Abstract

Several theoretical models for the prediction of liminal contrast are studied. One model that employs the known excitation and inhibition spread functions of the eye accurately predicts, at 10 mL, thresholds for circular targets, sine waves, and rectangles, including the line as a special case. Semiempirically, this model is extended to include thresholds at low light levels.

© 1971 Optical Society of America

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References

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  1. A. Rose, J. Opt. Soc. Am. 38, 196 (1948).
    [Crossref] [PubMed]
  2. J. Johnson, private communication.
  3. H. R. Blackwell, J. Opt. Soc. Am. 36, 624 (1946).
    [Crossref] [PubMed]
  4. The asterisk is used to designate the convolution operation.io(x,y,t)*ho(x,y,t)=∫dx′∫dy′∫dt′io(x′,y′,t′)ho[(x−x′),(y−y′),(t−t′)].
  5. J. L. Harris, J. Opt. Soc. Am. 56, 569 (1966).
    [Crossref]
  6. M. A. Bouman, J. J. Vos, and P. L. Walraven, J. Opt. Soc. Am. 53, 121 (1963).
    [Crossref] [PubMed]
  7. E. A. Trabka, J. Opt. Soc. Am. 59, 345 (1969).
    [Crossref] [PubMed]
  8. W. Feller, Probability Theory and its Applications (Wiley, New York, 1950), p. 249.
  9. Another approach that gives a similar final result is that of M. A. Bouman and H. A. van der Velden, J. Opt. Soc. Am. 37, 908 (1947) in terms of multistage cells and omission of dark noise.
    [PubMed]
  10. The terminology “signal density” is arbitrary and possibly confusing. The signal is related to ∫∫s2(x,y)dxdy, not ∫∫s(x,y) dxdy.
  11. A. van Meeteren and J. J. Vos, J. Opt. Soc. Am. 59, 1538 (1969).
  12. S. Hecht and E. U. Mintz, Physiol. 22, 593 (1939).
  13. O. Schade, J. Opt. Soc. Am. 46, 721 (1956).
    [Crossref] [PubMed]
  14. R. W. Gubisch, J. Opt. Soc. Am. 57, 407 (1967).
    [Crossref]
  15. G. A. Fry, in Progress in Optics VIII, edited by E. Wolf (North-Holland, Amsterdam, 1970).
  16. M. Davidson, J. Opt. Soc. Am. 58, 1300 (1968).
    [Crossref] [PubMed]
  17. Because the glance time tg is considerably greater than the response time of the eye, we approximate H(ω) to unity.
  18. D. E. Mitchell, R. D. Freeman, and G. Westheimer, J. Opt. Soc. Am. 57, 246 (1967).
    [Crossref] [PubMed]
  19. Ricco’s law states that the threshold contrast is proportional to the inverse of the area of small targets.
  20. A. B. Kristofferson and H. R. Blackwell, private communication.
  21. E. S. Lamar, S. Hecht, S. Shlaer, and C. D. Hendley, J. Opt. Soc. Am. 37, 531 (1947);J. Opt. Soc. Am. 38, 741 (1948).
    [Crossref] [PubMed]
  22. J. L. Brown, in Vision and Visual Perception, edited by C. H. Graham (Wiley, New York, 1965), p. 39.
  23. F. W. Campbell and D. G. Green, J. Physiol. (London) 181, 576 (1965).
  24. A. Watanabe, T. Mori, S. Nagata, and K. Hiwatashi, Vision Res. 8, 1245 (1968).
    [Crossref] [PubMed]
  25. P. Gaunt, Opt. Acta 15, 287 (1968).
    [Crossref]
  26. C. H. Graham, in Ref. 22, p. 68.
  27. S. Hecht, S. Ross, and C. G. Mueller, J. Opt. Soc. Am. 37, 500 (1947).
    [Crossref] [PubMed]
  28. J. L. Harris and S. Q. Duntley, in Performance of the Eye at Low Luminances, edited by M. A. Bouman and J. J. Vos (Excerpta Medica Foundation, Amsterdam and New York, 1966), p. 41.

