Abstract

Even the highest contrast sensitivities that humans can achieve for the detection of targets on uniform fields fall far short of ideal values. Recent theoretical formulations have attributed departures from ideal performance to two factors—the existence of internal noise within the observer and suboptimal stimulus information sampling by the observer. It has been postulated that the contributions of these two factors can be evaluated separately by measuring contrast-detection thresholds as a function of the level of externally added visual noise. We wished to determine whether a similar analysis could be applied to contrast discrimination and whether variation of the increment threshold with pedestal contrast is due to changes in internal noise or sampling efficiency. We measured contrast-increment thresholds as a function of noise spectral density for near-threshold and suprathreshold pedestal contrasts. The experiments were conducted separately for static and dynamic noise. Our findings indicate that the same formulation can be applied to contrast discrimination and that changes in the estimated values of internal noise, rather than changes in sampling efficiency, play the major role in determining properties of contrast discrimination. Implications for models of contrast coding in vision are discussed.

© 1987 Optical Society of America

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  1. J. Nachmias, R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
    [CrossRef] [PubMed]
  2. C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
    [CrossRef] [PubMed]
  3. G. E. Legge, J. M. Foley, “Contrast masking in human vision,”J. Opt. Soc. Am. 70, 1458–1471 (1980).
    [CrossRef] [PubMed]
  4. J. M. Foley, G. E. Legge, “Contrast detection and near-threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
    [CrossRef] [PubMed]
  5. G. E. Legge, “A power law for contrast discrimination,” Vision Res. 21, 457–467 (1981).
    [CrossRef] [PubMed]
  6. G. E. Legge, D. Kersten, “Light and dark bars: contrast discrimination,” Vision Res. 23, 473–483 (1983).
    [CrossRef]
  7. G. E. Legge, “Spatial frequency masking in human vision: binocular interactions,”J. Opt. Soc. Am. 69, 838–847 (1979).
    [CrossRef] [PubMed]
  8. H. R. Wilson, “A transducer function for threshold and suprathreshold spatial vision,” Biol. Cyb. 38, 171–178 (1980).
    [CrossRef]
  9. G. J. Burton, “Contrast discrimination by the human visual system,” Biol. Cyb. 40, 27–38 (1981).
    [CrossRef]
  10. H. B. Barlow, “Retinal and central factors in human vision limited by noise,” in Vertebrate Photoreception, H. B. Barlow, P. Fatt, eds. (Academic, New York, 1977).
  11. D. G. Pelli, “The effects of visual noise,” Ph.D. dissertation (Department of Physiology, Cambridge University, Cambridge, UK, 1981).
  12. A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
    [CrossRef] [PubMed]
  13. In the absence of a pedestal, when Lp= L0, S(x, y, t) is sometimes termed a contrast function.14
  14. E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal, New York, 1964).
  15. N. S. Nagaraja, “Effect of luminance noise on contrast thresholds,”J. Opt. Soc. Am. 54, 950–955 (1964).
    [CrossRef]
  16. Pelli11 wrote Eq. (1) with k expressed as (d′c2/J), as presented in Eq. (3). He used the term “calculation efficiency” for J rather: than “sampling efficiency.” Nagaraja15 used “squared normalized luminance” rather than “signal energy” and “variance” rather than “spectral density” having absorbed several values into k.
  17. W. A. Wickelgren, “Unidimensional strength theory and component analysis of noise in absolute and comparative judgments,”J. Math. Psychol. 5, 102–122 (1968).
    [CrossRef]
  18. W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
    [CrossRef]
  19. A. E. Burgess, H. B. Barlow, “The precision of numerosity discrimination in arrays of random dots,” Vision Res. 23, 811–820 (1983).
    [CrossRef] [PubMed]
  20. D. Kersten, “Spatial summation in visual noise,” Vision Res. 24, 1977–1990 (1984).
    [CrossRef] [PubMed]
  21. D. Kersten, H. B. Barlow, “Searching for the high-contrast patterns we see best,” Suppl. Invest. Ophthalmol. Vis. Sci. 25, 313 (1984).
  22. D. Kersten, H. B. Barlow, “Why are contrast thresholds so high?” Suppl. Invest. Ophthalmol. Vis. Sci. 26, 140 (1985).
  23. When the analysis is done in time and two spatial dimensions, the stochastic nature of the photon flux adds noise to the stimulus. If this noise is not included as part of the external noise N, it will appear as a nonzero value of equivalent noise for the ideal observer. Pelli11 has pointed out that the photon-noise spectral density is equal to the reciprocal of the photon flux.
  24. D. G. Pelli, “Uncertainty explains many aspects of visual contrast detection and discrimination,” J. Opt. Soc. Am. A 2, 1508–1532 (1985).
    [CrossRef] [PubMed]
  25. F. W. Campbell, D. G. Green, “Optical and retinal factors affecting visual resolution,”J. Physiol. (London) 181, 576–593 (1965).
