Abstract

The psychometric function for the detection of a foveal luminance increment was studied in human observers. The signal to be detected was a modulation of a 1300 cd/m2 6′ circular red target. For an ideal photodetector, the theory of signal detectability predicts that d′, the index of detectability, should rise linearly with the luminance, E, of the luminance increment unless the observer has some uncertainty concerning the parameters of the signal to be detected. Uncertainty is expressed by the parameter M that indicates the number of orthogonal signals possible. If M is greater than 1.0, (d′)2 = ln {1 − 1/M +(1/M) exp(E2/N2)} where N2 is the variance of the noise that obscures the signal. In addition, the theory predicts that the slope of the receiver operating characteristic (ROC curve) should decrease with increasing M. In one experiment, in which E was varied, a nonlinear psychometric function and an ROC curve of relatively low slope were found. In another experiment that included a pulsed background (pedestal) whether or not the signal was presented, the predicted linear M = 1 psychometric function was found. Finally, the ROC slope that was measured in rating experiments increased when the pedestal was used. Presumably, the pedestal provides the signal-parameter information that the observer could not remember. We conclude that the human observer acts like an ideal photodetector that has imperfect memory concerning the signal to be detected.

© 1974 Optical Society of America

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  1. S. Hecht, S. Shlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819 (1942).
  2. B. Sakitt, J. Physiol. 223, 131 (1972).
  3. W. P. Tanner and J. A. Swets, Psychol. Rev. 61, 401 (1954).
    [Crossref] [PubMed]
  4. W. W. Peterson, T. G. Birdsall, and W. C. Fox, IRE Trans. Inform. Theory 4, 171 (1954).
  5. J. Nachmias and E. C. Kocher, J. Opt. Soc. Am. 60, 382 (1970).
    [Crossref] [PubMed]
  6. D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).
  7. B. Leshowitz, H. B. Taub, and D. H. Raab, Percept. Psychophys. 4, 207 (1968).
    [Crossref]
  8. W. P. Tanner, Ann. N.Y. Acad. Sci. 89, 752 (1961).
    [Crossref]
  9. W. P. Tanner and R. Clark Jones, in Visual Research Problems, edited by A. Morris and E. P. Horne, Armed Forces NRC Committee on Vision (NAS–NRC Publication No. 712, Washington, D. C., 1960).
  10. L. W. Nolte and D. Jaarsma, J. Acoust. Soc. Am. 41, 497 (1967).
    [Crossref]
  11. T. G. Birdsall, Thesis (University of Michigan, 1966) (Univ. Microfilms, Ann Arbor, Mich., Order No. 67-08 217).
  12. T. E. Cohn, Am. J. Opt. 49, 1028 (1972).
    [Crossref]
  13. W. Volk, Applied Statistics for Engineers (McGraw–Hill, New York, 1969).
  14. S. Siegel, Nonparametric Statistics (McGraw–Hill, New York, 1956).
  15. T. E. Cohn, Thesis (University of Michigan, 1966) (Univ. Microfilms, Ann Arbor, Mich., Order No. 70-21 630).
  16. The label “ideal” must be qualified. We infer ideal performance from the linearity of the psychometric function, not from the value of its slope. The measured slope may be smaller than that achievable.
  17. R. FitzHugh, J. Gen. Physiol. 40, 925 (1957).
  18. W. P. Tanner, D. Main, and T. E. Cohn, in Current Developments in Optics and Vision, edited by W. Benson and M. Whitcomb, (NAS–NRC, Washington, D. C., 1968).
  19. T. E. Cohn and D. Lasley, Abstracts of 1974 A.R.V.O. Meeting (1974).

1974 (1)

T. E. Cohn and D. Lasley, Abstracts of 1974 A.R.V.O. Meeting (1974).

1972 (2)

B. Sakitt, J. Physiol. 223, 131 (1972).

T. E. Cohn, Am. J. Opt. 49, 1028 (1972).
[Crossref]

1970 (1)

1968 (1)

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

1967 (1)

L. W. Nolte and D. Jaarsma, J. Acoust. Soc. Am. 41, 497 (1967).
[Crossref]

1961 (1)

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

1957 (1)

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

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, IRE Trans. Inform. Theory 4, 171 (1954).

