Abstract

Some experimental studies of subthreshold summation between sinusoidal grating components have been interpreted as showing very narrow channel bandwidths in human visions. This paper discusses an alternative interpretation of these experiments based on consideration of probability-summation effects among spatially distributed detectors. We conclude that frequency-selective channels must still be hypothesized in order to fit the data, but the channel bandwidth may be much wider than earlier interpretations suggest.

© 1978 Optical Society of America

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References

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  1. F. W. Campbell and J. G. Robson, “Application of Fourier Analysis to the Visibility of Gratings,” J. Physiol. (London) 197, 551–556 (1968).
  2. C. Blakemore and 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).
  3. C. F. Stromeyer and B. Julesz, “Spatial-Frequency Masking in Vision: Critical Bands and Spread of Masking,” J. Opt. Soc. Am. 62, 1221–1232 (1972).
    [Crossref] [PubMed]
  4. M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-Frequency Channels in Human Vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
    [Crossref] [PubMed]
  5. J. J. Kulikowski and P. E. King-Smith, “Spatial Arrangement of Line, Edge, and Grating Detectors Revealed by Subthreshold Summation,” Vision Res. 13, 1455–1478 (1973).
    [Crossref] [PubMed]
  6. R. F. Quick and T. A. Reichert, “Spatial-Frequency Selectivity in Contrast Detection,” Vision Res. 15, 637–643 (1975).
    [Crossref] [PubMed]
  7. R. F. Quick, “A Vector-Magnitude Model of Contrast Detection,” Kybernetik 16, 65–67 (1974).
    [Crossref] [PubMed]
  8. C. F. Stromeyer and S. Klein, “Evidence Against Narrow-Band Spatial Frequency Channels in Human Vision: The Detectability of Frequency Modulated Gratings,” Vision Res. 15, 899–910 (1975).
    [Crossref] [PubMed]
  9. H. Mostafavi and D. J. Sakrison, “Structure and Properties of a Single Channel in the Human Visual System,” Vision Res. 16, 957–968 (1976).
    [Crossref] [PubMed]
  10. N. Graham, “Visual Detection of Aperiodic Spatial Stimuli by Probability Summation Among Multiple Channels,” Vision Res. (in press).
  11. The components actually used in the experiments were squarewaves; near threshold the harmonics have negligible effect (see Quick and Reichert6), so we have included only the fundamental components in this study.
  12. J. G. Robson, “Regional Variation of Contrast Sensitivity in the Visual Field,” presented to the Association for Research in Vision and Ophthalmology, April 1975 (unpublished).
  13. R. F. Quick, J. R. Hamerly, and T. A. Reichert, “The Absence of a Measurable ‘Critical Band’ at Low Suprathreshold Contrasts,” Vision Res. 16, 351–355 (1976).
    [Crossref]
  14. Strictly, the limit is the essential supremum of |f(x)|, which ignores pathological points on sets of zero measure. This does not matter if f(x) is continuous and of finite spatial extent.
  15. C. F. Stromeyer and S. Klein, “Spatial Frequency Channels in Human Vision as Asymmetric (Edge) Mechanisms,” Vision Res. 14, 1409–1420 (1974).
    [Crossref] [PubMed]
  16. We are aware that Dr. Stanley Klein obtained similar results in an earlier unpublished study. Apparently Klein’s results were obtained with a model similar to that of Stromeyer and Klein,8 which includes a spatial “pooling” mechanism in addition to probability summation. On the basis of the present study we do not believe that a pooling mechanism is necessary to explain our data.

