Abstract

Fifty years ago Birdsall, Tanner, and colleagues made rapid progress in developing signal detection theory into a powerful psychophysical tool. One of their major insights was the utility of adding external noise to the signals of interest. These methods have been enhanced in recent years by the addition of multipass and classification-image methods for opening up the black box. There remain a number of as yet unresolved issues. In particular, Birdsall developed a theorem that large amounts of external input noise can linearize nonlinear systems, and Tanner conjectured, with mathematical backup, that what had been previously thought of as a nonlinear system could actually be a linear system with uncertainty. Recent findings, both experimental and theoretical, have validated Birdsall’s theorem and Tanner’s conjecture. However, there have also been experimental and theoretical findings with the opposite outcome. In this paper we present new data and simulations in an attempt to sort out these issues. Our simulations and experiments plus data from others show that Birdsall’s theorem is quite robust. We argue that uncertainty can serve as an explanation for violations of Birdsall’s linearization by noise and also for reports of stochastic resonance. In addition, we modify present models to better handle detection of signals with both noise and pedestal backgrounds.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T.E.Cohn, ed., Visual Detection, Vol. 3 of Collected Works in Optics (Optical Society of America, 1993).
  2. D. J. Lasley and T. E. Cohn, “Why luminance discrimination may be better than detection,” Vision Res. 21, 273-278 (1981).
    [CrossRef] [PubMed]
  3. B. A. Dosher and L. L. Lu, “Mechanisms of perceptual attention in precuing of location,” Vision Res. 40, 1269-1292 (2000).
    [CrossRef] [PubMed]
  4. Z. L. Lu and B. A. Dosher, “Characterizing human perceptual inefficiencies with equivalent internal noise,” J. Opt. Soc. Am. A 16, 764-778 (1999).
    [CrossRef]
  5. Z. L. Lu and B. A. Dosher, “Characterizing observers using external noise and observer models: assessing internal representations with external noise,” Psychol. Rev. 115, 44-82 (2008).
    [CrossRef] [PubMed]
  6. J. Nachmias and R. V. Sansbury, “Grating contrast: Discrimination may be better than detection,” Vision Res. 14, 1039-1042 (1974).
    [CrossRef] [PubMed]
  7. C. F. Stromeyer and S. A. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409-1420 (1974).
    [CrossRef] [PubMed]
  8. L. L. Kontsevich, C. C. Chen, and C. W. Tyler, “Separating the effects of response nonlinearity and internal noise psychophysically,” Vision Res. 42, 1771-1784 (2002).
    [CrossRef] [PubMed]
  9. S. A. Klein, “Separating transducer nonlinearities and multiplicative noise in contrast discrimination,” Vision Res. 46, 4279-4293 (2006).
    [CrossRef] [PubMed]
  10. S. A. Klein, “A local measure for modeling contrast discrimination, Response to Katkov, Tsodyks and Sagi,” Vision Res. 47, 2912-2917 (2007).
    [CrossRef] [PubMed]
  11. M. Katkov, M. Tsodyks, and D. Sagi, “Singularities in the inverse modeling of 2AFC contrast discrimination data,” Vision Res. 46, 259-266 (2006).
    [CrossRef]
  12. D. M. Levi and S. A. Klein, “Classification images for detection and position discrimination in the fovea and parafovea,” J. Vision 2, 46-65 (2002).
    [CrossRef]
  13. D. M. Levi, S. A. Klein, and I. Chen, “The response of the amblyopic visual system to noise,” Vision Res. 47, 2531-2542 (2007).
    [CrossRef] [PubMed]
  14. D. M. Levi, S. A. Klein, and I. Chen, “What limits performance in the amblyopic visual system: seeing signals in noise with an amblyopic brain,” J. Vision 8, 1.1-23 (2008).
    [CrossRef]
  15. R. L. Goris, J. Wagemans, and F. A. Wichmann, “Modelling contrast discrimination data suggest both the pedestal effect and stochastic resonance to be caused by the same mechanism,” J. Vision 8, article 17, 1-21 (2008).