  26. P. Horowitz, W. Hill, The Art of Electronics (Cambridge U. Press, Cambridge, 1980).
  27. The rapid decay of the P31 phosphor—the intensity drops to 1% within 250 μ sec (Ref. 28)—is very short compared with the integration time of the eye. Therefore the 100-Hz frame rate of the display determines the temporal sampling characteristics relevant to the calculation of noise spectral density.
  28. R. A. Bell, “Principles of cathode-ray tubes, phosphors and high-speed oscillography,” Hewlett-Packard Application Note 115 (Hewlett-Packard, Colorado Springs, Colo., 1970).
  29. We derive this for one dimension. The three-dimensional case is a straightforward extension. We need to show that, in one dimension, the contrast power spectral density for pixel noise is(R1)N(f)=bcrms2 sinc2(bf),where b is the extent of the pixel (in degrees or seconds), crms2 is the contrast power, and f is the frequency (in cycles per degree or hertz). Consider a contrast noise sample of 1-sec duration:c(t)=∑n=0(1/b)-1c(nb) rect(t-nbb).By averaging the squared modulus of the Fourier transform of c(t), we getC(f)C*(f)¯=b2 sinc2(bf)∑n=0(1/b)-1c2(nb)¯,where we have assumed that adjacent pixel contrasts are uncorrelated. Further, assuming stationarity,N(f)=b2 sinc2(bf)crms21/b=b sinc2(bf)crms2,which is Eq. (R1). The signals in our experiments were at low spatiotemporal frequencies relative to the first zero of N(f), which occurs at f= 1/b(118 c/deg, 12 c/deg, and 100 Hz for the horizontal, vertical, and temporal frequencies). Thus the spectral density is approximately flat in the region of the signals and is given byN(f)=N(0)=crms2bor, in three dimensions, bycrms2bxbybt,where bx, by, and bt are the horizontal, vertical, and temporal sample sizes, respectively.
  30. D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Krieger, Huntington, N.Y., 1974).
  31. The quantization of the gray levels contributes additional noise. If we assume that the probabilities of the gray levels from the noise process are uniformly distributed across a quantization step, the quantization variance is Q2/12 per pixel, where Q is the gray-level step size.32 The noise spectral density equals rms contrast squared × pixel width × pixel height equals (1/128)2× (0.014)2× (1/12) equals 0.001 μ(deg2), which is negligible compared with the externally added noise.
  32. A. Burgess, “Effect of quantization noise on visual signal detection in noisy images,” J. Opt. Soc. Am. A 2, 1424–1428 (1985).
    [CrossRef] [PubMed]
  33. W. L. Hays, Statistics for Psychologists (Holt, New York, 1963).
  34. A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
    [CrossRef] [PubMed]
  35. The photopic quantum efficiencies found for intensity detection and discrimination of point sources36 are much higher (about 5%) than that of the optimal stimulus of Watson et al.34 (about 0.05%). However, a comparison in terms of contrast energy is problematic because contrast energy is not well defined for a point source.
  36. W. S. Geisler, K. D. Davila, “Ideal discriminators in spatial vision: two-point stimuli,” J. Opt. Soc. Am. A 2, 1483–1497 (1985).
    [CrossRef] [PubMed]
  37. D. Kersten, Department of Psychology, Brown University, Providence, R.I. 02912, and H. B. Barlow, Department of Physiology, Cambridge University, Cambridge CB2 3EG, UK (personal communication).
  38. For disks superimposed upon a uniform background, there is a nonlinear relationship between contrast defined as δL/L and Michelson contrast, (Lmax− Lmin)/(Lmax+ Lmin). The slopes cited in the text were calculated with contrast defined in the former way. The corresponding slopes for Michelson contrast are 0.45 and 0.62. The issue of contrast definition is dealt with in more detail by Legge and Kersten.6
  39. D. J. Lasley, T. E. Cohn, “Why luminance discrimination may be better than detection,” Vision Res. 21, 273–278 (1981).
    [CrossRef] [PubMed]
  40. The maximum-of-M-channels decision rule differs from the ideal decision rule for detection by a signal-uncertain observer.41 However, it is computationally simpler than the ideal rule and closely approximates it. For further discussion, see Pelli.24
  41. L. W. Nolte, D. Jaarsma, “More on the detection of one of M orthogonal signals,”J. Acoust. Soc. Am. 41, 497–505 (1967).
    [CrossRef]
  42. G. E. Legge, “Binocular contrast summation—II. Quadratic summation,” Vision Res. 24, 385–394 (1984).
    [CrossRef]
  43. D. Laming, Sensory Analysis (Academic, New York, 1986).
  44. C. Carlson, R. Cohen, “A nonlinear spatial frequency signal detection model for the human visual system,”J. Opt. Soc. Am. 68, 1379 (1978).
  45. D. J. Tolhurst, J. A. Movshon, I. D. Thompson, “The dependence of response amplitude and variance of cat visual cortical neurons on stimulus contrast,” Exp. Brain Res. 41, 414–419 (1981).
  46. D. G. Pelli, “The transduction efficiency of human vision,” Suppl. Invest. Ophthalmol. Vis. Sci. 21, 48 (1982).

