Abstract

The sensitivity of the visual system depends on ambient illumination: Sensitivity is reduced in the presence of a bright, uniform background. We asked how sensitivity is adjusted when the background is spatially detailed and therefore contains both luminance peaks and troughs in the neighborhood of a foreground object. A test flash was superimposed on a static sinusoidal grating. As the grating’s spatial frequency increased, sensitivity for flash detection declined, regardless of whether the flash was superimposed on a peak or a trough of the grating. We studied the mechanisms underlying this loss of sensitivity by delivering the test stimulus through one eye and the background through the other. The conclusion is that three mechanisms are involved. Luminance adaptation and a masking process adjust sensitivity at low- and mid-range spatial frequencies, respectively. The third mechanism, a contrast gain control, is localized (it occurs at spatial frequencies approaching the limit for resolution) and fast (complete in half a second), and it results from early processing in the visual pathway (it is absent during dichoptic viewing). This local adjustment of sensitivity may help to protect the clarity of even the smallest details in the visual scene.

© 1999 Optical Society of America

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References

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1998 (1)

H. R. Wilson, J. Kim, “Dynamics of a divisive gain control in human vision,” Vision Res. 38, 2735–2741 (1998).
[CrossRef] [PubMed]

1997 (1)

1996 (1)

A. W. Freeman, D. R. Badcock, “Visual adaptation is highly localised in the human retina,” Invest. Ophthalmol. Visual Sci. Suppl. 37, S726 (1996).

1995 (1)

A. W. Freeman, D. R. Badcock, V. A. Nguyen, K. C. McGuren, “The spatial spread of visual adaptation,” Austr. J. Psychol. 47, 8 (1995).

1994 (1)

1993 (2)

S. J. Waugh, D. M. Levi, T. Carney, “Orientation, masking, and vernier acuity for line targets,” Vision Res. 33, 1619–1638 (1993).
[CrossRef] [PubMed]

D. I. A. MacLeod, S. He, “Visible flicker from invisible patterns,” Nature (London) 361, 256–258 (1993).
[CrossRef]

1992 (2)

D. J. Heeger, “Normalization of cell responses in cat visual cortex,” Visual Neurosci. 9, 181–197 (1992).
[CrossRef]

E. A. Benardete, E. Kaplan, B. W. Knight, “Contrast gain control in the primate retina: P cells are not X-like, some M cells are,” Visual Neurosci. 8, 483–486 (1992).
[CrossRef]

1991 (1)

A. W. Freeman, “Spatial characteristics of the contrast gain control in the cat’s retina,” Vision Res. 31, 775–785 (1991).
[CrossRef]

1989 (2)

A. B. Bonds, “Role of inhibition in the specification of orientation selectivity of cells in the cat striate cortex,” Visual Neurosci. 2, 41–55 (1989).
[CrossRef]

G. Sclar, P. Lennie, D. D. DePriest, “Contrast adaptation in striate cortex of macaque,” Vision Res. 29, 747–755 (1989).
[CrossRef] [PubMed]

1986 (1)

A. M. Derrington, D. R. Badcock, “Detection of spatial beats: non-linearity or contrast increment detection?” Vision Res. 26, 343–348 (1986).
[CrossRef] [PubMed]

1985 (2)

I. Ohzawa, G. Sclar, R. D. Freeman, “Contrast gain control in the cat’s visual system,” J. Neurophysiol. 54, 651–667 (1985).
[PubMed]

D. C. Burr, J. Ross, M. C. Morrone, “Local regulation of luminance gain,” Vision Res. 25, 717–727 (1985).
[CrossRef] [PubMed]

1983 (2)

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1629 (1983).
[CrossRef] [PubMed]

1980 (1)

1979 (1)

J. A. Movshon, P. Lennie, “Pattern-selective adaptation in visual cortical neurones,” Nature (London) 278, 850–852 (1979).
[CrossRef]

1978 (1)

R. M. Shapley, J. D. Victor, “The effects of contrast on the transfer properties of cat retinal ganglion cells,” J. Physiol. (London) 285, 275–298 (1978).

