Uncertainty explains many aspects of visual contrast detection and discrimination

Denis G. Pelli

Denis G. Pelli

^{}Institute for Sensory Research, Syracuse University, Syracuse, New York, 13210 USA

^{}Some of these results appeared in my Ph.D. dissertation,62 and some were presented at the 1980 and 1984 Annual Meetings of the Optical Society of America in Chicago15 and San Diego63 and at the 1984 meeting of the European Conference on Visual Perception in Cambridge, England.64

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.

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Each trial was a Monte Carlo simulation of an observer with a psychometric function given by Eq. (4.9), with parameters M (as listed) and K = 1. The threshold α′ and steepness β are the parameters of the Weibull function for a maximum-likelihood fit42; the threshold
${c}_{0.76}^{\prime}$and steepness b are the parameters of the d′ power law [i.e., a straight line in Fig. 6(a)] for a maximum-likelihood fit. Each fit was made to the result of 100,000 Monte Carlo trials distributed over 10 contrasts. The table entries are the average of 10 replications. The Weibull parameter γ was fixed at 0.5 since these are 2afc trials.

Each trial was a Monte Carlo simulation with a hit rate given by Eq. (4.6), with parameters λ and M, as listed, and K = 1. The subjective criterion λ is listed in the left-hand column. The uncertainty M is listed in the bottom row. The threshold α′, steepness β, and false-alarm rate γ are the parameters of the Weibull function for a maximum-likelihood fit.42 Each fit was made to the result of 2,500 Monte Carlo trials distributed over five contrasts, including one near zero. Each table entry is the average of 10 replications. Only the significant digits are shown; α′ and β have no significant digits when γ is near 1.

Ref. 12. By conducting a rating-scale experiment at several contrasts, Nachmias obtained yes–no psychometric functions for three subjective criterion levels simultaneously. Furthermore, for some of the conditions, a 2afc psychometric function was measured under the same conditions. The table lists maximum-likelihood estimates of the parameters of a Weibull function fit to each psychometric function. For 2afc, γ was fixed at 0.5.

The second and third columns show estimates of the parameters A and M of the uncertainty model (with K = 1) to fit the yes–no data in Table 3. The last two columns show the ratios of the resulting 2afc predictions,
$\widehat{\alpha}$and
$\widehat{\beta}$, to the actual 2afc data, α and β.

Each trial was a Monte Carlo simulation of an observer with a psychometric function given by Eq. (4.9), with parameters M (as listed) and K = 1. The threshold α′ and steepness β are the parameters of the Weibull function for a maximum-likelihood fit42; the threshold
${c}_{0.76}^{\prime}$and steepness b are the parameters of the d′ power law [i.e., a straight line in Fig. 6(a)] for a maximum-likelihood fit. Each fit was made to the result of 100,000 Monte Carlo trials distributed over 10 contrasts. The table entries are the average of 10 replications. The Weibull parameter γ was fixed at 0.5 since these are 2afc trials.

Each trial was a Monte Carlo simulation with a hit rate given by Eq. (4.6), with parameters λ and M, as listed, and K = 1. The subjective criterion λ is listed in the left-hand column. The uncertainty M is listed in the bottom row. The threshold α′, steepness β, and false-alarm rate γ are the parameters of the Weibull function for a maximum-likelihood fit.42 Each fit was made to the result of 2,500 Monte Carlo trials distributed over five contrasts, including one near zero. Each table entry is the average of 10 replications. Only the significant digits are shown; α′ and β have no significant digits when γ is near 1.

Ref. 12. By conducting a rating-scale experiment at several contrasts, Nachmias obtained yes–no psychometric functions for three subjective criterion levels simultaneously. Furthermore, for some of the conditions, a 2afc psychometric function was measured under the same conditions. The table lists maximum-likelihood estimates of the parameters of a Weibull function fit to each psychometric function. For 2afc, γ was fixed at 0.5.

The second and third columns show estimates of the parameters A and M of the uncertainty model (with K = 1) to fit the yes–no data in Table 3. The last two columns show the ratios of the resulting 2afc predictions,
$\widehat{\alpha}$and
$\widehat{\beta}$, to the actual 2afc data, α and β.