1985

1984

G. E. Legge, “Binocular contrast summation—II. Quadratic summation,” Vision Res. 24, 385–394 (1984).
[CrossRef]

D. Kersten, “Spatial summation in visual noise,” Vision Res. 24, 1977–1990 (1984).
[CrossRef] [PubMed]

D. Kersten, H. B. Barlow, “Searching for the high-contrast patterns we see best,” Suppl. Invest. Ophthalmol. Vis. Sci. 25, 313 (1984).

1983

G. E. Legge, D. Kersten, “Light and dark bars: contrast discrimination,” Vision Res. 23, 473–483 (1983).
[CrossRef]

A. E. Burgess, H. B. Barlow, “The precision of numerosity discrimination in arrays of random dots,” Vision Res. 23, 811–820 (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]

1982

D. G. Pelli, “The transduction efficiency of human vision,” Suppl. Invest. Ophthalmol. Vis. Sci. 21, 48 (1982).

1981

D. J. Tolhurst, J. A. Movshon, I. D. Thompson, “The dependence of response amplitude and variance of cat visual cortical neurons on stimulus contrast,” Exp. Brain Res. 41, 414–419 (1981).

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

G. J. Burton, “Contrast discrimination by the human visual system,” Biol. Cyb. 40, 27–38 (1981).
[CrossRef]

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

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

1980

H. R. Wilson, “A transducer function for threshold and suprathreshold spatial vision,” Biol. Cyb. 38, 171–178 (1980).
[CrossRef]

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

1979

1978

C. Carlson, R. Cohen, “A nonlinear spatial frequency signal detection model for the human visual system,”J. Opt. Soc. Am. 68, 1379 (1978).

1974

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

1968

W. A. Wickelgren, “Unidimensional strength theory and component analysis of noise in absolute and comparative judgments,”J. Math. Psychol. 5, 102–122 (1968).
[CrossRef]

1967

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

1965

F. W. Campbell, D. G. Green, “Optical and retinal factors affecting visual resolution,”J. Physiol. (London) 181, 576–593 (1965).

1964

1958

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

Barlow, H. B.

D. Kersten, H. B. Barlow, “Why are contrast thresholds so high?” Suppl. Invest. Ophthalmol. Vis. Sci. 26, 140 (1985).

D. Kersten, H. B. Barlow, “Searching for the high-contrast patterns we see best,” Suppl. Invest. Ophthalmol. Vis. Sci. 25, 313 (1984).

A. E. Burgess, H. B. Barlow, “The precision of numerosity discrimination in arrays of random dots,” Vision Res. 23, 811–820 (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]

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

H. B. Barlow, “Retinal and central factors in human vision limited by noise,” in Vertebrate Photoreception, H. B. Barlow, P. Fatt, eds. (Academic, New York, 1977).

