Abstract

A modification and extension of Kortum and Geisler’s model [Vision Res. 35, 1595 (1995)] of early visual nonlinearities that incorporates an expansive nonlinearity (consistent with neurophysiological findings [Vision Res. 35, 2725 (1995)], a normalization based on a local average retinal illumination, similar to Mach’s proposal [F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, Calif., 1965)], and a subsequent compression suggested by Henning et al. [J. Opt. Soc. Am A 17, 1147 (2000)] captures a range of hitherto unexplained interactions between a sinusoidal grating of low spatial frequency and a contrast-modulated grating 2 octaves higher in spatial frequency.

© 2004 Optical Society of America

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References

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  1. F. Campbell, J. Robson, “Applications of Fourier analysis to the visibility of gratings,” J. Physiol. London 197, 551–566 (1968).
  2. C. Blakemore, F. Campbell, “Adaptation to spatial stimuli,” J. Physiol. London 200, 11P–13P (1969).
    [PubMed]
  3. G. Henning, B. Hertz, J. Hinton, “Effect of different hypothetical detection mechanisms on the shape of spatial-frequency filters inferred from masking experiments: noise masks,” J. Opt. Soc. Am. 71, 574–581 (1981).
    [CrossRef] [PubMed]
  4. G. Henning, “Spatial-frequency tuning as a function of temporal frequency and stimulus motion,” J. Opt. Soc. Am. A 5, 1362–1373 (1988).
    [CrossRef] [PubMed]
  5. C. Enroth-Cugell, J. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiology 258, 517–552 (1966).
  6. R. L. DeValois, R. E. Yund, N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531–544 (1982).
    [CrossRef]
  7. R. DeValois, D. Albrecht, L. Thorell, “Spatial frequencyselective cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
    [CrossRef]
  8. A. Derrington, G. Henning, “Some observations on the masking effects of two-dimensional stimuli,” Vision Res. 29, 241–246 (1989).
    [CrossRef] [PubMed]
  9. G. Henning, B. Hertz, D. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency,” Vision Res. 15, 887–897 (1975).
    [CrossRef] [PubMed]
  10. J. Nachmias, B. Rogowitz, “Masking by spatially modulated gratings,” Vision Res. 23, 1621–1630 (1983).
    [CrossRef]
  11. J. Nachmias, “Contrast modulated maskers: test of a late nonlinearity hypothesis,” Vision Res. 29, 137–142 (1989).
    [CrossRef] [PubMed]
  12. F. A. Wichmann, G. B. Henning, A. C. Ploghaus, “Non-linearities and the pedestal effect,” Perception 27, S86 (abstract) (1998).
  13. G. Henning, R. Millar, N. Hill, “Detection of incremental and decremental bars at different locations across Mach bands and related stimuli,” J. Opt. Soc. Am. A 17, 1147–1159 (2000).
    [CrossRef]
  14. S. Cropper, “Detection of chromatic and luminance amplitude modulation by the visual system,” J. Opt. Soc. Am. A 15, 1969–1986 (1998).
    [CrossRef]
  15. K.-I. Naka, W. Rushton, “S-potentials from luminosity units in the retina of fish (Cyprinicae),” J. Physiol. London 185, 587–599 (1966).
  16. D. Heeger, J. Nachmias, “A computer model of human retinal visual processing: effect of compressive nonlinearity on spatial frequency filters,” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1984), pp. 506–508.
  17. P. Kortum, W. S. Geisler, “Adaptation mechanisms in spatial vision II. Flash thresholds and background adaptation,” Vision Res. 35, 1595–1609 (1995).
    [CrossRef] [PubMed]
  18. W. S. Geisler, D. G. Albrecht, “Bayesian analysis of identification performance in monkey visual cortex: nonlinear mechanisms and stimulus certainty,” Vision Res. 35, 2723–2730 (1995).
    [CrossRef] [PubMed]
  19. I. Fine, “Mach bands,” undergraduate thesis (Oxford University, Oxford, UK, 1993).
  20. F. Van Nes, M. Bouman, “Spatial modulation transfer in the human eye,” J. Opt. Soc. Am. 57, 401–406 (1967).
    [CrossRef]
  21. R. DeValois, H. Morgan, D. Snodderly, “Psychophysical studies of monkey vision—III. Spatial luminance contrast sensitivity tests of macaque and human observers,” Vision Res. 14, 75–81 (1974).
    [CrossRef]
  22. E. Mach, “Uber die Wirkung der raumlichen Verteilung des Lichtreizes auf die Netzhaut,” translated in F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, 1965), pp. 45–79.
  23. D. Badcock, “Some aspects of ‘spatial phase’ coding in man,” D.Phil. (University of Oxford, Oxford, UK, 1983).
  24. D. Badcock, “Spatial phase or luminance profile discrimination?” Vision Res. 24, 613–623 (1984).
    [CrossRef] [PubMed]
  25. J. Nachmias, “How is a grating detected on a narrowband noise masker?” Vision Res. 39, 1133–1142 (1999).
    [CrossRef] [PubMed]
  26. T. Cornsweet, J. I. Yellott, “Intensity dependent spatial summation,” J. Opt. Soc. Am. A 2, 1769–1785 (1985).
    [CrossRef] [PubMed]
  27. T. Cornsweet, J. I. Yellott, “Intensity dependent spatial summation: errata,” J. Opt. Soc. Am. A 3, 165–165 (1986).
    [CrossRef]
  28. A. W. Freeman, D. Badcock, “Visual sensitivity in the presence of a patterned background,” J. Opt. Soc. Am. A 16, 979–986 (1999).
    [CrossRef]

