Abstract

Modulation thresholds have been measured for sine-wave gratings at various spatial frequencies. At low frequencies, the threshold is determined in part by the gradient between adjacent bright and dark bars. This is related to the effect of blurring the border between the bright and dark halves of a bipartite field; the results obtained with a sine-wave grating have been compared to the results obtained with a single border. An attempt has been made to isolate the roles played by optical blur and physiological spread of excitation and inhibition.

© 1969 Optical Society of America

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References

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  1. G. A. Fry, J. Opt. Soc. Am. 58, 1415 (1968).
    [Crossref]
  2. A. Arnulf and O. Dupuy, Compt. Rend. 250, 2757 (1960).
  3. S. Berger–Lheureux–Robardey, Rev. Opt. 44, 294 (1965).
  4. F. W. Campbell and D. G. Green, J. Physiol. (London) 181, 576 (1965).
  5. G. A. Fry, Blur of the Retinal Image (The Ohio State University Press, Columbus, Ohio, 1955), p. 83.
  6. G. A. Fry, J. Opt. Soc. Am. 55, 108 (1965).
    [Crossref]
  7. F. Ratliff, Mach Bands (Holden-Day, Inc., San Francisco, 1965).
  8. Y. Le Grand, Compt. Rend. 200, 490 (1935).
  9. G. Westheimer, J. Physiol. (London) 152, 67 (1960).
  10. C. Enroth–Cugell and J. G. Robson, J. Physiol. (London) 187, 517 (1966).
  11. F. Flamant, Rev. Opt. 34, 433 (1955).
  12. G. Westheimer and F. W. Campbell, J. Opt. Soc. Am. 52, 1040 (1962).
    [Crossref] [PubMed]
  13. J. Krauskopf, J. Opt. Soc. Am. 52, 1046 (1962).
    [Crossref]
  14. F. W. Campbell and R. W. Gubisch, J. Physiol. (London) 186, 558 (1966).
  15. A. S. Patel, J. Opt. Soc. Am. 56, 689 (1966).
    [Crossref] [PubMed]
  16. O. H. Schade, J. Opt. Soc. Am. 46, 721 (1956).
    [Crossref] [PubMed]
  17. J. Nachmias, J. Opt. Soc. Am. 58, 9 (1968).
    [Crossref] [PubMed]
  18. F. L. van Nes, “Experimental Studies in Spatiotemporal Contrast Transfer by the Human Eye” (doctoral dissertation), University of Utrecht (1968).
  19. D. G. Green, J. Physiol. (London) 196, 415 (1968).

1968 (3)

1966 (3)

F. W. Campbell and R. W. Gubisch, J. Physiol. (London) 186, 558 (1966).

A. S. Patel, J. Opt. Soc. Am. 56, 689 (1966).
[Crossref] [PubMed]

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

1965 (3)

S. Berger–Lheureux–Robardey, Rev. Opt. 44, 294 (1965).

F. W. Campbell and D. G. Green, J. Physiol. (London) 181, 576 (1965).

G. A. Fry, J. Opt. Soc. Am. 55, 108 (1965).
[Crossref]

1962 (2)

1960 (2)

G. Westheimer, J. Physiol. (London) 152, 67 (1960).

A. Arnulf and O. Dupuy, Compt. Rend. 250, 2757 (1960).

1956 (1)

1955 (1)

F. Flamant, Rev. Opt. 34, 433 (1955).

1935 (1)

Y. Le Grand, Compt. Rend. 200, 490 (1935).

Arnulf, A.

A. Arnulf and O. Dupuy, Compt. Rend. 250, 2757 (1960).

Berger–Lheureux–Robardey, S.

S. Berger–Lheureux–Robardey, Rev. Opt. 44, 294 (1965).

Campbell, F. W.

F. W. Campbell and R. W. Gubisch, J. Physiol. (London) 186, 558 (1966).

F. W. Campbell and D. G. Green, J. Physiol. (London) 181, 576 (1965).

G. Westheimer and F. W. Campbell, J. Opt. Soc. Am. 52, 1040 (1962).
[Crossref] [PubMed]

Dupuy, O.

A. Arnulf and O. Dupuy, Compt. Rend. 250, 2757 (1960).

Enroth–Cugell, C.

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

Flamant, F.

F. Flamant, Rev. Opt. 34, 433 (1955).

Fry, G. A.

G. A. Fry, J. Opt. Soc. Am. 58, 1415 (1968).
[Crossref]

G. A. Fry, J. Opt. Soc. Am. 55, 108 (1965).
[Crossref]

G. A. Fry, Blur of the Retinal Image (The Ohio State University Press, Columbus, Ohio, 1955), p. 83.

Green, D. G.

D. G. Green, J. Physiol. (London) 196, 415 (1968).

F. W. Campbell and D. G. Green, J. Physiol. (London) 181, 576 (1965).

Gubisch, R. W.

F. W. Campbell and R. W. Gubisch, J. Physiol. (London) 186, 558 (1966).

Krauskopf, J.

Le Grand, Y.

Y. Le Grand, Compt. Rend. 200, 490 (1935).

Nachmias, J.

Patel, A. S.

Ratliff, F.

F. Ratliff, Mach Bands (Holden-Day, Inc., San Francisco, 1965).

Robson, J. G.

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

Schade, O. H.

van Nes, F. L.

F. L. van Nes, “Experimental Studies in Spatiotemporal Contrast Transfer by the Human Eye” (doctoral dissertation), University of Utrecht (1968).