D. Kersten, Department of Psychology, Brown University, Providence, R.I. 02912, and H. B. Barlow, Department of Physiology, Cambridge University, Cambridge CB2 3EG, UK (personal communication).

Bell, R. A.

R. A. Bell, “Principles of cathode-ray tubes, phosphors and high-speed oscillography,” Hewlett-Packard Application Note 115 (Hewlett-Packard, Colorado Springs, Colo., 1970).

Birdsall, T. G.

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

Burgess, A.

Burgess, A. E.

A. E. Burgess, H. B. Barlow, “The precision of numerosity discrimination in arrays of random dots,” Vision Res. 23, 811–820 (1983).
[CrossRef] [PubMed]

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Burton, G. J.

G. J. Burton, “Contrast discrimination by the human visual system,” Biol. Cyb. 40, 27–38 (1981).
[CrossRef]

Campbell, F. W.

F. W. Campbell, D. G. Green, “Optical and retinal factors affecting visual resolution,”J. Physiol. (London) 181, 576–593 (1965).

Carlson, C.

C. Carlson, R. Cohen, “A nonlinear spatial frequency signal detection model for the human visual system,”J. Opt. Soc. Am. 68, 1379 (1978).

Cohen, R.

C. Carlson, R. Cohen, “A nonlinear spatial frequency signal detection model for the human visual system,”J. Opt. Soc. Am. 68, 1379 (1978).

Cohn, T. E.

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

Davila, K. D.

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]

Geisler, W. S.

Green, D. G.

F. W. Campbell, D. G. Green, “Optical and retinal factors affecting visual resolution,”J. Physiol. (London) 181, 576–593 (1965).

Green, D. M.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Krieger, Huntington, N.Y., 1974).

Hays, W. L.

W. L. Hays, Statistics for Psychologists (Holt, New York, 1963).

Hill, W.

P. Horowitz, W. Hill, The Art of Electronics (Cambridge U. Press, Cambridge, 1980).

Horowitz, P.

P. Horowitz, W. Hill, The Art of Electronics (Cambridge U. Press, Cambridge, 1980).

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]

Jennings, R. J.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Kersten, D.

D. Kersten, H. B. Barlow, “Why are contrast thresholds so high?” Suppl. Invest. Ophthalmol. Vis. Sci. 26, 140 (1985).

D. Kersten, “Spatial summation in visual noise,” Vision Res. 24, 1977–1990 (1984).
[CrossRef] [PubMed]

D. Kersten, H. B. Barlow, “Searching for the high-contrast patterns we see best,” Suppl. Invest. Ophthalmol. Vis. Sci. 25, 313 (1984).

G. E. Legge, D. Kersten, “Light and dark bars: contrast discrimination,” Vision Res. 23, 473–483 (1983).
[CrossRef]

D. Kersten, Department of Psychology, Brown University, Providence, R.I. 02912, and H. B. Barlow, Department of Physiology, Cambridge University, Cambridge CB2 3EG, UK (personal communication).

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]

Laming, D.

D. Laming, Sensory Analysis (Academic, New York, 1986).

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]

Legge, G. E.

G. E. Legge, “Binocular contrast summation—II. Quadratic summation,” Vision Res. 24, 385–394 (1984).
[CrossRef]

G. E. Legge, D. Kersten, “Light and dark bars: contrast discrimination,” Vision Res. 23, 473–483 (1983).
[CrossRef]

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, “A power law for contrast discrimination,” Vision Res. 21, 457–467 (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, “Spatial frequency masking in human vision: binocular interactions,”J. Opt. Soc. Am. 69, 838–847 (1979).
[CrossRef] [PubMed]

Linfoot, E. H.

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

Movshon, J. A.

D. J. Tolhurst, J. A. Movshon, I. D. Thompson, “The dependence of response amplitude and variance of cat visual cortical neurons on stimulus contrast,” Exp. Brain Res. 41, 414–419 (1981).

Nachmias, J.

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

Nagaraja, N. S.

Nolte, L. W.

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

Pelli, D. G.

D. G. Pelli, “Uncertainty explains many aspects of visual contrast detection and discrimination,” J. Opt. Soc. Am. A 2, 1508–1532 (1985).
[CrossRef] [PubMed]

D. G. Pelli, “The transduction efficiency of human vision,” Suppl. Invest. Ophthalmol. Vis. Sci. 21, 48 (1982).

D. G. Pelli, “The effects of visual noise,” Ph.D. dissertation (Department of Physiology, Cambridge University, Cambridge, UK, 1981).