2000 (1)

1999 (2)

J. Nachmias, “How is a grating detected on a narrowband noise masker?” Vision Res. 39, 1133–1142 (1999).
[CrossRef] [PubMed]

A. W. Freeman, D. Badcock, “Visual sensitivity in the presence of a patterned background,” J. Opt. Soc. Am. A 16, 979–986 (1999).
[CrossRef]

1998 (2)

S. Cropper, “Detection of chromatic and luminance amplitude modulation by the visual system,” J. Opt. Soc. Am. A 15, 1969–1986 (1998).
[CrossRef]

F. A. Wichmann, G. B. Henning, A. C. Ploghaus, “Non-linearities and the pedestal effect,” Perception 27, S86 (abstract) (1998).

1995 (2)

P. Kortum, W. S. Geisler, “Adaptation mechanisms in spatial vision II. Flash thresholds and background adaptation,” Vision Res. 35, 1595–1609 (1995).
[CrossRef] [PubMed]

W. S. Geisler, D. G. Albrecht, “Bayesian analysis of identification performance in monkey visual cortex: nonlinear mechanisms and stimulus certainty,” Vision Res. 35, 2723–2730 (1995).
[CrossRef] [PubMed]

1989 (2)

J. Nachmias, “Contrast modulated maskers: test of a late nonlinearity hypothesis,” Vision Res. 29, 137–142 (1989).
[CrossRef] [PubMed]

A. Derrington, G. Henning, “Some observations on the masking effects of two-dimensional stimuli,” Vision Res. 29, 241–246 (1989).
[CrossRef] [PubMed]

1988 (1)

1986 (1)

1985 (1)

1984 (1)

D. Badcock, “Spatial phase or luminance profile discrimination?” Vision Res. 24, 613–623 (1984).
[CrossRef] [PubMed]

1983 (1)

J. Nachmias, B. Rogowitz, “Masking by spatially modulated gratings,” Vision Res. 23, 1621–1630 (1983).
[CrossRef]

1982 (2)

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

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

1981 (1)

1975 (1)

G. Henning, B. Hertz, D. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency,” Vision Res. 15, 887–897 (1975).
[CrossRef] [PubMed]

1974 (1)

R. DeValois, H. Morgan, D. Snodderly, “Psychophysical studies of monkey vision—III. Spatial luminance contrast sensitivity tests of macaque and human observers,” Vision Res. 14, 75–81 (1974).
[CrossRef]

1969 (1)

C. Blakemore, F. Campbell, “Adaptation to spatial stimuli,” J. Physiol. London 200, 11P–13P (1969).
[PubMed]

1968 (1)

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

1967 (1)

1966 (2)

K.-I. Naka, W. Rushton, “S-potentials from luminosity units in the retina of fish (Cyprinicae),” J. Physiol. London 185, 587–599 (1966).

C. Enroth-Cugell, J. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiology 258, 517–552 (1966).

Albrecht, D.

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

Albrecht, D. G.

W. S. Geisler, D. G. Albrecht, “Bayesian analysis of identification performance in monkey visual cortex: nonlinear mechanisms and stimulus certainty,” Vision Res. 35, 2723–2730 (1995).
[CrossRef] [PubMed]

Badcock, D.

A. W. Freeman, D. Badcock, “Visual sensitivity in the presence of a patterned background,” J. Opt. Soc. Am. A 16, 979–986 (1999).
[CrossRef]

D. Badcock, “Spatial phase or luminance profile discrimination?” Vision Res. 24, 613–623 (1984).
[CrossRef] [PubMed]

D. Badcock, “Some aspects of ‘spatial phase’ coding in man,” D.Phil. (University of Oxford, Oxford, UK, 1983).

Blakemore, C.

C. Blakemore, F. Campbell, “Adaptation to spatial stimuli,” J. Physiol. London 200, 11P–13P (1969).
[PubMed]

Bouman, M.

Broadbent, D.

G. Henning, B. Hertz, D. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency,” Vision Res. 15, 887–897 (1975).
[CrossRef] [PubMed]

Campbell, F.

