Abstract

More than 20 years ago, Tanner [ Ann. N.Y. Acad. Sci. 89, 752 ( 1961)] noted that observers asked to detect a signal act as though they are uncertain about the physical characteristics of the signal to be detected. The popular assumptions of probability summation and decision variable, taken together, imply this uncertainty. This paper defines an uncertainty model of visual detection that assumes that the observer is uncertain among many signals and chooses the likeliest. With only four parameters, the uncertainty model explains why d′ is approximately a power function of contrast (“nonlinear transduction”) and accurately predicts effects of summation, facilitation, noise, subjective criterion, and task for near-threshold contrasts. Thus the uncertainty model offers a synthesis of much of our current understanding of visual contrast detection and discrimination.

© 1985 Optical Society of America

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  57. The probability of the maximum sample’s being a relevant sample can be estimated from the measured proportion correct P. In a 2afc trial a “relevant hit” means that a relevant sample on the signal interval was larger than all the other samples on both the signal and the blank intervals. If the uncertainty is high, K/M < 1%, we may correct for guessing to estimate the relevant-hit rate P* ≈ 2(P− 0.5). In general, for any uncertainty, the relevant-hit rate is P* = 1 − (1 − P)(2M− K)/M, which is greater than or equal to 2(P− 0.5). Threshold α is the contrast at which the hit rate is P= 82%, which corresponds to a relevant-hit rate P* greater than or equal to 64%. Thus most of the time (≧64%) at threshold contrast c = α the decision variable will be a relevant sample. At suprathreshold contrasts c > α the proportion will be even higher, and the irrelevant samples will have little effect on performance.
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1984 (4)

D. Yager, P. Kramer, M. Shaw, N. Graham, “Detection and identification of spatial frequency: models and data,” Vision Res. 24, 1021–1036 (1984).
[CrossRef] [PubMed]

W. S. Geisler, “Physical limits of acuity and hyperacuity,” J. Opt. Soc. Am. A 1, 775–782 (1984).
[CrossRef] [PubMed]

D. G. Pelli, “Uncertainty in visual detection and identification,” J. Opt. Soc. Am. A 1, 1240 (A) (1984).

D. G. Pelli, “Intrinsic uncertainty in visual detection,” Perception 13, 14 (A) (1984).

1983 (6)

J. P. Thomas, “Underlying psychometric function for detecting gratings and identifying spatial frequency,” J. Opt. Soc. Am. 73, 751–758 (1983).
[CrossRef] [PubMed]

A. B. Watson, D. G. Pelli, “quest: A Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–786 (1983).
[CrossRef] [PubMed]

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
[CrossRef] [PubMed]

N. Graham, P. Kramer, D. Yager, “Explaining uncertainty effects and probability summation,” Invest. Ophthal. Vis. Sci. Suppl. 24, 186 (A) (1983).

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position, or contrast of visual patterns,” Percept. Psychophys. 33, 20–28 (1983).
[CrossRef] [PubMed]

1982 (1)

1981 (5)

D. J. Lasley, T. E. Cohn, “Why luminance discrimination may be better than detection,” Vision Res. 21, 273–278 (1981).
[CrossRef] [PubMed]

J. M. Foley, G. E. Legge, “Contrast detection and near-threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef] [PubMed]

J. Nachmias, “On the psychometric function for contrast detection,” Vision Res. 21, 215–223 (1981).
[CrossRef] [PubMed]

G. E. Legge, “A power law for contrast discrimination,” Vision Res. 21, 457–467 (1981).
[CrossRef] [PubMed]

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 21, 409–418 (1981).
[CrossRef] [PubMed]

1980 (2)

G. E. Legge, J. M. Foley, “Contrast masking in human vision,” J. Opt. Soc. Am. 70, 1458–1471 (1980).
[CrossRef] [PubMed]

D. G. Pelli, “Channel uncertainty as a model of visual detection,” J. Opt. Soc. Am. 70, 1628–1629 (1980).

1979 (1)

A. B. Watson, “Probability summation over time,” Vision Res. 19, 515–522 (1979).
[CrossRef] [PubMed]

1978 (5)

G. E. Legge, “Sustained and transient mechanisms in human vision: temporal and spatial properties,” Vision Res. 18, 69–81 (1978).
[CrossRef] [PubMed]