Robson, J. G.

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

Sansbury, R. V.

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

Stromeyer, C. F.

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

Swets, J. A.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Krieger, Huntington, N.Y., 1974).

Tanner, W. P.

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

Thompson, I. D.

D. J. Tolhurst, J. A. Movshon, I. D. Thompson, “The dependence of response amplitude and variance of cat visual cortical neurons on stimulus contrast,” Exp. Brain Res. 41, 414–419 (1981).

Tolhurst, D. J.

D. J. Tolhurst, J. A. Movshon, I. D. Thompson, “The dependence of response amplitude and variance of cat visual cortical neurons on stimulus contrast,” Exp. Brain Res. 41, 414–419 (1981).

Wagner, R. F.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Watson, A. B.

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

Wickelgren, W. A.

W. A. Wickelgren, “Unidimensional strength theory and component analysis of noise in absolute and comparative judgments,”J. Math. Psychol. 5, 102–122 (1968).
[CrossRef]

Wilson, H. R.

H. R. Wilson, “A transducer function for threshold and suprathreshold spatial vision,” Biol. Cyb. 38, 171–178 (1980).
[CrossRef]

Biol. Cyb.

H. R. Wilson, “A transducer function for threshold and suprathreshold spatial vision,” Biol. Cyb. 38, 171–178 (1980).
[CrossRef]

G. J. Burton, “Contrast discrimination by the human visual system,” Biol. Cyb. 40, 27–38 (1981).
[CrossRef]

Exp. Brain Res.

D. J. Tolhurst, J. A. Movshon, I. D. Thompson, “The dependence of response amplitude and variance of cat visual cortical neurons on stimulus contrast,” Exp. Brain Res. 41, 414–419 (1981).

J. Acoust. Soc. Am.

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

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

J. Math. Psychol.

W. A. Wickelgren, “Unidimensional strength theory and component analysis of noise in absolute and comparative judgments,”J. Math. Psychol. 5, 102–122 (1968).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Physiol. (London)

F. W. Campbell, D. G. Green, “Optical and retinal factors affecting visual resolution,”J. Physiol. (London) 181, 576–593 (1965).

Nature

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

Science

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Suppl. Invest. Ophthalmol. Vis. Sci.

D. Kersten, H. B. Barlow, “Searching for the high-contrast patterns we see best,” Suppl. Invest. Ophthalmol. Vis. Sci. 25, 313 (1984).

D. Kersten, H. B. Barlow, “Why are contrast thresholds so high?” Suppl. Invest. Ophthalmol. Vis. Sci. 26, 140 (1985).

D. G. Pelli, “The transduction efficiency of human vision,” Suppl. Invest. Ophthalmol. Vis. Sci. 21, 48 (1982).

Vision Res.

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

G. E. Legge, “Binocular contrast summation—II. Quadratic summation,” Vision Res. 24, 385–394 (1984).
[CrossRef]

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]

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

G. E. Legge, D. Kersten, “Light and dark bars: contrast discrimination,” Vision Res. 23, 473–483 (1983).
[CrossRef]

A. E. Burgess, H. B. Barlow, “The precision of numerosity discrimination in arrays of random dots,” Vision Res. 23, 811–820 (1983).
[CrossRef] [PubMed]

D. Kersten, “Spatial summation in visual noise,” Vision Res. 24, 1977–1990 (1984).
[CrossRef] [PubMed]

Other

When the analysis is done in time and two spatial dimensions, the stochastic nature of the photon flux adds noise to the stimulus. If this noise is not included as part of the external noise N, it will appear as a nonzero value of equivalent noise for the ideal observer. Pelli11 has pointed out that the photon-noise spectral density is equal to the reciprocal of the photon flux.

P. Horowitz, W. Hill, The Art of Electronics (Cambridge U. Press, Cambridge, 1980).

The rapid decay of the P31 phosphor—the intensity drops to 1% within 250 μ sec (Ref. 28)—is very short compared with the integration time of the eye. Therefore the 100-Hz frame rate of the display determines the temporal sampling characteristics relevant to the calculation of noise spectral density.

R. A. Bell, “Principles of cathode-ray tubes, phosphors and high-speed oscillography,” Hewlett-Packard Application Note 115 (Hewlett-Packard, Colorado Springs, Colo., 1970).