C. Blakemore, F. Campbell, “Adaptation to spatial stimuli,” J. Physiol. London 200, 11P–13P (1969).
[PubMed]

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

Cornsweet, T.

Cropper, S.

Derrington, A.

A. Derrington, G. Henning, “Some observations on the masking effects of two-dimensional stimuli,” Vision Res. 29, 241–246 (1989).
[CrossRef] [PubMed]

DeValois, R.

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

R. DeValois, H. Morgan, D. Snodderly, “Psychophysical studies of monkey vision—III. Spatial luminance contrast sensitivity tests of macaque and human observers,” Vision Res. 14, 75–81 (1974).
[CrossRef]

DeValois, R. L.

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

Enroth-Cugell, C.

C. Enroth-Cugell, J. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiology 258, 517–552 (1966).

Fine, I.

I. Fine, “Mach bands,” undergraduate thesis (Oxford University, Oxford, UK, 1993).

Freeman, A. W.

Geisler, W. S.

P. Kortum, W. S. Geisler, “Adaptation mechanisms in spatial vision II. Flash thresholds and background adaptation,” Vision Res. 35, 1595–1609 (1995).
[CrossRef] [PubMed]

W. S. Geisler, D. G. Albrecht, “Bayesian analysis of identification performance in monkey visual cortex: nonlinear mechanisms and stimulus certainty,” Vision Res. 35, 2723–2730 (1995).
[CrossRef] [PubMed]

Heeger, D.

D. Heeger, J. Nachmias, “A computer model of human retinal visual processing: effect of compressive nonlinearity on spatial frequency filters,” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1984), pp. 506–508.

Henning, G.

Henning, G. B.

F. A. Wichmann, G. B. Henning, A. C. Ploghaus, “Non-linearities and the pedestal effect,” Perception 27, S86 (abstract) (1998).

Hepler, N.

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

Hertz, B.

G. Henning, B. Hertz, J. Hinton, “Effect of different hypothetical detection mechanisms on the shape of spatial-frequency filters inferred from masking experiments: noise masks,” J. Opt. Soc. Am. 71, 574–581 (1981).
[CrossRef] [PubMed]

G. Henning, B. Hertz, D. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency,” Vision Res. 15, 887–897 (1975).
[CrossRef] [PubMed]

Hill, N.

Hinton, J.

Kortum, P.

P. Kortum, W. S. Geisler, “Adaptation mechanisms in spatial vision II. Flash thresholds and background adaptation,” Vision Res. 35, 1595–1609 (1995).
[CrossRef] [PubMed]

Mach, E.

E. Mach, “Uber die Wirkung der raumlichen Verteilung des Lichtreizes auf die Netzhaut,” translated in F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, 1965), pp. 45–79.

Millar, R.

Morgan, H.

R. DeValois, H. Morgan, D. Snodderly, “Psychophysical studies of monkey vision—III. Spatial luminance contrast sensitivity tests of macaque and human observers,” Vision Res. 14, 75–81 (1974).
[CrossRef]

Nachmias, J.

J. Nachmias, “How is a grating detected on a narrowband noise masker?” Vision Res. 39, 1133–1142 (1999).
[CrossRef] [PubMed]

J. Nachmias, “Contrast modulated maskers: test of a late nonlinearity hypothesis,” Vision Res. 29, 137–142 (1989).
[CrossRef] [PubMed]

J. Nachmias, B. Rogowitz, “Masking by spatially modulated gratings,” Vision Res. 23, 1621–1630 (1983).
[CrossRef]

D. Heeger, J. Nachmias, “A computer model of human retinal visual processing: effect of compressive nonlinearity on spatial frequency filters,” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1984), pp. 506–508.

Naka, K.-I.

K.-I. Naka, W. Rushton, “S-potentials from luminosity units in the retina of fish (Cyprinicae),” J. Physiol. London 185, 587–599 (1966).

Ploghaus, A. C.

F. A. Wichmann, G. B. Henning, A. C. Ploghaus, “Non-linearities and the pedestal effect,” Perception 27, S86 (abstract) (1998).

Robson, J.

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

C. Enroth-Cugell, J. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiology 258, 517–552 (1966).

Rogowitz, B.

J. Nachmias, B. Rogowitz, “Masking by spatially modulated gratings,” Vision Res. 23, 1621–1630 (1983).
[CrossRef]

Rushton, W.

K.-I. Naka, W. Rushton, “S-potentials from luminosity units in the retina of fish (Cyprinicae),” J. Physiol. London 185, 587–599 (1966).

Snodderly, D.

R. DeValois, H. Morgan, D. Snodderly, “Psychophysical studies of monkey vision—III. Spatial luminance contrast sensitivity tests of macaque and human observers,” Vision Res. 14, 75–81 (1974).
[CrossRef]

Thorell, L.