D. M. Green, T. G. Birdsall, “Detection and recognition,” Psychol. Rev. 85, 192–206 (1978).
[CrossRef]

H. R. Wilson, J. Bergen, “A four-mechanism model for threshold spatial vision,” Vision Res. 19, 19–32 (1978).
[CrossRef]

N. Graham, J. G. Robson, J. Nachmias, “Grating summation in fovea and periphery,” Vision Res. 18, 815–826 (1978).
[CrossRef] [PubMed]

G. E. Legge, “Space domain properties of a spatial-frequency channel in human vision,” Vision Res. 18, 959–969 (1978).
[CrossRef]

1977 (1)

N. Graham, “Visual detection of aperiodic stimuli by probability summation among narrow band channels,” Vision Res. 17, 637–652 (1977).
[CrossRef]

1976 (1)

1975 (1)

D. M. Green, R. D. Luce, “Parallel psychometric functions from a set of independent detectors,” Psychol. Rev. 82, 483–486 (1975).
[CrossRef]

1974 (5)

R. F. Quick, “A vector magnitude model of contrast detection,” Kybernetik 16, 65–67 (1974).
[CrossRef]

T. E. Cohn, D. J. Lasley, “Detectability of a luminance increment: effect of spatial uncertainty,” J. Opt. Soc. Am. 64, 1715–1719 (1974).
[CrossRef] [PubMed]

T. E. Cohn, L. N. Thibos, R. N. Kleinstein, “Detectability of a luminance increment,” J. Opt. Soc. Am. 64, 1321–1327 (1974).
[CrossRef] [PubMed]

C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
[CrossRef] [PubMed]

J. Nachmias, R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
[CrossRef] [PubMed]

1971 (1)

1970 (1)

1968 (1)

B. Leshowitz, H. B. Taub, D. H. Raab, “Visual detection of signals in the presence of continuous and pulsed backgrounds,” Percept. Psychophys. 4, 207–213 (1968).
[CrossRef]

1967 (1)

L. W. Nolte, D. Jaarsma, “More on the detection of one of M orthogonal signals,” J. Acoust. Soc. Am. 41, 497–505 (1967).
[CrossRef]

1963 (1)

1961 (3)

J. A. Swets, W. P. Tanner, T. G. Birdsall, “Decision processes in perception,” Psychol. Rev. 68, 301–340 (1961).
[CrossRef] [PubMed]

D. M. Green, “Detection of auditory sinusoids of uncertain frequency,” J. Acoust. Soc. Am. 33, 897–903 (1961).
[CrossRef]

W. P. Tanner, “Physiological implications of psychophysical data,” Ann. N.Y. Acad. Sci. 89, 752–765 (1961).
[CrossRef] [PubMed]

1960 (1)

C. D. Creelman, “Detection of signals of uncertain frequency,” J. Acoust. Soc. Am. 32, 805–810 (1960).
[CrossRef]

1956 (1)

1954 (5)

W. W. Peterson, T. G. Birdsall, W. C. Fox, “Theory of signal detectability,” IRE Trans. Inf. Theory PGIT-4, 171–212 (1954).

W. P. Tanner, J. A. Swets, “The human use of information. I. Signal detection for the case of the signal known exactly,” IRE Trans. Inf. Theory PGIT-4, 213–221 (1954).

W. P. Tanner, R. Z. Norman, “The human use of information: II. Signal detection for the case of an unknown signal parameter,” IRE Trans. Inf. Theory PGIT-4, 222–227 (1954).

G. S. Brindley, “The order of coincidence required for visual thresholds,” Proc. Phys. Soc. London Ser. B 67, 673–676 (1954).
[CrossRef]

J. Lieblein, “Two early papers on the relation between extreme values and tensile strength,” Biometrika 41, 559–560 (1954).

1951 (1)

W. Weibull, “A statistical distribution function of wide applicability,” J. Appl. Mech. 18, 292–297 (1951).

1948 (1)

1943 (1)

H. de Vries, “The quantum character of light and its bearing upon threshold of vision, the differential sensitivity and visual acuity of the eye,” Physica 10, 553–564 (1943).
[CrossRef]

1927 (2)

L. L. Thurstone, “A law of comparative judgement,” Psychol. Rev. 34, 273–286 (1927).
[CrossRef]

L. L. Thurstone, “Psychophysical analysis,” Am. J. Psychol. 38, 368–389 (1927).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

Barker, R. A.

Barlow, H. B.