    [CrossRef]
  16. R. A. Smith and D. J. Swift, “Spatial-frequency masking and Birdsall's theorem,” J. Opt. Soc. Am. A 2, 1593-1599 (1985).
    [CrossRef] [PubMed]
  17. G. E. Legge and J. M. Foley, “Contrast masking in human vision,” J. Opt. Soc. Am. 70, 1458-1471 (1980).
    [CrossRef] [PubMed]
  18. J. M. Foley, “Human luminance pattern-vision mechanisms: masking experiments require a new model,” J. Opt. Soc. Am. A 11, 1710-1719 (1994).
    [CrossRef]
  19. G. E. Legge, D. Kersten, and A. E. Burgess, “Contrast discrimination in noise,” J. Opt. Soc. Am. A 4, 391-404 (1987).
    [CrossRef] [PubMed]
  20. J. M. Foley and G. M. Boynton, “Forward pattern masking and adaptation: effects of duration, interstimulus interval, contrast, and spatial and temporal frequency,” Vision Res. 33, 959-980 (1993).
    [CrossRef] [PubMed]
  21. A. E. Burgess and B. Colborne, “Visual signal detection. IV. Observer inconsistency,” J. Opt. Soc. Am. A 5, 617-27 (1988).
    [CrossRef] [PubMed]
  22. D. G. Pelli, “Uncertainty explains many aspects of visual contrast detection and discrimination,” J. Opt. Soc. Am. A 2, 1508-1532 (1985).
    [CrossRef] [PubMed]
  23. W. McIlhagga and A. Paakkonen, “Effects of contrast and length on vernier acuity explained with noisy templates,” Vision Res. 43, 707-716 (2003).
    [CrossRef] [PubMed]
  24. G. B. Henning and F. A. Wichmann, “Some observations on the pedestal effect,” J. Vision 7(1): 3, 1-15 (2009).
    [CrossRef]
  25. D. Kersten, “Spatial summation in visual noise,” Vision Res. 24, 1977-1990 (1984).
    [CrossRef] [PubMed]
  26. T. E. Cohn and D. J. Lasley, “Visual sensitivity,” Ann. Rev. Psychol. 37, 495-521 (1986).
    [CrossRef]
  27. R. L. Goris, F. A. Wichmann, and G. B. Henning, “A neurophysiologically plausible population code model for human contrast discrimination,” J. Vision 9(7), 15, 1-22 (2009).
    [CrossRef]
  28. R. L. Goris, P. Zaenen, and J. Wagemans, “Some observations on contrast detection in noise,” J. Vision 8(9), 4, 1-15 (2008).
    [CrossRef]
  29. T. E. Cohn, L. N. Thibos, and R. N. Kleinstein, “Detectability of a luminance increment,” J. Opt. Soc. Am. 64, 1321-1327 (1974).
    [CrossRef] [PubMed]
  30. W. P. Tanner, “Physiological implications of psychophysical data,” Ann. N.Y. Acad. Sci. 89, 752-65 (1961).
    [CrossRef] [PubMed]
  31. T. E. Cohn, “Detectability of a luminance increment: Effect of superimposed random luminance fluctuation,” J. Opt. Soc. Am. 66, 1426-1428 (1976).
    [CrossRef]
  32. M. P. Eckstein, A. J. Ahumada, and A. B. Watson, “Visual signal detection in structured backgrounds. II. Effects of contrast gain control, background variations, and white noise,” J. Opt. Soc. Am. A 14, 2406-2419 (1997).
    [CrossRef]
  33. K. T. Blackwell, “The effect of white and filtered noise on contrast detection thresholds,” Vision Res. 38, 267-280 (1998).
    [CrossRef] [PubMed]
  34. C. A. Perez, T. E. Cohn, L. E. Medina, and J. R. Donoso, “Coincidence-enhanced stochastic resonance: experimental evidence challenges the psychophysical theory behind stochastic resonance,” Neurosci. Lett. 424, 31-35 (2007).
    [CrossRef] [PubMed]
  35. D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Peninsula, 1966).
  36. W. S. Geisler and K. D. Davila, “Ideal discriminators in spatial vision: two-point stimuli,” J. Opt. Soc. Am. A 2, 1483-1497 (1985).
    [CrossRef] [PubMed]
  37. T. E. Cohn and W. Makous, “Detection and Identification,” (Special issue), J. Opt. Soc. Am. A 38, 1395-1610 (1985).
  38. A. E. Burgess, “Double Visual Signal detection: III. On Bayesian use of prior knowledge and cross correlation,” J. Opt. Soc. Am. A 2, 1498-1507 (1985).
    [CrossRef] [PubMed]
  39. J. P. Thomas, “Detection and identification: how are they related?” J. Opt. Soc. Am. A 2, 1457-1467 (1985).
    [CrossRef] [PubMed]
  40. S. A. Klein, “Double judgment psychophysics: problems and solutions,” J. Opt. Soc. Am. A 2, 1568-1585 (1985).
    [CrossRef]

