J. Physiol. London 283, 79–99 (1978).

1975

A. W. Goodwin, G. H. Henry, and P. O. Bishop, "Direction selectivity of simple striate cells: Properties and mechanism," J. Neurophysiol. 38, 1500–1523 (1975).

H. Ikeda and M. J. Wright, "Spatial and temporal properties of 'sustained' and 'transient' neurons in area 17 of the cat's visual cortex," Exp. Brain Res. 22, 363–383 (1975).

1973

P. O. Bishop, J. S. Coombs, and G. H. Henry, "Receptive fields of simple cells in the cat striate cortex," J. Physiol. London 231, 31–60 (1973).

L. Maffei and A. Fiorentini, "The visual cortex as a spatial frequency analyser," Vision Res. 13, 1255–1267 (1973).

1971

V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder, "On the completeness of the coherent states," Rep. Math. Phys. 2, 221–228 (1971).

1969

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, "The spatial selectivity of the visual cells of the cat," J. Physiol. London 203, 223–235 (1969).

1968

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

G. F. Cooper and J. G. Robson, "Successive transformation of spatial information in the visual system," IEE/NPL Conference on Pattern Recognition, IEE Conf. Publ. London 42, 134–143 (1968).

A. W. Rihaczek, "Signal energy distribution in time and frequency," IEEE Trans. Inf. Theory IT-14, 369–374(1968).

1967

L. K. Montgomery and I. S. Reed, "A generalization of the Gabor-Helstrom transform," IEEE Trans. Inf. Theory IT-13, 344–345 (1967).

1966

C. W. Helstrom, "An expansion of a signal in Gaussian elementary signals," IEEE Trans. Inf. Theory IT-12, 81–82 (1966).

1963

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766–2788 (1963).

1962

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

1946

D. Gabor, "Theory of communication," J. IEE London 93, 429–457 (1946).

Albrecht, D. G.

R. L. De Valois, D. G. Albrecht, and L. G. Thorell, "Cortical cells: Bar and edge detectors, or spatial frequency filters," in Frontiers in Visual Science, edited by S. J. Cool and E. L. Smith, III (Springer, New York, 1978).

Andrews, B. W.

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

Bargmann, V.

V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder, "On the completeness of the coherent states," Rep. Math. Phys. 2, 221–228 (1971).

Bishop, P. O.

G. A. Orban, H. Kato, and P. O. Bishop, "End-zone region in receptive fields of hypercomplex and other striate neurons in the cat," J. Neurophysiol. 42, 818–832 (1979).

A. W. Goodwin, G. H. Henry, and P. O. Bishop, "Direction selectivity of simple striate cells: Properties and mechanism," J. Neurophysiol. 38, 1500–1523 (1975).

P. O. Bishop, J. S. Coombs, and G. H. Henry, "Receptive fields of simple cells in the cat striate cortex," J. Physiol. London 231, 31–60 (1973).

Butera, P.

V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder, "On the completeness of the coherent states," Rep. Math. Phys. 2, 221–228 (1971).

Calvert, T. W.

T. Y. Young and T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974).

Campbell, F. W.

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, "The spatial selectivity of the visual cells of the cat," J. Physiol. London 203, 223–235 (1969).

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

Coombs, J. S.

P. O. Bishop, J. S. Coombs, and G. H. Henry, "Receptive fields of simple cells in the cat striate cortex," J. Physiol. London 231, 31–60 (1973).

Cooper, G. F.

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, "The spatial selectivity of the visual cells of the cat," J. Physiol. London 203, 223–235 (1969).

G. F. Cooper and J. G. Robson, "Successive transformation of spatial information in the visual system," IEE/NPL Conference on Pattern Recognition, IEE Conf. Publ. London 42, 134–143 (1968).

De Valois, R. L.

R. L. De Valois, D. G. Albrecht, and L. G. Thorell, "Cortical cells: Bar and edge detectors, or spatial frequency filters," in Frontiers in Visual Science, edited by S. J. Cool and E. L. Smith, III (Springer, New York, 1978).

Enroth-Cugell, C.

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, "The spatial selectivity of the visual cells of the cat," J. Physiol. London 203, 223–235 (1969).

Fiorentini, A.

L. Maffei and A. Fiorentini, "The visual cortex as a spatial frequency analyser," Vision Res. 13, 1255–1267 (1973).

Gabor, D.

D. Gabor, "Theory of communication," J. IEE London 93, 429–457 (1946).

Girardello, L.

V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder, "On the completeness of the coherent states," Rep. Math. Phys. 2, 221–228 (1971).

Glauber, R. J.

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766–2788 (1963).

Goodwin, A. W.

A. W. Goodwin, G. H. Henry, and P. O. Bishop, "Direction selectivity of simple striate cells: Properties and mechanism," J. Neurophysiol. 38, 1500–1523 (1975).