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
[CrossRef] [PubMed]

H. B. Barlow, “Retinal noise and absolute threshold,” J. Opt. Soc. Am. 46, 634–639 (1956).
[CrossRef] [PubMed]

Bergen, J.

H. R. Wilson, J. Bergen, “A four-mechanism model for threshold spatial vision,” Vision Res. 19, 19–32 (1978).
[CrossRef]

Birdsall, T. G.

D. M. Green, T. G. Birdsall, “Detection and recognition,” Psychol. Rev. 85, 192–206 (1978).
[CrossRef]

J. A. Swets, W. P. Tanner, T. G. Birdsall, “Decision processes in perception,” Psychol. Rev. 68, 301–340 (1961).
[CrossRef] [PubMed]

W. W. Peterson, T. G. Birdsall, W. C. Fox, “Theory of signal detectability,” IRE Trans. Inf. Theory PGIT-4, 171–212 (1954).

Brindley, G. S.

G. S. Brindley, “The order of coincidence required for visual thresholds,” Proc. Phys. Soc. London Ser. B 67, 673–676 (1954).
[CrossRef]

Cohn, T. E.

Creelman, C. D.

C. D. Creelman, “Detection of signals of uncertain frequency,” J. Acoust. Soc. Am. 32, 805–810 (1960).
[CrossRef]

David, H. A.

H. A. David, Order Statistics, 2nd ed. (Wiley, New York, 1981).

Davis, E. T.

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position, or contrast of visual patterns,” Percept. Psychophys. 33, 20–28 (1983).
[CrossRef] [PubMed]

de Vries, H.

H. de Vries, “The quantum character of light and its bearing upon threshold of vision, the differential sensitivity and visual acuity of the eye,” Physica 10, 553–564 (1943).
[CrossRef]

Dean, A. F.

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–786 (1983).
[CrossRef] [PubMed]

Foley, J. M.

J. M. Foley, G. E. Legge, “Contrast detection and near-threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef] [PubMed]

G. E. Legge, J. M. Foley, “Contrast masking in human vision,” J. Opt. Soc. Am. 70, 1458–1471 (1980).
[CrossRef] [PubMed]

Fox, W. C.

W. W. Peterson, T. G. Birdsall, W. C. Fox, “Theory of signal detectability,” IRE Trans. Inf. Theory PGIT-4, 171–212 (1954).

Galambos, J.

J. Galambos, The Asymptotic Theory of Extreme Order Statistics (Wiley, New York, 1978).

Geisler, W. S.

Gille, J.

Graham, N.

D. Yager, P. Kramer, M. Shaw, N. Graham, “Detection and identification of spatial frequency: models and data,” Vision Res. 24, 1021–1036 (1984).
[CrossRef] [PubMed]

N. Graham, P. Kramer, D. Yager, “Explaining uncertainty effects and probability summation,” Invest. Ophthal. Vis. Sci. Suppl. 24, 186 (A) (1983).

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position, or contrast of visual patterns,” Percept. Psychophys. 33, 20–28 (1983).
[CrossRef] [PubMed]

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 21, 409–418 (1981).
[CrossRef] [PubMed]

N. Graham, J. G. Robson, J. Nachmias, “Grating summation in fovea and periphery,” Vision Res. 18, 815–826 (1978).
[CrossRef] [PubMed]

N. Graham, “Visual detection of aperiodic stimuli by probability summation among narrow band channels,” Vision Res. 17, 637–652 (1977).
[CrossRef]

Green, D. M.

D. M. Green, T. G. Birdsall, “Detection and recognition,” Psychol. Rev. 85, 192–206 (1978).
[CrossRef]

D. M. Green, R. D. Luce, “Parallel psychometric functions from a set of independent detectors,” Psychol. Rev. 82, 483–486 (1975).
[CrossRef]

D. M. Green, “Detection of auditory sinusoids of uncertain frequency,” J. Acoust. Soc. Am. 33, 897–903 (1961).
[CrossRef]

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

Jaarsma, D.

L. W. Nolte, D. Jaarsma, “More on the detection of one of M orthogonal signals,” J. Acoust. Soc. Am. 41, 497–505 (1967).
[CrossRef]

Klein, S.

C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
[CrossRef] [PubMed]

Kleinstein, R. N.

Kocher, E. C.

Kramer, P.