2009 (2)

G. B. Henning and F. A. Wichmann, “Some observations on the pedestal effect,” J. Vision 7(1): 3, 1-15 (2009).
[CrossRef]

R. L. Goris, F. A. Wichmann, and G. B. Henning, “A neurophysiologically plausible population code model for human contrast discrimination,” J. Vision 9(7), 15, 1-22 (2009).
[CrossRef]

2008 (4)

R. L. Goris, P. Zaenen, and J. Wagemans, “Some observations on contrast detection in noise,” J. Vision 8(9), 4, 1-15 (2008).
[CrossRef]

Z. L. Lu and B. A. Dosher, “Characterizing observers using external noise and observer models: assessing internal representations with external noise,” Psychol. Rev. 115, 44-82 (2008).
[CrossRef] [PubMed]

D. M. Levi, S. A. Klein, and I. Chen, “What limits performance in the amblyopic visual system: seeing signals in noise with an amblyopic brain,” J. Vision 8, 1.1-23 (2008).
[CrossRef]

R. L. Goris, J. Wagemans, and F. A. Wichmann, “Modelling contrast discrimination data suggest both the pedestal effect and stochastic resonance to be caused by the same mechanism,” J. Vision 8, article 17, 1-21 (2008).
[CrossRef]

2007 (3)

D. M. Levi, S. A. Klein, and I. Chen, “The response of the amblyopic visual system to noise,” Vision Res. 47, 2531-2542 (2007).
[CrossRef] [PubMed]

S. A. Klein, “A local measure for modeling contrast discrimination, Response to Katkov, Tsodyks and Sagi,” Vision Res. 47, 2912-2917 (2007).
[CrossRef] [PubMed]

C. A. Perez, T. E. Cohn, L. E. Medina, and J. R. Donoso, “Coincidence-enhanced stochastic resonance: experimental evidence challenges the psychophysical theory behind stochastic resonance,” Neurosci. Lett. 424, 31-35 (2007).
[CrossRef] [PubMed]

2006 (2)

M. Katkov, M. Tsodyks, and D. Sagi, “Singularities in the inverse modeling of 2AFC contrast discrimination data,” Vision Res. 46, 259-266 (2006).
[CrossRef]

S. A. Klein, “Separating transducer nonlinearities and multiplicative noise in contrast discrimination,” Vision Res. 46, 4279-4293 (2006).
[CrossRef] [PubMed]

2003 (1)

W. McIlhagga and A. Paakkonen, “Effects of contrast and length on vernier acuity explained with noisy templates,” Vision Res. 43, 707-716 (2003).
[CrossRef] [PubMed]

2002 (2)

D. M. Levi and S. A. Klein, “Classification images for detection and position discrimination in the fovea and parafovea,” J. Vision 2, 46-65 (2002).
[CrossRef]

L. L. Kontsevich, C. C. Chen, and C. W. Tyler, “Separating the effects of response nonlinearity and internal noise psychophysically,” Vision Res. 42, 1771-1784 (2002).
[CrossRef] [PubMed]

2000 (1)

B. A. Dosher and L. L. Lu, “Mechanisms of perceptual attention in precuing of location,” Vision Res. 40, 1269-1292 (2000).
[CrossRef] [PubMed]

1999 (1)

1998 (1)

K. T. Blackwell, “The effect of white and filtered noise on contrast detection thresholds,” Vision Res. 38, 267-280 (1998).
[CrossRef] [PubMed]

1997 (1)

1994 (1)

1993 (1)

J. M. Foley and G. M. Boynton, “Forward pattern masking and adaptation: effects of duration, interstimulus interval, contrast, and spatial and temporal frequency,” Vision Res. 33, 959-980 (1993).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

1986 (1)

T. E. Cohn and D. J. Lasley, “Visual sensitivity,” Ann. Rev. Psychol. 37, 495-521 (1986).
[CrossRef]

1985 (7)

1984 (1)

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

1981 (1)

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

1980 (1)

1976 (1)

1974 (3)

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

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

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

1961 (1)

W. P. Tanner, “Physiological implications of psychophysical data,” Ann. N.Y. Acad. Sci. 89, 752-65 (1961).
[CrossRef] [PubMed]

Ahumada, A. J.

Blackwell, K. T.

K. T. Blackwell, “The effect of white and filtered noise on contrast detection thresholds,” Vision Res. 38, 267-280 (1998).
[CrossRef] [PubMed]

Boynton, G. M.

J. M. Foley and G. M. Boynton, “Forward pattern masking and adaptation: effects of duration, interstimulus interval, contrast, and spatial and temporal frequency,” Vision Res. 33, 959-980 (1993).
[CrossRef] [PubMed]

Burgess, A. E.

Chen, C. C.

L. L. Kontsevich, C. C. Chen, and C. W. Tyler, “Separating the effects of response nonlinearity and internal noise psychophysically,” Vision Res. 42, 1771-1784 (2002).
[CrossRef] [PubMed]

Chen, I.