1942 (1)

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

Birdsall, T. G.

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

T. G. Birdsall, Thesis (University of Michigan, 1966) (Univ. Microfilms, Ann Arbor, Mich., Order No. 67-08 217).

Clark Jones, R.

W. P. Tanner and R. Clark Jones, in Visual Research Problems, edited by A. Morris and E. P. Horne, Armed Forces NRC Committee on Vision (NAS–NRC Publication No. 712, Washington, D. C., 1960).

Cohn, T. E.

T. E. Cohn and D. Lasley, Abstracts of 1974 A.R.V.O. Meeting (1974).

T. E. Cohn, Am. J. Opt. 49, 1028 (1972).
[Crossref]

T. E. Cohn, Thesis (University of Michigan, 1966) (Univ. Microfilms, Ann Arbor, Mich., Order No. 70-21 630).

W. P. Tanner, D. Main, and T. E. Cohn, in Current Developments in Optics and Vision, edited by W. Benson and M. Whitcomb, (NAS–NRC, Washington, D. C., 1968).

FitzHugh, R.

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

Fox, W. C.

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

Green, D. M.

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

Hecht, S.

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

Jaarsma, D.

L. W. Nolte and D. Jaarsma, J. Acoust. Soc. Am. 41, 497 (1967).
[Crossref]

Kocher, E. C.

Lasley, D.

T. E. Cohn and D. Lasley, Abstracts of 1974 A.R.V.O. Meeting (1974).

Leshowitz, B.

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

Main, D.

W. P. Tanner, D. Main, and T. E. Cohn, in Current Developments in Optics and Vision, edited by W. Benson and M. Whitcomb, (NAS–NRC, Washington, D. C., 1968).

Nachmias, J.

Nolte, L. W.

L. W. Nolte and D. Jaarsma, J. Acoust. Soc. Am. 41, 497 (1967).
[Crossref]

Peterson, W. W.

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

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]

Sakitt, B.

B. Sakitt, J. Physiol. 223, 131 (1972).

Shlaer, S.

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

Siegel, S.

S. Siegel, Nonparametric Statistics (McGraw–Hill, New York, 1956).

Swets, J. A.

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 (Wiley, New York, 1966).

Tanner, W. P.

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

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

W. P. Tanner and R. Clark Jones, in Visual Research Problems, edited by A. Morris and E. P. Horne, Armed Forces NRC Committee on Vision (NAS–NRC Publication No. 712, Washington, D. C., 1960).

W. P. Tanner, D. Main, and T. E. Cohn, in Current Developments in Optics and Vision, edited by W. Benson and M. Whitcomb, (NAS–NRC, Washington, D. C., 1968).

Taub, H. B.

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

Volk, W.

W. Volk, Applied Statistics for Engineers (McGraw–Hill, New York, 1969).

Abstracts of 1974 A.R.V.O. Meeting (1)

T. E. Cohn and D. Lasley, Abstracts of 1974 A.R.V.O. Meeting (1974).

Am. J. Opt. (1)

T. E. Cohn, Am. J. Opt. 49, 1028 (1972).
[Crossref]

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

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

IRE Trans. Inform. Theory (1)

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

J. Acoust. Soc. Am. (1)

L. W. Nolte and D. Jaarsma, J. Acoust. Soc. Am. 41, 497 (1967).
[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. Opt. Soc. Am. (1)

J. Physiol. (1)

B. Sakitt, J. Physiol. 223, 131 (1972).

Percept. Psychophys. (1)

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

Psychol. Rev. (1)

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

Other (8)

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

T. G. Birdsall, Thesis (University of Michigan, 1966) (Univ. Microfilms, Ann Arbor, Mich., Order No. 67-08 217).

W. P. Tanner and R. Clark Jones, in Visual Research Problems, edited by A. Morris and E. P. Horne, Armed Forces NRC Committee on Vision (NAS–NRC Publication No. 712, Washington, D. C., 1960).

W. P. Tanner, D. Main, and T. E. Cohn, in Current Developments in Optics and Vision, edited by W. Benson and M. Whitcomb, (NAS–NRC, Washington, D. C., 1968).