1975 (1)

G. B. Henning, B. G. Hertz, D. E. 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]

1971 (1)

C. Blakemore, J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. (London) 213, 157–174 (1971).

1969 (1)

C. B. Blakemore, F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

1968 (1)

A. Pantle, R. Sekuler, “Size detecting mechanisms in human vision,” Science 162, 1146–1148 (1968).
[CrossRef] [PubMed]

1967 (1)

1966 (1)

C. Enroth-Cugell, J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. (London) 187, 517–552 (1966).

1965 (1)

W. A. H. Rushton, “Visual adaptation,” Proc. R. Soc. London, Ser. B 162, 20–46 (1965).
[CrossRef]

1960 (1)

1954 (1)

1947 (1)

B. H. Crawford, “Visual adaptation in relation to brief conditioning stimuli,” Proc. R. Soc. London Ser. B 134, 283–302 (1947).
[CrossRef]

Armington, J. C.

Badcock, D. R.

A. W. Freeman, D. R. Badcock, “Visual adaptation is highly localised in the human retina,” Invest. Ophthalmol. Visual Sci. Suppl. 37, S726 (1996).

A. W. Freeman, D. R. Badcock, V. A. Nguyen, K. C. McGuren, “The spatial spread of visual adaptation,” Austr. J. Psychol. 47, 8 (1995).

A. M. Derrington, D. R. Badcock, “Detection of spatial beats: non-linearity or contrast increment detection?” Vision Res. 26, 343–348 (1986).
[CrossRef] [PubMed]

Benardete, E. A.

E. A. Benardete, E. Kaplan, B. W. Knight, “Contrast gain control in the primate retina: P cells are not X-like, some M cells are,” Visual Neurosci. 8, 483–486 (1992).
[CrossRef]

Blakemore, C.

C. Blakemore, J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. (London) 213, 157–174 (1971).

Blakemore, C. B.

C. B. Blakemore, F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

Bonds, A. B.

A. B. Bonds, “Role of inhibition in the specification of orientation selectivity of cells in the cat striate cortex,” Visual Neurosci. 2, 41–55 (1989).
[CrossRef]

Bouman, M. A.

Broadbent, D. E.

G. B. Henning, B. G. Hertz, D. E. 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]

Burr, D. C.

D. C. Burr, J. Ross, M. C. Morrone, “Local regulation of luminance gain,” Vision Res. 25, 717–727 (1985).
[CrossRef] [PubMed]

Campbell, F. W.

C. B. Blakemore, F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

Carney, T.

S. J. Waugh, D. M. Levi, T. Carney, “Orientation, masking, and vernier acuity for line targets,” Vision Res. 33, 1619–1638 (1993).
[CrossRef] [PubMed]

Crawford, B. H.

B. H. Crawford, “Visual adaptation in relation to brief conditioning stimuli,” Proc. R. Soc. London Ser. B 134, 283–302 (1947).
[CrossRef]

DePriest, D. D.

G. Sclar, P. Lennie, D. D. DePriest, “Contrast adaptation in striate cortex of macaque,” Vision Res. 29, 747–755 (1989).
[CrossRef] [PubMed]

Derrington, A. M.

A. M. Derrington, D. R. Badcock, “Detection of spatial beats: non-linearity or contrast increment detection?” Vision Res. 26, 343–348 (1986).
[CrossRef] [PubMed]

Enroth-Cugell, C.

C. Enroth-Cugell, J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. (London) 187, 517–552 (1966).

R. M. Shapley, C. Enroth-Cugell, “Visual adaptation and retinal gain controls,” in Progress in Retinal Research, N. Osborne, G. Chader, eds. (Pergamon, Oxford, UK, 1984), pp. 263–346.

Foley, J. M.

Freeman, A. W.

A. W. Freeman, D. R. Badcock, “Visual adaptation is highly localised in the human retina,” Invest. Ophthalmol. Visual Sci. Suppl. 37, S726 (1996).

A. W. Freeman, D. R. Badcock, V. A. Nguyen, K. C. McGuren, “The spatial spread of visual adaptation,” Austr. J. Psychol. 47, 8 (1995).

A. W. Freeman, “Spatial characteristics of the contrast gain control in the cat’s retina,” Vision Res. 31, 775–785 (1991).
[CrossRef]

Freeman, R. D.

I. Ohzawa, G. Sclar, R. D. Freeman, “Contrast gain control in the cat’s visual system,” J. Neurophysiol. 54, 651–667 (1985).
[PubMed]

He, S.

D. I. A. MacLeod, S. He, “Visible flicker from invisible patterns,” Nature (London) 361, 256–258 (1993).
[CrossRef]

Heeger, D. J.

D. J. Heeger, “Normalization of cell responses in cat visual cortex,” Visual Neurosci. 9, 181–197 (1992).
[CrossRef]

Henning, G. B.