D. M. Levi, S. A. Klein, and I. Chen, “What limits performance in the amblyopic visual system: seeing signals in noise with an amblyopic brain,” J. Vision 8, 1.1-23 (2008).
[CrossRef]

D. M. Levi, S. A. Klein, and I. Chen, “The response of the amblyopic visual system to noise,” Vision Res. 47, 2531-2542 (2007).
[CrossRef] [PubMed]

Cohn, T. E.

C. A. Perez, T. E. Cohn, L. E. Medina, and J. R. Donoso, “Coincidence-enhanced stochastic resonance: experimental evidence challenges the psychophysical theory behind stochastic resonance,” Neurosci. Lett. 424, 31-35 (2007).
[CrossRef] [PubMed]

T. E. Cohn and D. J. Lasley, “Visual sensitivity,” Ann. Rev. Psychol. 37, 495-521 (1986).
[CrossRef]

T. E. Cohn and W. Makous, “Detection and Identification,” (Special issue), J. Opt. Soc. Am. A 38, 1395-1610 (1985).

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

T. E. Cohn, “Detectability of a luminance increment: Effect of superimposed random luminance fluctuation,” J. Opt. Soc. Am. 66, 1426-1428 (1976).
[CrossRef]

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

Colborne, B.

Davila, K. D.

Donoso, J. R.

C. A. Perez, T. E. Cohn, L. E. Medina, and J. R. Donoso, “Coincidence-enhanced stochastic resonance: experimental evidence challenges the psychophysical theory behind stochastic resonance,” Neurosci. Lett. 424, 31-35 (2007).
[CrossRef] [PubMed]

Dosher, B. A.

Z. L. Lu and B. A. Dosher, “Characterizing observers using external noise and observer models: assessing internal representations with external noise,” Psychol. Rev. 115, 44-82 (2008).
[CrossRef] [PubMed]

B. A. Dosher and L. L. Lu, “Mechanisms of perceptual attention in precuing of location,” Vision Res. 40, 1269-1292 (2000).
[CrossRef] [PubMed]

Z. L. Lu and B. A. Dosher, “Characterizing human perceptual inefficiencies with equivalent internal noise,” J. Opt. Soc. Am. A 16, 764-778 (1999).
[CrossRef]

Eckstein, M. P.

Foley, J. M.

Geisler, W. S.

Goris, R. L.

R. L. Goris, F. A. Wichmann, and G. B. Henning, “A neurophysiologically plausible population code model for human contrast discrimination,” J. Vision 9(7), 15, 1-22 (2009).
[CrossRef]

R. L. Goris, P. Zaenen, and J. Wagemans, “Some observations on contrast detection in noise,” J. Vision 8(9), 4, 1-15 (2008).
[CrossRef]

R. L. Goris, J. Wagemans, and F. A. Wichmann, “Modelling contrast discrimination data suggest both the pedestal effect and stochastic resonance to be caused by the same mechanism,” J. Vision 8, article 17, 1-21 (2008).
[CrossRef]

Green, D. M.

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

Henning, G. B.

R. L. Goris, F. A. Wichmann, and G. B. Henning, “A neurophysiologically plausible population code model for human contrast discrimination,” J. Vision 9(7), 15, 1-22 (2009).
[CrossRef]

G. B. Henning and F. A. Wichmann, “Some observations on the pedestal effect,” J. Vision 7(1): 3, 1-15 (2009).
[CrossRef]

Katkov, M.

M. Katkov, M. Tsodyks, and D. Sagi, “Singularities in the inverse modeling of 2AFC contrast discrimination data,” Vision Res. 46, 259-266 (2006).
[CrossRef]

Kersten, D.

Klein, S. A.

D. M. Levi, S. A. Klein, and I. Chen, “What limits performance in the amblyopic visual system: seeing signals in noise with an amblyopic brain,” J. Vision 8, 1.1-23 (2008).
[CrossRef]

S. A. Klein, “A local measure for modeling contrast discrimination, Response to Katkov, Tsodyks and Sagi,” Vision Res. 47, 2912-2917 (2007).
[CrossRef] [PubMed]

D. M. Levi, S. A. Klein, and I. Chen, “The response of the amblyopic visual system to noise,” Vision Res. 47, 2531-2542 (2007).
[CrossRef] [PubMed]

S. A. Klein, “Separating transducer nonlinearities and multiplicative noise in contrast discrimination,” Vision Res. 46, 4279-4293 (2006).
[CrossRef] [PubMed]

D. M. Levi and S. A. Klein, “Classification images for detection and position discrimination in the fovea and parafovea,” J. Vision 2, 46-65 (2002).
[CrossRef]

S. A. Klein, “Double judgment psychophysics: problems and solutions,” J. Opt. Soc. Am. A 2, 1568-1585 (1985).
[CrossRef]

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

Kleinstein, R. N.

Kontsevich, L. L.

