Abstract

Discrimination thresholds for spatial frequency and contrast were measured as a function of (1) the spatial-frequency bandwidth of the stimuli, (2) the reference contrast of the stimuli, and (3) whether the observer knew on which dimension, spatial frequency or contrast, the stimuli would differ. Relative bandwidth was manipulated by a one-dimensional Gaussian window and varied from 0.25 to 2.0 octaves. The results of two different tasks were compared. In a single-judgment task the observer knew on which dimension the stimuli would differ, and discrimination thresholds were measured in a standard two-interval forced-choice staircase procedure. In a dual-judgment task the dimension on which the stimuli differed varied from trial to trial. In this task the observer made two judgments on each trial in a modified two-interval forced-choice procedure: (1) on which dimension the stimuli differed and (2) in which interval the stimulus with the higher spatial frequency or contrast appeared. Thresholds were measured by five interleaved staircases for five reference contrast levels that varied from 0.02 to 0.32. Thresholds for both spatial frequency and contrast increased with increasing stimulus bandwidth and decreased with increasing stimulus contrast. Thresholds were, on average, 1.8 times higher in the dual-judgment task than in the single-judgment task, independent of the stimulus contrast and the bandwidth. A stimulus-uncertainty model is described that accounts for the difference between thresholds in the single- and the dual-judgment tasks.

© 1993 Optical Society of America

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References

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  23. The error distribution was tested with the χ2 statistic(2)χ2=∑i=1n((fobserved−fexpected)2fexpected),with df = 1, when we compared errors made with respect to the interval to those made with respect to the dimension (summed over correct and incorrect interval judgments). In its more general form the χ2 statistic can be used to test the frequency distribution in an n × m contingency table,(3)c2=∑i=1n∑j=1m((fobserved−fexpected)2fexpected),where fexpected = NiXj/N and df = (n − 1)(m − 1). This more general form of the χ2 statistic was used to test the effects of stimulus contrast and bandwidth on the error distributions.
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1992 (1)

M. W. Greenlee, “Spatial frequency discrimination of band-limited periodic targets: effects of stimulus contrast, bandwidth and retinal eccentricity,” Vis. Res. 32, 275–283 (1992).
[CrossRef] [PubMed]

1990 (1)

J. P. Thomas, L. A. Olzak, “Cue summation in spatial discriminations,” Vis. Res. 30, 1865–1875 (1990).
[CrossRef] [PubMed]

1989 (1)

1986 (1)

1985 (2)

1983 (2)

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

1982 (3)

1981 (2)

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vis. Res. 21, 1115–1122 (1981).
[CrossRef] [PubMed]

E. T. Davis, N. Graham, “Spatial frequency uncertainty effects in the detection of sinusoidal gratings,” Vis. Res. 21, 705–712 (1981).
[CrossRef] [PubMed]

1980 (1)

1978 (1)

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

1977 (1)

C. S. Furchner, J. P. Thomas, F. W. Campbell, “Detection and discrimination of simple and complex patterns at low spatial frequencies,” Vis. Res. 17, 827–836 (1977).
[CrossRef] [PubMed]

1974 (1)

1971 (1)

N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vis. Res. 11, 251–259 (1971).
[CrossRef]

1970 (1)

1956 (1)

W. P. Tanner, “Theory of recognition,” J. Acoust. Soc. Am. 28, 882–888 (1956).
[CrossRef]

Barker, R. A.

Campbell, F. W.

C. S. Furchner, J. P. Thomas, F. W. Campbell, “Detection and discrimination of simple and complex patterns at low spatial frequencies,” Vis. Res. 17, 827–836 (1977).
[CrossRef] [PubMed]

F. W. Campbell, J. Nachmias, J. Jukes, “Spatial frequency discrimination in human vision,” J. Opt. Soc. Am. 60, 555–559 (1970).
[CrossRef] [PubMed]

Cohn, T.

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]

E. T. Davis, N. Graham, “Spatial frequency uncertainty effects in the detection of sinusoidal gratings,” Vis. Res. 21, 705–712 (1981).
[CrossRef] [PubMed]

Dosher, B. A.

G. Sperling, B. A. Dosher, “Strategy and optimization in human information processing,” in Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, J. P. Thomas, eds. (Wiley, New York, 1986).

Furchner, C. S.

C. S. Furchner, J. P. Thomas, F. W. Campbell, “Detection and discrimination of simple and complex patterns at low spatial frequencies,” Vis. Res. 17, 827–836 (1977).
[CrossRef] [PubMed]

Gille, J.

Gouled Smith, B.

Graham, N.

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]

E. T. Davis, N. Graham, “Spatial frequency uncertainty effects in the detection of sinusoidal gratings,” Vis. Res. 21, 705–712 (1981).
[CrossRef] [PubMed]

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

N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vis. Res. 11, 251–259 (1971).
[CrossRef]

N. Graham, Visual Pattern Analyzers (Oxford U. Press, New York, 1989).
[CrossRef]

Greenlee, M. W.

M. W. Greenlee, “Spatial frequency discrimination of band-limited periodic targets: effects of stimulus contrast, bandwidth and retinal eccentricity,” Vis. Res. 32, 275–283 (1992).
[CrossRef] [PubMed]

Hirsch, J.