We derive this for one dimension. The three-dimensional case is a straightforward extension. We need to show that, in one dimension, the contrast power spectral density for pixel noise is(R1)N(f)=bcrms2 sinc2(bf),where b is the extent of the pixel (in degrees or seconds), crms2 is the contrast power, and f is the frequency (in cycles per degree or hertz). Consider a contrast noise sample of 1-sec duration:c(t)=∑n=0(1/b)-1c(nb) rect(t-nbb).By averaging the squared modulus of the Fourier transform of c(t), we getC(f)C*(f)¯=b2 sinc2(bf)∑n=0(1/b)-1c2(nb)¯,where we have assumed that adjacent pixel contrasts are uncorrelated. Further, assuming stationarity,N(f)=b2 sinc2(bf)crms21/b=b sinc2(bf)crms2,which is Eq. (R1). The signals in our experiments were at low spatiotemporal frequencies relative to the first zero of N(f), which occurs at f= 1/b(118 c/deg, 12 c/deg, and 100 Hz for the horizontal, vertical, and temporal frequencies). Thus the spectral density is approximately flat in the region of the signals and is given byN(f)=N(0)=crms2bor, in three dimensions, bycrms2bxbybt,where bx, by, and bt are the horizontal, vertical, and temporal sample sizes, respectively.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Krieger, Huntington, N.Y., 1974).

The quantization of the gray levels contributes additional noise. If we assume that the probabilities of the gray levels from the noise process are uniformly distributed across a quantization step, the quantization variance is Q2/12 per pixel, where Q is the gray-level step size.32 The noise spectral density equals rms contrast squared × pixel width × pixel height equals (1/128)2× (0.014)2× (1/12) equals 0.001 μ(deg2), which is negligible compared with the externally added noise.

In the absence of a pedestal, when Lp= L0, S(x, y, t) is sometimes termed a contrast function.14

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

Pelli11 wrote Eq. (1) with k expressed as (d′c2/J), as presented in Eq. (3). He used the term “calculation efficiency” for J rather: than “sampling efficiency.” Nagaraja15 used “squared normalized luminance” rather than “signal energy” and “variance” rather than “spectral density” having absorbed several values into k.

H. B. Barlow, “Retinal and central factors in human vision limited by noise,” in Vertebrate Photoreception, H. B. Barlow, P. Fatt, eds. (Academic, New York, 1977).

D. G. Pelli, “The effects of visual noise,” Ph.D. dissertation (Department of Physiology, Cambridge University, Cambridge, UK, 1981).

W. L. Hays, Statistics for Psychologists (Holt, New York, 1963).

The photopic quantum efficiencies found for intensity detection and discrimination of point sources36 are much higher (about 5%) than that of the optimal stimulus of Watson et al.34 (about 0.05%). However, a comparison in terms of contrast energy is problematic because contrast energy is not well defined for a point source.

D. Kersten, Department of Psychology, Brown University, Providence, R.I. 02912, and H. B. Barlow, Department of Physiology, Cambridge University, Cambridge CB2 3EG, UK (personal communication).

For disks superimposed upon a uniform background, there is a nonlinear relationship between contrast defined as δL/L and Michelson contrast, (Lmax− Lmin)/(Lmax+ Lmin). The slopes cited in the text were calculated with contrast defined in the former way. The corresponding slopes for Michelson contrast are 0.45 and 0.62. The issue of contrast definition is dealt with in more detail by Legge and Kersten.6

The maximum-of-M-channels decision rule differs from the ideal decision rule for detection by a signal-uncertain observer.41 However, it is computationally simpler than the ideal rule and closely approximates it. For further discussion, see Pelli.24

D. Laming, Sensory Analysis (Academic, New York, 1986).

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Figures (7)

Fig. 1
Fig. 1

Some special cases of the linear model of Eq. (1). Threshold signal energy is plotted as a function of noise spectral density, both in the same units. Threshold is defined by a criterion detectability of d′ = 1. The ideal observer exhibits a sampling efficiency of 1 and an equivalent noise of zero. Curve A has a sampling efficiency of 1 and an equivalent noise of 1. Curve B has a sampling efficiency of 1 and an equivalent noise of 2. Curve C has a sampling efficiency of 0.5 and an equivalent noise of 1.

