Abstract

Psychometric functions were determined concurrently for detection of simple gratings (luminance sinusoidally modulated with spatial frequency f) and complex gratings (luminance modulated by the sum of two sinusoids, with frequencies f and f′). Results were used to test the hypothesis that the two components of a complex grating may be detected independently. In an extensive experiment with f = 14 cycles/deg, the independence hypothesis was consistently rejected only when f/f=54 or 45, but rarely rejected when the value of f/f′ lay outside this range. In other experiments, f was between 1.9 and 22.4 cycles/deg. All results are compatible with the assumption that the human visual system contains sensory channels, each selectively sensitive to different narrow ranges of spatial frequencies, whose outputs are detected independently.

© 1971 Optical Society of America

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References

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  1. O. H. Schade, J. Opt. Soc. Am. 46, 721 (1956).
    [Crossref] [PubMed]
  2. H. deLange, J. Opt. Soc. Am. 48, 777 (1958).
    [Crossref]
  3. J. G. Robson, J. Opt. Soc. Am. 56, 1141 (1966).
    [Crossref]
  4. F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, 1965).
  5. F. W. Campbell and J. G. Robson, J. Physiol. (London) 197, 551 (1968).
  6. C. Enroth-Cugell and J. G. Robson, J. Physiol. (London) 187, 517 (1966).
  7. F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, J. Physiol. (London) 204, 120 (1969).
  8. A. Pantle and R. Sekuler, Science 162, 1146 (1968).
    [Crossref] [PubMed]
  9. C. Blakemore and F. W. Campbell, J. Physiol (London) 203, 237 (1969).
  10. C. Blakemore and P. Sutton, Science 166, 245 (1969).
    [Crossref] [PubMed]
  11. Whenever f/f′ can be expressed as a ratio of odd integers, then there is some value of (ϕ−ϕ′) for which the largest peak-to-trough difference in the input complex waveform will be equal to exactly 2L0(m+m′). The corresponding peak-to-trough value for the output of the single channel (assuming a symmetrical spread function) will be proportional to (am+a′m′). Hence, for a peak-to-trough detector, the functions G and G′ in Eq. (3) are linear in m and m′, respectively. For other values of f/f′, the largest peak-to-trough difference in the luminance waveform will be less than 2L0(m+m′) no matter what the phase of the component sinusoids.
  12. D. J. Finney, Probit Analysis (Cambridge U. P., England, 1964), Appendix II.
  13. A. M. Mood and F. A. Graybill, Introduction to the Theory of Statistics. 2nd ed. (McGraw–Hill, New York, 1963), Ch. 12.
  14. R. W. Rodieck and J. Stone, J. Neurophysiol. 28, 833 (1965).
  15. F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol (London) 203, 223 (1969).
  16. F. W. Campbell, J. Nachmias, and J. Jukes, J. Opt. Soc. Am. 60, 555 (1970).
    [Crossref] [PubMed]
  17. N. Graham and J. Nachmias, Vision Res. 11, 251 (1971).
    [Crossref] [PubMed]
  18. R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra(Dover, New York, 1958).
  19. D. D. McCracken and W. S. Dorn, Numerical Methods and Fortran Programming (Wiley, New York, 1964), Ch. 5.

1971 (1)

N. Graham and J. Nachmias, Vision Res. 11, 251 (1971).
[Crossref] [PubMed]

1970 (1)

1969 (4)

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol (London) 203, 223 (1969).

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, J. Physiol. (London) 204, 120 (1969).

C. Blakemore and F. W. Campbell, J. Physiol (London) 203, 237 (1969).

C. Blakemore and P. Sutton, Science 166, 245 (1969).
[Crossref] [PubMed]

1968 (2)

A. Pantle and R. Sekuler, Science 162, 1146 (1968).
[Crossref] [PubMed]

F. W. Campbell and J. G. Robson, J. Physiol. (London) 197, 551 (1968).

1966 (2)

C. Enroth-Cugell and J. G. Robson, J. Physiol. (London) 187, 517 (1966).

J. G. Robson, J. Opt. Soc. Am. 56, 1141 (1966).
[Crossref]

1965 (1)

R. W. Rodieck and J. Stone, J. Neurophysiol. 28, 833 (1965).

1958 (1)

1956 (1)

Blackman, R. B.

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra(Dover, New York, 1958).

Blakemore, C.

C. Blakemore and F. W. Campbell, J. Physiol (London) 203, 237 (1969).

C. Blakemore and P. Sutton, Science 166, 245 (1969).
[Crossref] [PubMed]

Campbell, F. W.

F. W. Campbell, J. Nachmias, and J. Jukes, J. Opt. Soc. Am. 60, 555 (1970).
[Crossref] [PubMed]

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol (London) 203, 223 (1969).

C. Blakemore and F. W. Campbell, J. Physiol (London) 203, 237 (1969).

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, J. Physiol. (London) 204, 120 (1969).

F. W. Campbell and J. G. Robson, J. Physiol. (London) 197, 551 (1968).

Cooper, G. F.