Howard, E.

E. Howard, Perception of Sinusoidal Gratings as a Function of Gaussian Truncation (U. California Press, Los Angeles, Calif., 1989).

Hylton, R.

Jukes, J.

Kim, C. B. Y.

Klein, S. A.

Kramer, P.

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]

Lasley, E. D. J.

Lieberman, H.

H. Lieberman, A. P. Pentland, “Microcomputer-based estimation of psychophysical thresholds: the best PEST,” Behav. Res. Methods Instrum. Computers 14, 21–25 (1982).
[CrossRef]

Marcelja, S.

Mayer, M. J.

Nachmias, J.

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

N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vis. Res. 11, 251–259 (1971).
[CrossRef]

F. W. Campbell, J. Nachmias, J. Jukes, “Spatial frequency discrimination in human vision,” J. Opt. Soc. Am. 60, 555–559 (1970).
[CrossRef] [PubMed]

Olzak, L. A.

J. P. Thomas, L. A. Olzak, “Cue summation in spatial discriminations,” Vis. Res. 30, 1865–1875 (1990).
[CrossRef] [PubMed]

J. P. Thomas, L. A. Olzak, “Simultaneous detection and identification,” in Multidimensional Models of Perception and Cognition, F. G. Ashby, ed. (Erlbaum, Hillsdale, N.J., to be published).

Pelli, D. G.

Pentland, A. P.

H. Lieberman, A. P. Pentland, “Microcomputer-based estimation of psychophysical thresholds: the best PEST,” Behav. Res. Methods Instrum. Computers 14, 21–25 (1982).
[CrossRef]

Robson, J. G.

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vis. Res. 21, 1115–1122 (1981).
[CrossRef] [PubMed]

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

Sperling, G.

G. Sperling, B. A. Dosher, “Strategy and optimization in human information processing,” in Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, J. P. Thomas, eds. (Wiley, New York, 1986).

Tanner, W. P.

W. P. Tanner, “Theory of recognition,” J. Acoust. Soc. Am. 28, 882–888 (1956).
[CrossRef]

Thomas, J. P.

J. P. Thomas, L. A. Olzak, “Cue summation in spatial discriminations,” Vis. Res. 30, 1865–1875 (1990).
[CrossRef] [PubMed]

B. Gouled Smith, J. P. Thomas, “Why are some spatial discriminations independent of contrast?” J. Opt. Soc. Am. A 6, 713–724 (1989).
[CrossRef]

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

J. P. Thomas, J. Gille, R. A. Barker, “Simultaneous visual detection and identification,” J. Opt. Soc. Am. 72, 1642–1651 (1982).
[CrossRef] [PubMed]

C. S. Furchner, J. P. Thomas, F. W. Campbell, “Detection and discrimination of simple and complex patterns at low spatial frequencies,” Vis. Res. 17, 827–836 (1977).
[CrossRef] [PubMed]

J. P. Thomas, L. A. Olzak, “Simultaneous detection and identification,” in Multidimensional Models of Perception and Cognition, F. G. Ashby, ed. (Erlbaum, Hillsdale, N.J., to be published).

Watson, A. B.

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vis. Res. 21, 1115–1122 (1981).
[CrossRef] [PubMed]

Behav. Res. Methods Instrum. Computers (1)

H. Lieberman, A. P. Pentland, “Microcomputer-based estimation of psychophysical thresholds: the best PEST,” Behav. Res. Methods Instrum. Computers 14, 21–25 (1982).
[CrossRef]

J. Acoust. Soc. Am. (1)

W. P. Tanner, “Theory of recognition,” J. Acoust. Soc. Am. 28, 882–888 (1956).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (4)

Percept. Psychophys. (1)

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]

Vis. Res. (7)

E. T. Davis, N. Graham, “Spatial frequency uncertainty effects in the detection of sinusoidal gratings,” Vis. Res. 21, 705–712 (1981).
[CrossRef] [PubMed]

N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vis. Res. 11, 251–259 (1971).
[CrossRef]

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

M. W. Greenlee, “Spatial frequency discrimination of band-limited periodic targets: effects of stimulus contrast, bandwidth and retinal eccentricity,” Vis. Res. 32, 275–283 (1992).
[CrossRef] [PubMed]

C. S. Furchner, J. P. Thomas, F. W. Campbell, “Detection and discrimination of simple and complex patterns at low spatial frequencies,” Vis. Res. 17, 827–836 (1977).
[CrossRef] [PubMed]

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vis. Res. 21, 1115–1122 (1981).
[CrossRef] [PubMed]

J. P. Thomas, L. A. Olzak, “Cue summation in spatial discriminations,” Vis. Res. 30, 1865–1875 (1990).
[CrossRef] [PubMed]

Other (5)

N. Graham, Visual Pattern Analyzers (Oxford U. Press, New York, 1989).
[CrossRef]

G. Sperling, B. A. Dosher, “Strategy and optimization in human information processing,” in Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, J. P. Thomas, eds. (Wiley, New York, 1986).