L. L. Kontsevich, C. C. Chen, and C. W. Tyler, “Separating the effects of response nonlinearity and internal noise psychophysically,” Vision Res. 42, 1771-1784 (2002).
[CrossRef] [PubMed]

Lasley, D. J.

T. E. Cohn and D. J. Lasley, “Visual sensitivity,” Ann. Rev. Psychol. 37, 495-521 (1986).
[CrossRef]

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

Legge, G. E.

Levi, D. M.

D. M. Levi, S. A. Klein, and I. Chen, “What limits performance in the amblyopic visual system: seeing signals in noise with an amblyopic brain,” J. Vision 8, 1.1-23 (2008).
[CrossRef]

D. M. Levi, S. A. Klein, and I. Chen, “The response of the amblyopic visual system to noise,” Vision Res. 47, 2531-2542 (2007).
[CrossRef] [PubMed]

D. M. Levi and S. A. Klein, “Classification images for detection and position discrimination in the fovea and parafovea,” J. Vision 2, 46-65 (2002).
[CrossRef]

Lu, L. L.

B. A. Dosher and L. L. Lu, “Mechanisms of perceptual attention in precuing of location,” Vision Res. 40, 1269-1292 (2000).
[CrossRef] [PubMed]

Lu, Z. L.

Z. L. Lu and B. A. Dosher, “Characterizing observers using external noise and observer models: assessing internal representations with external noise,” Psychol. Rev. 115, 44-82 (2008).
[CrossRef] [PubMed]

Z. L. Lu and B. A. Dosher, “Characterizing human perceptual inefficiencies with equivalent internal noise,” J. Opt. Soc. Am. A 16, 764-778 (1999).
[CrossRef]

Makous, W.

T. E. Cohn and W. Makous, “Detection and Identification,” (Special issue), J. Opt. Soc. Am. A 38, 1395-1610 (1985).

McIlhagga, W.

W. McIlhagga and A. Paakkonen, “Effects of contrast and length on vernier acuity explained with noisy templates,” Vision Res. 43, 707-716 (2003).
[CrossRef] [PubMed]

Medina, L. E.

C. A. Perez, T. E. Cohn, L. E. Medina, and J. R. Donoso, “Coincidence-enhanced stochastic resonance: experimental evidence challenges the psychophysical theory behind stochastic resonance,” Neurosci. Lett. 424, 31-35 (2007).
[CrossRef] [PubMed]

Nachmias, J.

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

Paakkonen, A.

W. McIlhagga and A. Paakkonen, “Effects of contrast and length on vernier acuity explained with noisy templates,” Vision Res. 43, 707-716 (2003).
[CrossRef] [PubMed]

Pelli, D. G.

Perez, C. A.

C. A. Perez, T. E. Cohn, L. E. Medina, and J. R. Donoso, “Coincidence-enhanced stochastic resonance: experimental evidence challenges the psychophysical theory behind stochastic resonance,” Neurosci. Lett. 424, 31-35 (2007).
[CrossRef] [PubMed]

Sagi, D.

M. Katkov, M. Tsodyks, and D. Sagi, “Singularities in the inverse modeling of 2AFC contrast discrimination data,” Vision Res. 46, 259-266 (2006).
[CrossRef]

Sansbury, R. V.

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

Smith, R. A.

Stromeyer, C. F.

C. F. Stromeyer and S. A. 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 and J. A. Swets, Signal Detection Theory and Psychophysics (Peninsula, 1966).

Swift, D. J.

Tanner, W. P.

W. P. Tanner, “Physiological implications of psychophysical data,” Ann. N.Y. Acad. Sci. 89, 752-65 (1961).
[CrossRef] [PubMed]

Thibos, L. N.

Thomas, J. P.

Tsodyks, M.

M. Katkov, M. Tsodyks, and D. Sagi, “Singularities in the inverse modeling of 2AFC contrast discrimination data,” Vision Res. 46, 259-266 (2006).
[CrossRef]

Tyler, C. W.

L. L. Kontsevich, C. C. Chen, and C. W. Tyler, “Separating the effects of response nonlinearity and internal noise psychophysically,” Vision Res. 42, 1771-1784 (2002).
[CrossRef] [PubMed]

Wagemans, J.

R. L. Goris, J. Wagemans, and F. A. Wichmann, “Modelling contrast discrimination data suggest both the pedestal effect and stochastic resonance to be caused by the same mechanism,” J. Vision 8, article 17, 1-21 (2008).
[CrossRef]

R. L. Goris, P. Zaenen, and J. Wagemans, “Some observations on contrast detection in noise,” J. Vision 8(9), 4, 1-15 (2008).
[CrossRef]

Watson, A. B.

Wichmann, F. A.