1969 (2)

E. A. Trabka, J. Opt. Soc. Am. 59, 345 (1969).
[Crossref] [PubMed]

A. van Meeteren and J. J. Vos, J. Opt. Soc. Am. 59, 1538 (1969).

1968 (3)

M. Davidson, J. Opt. Soc. Am. 58, 1300 (1968).
[Crossref] [PubMed]

A. Watanabe, T. Mori, S. Nagata, and K. Hiwatashi, Vision Res. 8, 1245 (1968).
[Crossref] [PubMed]

P. Gaunt, Opt. Acta 15, 287 (1968).
[Crossref]

1967 (2)

1966 (1)

1965 (1)

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

1963 (1)

1956 (1)

1948 (1)

1947 (3)

1946 (1)

1939 (1)

S. Hecht and E. U. Mintz, Physiol. 22, 593 (1939).

Blackwell, H. R.

H. R. Blackwell, J. Opt. Soc. Am. 36, 624 (1946).
[Crossref] [PubMed]

A. B. Kristofferson and H. R. Blackwell, private communication.

Bouman, M. A.

Brown, J. L.

J. L. Brown, in Vision and Visual Perception, edited by C. H. Graham (Wiley, New York, 1965), p. 39.

Campbell, F. W.

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

Davidson, M.

Duntley, S. Q.

J. L. Harris and S. Q. Duntley, in Performance of the Eye at Low Luminances, edited by M. A. Bouman and J. J. Vos (Excerpta Medica Foundation, Amsterdam and New York, 1966), p. 41.

Feller, W.

W. Feller, Probability Theory and its Applications (Wiley, New York, 1950), p. 249.

Freeman, R. D.

Fry, G. A.

G. A. Fry, in Progress in Optics VIII, edited by E. Wolf (North-Holland, Amsterdam, 1970).

Gaunt, P.

P. Gaunt, Opt. Acta 15, 287 (1968).
[Crossref]

Graham, C. H.

C. H. Graham, in Ref. 22, p. 68.

Green, D. G.

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

Gubisch, R. W.

Harris, J. L.

J. L. Harris, J. Opt. Soc. Am. 56, 569 (1966).
[Crossref]

J. L. Harris and S. Q. Duntley, in Performance of the Eye at Low Luminances, edited by M. A. Bouman and J. J. Vos (Excerpta Medica Foundation, Amsterdam and New York, 1966), p. 41.

Hecht, S.

Hendley, C. D.

Hiwatashi, K.

A. Watanabe, T. Mori, S. Nagata, and K. Hiwatashi, Vision Res. 8, 1245 (1968).
[Crossref] [PubMed]

Johnson, J.

J. Johnson, private communication.

Kristofferson, A. B.

A. B. Kristofferson and H. R. Blackwell, private communication.

Lamar, E. S.

Mintz, E. U.

S. Hecht and E. U. Mintz, Physiol. 22, 593 (1939).

Mitchell, D. E.

Mori, T.

A. Watanabe, T. Mori, S. Nagata, and K. Hiwatashi, Vision Res. 8, 1245 (1968).
[Crossref] [PubMed]

Mueller, C. G.

Nagata, S.

A. Watanabe, T. Mori, S. Nagata, and K. Hiwatashi, Vision Res. 8, 1245 (1968).
[Crossref] [PubMed]

Rose, A.

Ross, S.

Schade, O.

Shlaer, S.

Trabka, E. A.

van der Velden, H. A.

van Meeteren, A.

A. van Meeteren and J. J. Vos, J. Opt. Soc. Am. 59, 1538 (1969).

Vos, J. J.

A. van Meeteren and J. J. Vos, J. Opt. Soc. Am. 59, 1538 (1969).

M. A. Bouman, J. J. Vos, and P. L. Walraven, J. Opt. Soc. Am. 53, 121 (1963).
[Crossref] [PubMed]

Walraven, P. L.