E. Howard, Perception of Sinusoidal Gratings as a Function of Gaussian Truncation (U. California Press, Los Angeles, Calif., 1989).

J. P. Thomas, L. A. Olzak, “Simultaneous detection and identification,” in Multidimensional Models of Perception and Cognition, F. G. Ashby, ed. (Erlbaum, Hillsdale, N.J., to be published).

The error distribution was tested with the χ2 statistic(2)χ2=∑i=1n((fobserved−fexpected)2fexpected),with df = 1, when we compared errors made with respect to the interval to those made with respect to the dimension (summed over correct and incorrect interval judgments). In its more general form the χ2 statistic can be used to test the frequency distribution in an n × m contingency table,(3)c2=∑i=1n∑j=1m((fobserved−fexpected)2fexpected),where fexpected = NiXj/N and df = (n − 1)(m − 1). This more general form of the χ2 statistic was used to test the effects of stimulus contrast and bandwidth on the error distributions.

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Figures (8)

Fig. 1
Fig. 1

Discrimination thresholds as a function of reference contrast for observer EMM. Symbols represent thresholds measured for bandwidths of different spatial frequencies. Data points are the mean values of three independent runs. Error bars show ±1 standard error of the mean, a, Spatial-frequency discrimination thresholds (Δf/f) measured in the single-judgment task; b, spatial-frequency discrimination thresholds (Δf/f) measured in the dual-judgment task; c, contrast-discrimination thresholds (Δc/c) measured in the single-judgment task; d, contrast-discrimination thresholds (Δc/c) measured in the dual-judgment task. Note the use of different scales for the spatial-frequency and the contrast-discrimination data.

Fig. 2
Fig. 2

Results for the observer MWG; otherwise as in Fig. 1.

Fig. 3
Fig. 3

Logarithms of individual thresholds measured in the dual-judgment task for observer EMM are plotted against thresholds measured in the single-judgment task for the same stimulus contrast and bandwidth condition. Symbols represent the results for the different stimulus bandwidth conditions and represent the mean value of three independent runs. Error bars show +1 standard error of the mean for single (horizontal bars) and dual (vertical bars) judgments, a, Spatial-frequency discrimination; b, contrast discrimination. Different scales are used for spatial frequency and contrast discrimination data. The dashed oblique lines correspond to the predicted relationship based on the stimulus uncertainty model (see Section 4).

Fig. 4
Fig. 4

Results for observer MWG; otherwise as in Fig. 3.

Fig. 5
Fig. 5

Results of the four-way analysis of variance performed to partition the variance resulting from the different main effects and their interactions. The portion of residual variance is shown after the variance that is due to dimension difference D has been discounted. The main effects of task complexity (A), bandwidth (B), and contrast (C) and their interactions with one another and with stimulus dimension D are shown. Levels of significance are denoted by stars (p < 0.01*; p < 0.001**; p < 0.0001***).

Fig. 6
Fig. 6

Schematic illustration of the two-dimensional response space proposed to account for the elevations in discrimination thresholds in the dual-judgment task. The axes represent the estimated difference in contrast (ordinate) and spatial frequency (abscissa) of the stimuli that were presented in the first and second intervals. Each of the two centered dots represents the mean of unit-variance Gaussian probability distributions.

Fig. 7
Fig. 7

Schematic illustration of the two-dimensional response space and the corresponding decision criterion (heavy line) required for a single-judgment task when the differences are in spatial frequency. The decision-criterion line is positioned to partition the probability distribution into 75% right of the line and 25% left of the line and thus corresponds to the threshold criterion of 75% used in the single-judgment tasks. The observer should respond “first interval” for any estimate falling to the right of this line.

Fig. 8
Fig. 8

Schematic illustration of the two-dimensional response space and the corresponding decision criteria (heavy lines) required for the dual-judgment task when the differences are in spatial frequency. The mean of this probability distribution is positioned so that 62.5% of the density function falls to the right of both decision lines. The observer should respond “frequency difference, first interval” for estimates that fall within this quadrant.

Tables (2)

Tables Icon

Table 1 Error Distribution for Observer EMM on Dual-Judgment Task (Collapsed over Stimulus Dimensions) for Five Reference Contrasts

Tables Icon

Table 2 Error Distribution for Observer EMM on Dual-Judgment Task (Collapsed over Stimulus Dimensions) for Four Stimulus Bandwidths

Equations (7)

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L ( x ) = L 0 ( 1 + C { sin [ 2 π f ( x x 0 ) + δ ] } ) = × exp [ ( x x 0 ) 2 / 2 σ x 2 ] exp [ ( t t 0 ) 2 / 2 σ t 2 ]
if ( f 1 f 2 ) > 0 select first interval , else select second interval .
z 1 ( 0.75 ) = Δ f = 0.674 ,
if | f 1 f 2 | > | c 1 c 2 | select frequency difference , else select contrast difference and if the selected quantity > 0 select first interval , else select second interval .
z 1 ( 0.791 ) = Δ f / 2 , Δ f = 1.141 .
χ2=i=1n((fobservedfexpected)2fexpected),
c2=i=1nj=1m((fobservedfexpected)2fexpected),

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