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

Van Nes, F.

Wichmann, F. A.

F. A. Wichmann, G. B. Henning, A. C. Ploghaus, “Non-linearities and the pedestal effect,” Perception 27, S86 (abstract) (1998).

Yellott, J. I.

Yund, R. E.

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

J. Physiol. London (3)

K.-I. Naka, W. Rushton, “S-potentials from luminosity units in the retina of fish (Cyprinicae),” J. Physiol. London 185, 587–599 (1966).

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

C. Blakemore, F. Campbell, “Adaptation to spatial stimuli,” J. Physiol. London 200, 11P–13P (1969).
[PubMed]

J. Physiology (1)

C. Enroth-Cugell, J. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiology 258, 517–552 (1966).

Perception (1)

F. A. Wichmann, G. B. Henning, A. C. Ploghaus, “Non-linearities and the pedestal effect,” Perception 27, S86 (abstract) (1998).

Vision Res. (11)

D. Badcock, “Spatial phase or luminance profile discrimination?” Vision Res. 24, 613–623 (1984).
[CrossRef] [PubMed]

J. Nachmias, “How is a grating detected on a narrowband noise masker?” Vision Res. 39, 1133–1142 (1999).
[CrossRef] [PubMed]

R. DeValois, H. Morgan, D. Snodderly, “Psychophysical studies of monkey vision—III. Spatial luminance contrast sensitivity tests of macaque and human observers,” Vision Res. 14, 75–81 (1974).
[CrossRef]

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

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

A. Derrington, G. Henning, “Some observations on the masking effects of two-dimensional stimuli,” Vision Res. 29, 241–246 (1989).
[CrossRef] [PubMed]

G. Henning, B. Hertz, D. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency,” Vision Res. 15, 887–897 (1975).
[CrossRef] [PubMed]

J. Nachmias, B. Rogowitz, “Masking by spatially modulated gratings,” Vision Res. 23, 1621–1630 (1983).
[CrossRef]

J. Nachmias, “Contrast modulated maskers: test of a late nonlinearity hypothesis,” Vision Res. 29, 137–142 (1989).
[CrossRef] [PubMed]

P. Kortum, W. S. Geisler, “Adaptation mechanisms in spatial vision II. Flash thresholds and background adaptation,” Vision Res. 35, 1595–1609 (1995).
[CrossRef] [PubMed]

W. S. Geisler, D. G. Albrecht, “Bayesian analysis of identification performance in monkey visual cortex: nonlinear mechanisms and stimulus certainty,” Vision Res. 35, 2723–2730 (1995).
[CrossRef] [PubMed]

Other (4)

I. Fine, “Mach bands,” undergraduate thesis (Oxford University, Oxford, UK, 1993).

D. Heeger, J. Nachmias, “A computer model of human retinal visual processing: effect of compressive nonlinearity on spatial frequency filters,” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1984), pp. 506–508.

E. Mach, “Uber die Wirkung der raumlichen Verteilung des Lichtreizes auf die Netzhaut,” translated in F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, 1965), pp. 45–79.

D. Badcock, “Some aspects of ‘spatial phase’ coding in man,” D.Phil. (University of Oxford, Oxford, UK, 1983).

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Figures (2)

Fig. 1
Fig. 1

A, cross-sectional luminance profile of a sinusoidal (carrier) grating [Eq. (1)] of spatial frequency 10 c/deg and a contrast of 6%. B, the same carrier when sinusoidally contrast modulated [Eq. (2)] at a rate of 2 c/deg to a depth of 17%. C, response of the nonlinear system [Eq. (3)] to a 2-c/deg sinusoid of 17.4% contrast. D and E, line-spread function and attenuation characteristic of the 10-c/deg channel that characterizes human spatial-frequency selectivity at that frequency.3

Fig. 2
Fig. 2

Responses to channels driven by the nonlinear system of Eq. (3). A, response of a 10-c/deg channel to an unmodulated carrier (thin black curve) and to a carrier of the same spatial frequency the contrast of which was sinsoidally modulated at a rate of 2 c/deg and having the same phase as the contrast modulation. B, response to the same two stimuli in the presence of a 2-c/deg sinusoidal masking stimulus with a contrast of 17.4% added in the same phase as the contrast modulation. C, same as B save that the 2-c/deg masker was added 90° out of phase with the modulation. D and E, response to a channel at 2 c/deg to stimuli described in the text.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

Lc(x)=L[1+c sin(2πfcx+ϕc)],
Lcm(x)=L{1+c[1+m cos(2πfmx+ϕm)]×sin(2πfcx+ϕc)},
R[I(x)]=Rmax[I(x)]n[I(x)]n+αn.
R[I(x)]=Rmax{m[I(x)-s]}n{m[Iav(x)-s]}n+αn,

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