Abstract

A rating-scale psychophysical method was employed to obtain receiver operating characteristics for detection and for discrimination of luminance increments. These curves were used to estimate the parameters of a normal probability-density function which was assumed to describe the relevant internal effects of the test flash. The over-all results strongly indicate that the mean of this distribution is a positively accelerated function of luminance of the test flash, and suggest that the standard deviation is a non-monotonic function of luminance. It is argued that quantum fluctuations and criterion fluctuations cannot account for these findings, whereas channel uncertainty may be at the root of some of them.

© 1970 Optical Society of America

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References

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  1. D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (John Wiley & Sons, Inc., New York, 1966).
  2. R. D. Luce, R. R. Bush, and E. Galanter, Eds., Handbook of Mathematical Psychology (John Wiley & Sons, Inc., New York, 1963), Vol. 1, p. 103.
  3. D. Krantz, Psychol. Rev. 76, 308 (1969).
    [Crossref] [PubMed]
  4. J. Nachmias and R. M. Steinman, J. Opt. Soc. Am. 53, 1206 (1963); J. Opt. Soc. Am. 54, 554E (1964).
    [Crossref]
  5. J. Nachmias and R. M. Steinman, Vision Res. 5, 545 (1965).
    [Crossref] [PubMed]
  6. J. Nachmias, Percept. Psychophys. 3, 151 (1968).
  7. S. Hecht, S. Shlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819 (1942).
  8. J. Nachmias and R. M. Steinman [Vision Res. 5, 545 (1965)] reported that ROC curves for discrimination between flashes of high luminance had a tendency to be convex upward when plotted on double-gaussian coordinates. A similar tendency was barely discernible in the data from some observers in the present study, though it was quite pronounced in the data from one observer (J. B.). Her more aberrant results will not be included in this paper.
    [Crossref] [PubMed]
  9. C. D. Creelman, J. Acoust. Soc. Am. 35, 1201 (1963).
    [Crossref]
  10. J. A. Swets, W. P. Tanner, and T. G. Birdsall, Psychol. Rev. 68, 301 (1961).
    [Crossref] [PubMed]
  11. W. P. Tanner, Ann. N. Y. Acad. Sci. 89, 752 (1961).
    [Crossref] [PubMed]
  12. H. R. Blackwell, J. Opt. Soc. Am. 53, 129 (1963).
    [Crossref] [PubMed]
  13. W. P. Tanner and J. A. Swets, Psychol. Rev. 61, 401 (1954).
    [Crossref] [PubMed]
  14. B. Leshowitz, H. B. Taub, and D. H. Raab, Percept. Psychophys. 4, 207 (1968).
    [Crossref]
  15. P. L. Emerson, Percept. Psychophys. 5, 2 (1969).
    [Crossref]
  16. H. B. Barlow, J. Physiol. (London) 160, 169 (1962).
  17. T. N. Cornsweet and H. M. Pinsker, J. Physiol. (London) 176, 294 (1965).
  18. H. B. Barlow, J. Opt. Soc. Am. 46, 634 (1956).
    [Crossref] [PubMed]
  19. H. B. Barlow, J. Physiol. (London) 136, 469 (1957).
  20. See also W. A. Wickelgren, J. Math. Psychol. 5, 102 (1968).
    [Crossref]
  21. R. FitzHugh, J. Gen. Physiol. 40, 925 (1957).
  22. H. B. Barlow and W. R. Levick, J. Physiol. (London) 200, 1 (1969).
  23. J. Stone and M. Fabian, Vision Res. 8, 1023 (1968).
    [Crossref] [PubMed]
  24. W. W. Peterson, T. G. Birdsall, and W. C. Fox, Trans. IRE PGIT 4, 171 (1954).
  25. C. I. Howarth and G. Lowe, Nature 212, 324 (1966).
    [Crossref] [PubMed]
  26. G. Lowe, Percept. Psychophys. 2, 278 (1967).
    [Crossref]

1969 (3)

D. Krantz, Psychol. Rev. 76, 308 (1969).
[Crossref] [PubMed]

P. L. Emerson, Percept. Psychophys. 5, 2 (1969).
[Crossref]

H. B. Barlow and W. R. Levick, J. Physiol. (London) 200, 1 (1969).

1968 (4)

J. Stone and M. Fabian, Vision Res. 8, 1023 (1968).
[Crossref] [PubMed]

See also W. A. Wickelgren, J. Math. Psychol. 5, 102 (1968).
[Crossref]

B. Leshowitz, H. B. Taub, and D. H. Raab, Percept. Psychophys. 4, 207 (1968).
[Crossref]

J. Nachmias, Percept. Psychophys. 3, 151 (1968).

1967 (1)

G. Lowe, Percept. Psychophys. 2, 278 (1967).
[Crossref]

1966 (1)