Watanabe, A.

A. Watanabe, T. Mori, S. Nagata, and K. Hiwatashi, Vision Res. 8, 1245 (1968).
[Crossref] [PubMed]

Westheimer, G.

J. Opt. Soc. Am. (13)

J. Physiol. (London) (1)

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

Opt. Acta (1)

P. Gaunt, Opt. Acta 15, 287 (1968).
[Crossref]

Physiol. (1)

S. Hecht and E. U. Mintz, Physiol. 22, 593 (1939).

Vision Res. (1)

A. Watanabe, T. Mori, S. Nagata, and K. Hiwatashi, Vision Res. 8, 1245 (1968).
[Crossref] [PubMed]

Other (11)

C. H. Graham, in Ref. 22, p. 68.

J. L. Brown, in Vision and Visual Perception, edited by C. H. Graham (Wiley, New York, 1965), p. 39.

Ricco’s law states that the threshold contrast is proportional to the inverse of the area of small targets.

A. B. Kristofferson and H. R. Blackwell, private communication.

J. L. Harris and S. Q. Duntley, in Performance of the Eye at Low Luminances, edited by M. A. Bouman and J. J. Vos (Excerpta Medica Foundation, Amsterdam and New York, 1966), p. 41.

The terminology “signal density” is arbitrary and possibly confusing. The signal is related to ∫∫s2(x,y)dxdy, not ∫∫s(x,y) dxdy.

Because the glance time tg is considerably greater than the response time of the eye, we approximate H(ω) to unity.

G. A. Fry, in Progress in Optics VIII, edited by E. Wolf (North-Holland, Amsterdam, 1970).

W. Feller, Probability Theory and its Applications (Wiley, New York, 1950), p. 249.

The asterisk is used to designate the convolution operation.io(x,y,t)*ho(x,y,t)=∫dx′∫dy′∫dt′io(x′,y′,t′)ho[(x−x′),(y−y′),(t−t′)].

J. Johnson, private communication.

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Figures (9)

F. 1
F. 1

Schematic diagram of the eye–brain system.

F. 2
F. 2

Comparison of the predictions of the proposed models with Blackwell’s 10.8-mL, forced-choice, long-observation-time circle data. (The solid curve shows experimental data, whereas the dash–double-dot curve shows the point-radiance model, the dash– dot curve the area-radiance model, and the dashed curve the excitation–inhibition model.)

F. 3
F. 3

Comparison of the predictions of the excitation–inhibition model to three sets of Blackwell’s 10.8-mL circle data. (Curve 1 shows the forced-choice, long-time data, curve 2 shows the 6-position, 6-s-search data, and curve 3 represents the 0.01-s projection-time data.

F. 4
F. 4

Comparison of the predictions of the excitation–inhibition model to Lamar, Hecht, Shlaer, and Hendley’s 19-mL, 2-mm artificial-pupil rectangle data (prediction plotted as dotted line, shifted 1 log unit).

F. 5
F. 5

Comparison of the predictions of the excitation–inhibition model to Fry’s 23.4-mL, 2-mm artificial-pupil, sine-wave and degraded-border data (dashed, uncorrected; dotted, corrected fit).

F. 6
F. 6

Agreement of the excitation–inhibition model with Black-well’s forced-choice, long-observation-time data at all luminances.

F. 7
F. 7

The luminance-dependent phenomenological transfer functions and comparison of the predicted and Schade’s experimentally established frequencies of highest contrast sensitivity.

F. 8
F. 8

Agreement of the excitation–inhibition model with the Hecht et al. liminal line widths at all light levels.

F. 9
F. 9

Adjustment of the excitation–inhibition model to include the effects of inhomogeneous retinal sensitivity, and comparison of its amended predictions with Blackwell’s 10.8-mL, forced-choice, long-observation-time, circle data.