D. Yager, P. Kramer, M. Shaw, N. Graham, “Detection and identification of spatial frequency: models and data,” Vision Res. 24, 1021–1036 (1984).
[CrossRef] [PubMed]

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position, or contrast of visual patterns,” Percept. Psychophys. 33, 20–28 (1983).
[CrossRef] [PubMed]

N. Graham, P. Kramer, D. Yager, “Explaining uncertainty effects and probability summation,” Invest. Ophthal. Vis. Sci. Suppl. 24, 186 (A) (1983).

Lasley, D. J.

D. J. Lasley, T. E. Cohn, “Why luminance discrimination may be better than detection,” Vision Res. 21, 273–278 (1981).
[CrossRef] [PubMed]

T. E. Cohn, D. J. Lasley, “Detectability of a luminance increment: effect of spatial uncertainty,” J. Opt. Soc. Am. 64, 1715–1719 (1974).
[CrossRef] [PubMed]

Legge, G. E.

G. E. Legge, “A power law for contrast discrimination,” Vision Res. 21, 457–467 (1981).
[CrossRef] [PubMed]

J. M. Foley, G. E. Legge, “Contrast detection and near-threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef] [PubMed]

G. E. Legge, J. M. Foley, “Contrast masking in human vision,” J. Opt. Soc. Am. 70, 1458–1471 (1980).
[CrossRef] [PubMed]

G. E. Legge, “Sustained and transient mechanisms in human vision: temporal and spatial properties,” Vision Res. 18, 69–81 (1978).
[CrossRef] [PubMed]

G. E. Legge, “Space domain properties of a spatial-frequency channel in human vision,” Vision Res. 18, 959–969 (1978).
[CrossRef]

Leshowitz, B.

B. Leshowitz, H. B. Taub, D. H. Raab, “Visual detection of signals in the presence of continuous and pulsed backgrounds,” Percept. Psychophys. 4, 207–213 (1968).
[CrossRef]

Lieblein, J.

J. Lieblein, “Two early papers on the relation between extreme values and tensile strength,” Biometrika 41, 559–560 (1954).

Linfoot, E. H.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal, New York, 1964).

Luce, R. D.

D. M. Green, R. D. Luce, “Parallel psychometric functions from a set of independent detectors,” Psychol. Rev. 82, 483–486 (1975).
[CrossRef]

Movshon, J. A.

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–786 (1983).
[CrossRef] [PubMed]

Nachmias, J.

J. Nachmias, “On the psychometric function for contrast detection,” Vision Res. 21, 215–223 (1981).
[CrossRef] [PubMed]

N. Graham, J. G. Robson, J. Nachmias, “Grating summation in fovea and periphery,” Vision Res. 18, 815–826 (1978).
[CrossRef] [PubMed]

J. Nachmias, R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
[CrossRef] [PubMed]

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W. Weibull, “A statistical distribution function of wide applicability,” J. Appl. Mech. 18, 292–297 (1951).

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[CrossRef]

A. B. Watson, D. G. Pelli, “quest: A Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

Perception (1)

D. G. Pelli, “Intrinsic uncertainty in visual detection,” Perception 13, 14 (A) (1984).

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D. M. Green, R. D. Luce, “Parallel psychometric functions from a set of independent detectors,” Psychol. Rev. 82, 483–486 (1975).
[CrossRef]

D. M. Green, T. G. Birdsall, “Detection and recognition,” Psychol. Rev. 85, 192–206 (1978).
[CrossRef]

L. L. Thurstone, “A law of comparative judgement,” Psychol. Rev. 34, 273–286 (1927).
[CrossRef]

J. A. Swets, W. P. Tanner, T. G. Birdsall, “Decision processes in perception,” Psychol. Rev. 68, 301–340 (1961).
[CrossRef] [PubMed]

Vision Res. (15)

D. Yager, P. Kramer, M. Shaw, N. Graham, “Detection and identification of spatial frequency: models and data,” Vision Res. 24, 1021–1036 (1984).
[CrossRef] [PubMed]

D. J. Lasley, T. E. Cohn, “Why luminance discrimination may be better than detection,” Vision Res. 21, 273–278 (1981).
[CrossRef] [PubMed]

C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
[CrossRef] [PubMed]

J. Nachmias, R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
[CrossRef] [PubMed]

J. M. Foley, G. E. Legge, “Contrast detection and near-threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef] [PubMed]