C. I. Howarth and G. Lowe, Nature 212, 324 (1966).
[Crossref] [PubMed]

1965 (3)

J. Nachmias and R. M. Steinman [Vision Res. 5, 545 (1965)] reported that ROC curves for discrimination between flashes of high luminance had a tendency to be convex upward when plotted on double-gaussian coordinates. A similar tendency was barely discernible in the data from some observers in the present study, though it was quite pronounced in the data from one observer (J. B.). Her more aberrant results will not be included in this paper.
[Crossref] [PubMed]

J. Nachmias and R. M. Steinman, Vision Res. 5, 545 (1965).
[Crossref] [PubMed]

T. N. Cornsweet and H. M. Pinsker, J. Physiol. (London) 176, 294 (1965).

1963 (3)

1962 (1)

H. B. Barlow, J. Physiol. (London) 160, 169 (1962).

1961 (2)

J. A. Swets, W. P. Tanner, and T. G. Birdsall, Psychol. Rev. 68, 301 (1961).
[Crossref] [PubMed]

W. P. Tanner, Ann. N. Y. Acad. Sci. 89, 752 (1961).
[Crossref] [PubMed]

1957 (2)

H. B. Barlow, J. Physiol. (London) 136, 469 (1957).

R. FitzHugh, J. Gen. Physiol. 40, 925 (1957).

1956 (1)

1954 (2)

W. P. Tanner and J. A. Swets, Psychol. Rev. 61, 401 (1954).
[Crossref] [PubMed]

W. W. Peterson, T. G. Birdsall, and W. C. Fox, Trans. IRE PGIT 4, 171 (1954).

1942 (1)

S. Hecht, S. Shlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819 (1942).

Barlow, H. B.

H. B. Barlow and W. R. Levick, J. Physiol. (London) 200, 1 (1969).

H. B. Barlow, J. Physiol. (London) 160, 169 (1962).

H. B. Barlow, J. Physiol. (London) 136, 469 (1957).

H. B. Barlow, J. Opt. Soc. Am. 46, 634 (1956).
[Crossref] [PubMed]

Birdsall, T. G.

J. A. Swets, W. P. Tanner, and T. G. Birdsall, Psychol. Rev. 68, 301 (1961).
[Crossref] [PubMed]

W. W. Peterson, T. G. Birdsall, and W. C. Fox, Trans. IRE PGIT 4, 171 (1954).

Blackwell, H. R.

Cornsweet, T. N.

T. N. Cornsweet and H. M. Pinsker, J. Physiol. (London) 176, 294 (1965).

Creelman, C. D.

C. D. Creelman, J. Acoust. Soc. Am. 35, 1201 (1963).
[Crossref]

Emerson, P. L.

P. L. Emerson, Percept. Psychophys. 5, 2 (1969).
[Crossref]

Fabian, M.

J. Stone and M. Fabian, Vision Res. 8, 1023 (1968).
[Crossref] [PubMed]

FitzHugh, R.

R. FitzHugh, J. Gen. Physiol. 40, 925 (1957).

Fox, W. C.

W. W. Peterson, T. G. Birdsall, and W. C. Fox, Trans. IRE PGIT 4, 171 (1954).

Green, D. M.

D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (John Wiley & Sons, Inc., New York, 1966).

Hecht, S.

S. Hecht, S. Shlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819 (1942).

Howarth, C. I.

C. I. Howarth and G. Lowe, Nature 212, 324 (1966).
[Crossref] [PubMed]

Krantz, D.

D. Krantz, Psychol. Rev. 76, 308 (1969).
[Crossref] [PubMed]

Leshowitz, B.

B. Leshowitz, H. B. Taub, and D. H. Raab, Percept. Psychophys. 4, 207 (1968).
[Crossref]

Levick, W. R.

H. B. Barlow and W. R. Levick, J. Physiol. (London) 200, 1 (1969).

Lowe, G.

G. Lowe, Percept. Psychophys. 2, 278 (1967).
[Crossref]

C. I. Howarth and G. Lowe, Nature 212, 324 (1966).
[Crossref] [PubMed]

Nachmias, J.

J. Nachmias, Percept. Psychophys. 3, 151 (1968).

J. Nachmias and R. M. Steinman [Vision Res. 5, 545 (1965)] reported that ROC curves for discrimination between flashes of high luminance had a tendency to be convex upward when plotted on double-gaussian coordinates. A similar tendency was barely discernible in the data from some observers in the present study, though it was quite pronounced in the data from one observer (J. B.). Her more aberrant results will not be included in this paper.
[Crossref] [PubMed]

J. Nachmias and R. M. Steinman, Vision Res. 5, 545 (1965).
[Crossref] [PubMed]

J. Nachmias and R. M. Steinman, J. Opt. Soc. Am. 53, 1206 (1963); J. Opt. Soc. Am. 54, 554E (1964).
[Crossref]

Peterson, W. W.