1976 (2)

H. Mostafavi and D. J. Sakrison, “Structure and Properties of a Single Channel in the Human Visual System,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

R. F. Quick, J. R. Hamerly, and T. A. Reichert, “The Absence of a Measurable ‘Critical Band’ at Low Suprathreshold Contrasts,” Vision Res. 16, 351–355 (1976).
[Crossref]

1975 (2)

R. F. Quick and T. A. Reichert, “Spatial-Frequency Selectivity in Contrast Detection,” Vision Res. 15, 637–643 (1975).
[Crossref] [PubMed]

C. F. Stromeyer and S. Klein, “Evidence Against Narrow-Band Spatial Frequency Channels in Human Vision: The Detectability of Frequency Modulated Gratings,” Vision Res. 15, 899–910 (1975).
[Crossref] [PubMed]

1974 (2)

R. F. Quick, “A Vector-Magnitude Model of Contrast Detection,” Kybernetik 16, 65–67 (1974).
[Crossref] [PubMed]

C. F. Stromeyer and S. Klein, “Spatial Frequency Channels in Human Vision as Asymmetric (Edge) Mechanisms,” Vision Res. 14, 1409–1420 (1974).
[Crossref] [PubMed]

1973 (1)

J. J. Kulikowski and P. E. King-Smith, “Spatial Arrangement of Line, Edge, and Grating Detectors Revealed by Subthreshold Summation,” Vision Res. 13, 1455–1478 (1973).
[Crossref] [PubMed]

1972 (1)

1971 (1)

1969 (1)

C. Blakemore and 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)

F. W. Campbell and J. G. Robson, “Application of Fourier Analysis to the Visibility of Gratings,” J. Physiol. (London) 197, 551–556 (1968).

Blakemore, C.

C. Blakemore and 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).

Campbell, F. W.

C. Blakemore and 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).

F. W. Campbell and J. G. Robson, “Application of Fourier Analysis to the Visibility of Gratings,” J. Physiol. (London) 197, 551–556 (1968).

Graham, N.

N. Graham, “Visual Detection of Aperiodic Spatial Stimuli by Probability Summation Among Multiple Channels,” Vision Res. (in press).

Hamerly, J. R.

R. F. Quick, J. R. Hamerly, and T. A. Reichert, “The Absence of a Measurable ‘Critical Band’ at Low Suprathreshold Contrasts,” Vision Res. 16, 351–355 (1976).
[Crossref]

Julesz, B.

King-Smith, P. E.

J. J. Kulikowski and P. E. King-Smith, “Spatial Arrangement of Line, Edge, and Grating Detectors Revealed by Subthreshold Summation,” Vision Res. 13, 1455–1478 (1973).
[Crossref] [PubMed]

Klein, S.

C. F. Stromeyer and S. Klein, “Evidence Against Narrow-Band Spatial Frequency Channels in Human Vision: The Detectability of Frequency Modulated Gratings,” Vision Res. 15, 899–910 (1975).
[Crossref] [PubMed]

C. F. Stromeyer and S. Klein, “Spatial Frequency Channels in Human Vision as Asymmetric (Edge) Mechanisms,” Vision Res. 14, 1409–1420 (1974).
[Crossref] [PubMed]

Kulikowski, J. J.

J. J. Kulikowski and P. E. King-Smith, “Spatial Arrangement of Line, Edge, and Grating Detectors Revealed by Subthreshold Summation,” Vision Res. 13, 1455–1478 (1973).
[Crossref] [PubMed]

Mostafavi, H.

H. Mostafavi and D. J. Sakrison, “Structure and Properties of a Single Channel in the Human Visual System,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

Nachmias, J.

Quick, R. F.

R. F. Quick, J. R. Hamerly, and T. A. Reichert, “The Absence of a Measurable ‘Critical Band’ at Low Suprathreshold Contrasts,” Vision Res. 16, 351–355 (1976).
[Crossref]

R. F. Quick and T. A. Reichert, “Spatial-Frequency Selectivity in Contrast Detection,” Vision Res. 15, 637–643 (1975).
[Crossref] [PubMed]

R. F. Quick, “A Vector-Magnitude Model of Contrast Detection,” Kybernetik 16, 65–67 (1974).
[Crossref] [PubMed]

Reichert, T. A.

R. F. Quick, J. R. Hamerly, and T. A. Reichert, “The Absence of a Measurable ‘Critical Band’ at Low Suprathreshold Contrasts,” Vision Res. 16, 351–355 (1976).
[Crossref]

R. F. Quick and T. A. Reichert, “Spatial-Frequency Selectivity in Contrast Detection,” Vision Res. 15, 637–643 (1975).
[Crossref] [PubMed]

Robson, J. G.