J. Nachmias, “On the psychometric function for contrast detection,” Vision Res. 21, 215–223 (1981).
[CrossRef] [PubMed]

G. E. Legge, “Sustained and transient mechanisms in human vision: temporal and spatial properties,” Vision Res. 18, 69–81 (1978).
[CrossRef] [PubMed]

A. B. Watson, “Probability summation over time,” Vision Res. 19, 515–522 (1979).
[CrossRef] [PubMed]

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–786 (1983).
[CrossRef] [PubMed]

N. Graham, “Visual detection of aperiodic stimuli by probability summation among narrow band channels,” Vision Res. 17, 637–652 (1977).
[CrossRef]

H. R. Wilson, J. Bergen, “A four-mechanism model for threshold spatial vision,” Vision Res. 19, 19–32 (1978).
[CrossRef]

N. Graham, J. G. Robson, J. Nachmias, “Grating summation in fovea and periphery,” Vision Res. 18, 815–826 (1978).
[CrossRef] [PubMed]

G. E. Legge, “Space domain properties of a spatial-frequency channel in human vision,” Vision Res. 18, 959–969 (1978).
[CrossRef]

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 21, 409–418 (1981).
[CrossRef] [PubMed]

G. E. Legge, “A power law for contrast discrimination,” Vision Res. 21, 457–467 (1981).
[CrossRef] [PubMed]

Other (12)

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

D. G. Pelli, “Effects of visual noise,” Ph.D. dissertation (Cambridge University, Cambridge, 1981).

The probability of the maximum sample’s being a relevant sample can be estimated from the measured proportion correct P. In a 2afc trial a “relevant hit” means that a relevant sample on the signal interval was larger than all the other samples on both the signal and the blank intervals. If the uncertainty is high, K/M < 1%, we may correct for guessing to estimate the relevant-hit rate P* ≈ 2(P− 0.5). In general, for any uncertainty, the relevant-hit rate is P* = 1 − (1 − P)(2M− K)/M, which is greater than or equal to 2(P− 0.5). Threshold α is the contrast at which the hit rate is P= 82%, which corresponds to a relevant-hit rate P* greater than or equal to 64%. Thus most of the time (≧64%) at threshold contrast c = α the decision variable will be a relevant sample. At suprathreshold contrasts c > α the proportion will be even higher, and the irrelevant samples will have little effect on performance.

H. A. David, Order Statistics, 2nd ed. (Wiley, New York, 1981).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

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

J. Nachmias, “Signal detection theory and its applications to problems in vision,” in Handbook of Sensory Physiology VII/4, Psychophysics, D. Jameson, L. Hurvich, eds. (Springer-Verlag, Berlin, 1972), pp. 56–77.
[CrossRef]

A. B. Watson, “Detection and recognition of simple spatial forms,” NASA Tech. Memorandum84353 (1983).

L. A. Wainstein, V. D. Zubakov, Extraction of Signals from Noise (Prentice-Hall, Englewood Cliffs, N.J., 1962).

J. Galambos, The Asymptotic Theory of Extreme Order Statistics (Wiley, New York, 1978).

H. L. van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal, New York, 1964).

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Figures (11)

Fig. 1
Fig. 1

The probability-summation assumption. The stimulus, either a signal or a blank, is a function of space and time. The arrows on the left are broad to indicate the transmission of an entire function. The thin arrows on the right designate transmission of simple scalar quantities, in this case yes or no. The stimulus may cause a yes for any of many relevant reasons. A yes may also occur for any of many irrelevant reasons. The long box labeled “or” will cause the observer to say “yes” if any of the reasons produces a yes.

Fig. 2
Fig. 2

The decision-variable assumption. The stimulus is either a signal or a blank, a function of space and time. All the information about the stimulus is somehow reduced to a scalar quantity L, which is the decision variable. The observer says “yes” if the decision variable L exceeds the subjective criterion λ and says “no” otherwise.

Fig. 3
Fig. 3

The probability-summation and decision-variable assumptions. The stimulus is a signal or a blank. It affects the relevant variables, L1, …, LK. Irrelevant variables, LK+1, …, LM, also produce values on each presentation. The maximum value is passed by the long box to become the decision variable L.