W. W. Peterson, T. G. Birdsall, and W. C. Fox, Trans. IRE PGIT 4, 171 (1954).

Pinsker, H. M.

T. N. Cornsweet and H. M. Pinsker, J. Physiol. (London) 176, 294 (1965).

Pirenne, M. H.

S. Hecht, S. Shlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819 (1942).

Raab, D. H.

B. Leshowitz, H. B. Taub, and D. H. Raab, Percept. Psychophys. 4, 207 (1968).
[Crossref]

Shlaer, S.

S. Hecht, S. Shlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819 (1942).

Steinman, R. M.

J. Nachmias and R. M. Steinman, Vision Res. 5, 545 (1965).
[Crossref] [PubMed]

J. Nachmias and R. M. Steinman [Vision Res. 5, 545 (1965)] reported that ROC curves for discrimination between flashes of high luminance had a tendency to be convex upward when plotted on double-gaussian coordinates. A similar tendency was barely discernible in the data from some observers in the present study, though it was quite pronounced in the data from one observer (J. B.). Her more aberrant results will not be included in this paper.
[Crossref] [PubMed]

J. Nachmias and R. M. Steinman, J. Opt. Soc. Am. 53, 1206 (1963); J. Opt. Soc. Am. 54, 554E (1964).
[Crossref]

Stone, J.

J. Stone and M. Fabian, Vision Res. 8, 1023 (1968).
[Crossref] [PubMed]

Swets, J. A.

J. A. Swets, W. P. Tanner, and T. G. Birdsall, Psychol. Rev. 68, 301 (1961).
[Crossref] [PubMed]

W. P. Tanner and J. A. Swets, Psychol. Rev. 61, 401 (1954).
[Crossref] [PubMed]

D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (John Wiley & Sons, Inc., New York, 1966).

Tanner, W. P.

J. A. Swets, W. P. Tanner, and T. G. Birdsall, Psychol. Rev. 68, 301 (1961).
[Crossref] [PubMed]

W. P. Tanner, Ann. N. Y. Acad. Sci. 89, 752 (1961).
[Crossref] [PubMed]

W. P. Tanner and J. A. Swets, Psychol. Rev. 61, 401 (1954).
[Crossref] [PubMed]

Taub, H. B.

B. Leshowitz, H. B. Taub, and D. H. Raab, Percept. Psychophys. 4, 207 (1968).
[Crossref]

Wickelgren, W. A.

See also W. A. Wickelgren, J. Math. Psychol. 5, 102 (1968).
[Crossref]

Ann. N. Y. Acad. Sci. (1)

W. P. Tanner, Ann. N. Y. Acad. Sci. 89, 752 (1961).
[Crossref] [PubMed]

J. Acoust. Soc. Am. (1)

C. D. Creelman, J. Acoust. Soc. Am. 35, 1201 (1963).
[Crossref]

J. Gen. Physiol. (2)

S. Hecht, S. Shlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819 (1942).

R. FitzHugh, J. Gen. Physiol. 40, 925 (1957).

J. Math. Psychol. (1)

See also W. A. Wickelgren, J. Math. Psychol. 5, 102 (1968).
[Crossref]

J. Opt. Soc. Am. (3)

J. Physiol. (London) (4)

H. B. Barlow, J. Physiol. (London) 160, 169 (1962).

T. N. Cornsweet and H. M. Pinsker, J. Physiol. (London) 176, 294 (1965).

H. B. Barlow, J. Physiol. (London) 136, 469 (1957).

H. B. Barlow and W. R. Levick, J. Physiol. (London) 200, 1 (1969).

Nature (1)

C. I. Howarth and G. Lowe, Nature 212, 324 (1966).
[Crossref] [PubMed]

Percept. Psychophys. (4)

G. Lowe, Percept. Psychophys. 2, 278 (1967).
[Crossref]

B. Leshowitz, H. B. Taub, and D. H. Raab, Percept. Psychophys. 4, 207 (1968).
[Crossref]

P. L. Emerson, Percept. Psychophys. 5, 2 (1969).
[Crossref]

J. Nachmias, Percept. Psychophys. 3, 151 (1968).

Psychol. Rev. (3)

D. Krantz, Psychol. Rev. 76, 308 (1969).
[Crossref] [PubMed]

W. P. Tanner and J. A. Swets, Psychol. Rev. 61, 401 (1954).
[Crossref] [PubMed]

J. A. Swets, W. P. Tanner, and T. G. Birdsall, Psychol. Rev. 68, 301 (1961).
[Crossref] [PubMed]

Trans. IRE PGIT (1)