G. B. Henning, B. G. Hertz, D. E. 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]

Hertz, B. G.

G. B. Henning, B. G. Hertz, D. E. 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]

Kaplan, E.

E. A. Benardete, E. Kaplan, B. W. Knight, “Contrast gain control in the primate retina: P cells are not X-like, some M cells are,” Visual Neurosci. 8, 483–486 (1992).
[CrossRef]

Kim, J.

H. R. Wilson, J. Kim, “Dynamics of a divisive gain control in human vision,” Vision Res. 38, 2735–2741 (1998).
[CrossRef] [PubMed]

Knight, B. W.

E. A. Benardete, E. Kaplan, B. W. Knight, “Contrast gain control in the primate retina: P cells are not X-like, some M cells are,” Visual Neurosci. 8, 483–486 (1992).
[CrossRef]

Legge, G. E.

Lennie, P.

G. Sclar, P. Lennie, D. D. DePriest, “Contrast adaptation in striate cortex of macaque,” Vision Res. 29, 747–755 (1989).
[CrossRef] [PubMed]

J. A. Movshon, P. Lennie, “Pattern-selective adaptation in visual cortical neurones,” Nature (London) 278, 850–852 (1979).
[CrossRef]

Levi, D. M.

S. J. Waugh, D. M. Levi, T. Carney, “Orientation, masking, and vernier acuity for line targets,” Vision Res. 33, 1619–1638 (1993).
[CrossRef] [PubMed]

MacLeod, D. I. A.

D. I. A. MacLeod, S. He, “Visible flicker from invisible patterns,” Nature (London) 361, 256–258 (1993).
[CrossRef]

McFarlane, D. K.

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

McGuren, K. C.

A. W. Freeman, D. R. Badcock, V. A. Nguyen, K. C. McGuren, “The spatial spread of visual adaptation,” Austr. J. Psychol. 47, 8 (1995).

Morrone, M. C.

D. C. Burr, J. Ross, M. C. Morrone, “Local regulation of luminance gain,” Vision Res. 25, 717–727 (1985).
[CrossRef] [PubMed]

Movshon, J. A.

J. A. Movshon, P. Lennie, “Pattern-selective adaptation in visual cortical neurones,” Nature (London) 278, 850–852 (1979).
[CrossRef]

Nachmias, J.

J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1629 (1983).
[CrossRef] [PubMed]

C. Blakemore, J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. (London) 213, 157–174 (1971).

Neihl, E. W.

Nguyen, V. A.

A. W. Freeman, D. R. Badcock, V. A. Nguyen, K. C. McGuren, “The spatial spread of visual adaptation,” Austr. J. Psychol. 47, 8 (1995).

Ohzawa, I.

I. Ohzawa, G. Sclar, R. D. Freeman, “Contrast gain control in the cat’s visual system,” J. Neurophysiol. 54, 651–667 (1985).
[PubMed]

Pantle, A.

A. Pantle, R. Sekuler, “Size detecting mechanisms in human vision,” Science 162, 1146–1148 (1968).
[CrossRef] [PubMed]

Phillips, G. C.

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

Poot, L.

Ratliff, F.

Riggs, L. A.

Robson, J. G.

C. Enroth-Cugell, J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. (London) 187, 517–552 (1966).

Rogowitz, B. E.

J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1629 (1983).
[CrossRef] [PubMed]

Ross, J.

D. C. Burr, J. Ross, M. C. Morrone, “Local regulation of luminance gain,” Vision Res. 25, 717–727 (1985).
[CrossRef] [PubMed]

Rushton, W. A. H.

W. A. H. Rushton, “Visual adaptation,” Proc. R. Soc. London, Ser. B 162, 20–46 (1965).
[CrossRef]

Sclar, G.

G. Sclar, P. Lennie, D. D. DePriest, “Contrast adaptation in striate cortex of macaque,” Vision Res. 29, 747–755 (1989).
[CrossRef] [PubMed]

I. Ohzawa, G. Sclar, R. D. Freeman, “Contrast gain control in the cat’s visual system,” J. Neurophysiol. 54, 651–667 (1985).
[PubMed]

Sekuler, R.

A. Pantle, R. Sekuler, “Size detecting mechanisms in human vision,” Science 162, 1146–1148 (1968).
[CrossRef] [PubMed]

Shapley, R. M.

