Abstract

This Discussion Paper seeks to kill off probability summation, specifically the high-threshold assumption, as an explanatory idea in visual science. In combination with a Weibull function of a parameter of about 4, probability summation can accommodate, to within the limits of experimental error, the shape of the detectability function for contrast, the reduction in threshold that results from the combination of widely separated grating components, summation with respect to duration at threshold, and some instances, but not all, of spatial summation. But it has repeated difficulty with stimuli below threshold, because it denies the availability of input from such stimuli. All the phenomena listed above, and many more, can be accommodated equally accurately by signal-detection theory combined with an accelerated nonlinear transform of small, near-threshold, contrasts. This is illustrated with a transform that is the fourth power for the smallest contrasts, but tends to linear above threshold. Moreover, this particular transform can be derived from elementary properties of sensory neurons. Probability summation cannot be regarded as a special case of a more general theory, because it depends essentially on the 19th-century notion of a high fixed threshold. It is simply an obstruction to further progress.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. The argument by Sachs et al. ([4]; see their Fig. 4) looks compelling, except that they have assumed the psychometric function for detection of a grating to be normal with respect to contrast. They could alternatively have accommodated their experimental results by supposing the psychometric function to be normal with respect to some power of contrast (as in Fig. 1 here), with detectability depending on the summation of that power over all grating components. There would then have been no need for probability summation, nor for a fixed threshold. The evidence, as at that time, pointing to a power-law transform of small near-threshold stimuli had already been summarized by Nachmias and Kocher [6]. This article shows that the idea that Sachs et al. did not explore, that is, of detectability depending on the summation of power-law transforms over all grating components, provides a more comprehensive account of visual sensitivity.
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  30. J. Nachmias and R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vis. Res. 14, 1039–1042 (1974).
    [CrossRef]
  31. G. B. Weatherill and H. Levitt, “Sequential estimation of points on a psychometric function,” Brit. J. Math. Statist. Psychol. 18, 1–9 (1965).
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    [CrossRef]
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    [CrossRef]
  38. H. E. Smithson, G. B. Henning, D. I. A. MacLeod, and A. Stockman, “The effect of notched noise on flicker detection and discrimination,” J. Vis. 9(5):21, 1–18 (2009).
    [CrossRef]
  39. F. W. Campbell and D. G. Green, “Monocular versus binocular visual acuity,” Nature 208, 191–192 (1965).
    [CrossRef]
  40. G. E. Legge, “Binocular contrast summation—II. Quadratic summation,” Vis. Res. 24, 385–394 (1984).
    [CrossRef]
  41. G. E. Legge, “Spatial frequency masking in human vision: binocular interactions,” J. Opt. Soc. Am. 69, 838–847 (1979).
    [CrossRef]
  42. D. Laming, “Spatial frequency channels,” in Vision and Visual Dysfunction, Vol 5: Limits of Visual Perception, J. J. Kulikowski, V. Walsh, and I. J. Murray, eds. (Macmillan, 1991), pp. 97–105.
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    [CrossRef]
  44. G. J. Burton, “Contrast discrimination by the human visual system,” Biol. Cybern. 40, 27–38 (1981).
    [CrossRef]
  45. F. W. Campbell and J. J. Kulikowski, “Orientational selectivity of the human visual system,” J. Physiol. 187, 437–445 (1966).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  53. E. R. Howell and R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vis. Res. 18, 369–374 (1978).
    [CrossRef]
  54. Detection of a grating in a dark surround under continuous inspection needs further comment. The only cue to detection of sinusoidal modulation in such a case is the variation in input to individual units as the eye moves laterally with respect to the bars of the grating. Temporal modulations generate a square-law perturbation [27, Fig. 7.4, p. 112], in this case over the face of the grating. Moreover, temporal sensitivity is maintained in the peripheral retina, so that the extreme extent of the grating in Fig. 11 does not matter. Ordinarily, comparison with a matched surround provides the more sensitive cue. However, thresholds for (0.1  c/deg) gratings with and without a surround both reach a lower limit at a grating height of 3.2λ, whereafter they do not noticeably differ. This suggests that the same limit to summation applies to both (as one should expect) and, at that limit, spatio-temporal modulations across the boundary with a matched surround are no more informative than modulations within the grating field.
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    [CrossRef]
  56. J. Rovamo, J. Mustonen, and R. Naesaenen, “Modelling contrast sensitivity as a function of retinal illuminance and grating area,” Vis. Res. 34, 1301–1314 (1994).
    [CrossRef]
  57. N. v. S. Graham, Visual Pattern Analyzers (Oxford, 1989).
  58. J. M. Foley, “Human luminance pattern-vision mechanisms: masking experiments require a new model,” J. Opt. Soc. Am. A 11, 1710–1719 (1994).
    [CrossRef]
  59. R. L. T. Goris, F. A. Wichmann, and G. B. Henning, “A neurophysiologically plausible population-code model for human contrast discrimination,” J. Vis. 9(7):15, 1–22 (2009).
    [CrossRef]
  60. L. Itti, C. Koch, and J. Braun, “Revisiting spatial vision: toward a unifying model,” J. Opt. Soc. Am. A 17, 1899–1917 (2000).
    [CrossRef]
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2011 (1)