Helstrom, C. W.

C. W. Helstrom, "An expansion of a signal in Gaussian elementary signals," IEEE Trans. Inf. Theory IT-12, 81–82 (1966).

Henry, G. H.

A. W. Goodwin, G. H. Henry, and P. O. Bishop, "Direction selectivity of simple striate cells: Properties and mechanism," J. Neurophysiol. 38, 1500–1523 (1975).

P. O. Bishop, J. S. Coombs, and G. H. Henry, "Receptive fields of simple cells in the cat striate cortex," J. Physiol. London 231, 31–60 (1973).

Hubel, D. G.

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

Ikeda, H.

H. Ikeda and M. J. Wright, "Spatial and temporal properties of 'sustained' and 'transient' neurons in area 17 of the cat's visual cortex," Exp. Brain Res. 22, 363–383 (1975).

Kato, H.

G. A. Orban, H. Kato, and P. O. Bishop, "End-zone region in receptive fields of hypercomplex and other striate neurons in the cat," J. Neurophysiol. 42, 818–832 (1979).

Klauder, J. R.

V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder, "On the completeness of the coherent states," Rep. Math. Phys. 2, 221–228 (1971).

Lerner, R. M.

R. M. Lerner, "Representation of signals," in Lectures on Communication System Theory, edited by E. Baghdady (McGraw-Hill, New York, 1961).

Maffei, L.

L. Maffei, C. Morrone, M. Pirchio, and G. Sandini, "Responses of visual cortical cells to periodic and non-periodic stimuli," J. Physiol. London 296, 27–47 (1979).

L. Maffei and A. Fiorentini, "The visual cortex as a spatial frequency analyser," Vision Res. 13, 1255–1267 (1973).

Montgomery, L. K.

L. K. Montgomery and I. S. Reed, "A generalization of the Gabor-Helstrom transform," IEEE Trans. Inf. Theory IT-13, 344–345 (1967).

Morrone, C.

L. Maffei, C. Morrone, M. Pirchio, and G. Sandini, "Responses of visual cortical cells to periodic and non-periodic stimuli," J. Physiol. London 296, 27–47 (1979).

Movshon, J. A.

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Receptive field organization of complex cells in the cat's striate cortex," J. Physiol. London 283, 79–99 (1978).

J. A. Movshon, I. D. Thompson, and 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).

Neumann, J. von

J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University, Princeton, 1955).

Orban, G. A.

G. A. Orban, H. Kato, and P. O. Bishop, "End-zone region in receptive fields of hypercomplex and other striate neurons in the cat," J. Neurophysiol. 42, 818–832 (1979).

Pirchio, M.

L. Maffei, C. Morrone, M. Pirchio, and G. Sandini, "Responses of visual cortical cells to periodic and non-periodic stimuli," J. Physiol. London 296, 27–47 (1979).

Pollen, D. A.

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

Reed, I. S.

L. K. Montgomery and I. S. Reed, "A generalization of the Gabor-Helstrom transform," IEEE Trans. Inf. Theory IT-13, 344–345 (1967).

Rihaczek, A. W.

A. W. Rihaczek, "Signal energy distribution in time and frequency," IEEE Trans. Inf. Theory IT-14, 369–374(1968).

Robson, J. G.

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

G. F. Cooper and J. G. Robson, "Successive transformation of spatial information in the visual system," IEE/NPL Conference on Pattern Recognition, IEE Conf. Publ. London 42, 134–143 (1968).

J. G. Robson, "Receptive fields: Neural representation of the spatial and intensive attributes of the visual image," in Handbook of Perception, edited by E. C. Carterette and M. P. Friedman (Academic, New York, 1975), Vol. V.

Sandini, G.

L. Maffei, C. Morrone, M. Pirchio, and G. Sandini, "Responses of visual cortical cells to periodic and non-periodic stimuli," J. Physiol. London 296, 27–47 (1979).

Thompson, I. D.

J. A. Movshon, I. D. Thompson, and 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, I. D. Thompson, and D. J. Tolhurst, "Receptive field organization of complex cells in the cat's striate cortex," J. Physiol. London 283, 79–99 (1978).

Thorell, L. G.

R. L. De Valois, D. G. Albrecht, and L. G. Thorell, "Cortical cells: Bar and edge detectors, or spatial frequency filters," in Frontiers in Visual Science, edited by S. J. Cool and E. L. Smith, III (Springer, New York, 1978).

Tolhurst, D. J.

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Receptive field organization of complex cells in the cat's striate cortex," J. Physiol. London 283, 79–99 (1978).