R. M. Shapley, J. D. Victor, “The effects of contrast on the transfer properties of cat retinal ganglion cells,” J. Physiol. (London) 285, 275–298 (1978).

R. M. Shapley, C. Enroth-Cugell, “Visual adaptation and retinal gain controls,” in Progress in Retinal Research, N. Osborne, G. Chader, eds. (Pergamon, Oxford, UK, 1984), pp. 263–346.

Snippe, H. P.

van Hateren, J. H.

van Nes, F. L.

Victor, J. D.

R. M. Shapley, J. D. Victor, “The effects of contrast on the transfer properties of cat retinal ganglion cells,” J. Physiol. (London) 285, 275–298 (1978).

Waugh, S. J.

S. J. Waugh, D. M. Levi, T. Carney, “Orientation, masking, and vernier acuity for line targets,” Vision Res. 33, 1619–1638 (1993).
[CrossRef] [PubMed]

Wilson, H. R.

H. R. Wilson, J. Kim, “Dynamics of a divisive gain control in human vision,” Vision Res. 38, 2735–2741 (1998).
[CrossRef] [PubMed]

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

Austr. J. Psychol. (1)

A. W. Freeman, D. R. Badcock, V. A. Nguyen, K. C. McGuren, “The spatial spread of visual adaptation,” Austr. J. Psychol. 47, 8 (1995).

Invest. Ophthalmol. Visual Sci. Suppl. (1)

A. W. Freeman, D. R. Badcock, “Visual adaptation is highly localised in the human retina,” Invest. Ophthalmol. Visual Sci. Suppl. 37, S726 (1996).

J. Neurophysiol. (1)

I. Ohzawa, G. Sclar, R. D. Freeman, “Contrast gain control in the cat’s visual system,” J. Neurophysiol. 54, 651–667 (1985).
[PubMed]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

J. Physiol. (London) (4)

R. M. Shapley, J. D. Victor, “The effects of contrast on the transfer properties of cat retinal ganglion cells,” J. Physiol. (London) 285, 275–298 (1978).

C. Enroth-Cugell, J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. (London) 187, 517–552 (1966).

C. Blakemore, J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. (London) 213, 157–174 (1971).

C. B. Blakemore, F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

Nature (London) (2)

D. I. A. MacLeod, S. He, “Visible flicker from invisible patterns,” Nature (London) 361, 256–258 (1993).
[CrossRef]

J. A. Movshon, P. Lennie, “Pattern-selective adaptation in visual cortical neurones,” Nature (London) 278, 850–852 (1979).
[CrossRef]

Proc. R. Soc. London Ser. B (1)

B. H. Crawford, “Visual adaptation in relation to brief conditioning stimuli,” Proc. R. Soc. London Ser. B 134, 283–302 (1947).
[CrossRef]

Proc. R. Soc. London, Ser. B (1)

W. A. H. Rushton, “Visual adaptation,” Proc. R. Soc. London, Ser. B 162, 20–46 (1965).
[CrossRef]

Science (1)

A. Pantle, R. Sekuler, “Size detecting mechanisms in human vision,” Science 162, 1146–1148 (1968).
[CrossRef] [PubMed]

Vision Res. (9)

A. W. Freeman, “Spatial characteristics of the contrast gain control in the cat’s retina,” Vision Res. 31, 775–785 (1991).
[CrossRef]

G. Sclar, P. Lennie, D. D. DePriest, “Contrast adaptation in striate cortex of macaque,” Vision Res. 29, 747–755 (1989).
[CrossRef] [PubMed]

H. R. Wilson, J. Kim, “Dynamics of a divisive gain control in human vision,” Vision Res. 38, 2735–2741 (1998).
[CrossRef] [PubMed]

S. J. Waugh, D. M. Levi, T. Carney, “Orientation, masking, and vernier acuity for line targets,” Vision Res. 33, 1619–1638 (1993).
[CrossRef] [PubMed]

D. C. Burr, J. Ross, M. C. Morrone, “Local regulation of luminance gain,” Vision Res. 25, 717–727 (1985).
[CrossRef] [PubMed]

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

G. B. Henning, B. G. Hertz, D. E. 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. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1629 (1983).
[CrossRef] [PubMed]

A. M. Derrington, D. R. Badcock, “Detection of spatial beats: non-linearity or contrast increment detection?” Vision Res. 26, 343–348 (1986).
[CrossRef] [PubMed]

Visual Neurosci. (3)