D. Koenig and H. Hofer, “The absolute threshold of cone vision,” J. Vis. 11(1):21, 1–24 (2011).
[CrossRef]

2010 (1)

D. Laming, “Fechner’s law: where does the log transform come from?” Seeing Perceiving 23, 155–171 (2010).

2009 (2)

H. E. Smithson, G. B. Henning, D. I. A. MacLeod, and A. Stockman, “The effect of notched noise on flicker detection and discrimination,” J. Vis. 9(5):21, 1–18 (2009).
[CrossRef]

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

2007 (1)

G. B. Henning and F. A. Wichmann, “Some observations on the pedestal effect,” J. Vis. 7(1):3, 1–15 (2007).
[CrossRef]

2005 (2)

A. B. Watson and A. J. Ahumada, “A standard model for foveal detection of spatial contrast,” J. Vis. 5(9):6, 1–23 (2005).
[CrossRef]

N. Stewart, G. D. A. Brown, and N. Chater, “Absolute identification by relative judgment,” Psychol. Rev. 112, 881–911 (2005).
[CrossRef]

2002 (1)

2000 (1)

1995 (1)

J. Yang and W. Makous, “Modeling pedestal experiments with amplitude instead of contrast,” Vis. Res. 35, 1979–1989 (1995).
[CrossRef]

1994 (2)

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

J. Rovamo, J. Mustonen, and R. Naesaenen, “Modelling contrast sensitivity as a function of retinal illuminance and grating area,” Vis. Res. 34, 1301–1314 (1994).
[CrossRef]

1993 (1)

J. Rovamo, O. Luntinen, and R. Naesaenen, “Modelling the dependence of contrast sensitivity on grating area and spatial frequency,” Vis. Res. 33, 2773–2788 (1993).
[CrossRef]

1992 (1)

H. D. Speed and J. Ross, “Spatial frequency tuning of facilitation by masks,” Vis. Res. 32, 1143–1148 (1992).
[CrossRef]

1985 (1)

1984 (1)

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

1981 (4)

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

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

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

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

1979 (2)

1978 (1)

E. R. Howell and R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vis. Res. 18, 369–374 (1978).
[CrossRef]

1977 (1)

D. H. Hubel and T. N. Wiesel, “Functional architecture of macaque monkey visual cortex,” Proc. R. Soc. Lond. B 198, 1–59 (1977).
[CrossRef]

1976 (1)

J. J. Kulikowski, “Effective contrast constancy and linearity of contrast sensation,” Vis. Res. 16, 1419–1431 (1976).

1974 (2)

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

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

1971 (1)

1970 (2)

J. Nachmias and E. C. Kocher, “Visual detection and discrimination of luminance increments,” J. Opt. Soc. Am. 60, 382–389 (1970).
[CrossRef]

T. A. Tanner, J. A. Rauk, and R. C. Atkinson, “Signal recognition as influenced by information feedback,” J. Math. Psychol. 7, 259–274 (1970).
[CrossRef]

1969 (2)

J. J. Kulikowski, “Limiting conditions of visual perception,” Prace Inst. Automat. PAN (Warsaw) 77, 1–133 (1969). (English translation)

D. H. Kelly, “Flickering patterns and lateral inhibition,” J. Opt. Soc. Am. 59, 1361–1370 (1969).
[CrossRef]

1968 (1)

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

1966 (2)

C. Enroth-Cugell and J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. 187, 517–552 (1966).

F. W. Campbell and J. J. Kulikowski, “Orientational selectivity of the human visual system,” J. Physiol. 187, 437–445 (1966).

1965 (3)

F. W. Campbell and D. G. Green, “Monocular versus binocular visual acuity,” Nature 208, 191–192 (1965).
[CrossRef]

G. B. Weatherill and H. Levitt, “Sequential estimation of points on a psychometric function,” Brit. J. Math. Statist. Psychol. 18, 1–9 (1965).
[CrossRef]

J. Nachmias and R. M. Steinman, “Brightness and discriminability of light flashes,” Vis. Res. 5, 545–557 (1965).
[CrossRef]

1962 (2)

S. M. Pfafflin and M. V. Mathews, “Energy-detection model for monaural auditory detection,” J. Acoust. Soc. Am. 34, 1842–1853 (1962).
[CrossRef]

L. Matin, “Binocular summation at the absolute threshold of peripheral vision,” J. Opt. Soc. Am. 52, 1276–1286 (1962).
[CrossRef]