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, J. Physiol. (London) 204, 120 (1969).

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol (London) 203, 223 (1969).

deLange, H.

Dorn, W. S.

D. D. McCracken and W. S. Dorn, Numerical Methods and Fortran Programming (Wiley, New York, 1964), Ch. 5.

Enroth-Cugell, C.

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol (London) 203, 223 (1969).

C. Enroth-Cugell and J. G. Robson, J. Physiol. (London) 187, 517 (1966).

Finney, D. J.

D. J. Finney, Probit Analysis (Cambridge U. P., England, 1964), Appendix II.

Graham, N.

N. Graham and J. Nachmias, Vision Res. 11, 251 (1971).
[Crossref] [PubMed]

Graybill, F. A.

A. M. Mood and F. A. Graybill, Introduction to the Theory of Statistics. 2nd ed. (McGraw–Hill, New York, 1963), Ch. 12.

Jukes, J.

McCracken, D. D.

D. D. McCracken and W. S. Dorn, Numerical Methods and Fortran Programming (Wiley, New York, 1964), Ch. 5.

Mood, A. M.

A. M. Mood and F. A. Graybill, Introduction to the Theory of Statistics. 2nd ed. (McGraw–Hill, New York, 1963), Ch. 12.

Nachmias, J.

Pantle, A.

A. Pantle and R. Sekuler, Science 162, 1146 (1968).
[Crossref] [PubMed]

Ratliff, F.

F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, 1965).

Robson, J. G.

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, J. Physiol. (London) 204, 120 (1969).

F. W. Campbell and J. G. Robson, J. Physiol. (London) 197, 551 (1968).

C. Enroth-Cugell and J. G. Robson, J. Physiol. (London) 187, 517 (1966).

J. G. Robson, J. Opt. Soc. Am. 56, 1141 (1966).
[Crossref]

Rodieck, R. W.

R. W. Rodieck and J. Stone, J. Neurophysiol. 28, 833 (1965).

Sachs, M. B.

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, J. Physiol. (London) 204, 120 (1969).

Schade, O. H.

Sekuler, R.

A. Pantle and R. Sekuler, Science 162, 1146 (1968).
[Crossref] [PubMed]

Stone, J.

R. W. Rodieck and J. Stone, J. Neurophysiol. 28, 833 (1965).

Sutton, P.

C. Blakemore and P. Sutton, Science 166, 245 (1969).
[Crossref] [PubMed]

Tukey, J. W.

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra(Dover, New York, 1958).

J. Neurophysiol. (1)

R. W. Rodieck and J. Stone, J. Neurophysiol. 28, 833 (1965).

J. Opt. Soc. Am. (4)

J. Physiol (London) (2)

C. Blakemore and F. W. Campbell, J. Physiol (London) 203, 237 (1969).

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol (London) 203, 223 (1969).

J. Physiol. (London) (3)

F. W. Campbell and J. G. Robson, J. Physiol. (London) 197, 551 (1968).

C. Enroth-Cugell and J. G. Robson, J. Physiol. (London) 187, 517 (1966).

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, J. Physiol. (London) 204, 120 (1969).

Science (2)

A. Pantle and R. Sekuler, Science 162, 1146 (1968).
[Crossref] [PubMed]

C. Blakemore and P. Sutton, Science 166, 245 (1969).
[Crossref] [PubMed]

Vision Res. (1)

N. Graham and J. Nachmias, Vision Res. 11, 251 (1971).
[Crossref] [PubMed]

Other (6)

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra(Dover, New York, 1958).

D. D. McCracken and W. S. Dorn, Numerical Methods and Fortran Programming (Wiley, New York, 1964), Ch. 5.

Whenever f/f′ can be expressed as a ratio of odd integers, then there is some value of (ϕ−ϕ′) for which the largest peak-to-trough difference in the input complex waveform will be equal to exactly 2L0(m+m′). The corresponding peak-to-trough value for the output of the single channel (assuming a symmetrical spread function) will be proportional to (am+a′m′). Hence, for a peak-to-trough detector, the functions G and G′ in Eq. (3) are linear in m and m′, respectively. For other values of f/f′, the largest peak-to-trough difference in the luminance waveform will be less than 2L0(m+m′) no matter what the phase of the component sinusoids.

D. J. Finney, Probit Analysis (Cambridge U. P., England, 1964), Appendix II.

A. M. Mood and F. A. Graybill, Introduction to the Theory of Statistics. 2nd ed. (McGraw–Hill, New York, 1963), Ch. 12.

F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, 1965).

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Figures (13)

Fig. 1
Fig. 1

Simple gratings (a) and (b), complex grating (c), and corresponding luminance waveforms. The amplitudes of the two components of the complex gratings are one half the amplitudes for the corresponding simple gratings so that the peak-to-trough amplitudes of all three patterns are the same. In our experiments, the frequencies (cycles/deg) of these gratings were: (a), f = 14; (b), f′ = 2.8.