A. B. Bonds, “Role of inhibition in the specification of orientation selectivity of cells in the cat striate cortex,” Visual Neurosci. 2, 41–55 (1989).
[CrossRef]

D. J. Heeger, “Normalization of cell responses in cat visual cortex,” Visual Neurosci. 9, 181–197 (1992).
[CrossRef]

E. A. Benardete, E. Kaplan, B. W. Knight, “Contrast gain control in the primate retina: P cells are not X-like, some M cells are,” Visual Neurosci. 8, 483–486 (1992).
[CrossRef]

Other (1)

R. M. Shapley, C. Enroth-Cugell, “Visual adaptation and retinal gain controls,” in Progress in Retinal Research, N. Osborne, G. Chader, eds. (Pergamon, Oxford, UK, 1984), pp. 263–346.

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Figures (6)

Fig. 1
Fig. 1

Stimulus spatial layout (left and center) and timing (right). The stimulus comprised a steady sinusoidal grating and a brief test stimulus superimposed on the grating. The test stimulus was presented with either its left or its right edge at the vertical midline (only the latter is shown). The subject’s task was to signal the side on which the test appeared.

Fig. 2
Fig. 2

Contrast sensitivity to a flashed test stimulus superimposed on the trough or the peak of a background grating. Data for one subject (KM) are shown. The dashed line shows the contrast sensitivity obtained when the grating was absent. At low spatial frequencies of the grating (which result in an almost uniform change in background luminance), sensitivity varies in approximately inverse proportion to luminance. At the highest spatial frequencies, the effect of the background grating disappears.

Fig. 3
Fig. 3

Contrast sensitivities of subjects KM and VN for a test stimulus superimposed on a background grating. The test appeared either 0.5 or 1 s after grating onset. The test stimulus was centered on a peak (○) or a trough (●) of the grating. All values have been normalized by the sensitivity (shown by the dashed lines) obtained in the absence of a grating. Each error bar has a length equal to twice the standard error of the mean. Results for the two onset asynchronies are very similar, indicating that sensitivity changes are complete in half a second.

Fig. 4
Fig. 4

Contrast sensitivities of subjects KM and VN for a test stimulus presented to the right eye and a background grating to the left. The test appeared either 0.5 or 1 s after grating onset. In the fused image, the test appeared centered on a peak (○) or a trough (●) of the grating. Sensitivities have been normalized by the value (shown by the dashed lines) obtained in the absence of the grating. Each error bar has a length equal to twice the standard error of the mean. A loss of sensitivity is found only at those spatial frequencies for which the test stimulus can be confused with a grating bar.

Fig. 5
Fig. 5

Contrast sensitivities reflecting responses in the monocular portions of the visual pathway. Data are shown for subjects KM and VN stimulated with a test stimulus superimposed on a peak (○) or a trough (●) of a background grating. The test appeared either 0.5 s or 1 s after grating onset; data have been averaged across these two onset asynchronies. Sensitivities have been normalized by the value (shown by the dashed lines) obtained in the absence of the grating. Sensitivities have also been divided by the value obtained with dichoptic stimulation; the data therefore reflect sensitivity changes that occur before the combination of information from the two eyes. The length of the error bars is 2 standard errors of the mean. The results show that the background grating influences monocular responses for frequencies at least as high as 20 cycles/deg.

Fig. 6
Fig. 6

Decomposition of the results into components due to three mechanisms. C shows the predictions of the contrast gain control model described in Appendix A. Predictions are shown for each of the subjects KM and VN. A shows the changes in sensitivity that are due to luminance adaptation; it was obtained by dividing the monocular response (Fig. 5) by the model predictions in C. B gives sensitivity changes that are due to binocular portions of the visual pathway. It was computed from the results of dichoptic stimulation (Fig. 4) by averaging across the 0.5- and 1-s onset asynchronies. Error bars have a length equal to 2 standard errors of the mean. The three mechanisms shown in A, B, and C have their peak effects at low, medium, and high spatial frequencies, respectively.

Equations (6)

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stimulusluminance=Lm[1+c cos(2πux+ϕ)],
Lmc[1-cos(2πud)]
Lmc{1-cos[minimum(2πud, π)]}.
cdrive=c exp[-(πau)2],
Lmc exp[-(πau)2]{1-cos[minimum(2πud, π)]}.
sensitivity=1/(1+k exp[-(πau)2]×{1-cos[minimum(2πud, π)]}),

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