Fig. 4
Fig. 4

The uncertainty model. Each receptive field multiplies the input (stimulus plus noise) with an expected signal si(x, y, t) and integrates the result (i.e., it evaluates the cross correlation), yielding a single number Li on each presentation. The largest Li is passed to the decision maker on the right. The box says “choose largest” to indicate that the identity as well as the value of the most active receptive field is preserved. This allows the model to be applied to identification experiments. Adapted from van Trees28Fig. 4.22.

Fig. 5
Fig. 5

The probability distribution of the decision variable L in the uncertainty model. Each curve in (a) shows the probability density for the decision variable L equaling the subjective criterion λ, L = λ, as a function of λ. Each curve in (b) shows the yes–no hit rate P(c′), which is the probability of the decision variable’s exceeding the subjective criterion λ, L > λ, as a function of λ. Each figure has separate panels for five degrees of uncertainty, M = 1, 10, 100, 1,000, 10,000. Each panel has separate curves for five normalized contrasts, c′ = 0, 2, 4, 6, 8. K is 1.

Fig. 6
Fig. 6

2afc detection by the uncertainty model. The solid curves show the detectability d′ as a function of normalized contrast c′ and uncertainty M. K is 1. The dotted lines show the approximation of Peterson et al.35 for M = 100. (a) Shows that the curves are nearly straight in log–log coordinates. (b) Shows that in linear coordinates the curves tend to translate along the contrast axis and that above threshold (d′ > 1) they all have unit slope.

Fig. 7
Fig. 7

2afc detectability d′ as a function of contrast c of a grating. Proportion correct are indicated on the right. The ○’s (from Nachmias and Sansbury10) are for detection of a 9-c/deg grating, and the ×’s (from Legge41) are for detection of a 3-c/deg grating. There are 160 trials per point. The vertical bars represent 95% confidence intervals. The solid curves are fits by the uncertainty model with K = 1 and M = 800 (○’s) and M = 18 (×’s). The dashed lines are maximum-likelihood fits by Weibull functions with (β = 3.66 (○’s) and β = 2.36 (×’s).

Fig. 8
Fig. 8

Threshold contrast for a sinusoidal grating as a function of duration. The ×’s are 2afc thresholds (from Legge41), and the ○’s are yes–no thresholds (from Watson42). The dashed lines show the fit by the standard summation formula [relation (6.1)] as made by the original authors. The solid curves show the fit by the uncertainty model. For the ×’s, M = 60 and A = 0.00133. For the ○’s, λ = 3.5, A = 0.00304, and M = 3000. Both yes–no fits ignore the point at 100 msec, and both 2afc fits ignore the point at 1096 msec.

Fig. 9
Fig. 9

Analysis of yes–no performance. Each condition from Table 3 is represented in each figure by two or three points connected by dashed lines. The solid curves show the performance of the uncertainty model for values of M from 1 to 1,000,000. K is 1. (a) Normalized threshold contrast α′ versus β. Note that the curve is essentially independent of M. (b) False-alarm rate γ versus β. (c) False-alarm rate γ versus normalized threshold contrast α′.

Fig. 10
Fig. 10

2afc detection and discrimination of contrast of sinusoidal gratings. The solid curves are a fit of the uncertainty model to the detection data (circles) and a prediction of the discrimination data (squares). The dashed curves show the prediction of d′ additivity. The data are from Foley and Legge.11 The vertical bars in (a) indicate the 95% binomial confidence interval. (a) and (b) Show the 0.5- and 8-c/deg results, respectively. A similarly successful prediction was also made for Foley and Legge’s 2-c/deg results.

Fig. 11
Fig. 11

2afc contrast discrimination in noise. (a) The vertical scale is the just-detectable (82%-correct) increment in contrast Δc from a pedestal contrast of c, represented by the horizontal scale. The pattern was a 4-c/deg grating. The thresholds for detection are indicated by vertical arrows along the contrast axis. In two of the conditions (O and ×) white noise, uncorrelated over space and time, was added to the display. (b) Same as (a), but both scales have been normalized by the threshold contrast α.