W. W. Peterson, T. G. Birdsall, and W. C. Fox, Trans. IRE PGIT 4, 171 (1954).

Vision Res. (3)

J. Stone and M. Fabian, Vision Res. 8, 1023 (1968).
[Crossref] [PubMed]

J. Nachmias and R. M. Steinman [Vision Res. 5, 545 (1965)] reported that ROC curves for discrimination between flashes of high luminance had a tendency to be convex upward when plotted on double-gaussian coordinates. A similar tendency was barely discernible in the data from some observers in the present study, though it was quite pronounced in the data from one observer (J. B.). Her more aberrant results will not be included in this paper.
[Crossref] [PubMed]

J. Nachmias and R. M. Steinman, Vision Res. 5, 545 (1965).
[Crossref] [PubMed]

Other (2)

D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (John Wiley & Sons, Inc., New York, 1966).

R. D. Luce, R. R. Bush, and E. Galanter, Eds., Handbook of Mathematical Psychology (John Wiley & Sons, Inc., New York, 1963), Vol. 1, p. 103.

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Figures (14)

Fig. 1
Fig. 1

Estimates of d′ vs luminance of test flash, on log–log coordinates. No background luminance. Lines connect points obtained from the same observer. The 45° line represents the relation, d′∝ test-flash luminance.

Fig. 2
Fig. 2

Estimates of d′ vs luminance of test flash, on log–log coordinates. Lines connect points obtained from the same observer in consecutive experimental sessions.

Fig. 3
Fig. 3

Estimates of d′ from two observers vs luminance of test flash, on log–log coordinates. Background luminance, 0.68 mL. The bootstrap procedure (see text) was used to obtain the upper three points in each set.

Fig. 4
Fig. 4

The relation between estimates of σs/σn and d′. Results from experimental conditions II and III are plotted as empty circles, those from condition I as filled circles. The line represents the relation, σ s / σ n = 1 + 1 4 d .

Fig. 5
Fig. 5

Examples of ROC curves for discrimination with slopes greater than unity.

Fig. 6
Fig. 6

d′ for discrimination of a luminance difference of 0.07 mL, as a function of the luminance of the pedestal upon which the difference is added.

Fig. 7
Fig. 7

Hit rates vs false-alarm rates on double-gaussian coordinates. Broken lines near each set of points derived from best-fitting lines (not shown) to other two sets in same panel. Background luminance, 6.0 mL. ■ detection of 0.074 mL; ● detection of 0.15 mL; ○ discrimination between 0.074 and 0.15 mL.

Fig. 8
Fig. 8

Same as Fig. 7. Background luminance, 0.68 mL. ■ detection of 0.023 mL; ● detection of 0.046 mL; ○ discrimination between 0.023 and 0.046 mL.

Fig. 9
Fig. 9

Comparison of direct and indirect estimates of d2,3′. For explanation, see text.

Fig. 10
Fig. 10

Comparison of direct and indirect estimates of σ3/σ2. For explanation, see text.

Fig. 11
Fig. 11

Theoretical hit rates vs false-alarm rates based on Poisson distributions, plotted on double-gaussian coordinates. μn = 1.3; Δμ = 1.5 (●), 3.0 (○).

Fig. 12
Fig. 12

Same as Fig. 11, with μn = 40; Δμ = 8, 16, and 24.

Fig. 13
Fig. 13

The values of the x intercept of ROC curves such as those in Fig. 11 vs signal-to-noise ratio in the critical cell, on log–log coordinates. Left panel μn = 1.3; right panel, μn = 40. In m − 1 cells, μs = μn.

Fig. 14
Fig. 14

Same as Fig. 12, except that 16 cells are being monitored for 15 of which μs = μn = 40.

Equations (5)

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z ( S s ) = b [ z ( S n ) - d ] .
b = σ n / σ s             d = μ s - μ n / σ n d .
( d 1 , 3 - d 1 , 2 ) b 1 , 2 = ( μ 3 - μ 1 σ 1 - μ 2 - μ 1 σ 1 ) σ 1 σ 2 = μ 3 - μ 2 σ 2 = d ˆ 2 , 3 b 1 , 2 ÷ b 1 , 3 = σ 1 σ 2 ÷ σ 1 σ 3 = σ 3 σ 2 = 1 / b ˆ 2 , 3 .
P ( S s ) = x = x c ( μ s x / x ! ) exp ( - μ s ) x = x c p ( x ; μ s ) P ( S n ) = x = x c ( μ n x / x ! ) exp ( - μ n ) x = x c p ( x ; μ n ) }             x c = 0 , 1 , 2 , 3 .
P ( S s ) = 1 - [ x = 0 x c p ( x ; μ s ) ] [ x = 0 x c p ( x ; μ n ) ] ( m - 1 ) P ( S n ) = 1 - [ x = 0 x c p ( x ; μ n ) ] m .