J. A. Movshon, I. D. Thompson, and 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).

D. J. Tolhurst, "Symmetry and receptive fields," in Spatial Contrast, edited by H. Spekreijse and L. H. Van der Tweel (North-Holland, Amsterdam, 1977). The requirement of pure symmetric and pure antisymmetric elementary signals is not essential to the scheme. When the phase ø in Eq. (4) differs from zero, one obtains a pair of elementary signals that are a linear combination of the symmetric and antisymmetric Gabor signals. Such elementary signals may correspond to experimentally measured receptive fields that do not show perfect symmetry. I am grateful to J. J. Kulikowski, J. A. Movshon, and J. G. Robson for pointing out the need to accommodate receptive fields without perfect symmetry within the mathematical scheme.

Wiesel, T. N.

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

Wright, M. J.

H. Ikeda and M. J. Wright, "Spatial and temporal properties of 'sustained' and 'transient' neurons in area 17 of the cat's visual cortex," Exp. Brain Res. 22, 363–383 (1975).

Young, T. Y.

T. Y. Young and T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974).

Exp. Brain Res.

H. Ikeda and M. J. Wright, "Spatial and temporal properties of 'sustained' and 'transient' neurons in area 17 of the cat's visual cortex," Exp. Brain Res. 22, 363–383 (1975).

IEE Conf. Publ. London

G. F. Cooper and J. G. Robson, "Successive transformation of spatial information in the visual system," IEE/NPL Conference on Pattern Recognition, IEE Conf. Publ. London 42, 134–143 (1968).

IEEE Trans. Inf. Theory

C. W. Helstrom, "An expansion of a signal in Gaussian elementary signals," IEEE Trans. Inf. Theory IT-12, 81–82 (1966).

L. K. Montgomery and I. S. Reed, "A generalization of the Gabor-Helstrom transform," IEEE Trans. Inf. Theory IT-13, 344–345 (1967).

A. W. Rihaczek, "Signal energy distribution in time and frequency," IEEE Trans. Inf. Theory IT-14, 369–374(1968).

J. IEE London

D. Gabor, "Theory of communication," J. IEE London 93, 429–457 (1946).

J. Neurophysiol.

A. W. Goodwin, G. H. Henry, and P. O. Bishop, "Direction selectivity of simple striate cells: Properties and mechanism," J. Neurophysiol. 38, 1500–1523 (1975).

G. A. Orban, H. Kato, and P. O. Bishop, "End-zone region in receptive fields of hypercomplex and other striate neurons in the cat," J. Neurophysiol. 42, 818–832 (1979).

J. Physiol. London

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

P. O. Bishop, J. S. Coombs, and G. H. Henry, "Receptive fields of simple cells in the cat striate cortex," J. Physiol. London 231, 31–60 (1973).

J. A. Movshon, I. D. Thompson, and 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, I. D. Thompson, and D. J. Tolhurst, "Receptive field organization of complex cells in the cat's striate cortex," J. Physiol. London 283, 79–99 (1978).

L. Maffei, C. Morrone, M. Pirchio, and G. Sandini, "Responses of visual cortical cells to periodic and non-periodic stimuli," J. Physiol. London 296, 27–47 (1979).

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, "The spatial selectivity of the visual cells of the cat," J. Physiol. London 203, 223–235 (1969).

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

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

Phys. Rev.

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766–2788 (1963).

Rep. Math. Phys.

V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder, "On the completeness of the coherent states," Rep. Math. Phys. 2, 221–228 (1971).

Vision Res.

L. Maffei and A. Fiorentini, "The visual cortex as a spatial frequency analyser," Vision Res. 13, 1255–1267 (1973).

Other

J. G. Robson, "Receptive fields: Neural representation of the spatial and intensive attributes of the visual image," in Handbook of Perception, edited by E. C. Carterette and M. P. Friedman (Academic, New York, 1975), Vol. V.

J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University, Princeton, 1955).

D. J. Tolhurst, "Symmetry and receptive fields," in Spatial Contrast, edited by H. Spekreijse and L. H. Van der Tweel (North-Holland, Amsterdam, 1977). The requirement of pure symmetric and pure antisymmetric elementary signals is not essential to the scheme. When the phase ø in Eq. (4) differs from zero, one obtains a pair of elementary signals that are a linear combination of the symmetric and antisymmetric Gabor signals. Such elementary signals may correspond to experimentally measured receptive fields that do not show perfect symmetry. I am grateful to J. J. Kulikowski, J. A. Movshon, and J. G. Robson for pointing out the need to accommodate receptive fields without perfect symmetry within the mathematical scheme.

R. M. Lerner, "Representation of signals," in Lectures on Communication System Theory, edited by E. Baghdady (McGraw-Hill, New York, 1961).

R. L. De Valois, D. G. Albrecht, and L. G. Thorell, "Cortical cells: Bar and edge detectors, or spatial frequency filters," in Frontiers in Visual Science, edited by S. J. Cool and E. L. Smith, III (Springer, New York, 1978).

T. Y. Young and T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974).

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