Fig. 2
Fig. 2

Block diagram of the single-channel model.

Fig. 3
Fig. 3

Block diagram of the multiple-channel model.

Fig. 4
Fig. 4

Hypothetical complex psychometric functions resulting from the given simple psychometric function and (a), single-channel model; (b), the multi-channel model in the case where the two complex grating components are detected by completely separate channels; (c), the multiple-channel model in the case where one component is detected by two channels and the other component by only one of these channels. Dotted lines are simple psychometric functions and are the same for all cases; solid lines are complex psychometric functions for one value of m′.

Fig. 5
Fig. 5

Simple psychometric functions for f = 14 cycles/deg. Each symbol represents results obtained on a different day during the period 21 March–27 May 1969. Left panel, O:MS; right panel, O:MR.

Fig. 6
Fig. 6

Psychometric functions for simple gratings of various frequencies. Solid lines are fitted in accordance with Eq. (10). Frequencies are given in cycles/deg. Modulation scale is linear. Observer MS.

Fig. 7
Fig. 7

Modulation sensitivity as a function of spatial frequency. The circles are average estimates of a in Eq. (10) based on the results of observer MS. The solid curve is from Campbell and Robson,16 Fig. 3, and has been displaced vertically to fit the circles.

Fig. 8
Fig. 8

Simple and complex psychometric functions for f = 14 cycles/deg and various values of f′ (given in cycles/deg). The solid curves are fitted to the simple psychometric functions in accordance with Eq. (10). The solid curves fitted to the complex psychometric functions correspond to the multiple-channel model in the case where f and f′ are detected by completely separate channels (see text). Vertical bars correspond to 95% confidence intervals. Observer MR.

Fig. 9
Fig. 9

Simple and complex psychometric functions for f/f′ = 2 and several values of f′ (in cycles/deg). Observer MS.

Fig. 10
Fig. 10

Simple and complex psychometric functions for f = 2.8 and f′ = 1.4 and 0.93 cycles/deg. Observer MS.

Fig. 11
Fig. 11

Simple (×) and complex (○) psychometric functions for f = 14 cycles/deg and f′ = 2.8 cycles/deg, plotted on abscissa linear in m (upper panels) and linear in m2 (lower panels). As before, Eq. (10) has been fitted to the simple psychometric function. The curves were then displaced leftward to provide the best fit possible to the complex psychometric functions, in accordance with two versions of a single-channel model.

Fig. 12
Fig. 12

Relative sensitivity as a function of frequency for the channel most sensitive to f = 14 cycles/deg. ×’s are based on data from observer MR, ○’s from observer MS. Solid lines join mean values at each frequency tested. (See Appendix.)

Fig. 13
Fig. 13

A. Mean values of relative sensitivity from Fig. 12. B. Narrowest frequency-response function described by Eq. (11). C. Tenfold cascade of the system described by B.

Tables (2)

Tables Icon

Table I Probabilities from χ2 test of independence, for f = 14 cycles/deg.

Tables Icon

Table II Probabilities from χ2 test of independence, for f = 2f′.

Equations (14)

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L ( x ) = L 0 [ 1 + m sin ( 2 π f x + ϕ ) ] .
L ( x ) = L 0 [ 1 + m sin ( 2 π f x + ϕ ) + m sin ( 2 π f x + ϕ ) ] .
P ( Y m , m ) = F [ a G ( m ) + a G ( m ) ] ,
P ( Y m , 0 ) = F [ a G ( m ) + a G ( 0 ) ] .
P ( T 1 m , m ) = P ( T 1 m , 0 ) P ( T 1 m ) ,
P ( T 2 m , m ) = P ( T 2 0 , m ) P ( T 2 m ) ,
P ( N m , m ) = P ( T ¯ 1 m ) P ( T ¯ 2 m ) ( 1 - c )
P ( N m , 0 ) = P ( T ¯ 1 m ) P ( T ¯ 2 0 ) ( 1 - c )
P ( N 0 , m ) = P ( T ¯ 1 0 ) P ( T ¯ 2 m ) ( 1 - c ) .
P ( N m , m ) P ( N m , 0 ) = P ( T ¯ 2 m ) P ( T ¯ 2 0 ) = a constant 1.
P ( Y m ) = 1 - [ 1 ( 2 π ) 1 2 - K - G ( m ) exp ( - x 2 2 ) d x ] ( 1 - c ) ,
P ( Y m ) = 1 - [ 1 ( 2 π ) 1 2 - K - a m exp ( - x 2 2 ) d x ] ( 1 - c ) .
S ( ν ) = k π { k c r c 2 exp [ - ( π r c ν ) 2 ] - k s r s 2 exp [ - ( π r s ν ) 2 } .
P ( N m , m ) = [ 1 ( 2 π ) 1 2 - K - a 11 m 1 - a 12 m 2 exp ( - x 2 / 2 ) d x ] × [ 1 ( 2 π ) 1 2 - K - a 21 m 1 - a 22 m 2 exp ( - x 2 / 2 ) d x ] ( 1 - c ) ,