Fig. 2
Fig. 2

Illustrative psychometric functions. Detectability (left- hand ordinate) and percent correct (right-hand ordinate) are plotted as a function of increment contrast for fixed values of pedestal contrast and noise spectral density. A, Dynamic noise (Minnesota): the four symbols show replications in four separate sessions for one observer. Each point is based on about 100 trials. B, Static noise (UBC): the two symbols show psychometric functions for two observers under identical conditions.

Fig. 3
Fig. 3

Threshold signal energy as a function of noise spectral density in dynamic noise (Minnesota). The main graph compares data for pedestals having contrasts of zero and 0.25. The inset shows data for the 0.01-contrast pedestal and some of the data for the zero-contrast pedestal replotted from the main graph. Best-fitting straight lines (least-squares criterion) have been fitted to the three sets of data. The dashed line with slope 1.0 shows the performance of an ideal observer having a sampling efficiency of 1.0 and equivalent noise of zero. The relation between signal energy E and contrast C is E = 0.0192C2 deg2 sec. Accordingly, the 0.01- and 0.25-contrast pedestals have energies of 1.92 and 1200 ν(deg2 sec), respectively. ν(deg2 sec) means 10−6 deg2 sec. A and B show data for two observers.

Fig. 4
Fig. 4

Threshold signal energy as a function of noise spectral density in static noise (UBC). Data are shown for five pedestal contrasts ranging from 0.053 to 0.43. Best-fitting straight lines (least-squares criterion) have been fitted to the data. The dashed line with slope 1.0 shows the performance of an ideal observer. The relation between signal energy E and contrast C is E = 0.0403C2 deg2. A and B show data for two observers.

Fig. 5
Fig. 5

Sampling efficiency and equivalent noise as a function of pedestal contrast. The values of sampling efficiency and equivalent noise that were estimated from the data of Figs. 3 and 4 are plotted for dynamic noise (panels A and B) and static noise (panels C and D).

Fig. 6
Fig. 6

Summary of the properties of the signal-uncertainty model for discrimination in noise. The model’s behavior depends on the detectability d′ of the pedestal. For details, see the text.

Fig. 7
Fig. 7

Threshold signal energy as a function of noise spectral density in dynamic noise (Minnesota). In these experiments, the pedestal had a contrast of 0.024 with corresponding signal energy of 11 ν(deg2 sec). For observer MK, its detectability d′ is 1 for a noise spectral (density of 0.177 ν(deg2 sec). Panel A shows threshold signal energy for contrast increments when noise levels are higher than this, and panel B shows that for lower noise levels. Each point is the mean of four threshold estimates, each based on a 300-trial psychometric function. Best-fitting straight lines (least-squares criterion) have been fitted to the data.

Tables (1)

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Table 1 Mean Slopes of Psychometric Functions (Minnesota Data)a

Equations (20)

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S ( x , y , t ) = [ L s + p ( x , y , t ) - L p ( x , y , t ) ] / L 0 .
E = S 2 ( x , y , t ) d x d y d t .
E t = k ( N + N eq ) = k N + k N eq .
d c = ( E t / N )             or             E t = ( d c ) 2 N ( ideal observer ) .
J = ( d c ) 2 / k .
E t = [ ( d c ) 2 / J ] ( N + N eq ) .
E t = ( 1 / J ) ( N + N eq )             ( threshold criterion , d = 1 ) .
L ( x , y , t ) = L 0 [ 1 + m ( x , y , t ) q ( x ) ] ,
q ( x ) = C cos ( 2 π f x )
q ( x ) = ( C + Δ C ) cos ( 2 π f x )
m ( x , y , t ) = exp [ - ( x / s x ) 2 - ( y / s y ) 2 - ( t / s t ) 2 ] ,
d = ( C / C ) n .
F ( x ) = F ( A 0 ) + F ( A 0 ) ( x - A 0 ) + F ( A 0 ) ( x - A 0 ) 2 + .
F ( x ) g x + b ,
N(f)=bcrms2sinc2(bf),
c(t)=n=0(1/b)-1c(nb)rect(t-nbb).
C(f)C*(f)¯=b2sinc2(bf)n=0(1/b)-1c2(nb)¯,
N(f)=b2sinc2(bf)crms21/b=bsinc2(bf)crms2,
N(f)=N(0)=crms2b
crms2bxbybt,

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