M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-Frequency Channels in Human Vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
[Crossref] [PubMed]

F. W. Campbell and J. G. Robson, “Application of Fourier Analysis to the Visibility of Gratings,” J. Physiol. (London) 197, 551–556 (1968).

J. G. Robson, “Regional Variation of Contrast Sensitivity in the Visual Field,” presented to the Association for Research in Vision and Ophthalmology, April 1975 (unpublished).

Sachs, M. B.

Sakrison, D. J.

H. Mostafavi and D. J. Sakrison, “Structure and Properties of a Single Channel in the Human Visual System,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

Stromeyer, C. F.

C. F. Stromeyer and S. Klein, “Evidence Against Narrow-Band Spatial Frequency Channels in Human Vision: The Detectability of Frequency Modulated Gratings,” Vision Res. 15, 899–910 (1975).
[Crossref] [PubMed]

C. F. Stromeyer and S. Klein, “Spatial Frequency Channels in Human Vision as Asymmetric (Edge) Mechanisms,” Vision Res. 14, 1409–1420 (1974).
[Crossref] [PubMed]

C. F. Stromeyer and B. Julesz, “Spatial-Frequency Masking in Vision: Critical Bands and Spread of Masking,” J. Opt. Soc. Am. 62, 1221–1232 (1972).
[Crossref] [PubMed]

J. Opt. Soc. Am. (2)

J. Physiol. (London) (2)

F. W. Campbell and J. G. Robson, “Application of Fourier Analysis to the Visibility of Gratings,” J. Physiol. (London) 197, 551–556 (1968).

C. Blakemore and 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).

Kybernetik (1)

R. F. Quick, “A Vector-Magnitude Model of Contrast Detection,” Kybernetik 16, 65–67 (1974).
[Crossref] [PubMed]

Vision Res. (6)

C. F. Stromeyer and S. Klein, “Evidence Against Narrow-Band Spatial Frequency Channels in Human Vision: The Detectability of Frequency Modulated Gratings,” Vision Res. 15, 899–910 (1975).
[Crossref] [PubMed]

H. Mostafavi and D. J. Sakrison, “Structure and Properties of a Single Channel in the Human Visual System,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

J. J. Kulikowski and P. E. King-Smith, “Spatial Arrangement of Line, Edge, and Grating Detectors Revealed by Subthreshold Summation,” Vision Res. 13, 1455–1478 (1973).
[Crossref] [PubMed]

R. F. Quick and T. A. Reichert, “Spatial-Frequency Selectivity in Contrast Detection,” Vision Res. 15, 637–643 (1975).
[Crossref] [PubMed]

R. F. Quick, J. R. Hamerly, and T. A. Reichert, “The Absence of a Measurable ‘Critical Band’ at Low Suprathreshold Contrasts,” Vision Res. 16, 351–355 (1976).
[Crossref]

C. F. Stromeyer and S. Klein, “Spatial Frequency Channels in Human Vision as Asymmetric (Edge) Mechanisms,” Vision Res. 14, 1409–1420 (1974).
[Crossref] [PubMed]

Other (5)

We are aware that Dr. Stanley Klein obtained similar results in an earlier unpublished study. Apparently Klein’s results were obtained with a model similar to that of Stromeyer and Klein,8 which includes a spatial “pooling” mechanism in addition to probability summation. On the basis of the present study we do not believe that a pooling mechanism is necessary to explain our data.

Strictly, the limit is the essential supremum of |f(x)|, which ignores pathological points on sets of zero measure. This does not matter if f(x) is continuous and of finite spatial extent.

N. Graham, “Visual Detection of Aperiodic Spatial Stimuli by Probability Summation Among Multiple Channels,” Vision Res. (in press).

The components actually used in the experiments were squarewaves; near threshold the harmonics have negligible effect (see Quick and Reichert6), so we have included only the fundamental components in this study.

J. G. Robson, “Regional Variation of Contrast Sensitivity in the Visual Field,” presented to the Association for Research in Vision and Ophthalmology, April 1975 (unpublished).