Tables (4)

Tables Icon

Table 1 2afc Performance of the Uncertainty Modela

Tables Icon

Table 2 Yes–No Performance of the Uncertainty Modela

Tables Icon

Table 3 Data from Nachmiasa

Tables Icon

Table 4 Yes–No Fit Predicts 2afc Resultsa

Equations (114)

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P ( c ) = 1 ( 1 G ) [ 1 P * ( c ) ] .
P ( c ) = 1 i = K + 1 M ( 1 G ) i = 1 K ( 1 P i * ) ,
G = 1 i = K + 1 M ( 1 G i )
P * ( c ) = 1 i = 1 K ( 1 P i * ( c ) ) .
P * ( c ) 1 [ 1 P ( c ) ] / [ 1 P ( 0 ) ] .
P ( 0 ) G .
if G 1 then P * ( 0 ) 0 .
l * j = i l i , j ,
l i , * = i l i , j .
l * , 1 l * , 2 2 1 .
i l i , 1 i l i , 2 2 1 .
max i l i , 1 max l i , 2 , i 2 1
l i , j exp ( c i L i , j ) ,
i exp ( c i L i , 1 ) i exp ( c i L i , 2 ) 2 1 .
max i exp ( c i L i , 1 ) max i exp ( c i L i , 2 ) 2 1 ,
max i c i L i , 1 max i c i L i , 2 2 1 .
s ( x , y , t ) = L ( x , y , t ) / L av 1 .
E = s 2 ( x , y , t ) d x d y d t .
max i L i , 1 max i L i , 2 2 1 .
i L i , 1 i L i , 2 2 1 .
i exp ( c L i , 1 ) no yes λ .
( d ) 2 ln [ 1 1 M + 1 M exp ( c 2 ) ] .
max i L i , 1 no yes λ .
c = A c ,
Φ ( λ ) = 1 2 π λ exp ( t 2 / 2 t d ) .
P yn ( c ) = 1 ( 1 G yn ) [ 1 P yn * ( c ) ] ,
P yn * ( c ) = 1 Φ K ( λ c ) .
G yn = 1 Φ M K ( λ ) .
P yn ( c ) = 1 Φ M K ( λ ) Φ K ( λ c ) .
P yn ( 0 ) = 1 Φ M ( λ ) .
P fc ( c ) = λ = + λ = P yn ( c ) d P yn ( 0 ) .
P fc ( c ) = 1 M + Φ K ( λ c ) Φ 2 M K 1 ( λ ) × d Φ d λ ( λ ) d λ .
P ( c ) = 1 ( 1 γ ) exp [ ( c / α ) β ] ,
α yn / α yn = α fc / α fc = A .
α β .
β = 1.358 + 0.792 log M ± 0.043 .
log d b log ( c / c 0.76 ) ,
d ( c / c 0.76 ) b .
b = 0.80 β ± 0.15 .
β λ + 0.6 .
α extent 1 / β .
P i * ( c ) = 1 exp [ ( c / α i ) β ] ,
α = ( i = 1 K α i β ) 1 / β .
P ( c ) = 1 i 1 P i ( c ) ,
α = K 1 / β α i .
P ( c ) 1 ( 1 γ ) exp [ ( c / α ) β ] .
P 1 ( c ) 1 ( 1 γ 1 ) exp [ ( c / α 1 ) β 1 ] ,
P 1 * ( c ) 1 exp [ ( c / α 1 ) β 1 ] .
1 [ 1 P 1 * ( c ) ] K 1 exp [ ( K 1 / β 1 c / α 1 ) β 1 ] .
P * ( c ) 1 exp [ ( c / α ) β ] ,
α = α 1 K 1 / β 1
β = β 1 .
α β K 1 / β .
A α / ( β K 1 / β ) .
K = duration / ( 100 msec ) .
min A , K i = 1 , 3 [ log ( α i ) log ( A β i K 1 / β i ) ] 2 .
a i = log ( α i / β i ) , b i = 1 / β i
min A , K i = 1 , 3 [ a i log ( A ) b i log ( K ) ] 2 ,
α yn / α fc β yn / β fc = 4.2 / 3 = 1.4 ,
β ̂ = 1.358 + 0.792 log M ,
α ̂ = A β ̂ K 1 / β ̂ = A β ̂ ,
d 1 , 3 = d 1 , 2 + d 2 , 3 , c 1 d 1 , 2 c 2 d 2 , 3 c 3 c 1 d 1 , 3 c 3 .
P yn ( c 1 ) = 1 Φ ( λ c 1 ) Φ M 1 ( λ ) , P yn ( c 2 ) = 1 Φ ( λ c 2 ) Φ M 1 ( λ ) .