R. L. Goris, F. A. Wichmann, and G. B. Henning, “A neurophysiologically plausible population code model for human contrast discrimination,” J. Vision 9(7), 15, 1-22 (2009).
[CrossRef]

G. B. Henning and F. A. Wichmann, “Some observations on the pedestal effect,” J. Vision 7(1): 3, 1-15 (2009).
[CrossRef]

R. L. Goris, J. Wagemans, and F. A. Wichmann, “Modelling contrast discrimination data suggest both the pedestal effect and stochastic resonance to be caused by the same mechanism,” J. Vision 8, article 17, 1-21 (2008).
[CrossRef]

Zaenen, P.

R. L. Goris, P. Zaenen, and J. Wagemans, “Some observations on contrast detection in noise,” J. Vision 8(9), 4, 1-15 (2008).
[CrossRef]

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

W. P. Tanner, “Physiological implications of psychophysical data,” Ann. N.Y. Acad. Sci. 89, 752-65 (1961).
[CrossRef] [PubMed]

Ann. Rev. Psychol. (1)

T. E. Cohn and D. J. Lasley, “Visual sensitivity,” Ann. Rev. Psychol. 37, 495-521 (1986).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (12)

W. S. Geisler and K. D. Davila, “Ideal discriminators in spatial vision: two-point stimuli,” J. Opt. Soc. Am. A 2, 1483-1497 (1985).
[CrossRef] [PubMed]

T. E. Cohn and W. Makous, “Detection and Identification,” (Special issue), J. Opt. Soc. Am. A 38, 1395-1610 (1985).

A. E. Burgess, “Double Visual Signal detection: III. On Bayesian use of prior knowledge and cross correlation,” J. Opt. Soc. Am. A 2, 1498-1507 (1985).
[CrossRef] [PubMed]

J. P. Thomas, “Detection and identification: how are they related?” J. Opt. Soc. Am. A 2, 1457-1467 (1985).
[CrossRef] [PubMed]

S. A. Klein, “Double judgment psychophysics: problems and solutions,” J. Opt. Soc. Am. A 2, 1568-1585 (1985).
[CrossRef]

J. M. Foley, “Human luminance pattern-vision mechanisms: masking experiments require a new model,” J. Opt. Soc. Am. A 11, 1710-1719 (1994).
[CrossRef]

G. E. Legge, D. Kersten, and A. E. Burgess, “Contrast discrimination in noise,” J. Opt. Soc. Am. A 4, 391-404 (1987).
[CrossRef] [PubMed]

A. E. Burgess and B. Colborne, “Visual signal detection. IV. Observer inconsistency,” J. Opt. Soc. Am. A 5, 617-27 (1988).
[CrossRef] [PubMed]

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

M. P. Eckstein, A. J. Ahumada, and A. B. Watson, “Visual signal detection in structured backgrounds. II. Effects of contrast gain control, background variations, and white noise,” J. Opt. Soc. Am. A 14, 2406-2419 (1997).
[CrossRef]

Z. L. Lu and B. A. Dosher, “Characterizing human perceptual inefficiencies with equivalent internal noise,” J. Opt. Soc. Am. A 16, 764-778 (1999).
[CrossRef]

R. A. Smith and D. J. Swift, “Spatial-frequency masking and Birdsall's theorem,” J. Opt. Soc. Am. A 2, 1593-1599 (1985).
[CrossRef] [PubMed]

J. Vision (6)

D. M. Levi, S. A. Klein, and I. Chen, “What limits performance in the amblyopic visual system: seeing signals in noise with an amblyopic brain,” J. Vision 8, 1.1-23 (2008).
[CrossRef]

R. L. Goris, J. Wagemans, and F. A. Wichmann, “Modelling contrast discrimination data suggest both the pedestal effect and stochastic resonance to be caused by the same mechanism,” J. Vision 8, article 17, 1-21 (2008).
[CrossRef]

D. M. Levi and S. A. Klein, “Classification images for detection and position discrimination in the fovea and parafovea,” J. Vision 2, 46-65 (2002).
[CrossRef]

R. L. Goris, F. A. Wichmann, and G. B. Henning, “A neurophysiologically plausible population code model for human contrast discrimination,” J. Vision 9(7), 15, 1-22 (2009).
[CrossRef]

R. L. Goris, P. Zaenen, and J. Wagemans, “Some observations on contrast detection in noise,” J. Vision 8(9), 4, 1-15 (2008).
[CrossRef]

G. B. Henning and F. A. Wichmann, “Some observations on the pedestal effect,” J. Vision 7(1): 3, 1-15 (2009).
[CrossRef]

Neurosci. Lett. (1)