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Figures (4)

FIG. 1
FIG. 1

The multiple-channel model of contrast detection used in this study. Detectors having the same spatial-frequency selectivity characteristics are grouped together into a “channel.” The spatial filters in the channels model the selective frequency responses of the detectors. Filter outputs can be interpreted as the input to the detectors, as a function of detector location on the retina. Probability-summation effects among the spatially distributed detectors are produced by the addition of independent white noise in each channel prior to detection.

FIG. 2
FIG. 2

Results of a “single-channel” model of contrast detection, including the effects of probability summation among wideband detectors distributed over the visual field. (a) The curves illustrate results for different functional forms assumed for the decrease in contrast sensitivity with eccentricity, given that each function has an effective width (see text) of two degrees of visual angle. (b) Shows the result of changing the effective width. Probability summation effects were approximated by an integral of the fourth power of the detector inputs (see text). The sharp drop for small Δf is attributed to the effects of “beats” between the two frequency components of the grating.

FIG. 3
FIG. 3

Results of a “multiple-channel” model of contrast detection including the effects of probability summation among frequency-selective detectors distributed over spatial frequency and spatial position. The three curves illustrate the effect of changing the channel bandwidth, assuming Gaussian channel frequency response and a Gaussian decrease in sensitivity with eccentricity (2° effective width). The dotted, solid, and dashed curves are channel bandwidths of 2, 4, and 8 cycles per degree, respectively. Probability summation was approximated by an integral of the fourth power of the detector inputs (see text). The sharp drop for small Δf occurs as in the single-channel results, and is the result of spatial beats between frequency components. The curves drop off still further for Δf greater than the channel bandwidth, in a manner which approximates the channel frequency response curve.

FIG. 4
FIG. 4

(a) Comparison of model predictions with data from Quick and Reichert.6 Solid line, prediction of the multiple-channel model for an effective spatial width of 2°, channel bandwidth 6 cycles per degree and fourth-power approximation for probability summation. Dashed line, prediction of the singe-channel model (parameters as for the multiple-channel model except that only wideband detectors are included). The single-channel model does not give the correct asymptotic value as Δf increases. (b) Data obtained from new experiments at 5 cycle per degree center frequency. The solid line is the prediction of the multiple-channel model with a 3 cycle per degree bandwidth.

Equations (14)

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L ( x ) = L o { 1 + C [ cos 2 π ( f o - ½ Δ f ) x + p cos 2 π ( f o + ½ Δ f ) x ] } ,
e ( x ) = C 1 [ cos 2 π ( f o - ½ Δ f ) x + cos 2 π ( f o + ½ Δ f ) x ] .
e ( x ) = 2 C 1 cos ( π Δ f x ) cos ( 2 π f o x ) .
W ( x ) = e - x / T s ,
W ( x ) = e - x 2 / σ s 2 ,
W ( z ) = e - x 4 / S S 4 .
S = max x f K [ i W ( x i - x f ) e ( x i ) γ ] 1 / γ ,
S max x f K Δ x o - 1 / γ [ W ( x - x f ) e ( x ) γ d x ] 1 / γ .
lim γ [ f ( x ) γ ] 1 / γ = max x f ( x ) .
R f c ( f ) = { exp [ - f - f c / T f ]             or exp [ - ( f - f c ) 2 / 2 σ f 2 ] ,
e f c ( x ) = C 2 [ R f c ( f o - ½ Δ f ) cos 2 π ( f o - ½ Δ f ) x + R f c ( f o + ½ Δ f ) cos 2 π ( f o + ½ Δ f ) x ] .
A = R f c ( f o - ½ Δ f ) + R f c ( f o + ½ Δ f ) , Θ = 1 A R f c ( f o - ½ Δ f ) ,
e f c ( x ) = A C 2 [ Θ cos 2 π ( f o - ½ Δ f ) x + ( 1 - Θ ) cos 2 π ( f o + ½ Δ f ) x ] = A C 2 [ 1 - 4 Θ ( 1 - Θ ) sin 2 ( π Δ f x ) ] 1 / 2 cos [ 2 π f o x + ϕ ( x ) ] ,
S ( Δ f ) max x f K 1 [ W ( x - x f ) e f c ( x ) γ d x d f c ] 1 / γ .