Westheimer, G.

Compt. Rend. (2)

A. Arnulf and O. Dupuy, Compt. Rend. 250, 2757 (1960).

Y. Le Grand, Compt. Rend. 200, 490 (1935).

J. Opt. Soc. Am. (7)

J. Physiol. (London) (5)

D. G. Green, J. Physiol. (London) 196, 415 (1968).

F. W. Campbell and R. W. Gubisch, J. Physiol. (London) 186, 558 (1966).

G. Westheimer, J. Physiol. (London) 152, 67 (1960).

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

F. W. Campbell and D. G. Green, J. Physiol. (London) 181, 576 (1965).

Rev. Opt. (2)

S. Berger–Lheureux–Robardey, Rev. Opt. 44, 294 (1965).

F. Flamant, Rev. Opt. 34, 433 (1955).

Other (3)

G. A. Fry, Blur of the Retinal Image (The Ohio State University Press, Columbus, Ohio, 1955), p. 83.

F. Ratliff, Mach Bands (Holden-Day, Inc., San Francisco, 1965).

F. L. van Nes, “Experimental Studies in Spatiotemporal Contrast Transfer by the Human Eye” (doctoral dissertation), University of Utrecht (1968).

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Figures (14)

Fig. 1
Fig. 1

Modulation (M) of a sine-wave grating.

Fig. 2
Fig. 2

The effect of convolution of a square wave with a gaussian spread function. The σ/ s ¯ values represent the ratio of the standard deviation (σ) of the gaussian spread function to the center-to-center separation ( s ¯) between the bright bars.

Fig. 3
Fig. 3

Convolution of a square wave with a gaussian spread function. The center-to-center separation is constant at 4 min.

Fig. 4
Fig. 4

Apparatus.

Fig. 5
Fig. 5

Modulation threshold for a sine-wave grating. Data for Subject D. K. The average luminance of the grating was kept constant at 74.3 cd/m2.

Fig. 6
Fig. 6

Basis for comparing the modulation threshold for a single blurred border to the modulation threshold for a sine-wave grating.

Fig. 7
Fig. 7

Modulation thresholds for a single blurred border (A) and a sine-wave grating (B). Data for Subject D. K. Curve C is the modulation-threshold curve for a sine-wave grating corrected for the border-gradient effect.

Fig. 8
Fig. 8

Point spread function for optical blur of the retinal image. Point of white light (equal-energy spectrum) viewed through a 2-mm artificial pupil. Continuous curve is the spread function calculated from diffraction theory and chromatic-aberration data. The dotted curve is the gaussian approximation.

Fig. 9
Fig. 9

The effect of optical blur on the modulation of the retinal image of a sine-wave grating at the threshold of visibility.

Fig. 10
Fig. 10

Line spread function for physiological irradiation (R) at luminance levels near the cone threshold. The continuous curve conforms to Eq. (6) and the broken curve to Eq. (5).

Fig. 11
Fig. 11

The interrelations between the components contributing to the modulation transfer function of the human eye. The constants used in computing these curves are as follows: σE = 0.4, σI = 8, K = 0.8, and ψ/μ = 0.8.

Fig. 12
Fig. 12

Distributions of excitation (R) and inhibition (I) across the retina produced by a sine-wave grating of constant modulation (0.10) and variable frequency. Each half-cycle is 1.25 times wider than the preceding half-cycle. The numbers above and below the R-I curve indicate the lengths of the half-cycles in minutes.

Fig. 13
Fig. 13

Relation of the modulation threshold values (continuous curve) to the modulation transfer function of the eye (T) defined by Eq. (14). The reciprocal of T has been multiplied by 1 100 instead of 1 5 as indicated in the figure.

Fig. 14
Fig. 14

The image of a sharp contrast border (left) and the image of a sine-wave grating with a center-to-center spacing of 12 min (right) formed on the retina and transmitted through the retina to the level of the ganglion cells.

Equations (17)

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modualtion = ( L max - L min ) / ( L max + L min ) .
d L / d s = [ σ ( 2 π ) 1 2 ] - 1 ( L max - L min ) .
d L / d s = ( π / s ¯ ) ( L max - L min ) .
β = threshold value for a sharp border threshold value for a blurred border .
E = E 0 exp [ - 1 2 ( r / σ ) 2 ] ,
T E = exp [ - 2 ( π σ / s ¯ ) 2 ] .
R ~ ( r + 5 ) - 3 ,
R ~ 2 0 3 0 [ 5 + ( t 2 + h 2 ) 1 2 ] - 3 d h ,
R = R 0 exp [ - k t ] ,
R = μ E + μ Δ E [ 1 + ( 2 π / k s ¯ ) 2 ] - 1 cos ( 2 π s / s ¯ ) ,
T R = [ 1 + ( 2 π / k s ¯ ) 2 ] - 1 .
I = I 0 exp [ - 1 2 ( t / σ ) 2 ] .
I = ψ E + ψ Δ E exp [ - 2 ( π σ / s ¯ ) 2 ] cos ( 2 π s / s ¯ ) ,
T I = exp [ - 2 ( π σ / s ¯ ) 2 ] .
( R - I ) = ( μ - ψ ) E + Δ E ( μ T R - ψ T I ) cos ( 2 π s / s ¯ ) .
T ( R - I ) = ( 1 - ψ / μ ) - 1 [ T R - ( ψ / μ ) T I ] .
T = T E ( 1 - ψ / μ ) - 1 [ T R - ( ψ / μ ) T I ] .