C. A. Perez, T. E. Cohn, L. E. Medina, and J. R. Donoso, “Coincidence-enhanced stochastic resonance: experimental evidence challenges the psychophysical theory behind stochastic resonance,” Neurosci. Lett. 424, 31-35 (2007).
[CrossRef] [PubMed]

Psychol. Rev. (1)

Z. L. Lu and B. A. Dosher, “Characterizing observers using external noise and observer models: assessing internal representations with external noise,” Psychol. Rev. 115, 44-82 (2008).
[CrossRef] [PubMed]

Vision Res. (13)

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

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

L. L. Kontsevich, C. C. Chen, and C. W. Tyler, “Separating the effects of response nonlinearity and internal noise psychophysically,” Vision Res. 42, 1771-1784 (2002).
[CrossRef] [PubMed]

S. A. Klein, “Separating transducer nonlinearities and multiplicative noise in contrast discrimination,” Vision Res. 46, 4279-4293 (2006).
[CrossRef] [PubMed]

S. A. Klein, “A local measure for modeling contrast discrimination, Response to Katkov, Tsodyks and Sagi,” Vision Res. 47, 2912-2917 (2007).
[CrossRef] [PubMed]

M. Katkov, M. Tsodyks, and D. Sagi, “Singularities in the inverse modeling of 2AFC contrast discrimination data,” Vision Res. 46, 259-266 (2006).
[CrossRef]

D. M. Levi, S. A. Klein, and I. Chen, “The response of the amblyopic visual system to noise,” Vision Res. 47, 2531-2542 (2007).
[CrossRef] [PubMed]

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

B. A. Dosher and L. L. Lu, “Mechanisms of perceptual attention in precuing of location,” Vision Res. 40, 1269-1292 (2000).
[CrossRef] [PubMed]

K. T. Blackwell, “The effect of white and filtered noise on contrast detection thresholds,” Vision Res. 38, 267-280 (1998).
[CrossRef] [PubMed]

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

W. McIlhagga and A. Paakkonen, “Effects of contrast and length on vernier acuity explained with noisy templates,” Vision Res. 43, 707-716 (2003).
[CrossRef] [PubMed]

J. M. Foley and G. M. Boynton, “Forward pattern masking and adaptation: effects of duration, interstimulus interval, contrast, and spatial and temporal frequency,” Vision Res. 33, 959-980 (1993).
[CrossRef] [PubMed]

Other (2)

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

T.E.Cohn, ed., Visual Detection, Vol. 3 of Collected Works in Optics (Optical Society of America, 1993).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Result of simulations of the perceptual template model specified by Eqs. (4, 5, 6, 7). The model parameters are γ = 2 , γ t = 1.7 , and T ext = 1 and T test = 1.25 . The values for the M ext and M test vary for the three rows of the figure as indicated in the legends in the three (b) panels. (a) is a plot of d vs. test strength for four strengths of external noise, 0, 0.5%, 2%, and 8%. The d curve for zero external noise [from Eq. (10)] is much steeper than those for the simulations with substantial external noise (Birdsall linearization). (b) is from the same data as (a) but replotted as TvN curves (test contrast vs. external noise strength) for d levels of 0.5, 1, 2, and 4, corresponding to the four horizontal lines in (a). The three vertical lines in (b) correspond to the three nonzero values of external noise in (a). (c) shows the effective exponents of d psychometric functions. They correspond to the slopes of the log–log curves in (a). (d) has the same abscissa and ordinate as (b), but the contour lines are now iso % agreement showing the fraction of times the observer gives the identical response in a 2AFC method to trials with identical external noise. The dashed lines are replotted from (b) for d = 0.5 and 1.0.

Fig. 2
Fig. 2

Reanalysis of the data of Fig. 15 of Levi et al. [14] for five normal observers. (a) is similar to Fig. 1a, except that the external noise is specified in noise threshold units (NTU) (see the numbers above the data). NTU = 2 means the noise was twice its detection threshold at d = 1 . The abscissa and ordinate show that logarithmic axes are being used. (b) is the effective exponent of the d function and is given by the log–log slope of the data in (a). There is a dramatic reduction of the slope once external noise is present, as would be suggested by Birdsall’s theorem.

Fig. 3
Fig. 3

The double-pass % agreement data from Fig. 1c is replotted “inside-out” with axes following the Burgess and Colborne [21] diagram. The three panels have the same parameters as Fig. 1. The axes are the % agreement and % correct with contour lines being strength of test pattern and external noise. The leftmost parabola is independent of model parameters and represents the limiting case where the probability of a correct judgment does not depend on the particular external noise being shown. It is seen that as the external noise increases, the % agreement also increases. Also shown are lines of constant signal strength of c test = 0.5 , 1.0, and 2 as well as lines of constant external noise. The positive slope at the left portion of the c test = 0.5 curve shows the stochastic resonance effect. The horizontal line at 76% correct ( d = 1 ) shows how the % agreement can distinguish among the three values of M ext = 0 , 1.5, and 5.0 that are shown in the three panels.