W. Volk, Applied Statistics for Engineers (McGraw–Hill, New York, 1969).

S. Siegel, Nonparametric Statistics (McGraw–Hill, New York, 1956).

T. E. Cohn, Thesis (University of Michigan, 1966) (Univ. Microfilms, Ann Arbor, Mich., Order No. 70-21 630).

The label “ideal” must be qualified. We infer ideal performance from the linearity of the psychometric function, not from the value of its slope. The measured slope may be smaller than that achievable.

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Figures (5)

Fig. 1
Fig. 1

Two gaussian distributions of photon count for background alone (left-most) and background plus increment (right-most). Ordinate, probability. Abscissa, count.

Fig. 2
Fig. 2

Time course of stimuli used in the experiment. Ordinate, luminance (arbitrary units). The continuous (background) level was 500 cd/m2. Abscissa, time (stimulus duration was 3–4 ms.). The first experiment employed two stimulus intervals per trial. Their separation was determined by the subject and was approximately 0.2 s. The stimulus, an increment of luminance described in text, appeared in one of the two intervals (left-hand figure). In a pedestal run, another luminance increment of the same time course was added to both intervals (right-hand figure). In the second experiment, there was one stimulus interval per trial; the stimulus occurred during only half of the trials. The pedestal always occurred in the pedestal runs.

Fig. 3
Fig. 3

Results for a non-pedestal experiment that demonstrates very-poor performance for low signal luminance. Top figure: ordinate, d′; abscissa, luminance, E, of signal increment. Luminance units represent centimeters on oscilloscope monitor. 1 cm corresponds to 250 cd/m2. Bottom, loged′ vs logeE. These data do not support the hypothesis that d′ = KE, because a least-squares regression line (solid) intersects the energy axis to right of the origin (top) and because the slope of the regression line on logarithmic coordinates (bottom) is greater than 1. Previous investigators have concluded from similar data that the psychometric function is some nonlinear function like the dotted curve in the upper figure. To the contrary, we interpret these data as evidence of subject’s uncertainty for signal parameters. (Subject BK.)

Fig. 4
Fig. 4

Results of a pedestal experiment to verify the linear psychometric function, d′ = KE. Coordinates as in Fig. 3. (On abscissa, one division corresponds to 250 cd/m2.) Experiment was repeated three times (circles, triangles, squares) for each of the three subjects (upper = LT; middle = TC; bottom = BK). Solid curves are least-squares regression lines. There is no evidence of weak-signal suppression or of a power-low function, because the intercept on linear graphs is nearly at the origin and the slopes on log–log graphs are not significantly different from unity.

Fig. 5
Fig. 5

ROC curve for pedestal and no-pedestal experiments. Probability coordinates. Ordinate, hit rate. Abscissa, false-alarm rate. Solid points are for the pedestal condition; open points for no pedestal. Best (circles) and worst (squares) cases from eight experiments are shown. Theory predicts that the ROC slope will increase if subject uncertainty is removed by the pedestal; the prediction is supported by data in all eight cases (see text). (Subject LT.)

Tables (2)

Tables Icon

Table I Conversion of raw data to estimates of d′. 2 × 2 Stimulus/response matrix

Tables Icon

Table II Example of 2 × 4 stimulus–response matrix in rating-scale experiment. (Subject TC.)

Equations (11)

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Max { Pr ( Hit ) - w Pr ( False Alarm ) } ,
l ( x ) = f ( x / s ) f ( x / n ) .
signal             if             g ( l ( x ) ) > g ( w ) ,
g { l ( x ) } = ln l ( x ) = z
d = E / N .
l ( X ) = f ( X / S ) f ( X / n ) ,
l ( X ) = 1 M j f ( X / s j ) / f ( X / n ) .
f ( X / s j ) = i f ( x i / s j ) .
l ( X ) = f ( X / s ) f ( X / n ) = 1 M j i f ( x i / s j ) / i f ( x i / n ) .
ln l ( x ) = ln { ( 1 / M ) i e z i } ,
( d ) 2 = ln { 1 - 1 M + 1 M exp ( E 2 / N 2 ) } ,