P fc ( c ) = λ = + λ = P yn ( c 1 ) d P yn ( c 2 )
= 1 + Φ ( λ c 1 ) Φ 2 M 3 ( λ ) [ Φ ( λ ) d Φ d λ ( λ c 2 ) + ( M 1 ) Φ ( λ c 2 ) d Φ d λ ( λ ) ] d λ .
d = ( c / c 0.76 ) b
c 0.76
α ̂ / α
β ̂ / β
G = 1 ( 1 F ) M K .
P ( 0 ) = 1 ( 1 F ) M .
1 P ( 0 ) = ( 1 G ) M / ( M K ) .
1 P ( 0 ) / G M / ( M K ) 1.0101 ,
P ( c ) = 1 ( 1 G ) [ 1 P * ( c ) ] .
G = ( M K ) / ( 2 M K ) .
1 < P ( 0 ) / G 1.00505 ,
A = a 1 n 1 + a 2 n 2 + + a k n k , B = b 1 n i + b 2 n 2 + + b k n k .
a 1 b 1 + a 2 b 2 + + a k b k = 0 .
Av { A B } = Av { ( a 1 n 1 + a 2 n 2 + + a k n k ) ( b 1 n i + b 2 n 2 + + b k n k ) } = Av { i = 1 , k j = 1 , k a i n i b j n j } .
Av { A } Av { B } = Av { a 1 n 1 + a 2 n 2 + + a k n k } Av { b 1 n i + b 2 n 2 + + b k n k } = Av { i = 1 , k a i n i } Av { j = 1 , k b j n j } .
Av { A B } Av { A } Av { B } = Av { i = 1 , k j = 1 , k a i n i b j n j } Av { i = 1 , k a i n i } Av { j = 1 , k b j n j } .
= i = 1 , k j = 1 , k a i b j Av { n i n j } i = 1 , k a i Av { n i } j = 1 , k b j Av { n j } .
= i = 1 , k j = 1 , k a i b j Av { n i n j } i = 1 , k j = 1 , k a i b j μ 2 = i = 1 , k j = 1 , k a i b j ( Av { n i n j } μ 2 ) .
= i a i b j ( Av { n i 2 } μ 2 ) + i j a i b j ( Av { n i n j } μ 2 ) .
= ( σ 2 μ 2 ) a i b i + 0 .
= 0 .
log ( K + 1 ) log ( 2 M ) .
1 P * ( c ) = 1 P ( X > Y ) ,
if σ X 2 σ Y 2 , then P ( X > Y ) P ( X > μ Y ) .
σ X 2 σ Y 2 ln ( 2 M K + 1 ) / ln ( K + 1 ) .
1 P ( X > Y ) 1 P ( X > μ Y ) .
= i = 1 K 1 P ( X i > μ Y ) ,
1 P * ( c ) i = 1 K 1 P ( X i > μ Y ) .
P ( X i < x ) = Φ ( x ) = 1 / 2 π x exp ( x 2 / 2 x d ) ,
P ( Z n < x ) = P n ( X < x ) = Φ n ( x ) .
a n = ( 2 ln n ) 1 / 2 0.5 ( ln ln n + ln 4 π ) / ( 2 ln n ) 1 / 2 ,
b n = 1 / ( 2 ln n ) 1 / 2 .
lim n P ( Z n b n + a n < x ) = H ( x ) ,
lim n P { ( Z n a n ) / b n < x } ,
H ( x ) = exp ( e x ) .
H ( x ) = H n { x + ln ( n ) } for all n .
H { ( x a n ) / b n } P ( Z n < x ) for n 100 .
= P n ( X < x )
= [ P 100 ( X < x ) ] n / 100 .
[ H { ( x a 100 ) / b 100 } ] n / 100 .
= H { ( x a 100 ) / b 100 ln ( n / 100 ) } ,
H { ( x a n ) / b n } H { ( x a 100 ) / b 100 ln ( n / 100 ) } .
1 b n / b 100 0.64
| a n a 100 b 100 ln ( n / 100 ) | 0.08 .
Φ n ( x ) Φ 100 { x b 100 ln ( n / 100 ) } .
P yn ( c ) 1 Φ 100 ( λ ̂ ) Φ K ( λ ̂ { c b 100 ln [ ( M K ) / 100 ] } ) ,
P yn ( 0 ) 1 Φ 100 ( λ ̂ ) ,
P 2 ( c ) P 1 [ c 0.330 ln ( M 2 / M 1 ) ] ,
P 2 ( c ) P 1 [ c 0.330 A ln ( M 2 / M 1 ) ] ,

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