Fig. 4
Fig. 4

(a)–(c) are similar to Figs. 1a, 1b, 1c except that the analytic PTM of Eqs. (16, 17) (solid curves) are plotted on top of the stochastic PTM (dotted curves) of Eqs. (4, 5, 6, 7). Another difference from Fig. 1 is that γ t = 2 to agree with the original PTM model, giving a hard saturation of the d function as seen in (a). The remarkable feature of (b) is that the stochastic and analytic models are in excellent agreement for d = 1 and γ = 2 . Away from the d = 1 curve, one sees the effect of Birdsall linearization in that the d = 0.5 and 2.0 curves become separated by factors of 2 from the d = 1 curve (linearity) for large values of σ ext , whereas at small values of σ ext they are separated by sqrt(2) as expected from a γ = 2 nonlinearity. (c) shows the opposite pattern for the d exponent for the analytic model from what was seen for the stochastic model of Fig. 1c. The decrement in exponent at σ ext = 0 is because of the saturating d function seen in Fig. 4a. (d) and (e) show the signal detection ROC plots of the z scores of hit rate vs. false alarm rate for the parameter values indicated in the plot. The thick solid curve is the ROC curve based on the raw data from the simulations of the stochastic PTM. The dotted curve is for the assumptions of the analytic PTM. The dashed curve is based on a Gaussian approximation done by summarizing the simulations in terms of the expected values of the means and standard deviations of the simulations.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

y ( c test , σ ext ) = ( c test + σ ext R 1 ) γ + σ int R 2 ,
y ( c test , σ ext ) = ( c test T test + σ ext T ext R 1 ) γ + σ add R 2 + σ mult R 3 ,
d template = ( c test T test ) ( σ ext T ext ) .
y ( c test , 1 , σ ext ) = ( c test , 1 T test + σ ext T ext R 1 ) γ + R 2 + σ mult R 3 .
σ mult 2 = M test 2 ( c test T test ) 2 γ t + M ext 2 ( σ ext T ext ) 2 γ .
y 2 ( c test , 2 , σ ext ) = ( c test , 2 T test + σ ext T ext R 4 ) γ + R 5 + σ mult R 6 .
choose interval 1 if y 1 > y 2 , otherwise choose interval 2 .
σ tot = ( 1 + σ mult 2 ) 1 2 .
y ( c test , σ ext ) = ( c test T test + σ ext T ext R 1 ) γ σ tot + R 2 .
d = ( c test T test ) γ ( 1 + M test 2 ( c test T test ) 2 γ t ) 1 2 .
d = ( c test 1.25 ) γ ( 1 + M test 2 ( c test 1.25 ) 2 γ t ) 1 2 ,
γ eff ( c test ) = log ( d ( c test ) d ( c test 2 ) ) log ( 2 ) ,
d ideal = c test σ ext .
p agree - min = p 2 + ( 1 p ) 2 = 1 2 + 2 ( p 1 2 ) 2 for σ ext = 0 ,
y ( c test , 1 , σ ext ) = ( c test , 1 T test + σ ext T ext R 1 + σ ext M early R m ) γ + R 2 + σ mult R 3 .
d 2 = ( c test T test ) 2 γ [ k ( σ ext T ext ) 2 γ + 1 + M test 2 ( c test T test ) 2 γ t + M ext 2 ( σ ext T ext ) 2 γ ] .
( c test T test ) 2 γ = [ 1 + ( 1 + M ext 2 ) ( σ ext T ext ) 2 γ ] ( 1 d 2 M test 2 ) .
CC ̱ SJ ̱ WC ̱ γ = γ t 2.05 2.27 2.36 = γ , T test = 0.055 0.048 0.086 = σ add 1 γ β , T ext = 0.103 0.086 0.128 = σ add 1 γ , M ext = 1.23 0.70 1.60 = N mul , M test = 0.33 0.30 0.35 = N mul ( β 2 β ) γ sqrt ( 2 ) ,
c test = Th 0 C ( d ) F ( σ ext )
Th 0 = T test ( 1 M test 2 ) 1 2 γ = 0.061 , 0.053 , 0.096 for the three observers of Eq . ( 18 ) .
C ( d ) = ( ( 1 M test 2 ) ( 1 d 2 M test 2 ) ) 1 2 γ .
F ( σ ext ) = ( ( 1 + ( σ ext N eq ) 2 γ ) ) 1 2 γ ,
N eq = T ext ( 1 + M ext 2 ) 1 2 γ = 0.083 , 0.079 , 0.097 for the three observers of Eq . ( 18 ) .

Metrics