Tables (2)

Tables Icon

Table I Asymptotic results of the models. a = Target area, c = target circumference, a = line width, b = line length. For GB, GS, GL values see Table II.

Tables Icon

Table II Luminance-dependent constants used in the excitation–inhibition model.

Equations (33)

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h 0 ( x , y , t ) = h o s ( x , y ) δ ( t ) , ( i ) h r ( x , y , t ) = h r s ( x , y ) h r t ( t ) . ( ii )
i 1 = ( i o * h o * h r ) + N 1 + N D .
S ( ω x , ω y , ω ) = N H r 2 .
D 2 Δ _ d ω x d ω y d ω I s 2 ( ω x , ω y , ω ) / S ( ω x , ω y , ω ) .
p ( t 0 ) = λ 0 τ f ( τ ) exp [ λ ( t 0 τ ) ] d τ ,
t ¯ 0 = 0 τ p ( τ ) d τ = t r + 1 / λ ( i )
n ¯ = 1 / t ¯ 0 = λ / ( λ l r + 1 ) . ( ii )
Δ n ¯ = Δ λ / ( λ t r + 1 ) 2 ,
σ n 2 = λ / [ ( λ t r + 1 ) 3 t ] .
( S N ) = ( C 2 λ t 1 + λ t r ) 1 2 = C ( α A d t ) 1 2 ( N 1 + α A d N t r ) 1 2 .
i o = [ λ d d / ( λ t r + 1 ) ] [ C p ( x , y , t ) / ( λ t r + 1 ) + 1 ] ,
N = σ n 2 d d t = λ d d / [ ( λ t r + 1 ) 3 ] .
S p r = C η [ p s ( 0 , 0 ) * h o s * h r s ] = 1 4 C η π 2 d ω x d ω y P s ( ω x , ω y ) H o s H r s ,
S p r ( ω ) = 1 4 H r t 2 ( ω ) π 2 [ N d ω x d ω y H r s 2 ] + N D ,
D p r 2 = t g 2 ( 2 π ) 1 d ω S p r 2 sinc 2 ( 1 2 ω t g ) H r t 2 ( ω ) / S p r ( ω ) .
s a r ( x , y ) = C η [ p s * h o s * h h s ] .
D a r 2 = t g 2 8 π 3 d ω x d ω y d ω × C 2 P s 2 H r 2 sinc 2 ( 1 2 ω t g ) N H r 2 + N D η 2 H o 2 .
H p ( ω x , ω y ) = exp [ Γ ( | ω x | + | ω y | ) ] 0.64 exp [ 1 2 σ 2 ( ω x 2 + ω y 2 ) ] ,
H ( ω x , ω y ) = exp [ Γ ( | ω x | + | ω y | ) ] H I ( ω x ) H I ( ω y ) ,
H I ( ω x ) = exp [ ( S σ ω x + 1 2 T 2 σ 2 ω x 2 ) ] when ω x R / σ , ( i ) H I ( ω x ) = 0.8 exp [ 1 2 σ 2 ω x 2 ) ] when ω x R / σ . ( ii )
D = C ( α A d d d t g N 2 / N ) 1 2 F ,
N Δ _ N + α A d t r N 2 + N D .
C t h Δ _ D o K / F ,
H ( ω o , 0 ) C = K A o 1 2 D o ,
[ d y exp ( y 2 / r c 2 ) / C ] 1 2 = ( π 1 2 r c / C ) 1 2 .
a / ( π Γ ) 2
a 1 2
a / ( 2 π Γ )
( a b ) / ( 2 π Γ ) 1 2
[ ( 1 2 c ) 1 2 ( σ 2 Γ 2 ) 1 2 ] G B
[ a / ( 2 π Γ ) ] G S
[ ( a b ) / ( 2 π Γ ) 1 2 ] G L
io(x,y,t)*ho(x,y,t)=dxdydtio(x,y,t)ho[(xx),(yy),(tt)].