1961 (1)

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

1956 (1)

C. I. Howarth and M. G. Bulmer, “Non-random sequences in visual threshold experiments,” Q. J. Exp. Psychol. 8, 163–171 (1956).
[CrossRef]

1954 (2)

G. Collier, “Probability of response and interocular association as function of monocular and binocular stimulation,” J. Exp. Psychol. 47, 75–83 (1954).
[CrossRef]

W. W. Peterson, T. G. Birdsall, and W. C. Fox, “The theory of signal detectability,” IEEE Trans. Inf. Theory PGIT-4, 171–212 (1954). This paper is the ultimate source of signal-detection theory. The authors conducted their research at the University of Michigan, where Tanner and Swets learnt about it well in advance of their own publication. Tanner and Swets (1954) list a precursor of the Peterson, Birdsall, and Fox (1954) paper among their references.
[CrossRef]

1945 (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 24, 46–156 (1945).

1944 (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944).

1943 (1)

M. H. Pirenne, “Binocular and uniocular threshold of vision,” Nature 152, 698–699 (1943).
[CrossRef]

1933 (1)

J. Neyman and E. S. Pearson, “On the problem of the most efficient tests of statistical hypotheses,” Philos. Trans. R. Soc. Lond. A 231, 289–337 (1933).
[CrossRef]

Ahumada, A. J.

A. B. Watson and A. J. Ahumada, “A standard model for foveal detection of spatial contrast,” J. Vis. 5(9):6, 1–23 (2005).
[CrossRef]

Atkinson, R. C.

T. A. Tanner, J. A. Rauk, and R. C. Atkinson, “Signal recognition as influenced by information feedback,” J. Math. Psychol. 7, 259–274 (1970).
[CrossRef]

Bird, C. M.

Birdsall, T. G.

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

W. W. Peterson, T. G. Birdsall, and W. C. Fox, “The theory of signal detectability,” IEEE Trans. Inf. Theory PGIT-4, 171–212 (1954). This paper is the ultimate source of signal-detection theory. The authors conducted their research at the University of Michigan, where Tanner and Swets learnt about it well in advance of their own publication. Tanner and Swets (1954) list a precursor of the Peterson, Birdsall, and Fox (1954) paper among their references.
[CrossRef]

Braun, J.

Brown, G. D. A.

N. Stewart, G. D. A. Brown, and N. Chater, “Absolute identification by relative judgment,” Psychol. Rev. 112, 881–911 (2005).
[CrossRef]

Bulmer, M. G.

C. I. Howarth and M. G. Bulmer, “Non-random sequences in visual threshold experiments,” Q. J. Exp. Psychol. 8, 163–171 (1956).
[CrossRef]

Burton, G. J.

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

Campbell, F. W.

F. W. Campbell and J. J. Kulikowski, “Orientational selectivity of the human visual system,” J. Physiol. 187, 437–445 (1966).

F. W. Campbell and D. G. Green, “Monocular versus binocular visual acuity,” Nature 208, 191–192 (1965).
[CrossRef]

Chater, N.

N. Stewart, G. D. A. Brown, and N. Chater, “Absolute identification by relative judgment,” Psychol. Rev. 112, 881–911 (2005).
[CrossRef]

Collier, G.

G. Collier, “Probability of response and interocular association as function of monocular and binocular stimulation,” J. Exp. Psychol. 47, 75–83 (1954).
[CrossRef]

Enroth-Cugell, C.

C. Enroth-Cugell and J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. 187, 517–552 (1966).

Fechner, G. T.

G. T. Fechner, Elemente der Psychophysik (Breitkopf and Härtel, 1860).

Foley, J. M.

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

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

Fox, W. C.

W. W. Peterson, T. G. Birdsall, and W. C. Fox, “The theory of signal detectability,” IEEE Trans. Inf. Theory PGIT-4, 171–212 (1954). This paper is the ultimate source of signal-detection theory. The authors conducted their research at the University of Michigan, where Tanner and Swets learnt about it well in advance of their own publication. Tanner and Swets (1954) list a precursor of the Peterson, Birdsall, and Fox (1954) paper among their references.
[CrossRef]

Goris, R. L. T.

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

Graham, N. v. S.

N. v. S. Graham, Visual Pattern Analyzers (Oxford, 1989).

N. v. S. Graham, “Spatial-frequency channels in human vision: detecting edges without edge detectors,” in Visual Coding and Adaptability, C. S. Harris, ed. (Erlbaum, 1980), pp. 215–262.

Green, D. G.

F. W. Campbell and D. G. Green, “Monocular versus binocular visual acuity,” Nature 208, 191–192 (1965).
[CrossRef]

Green, D. M.

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

Henning, G. B.

H. E. Smithson, G. B. Henning, D. I. A. MacLeod, and A. Stockman, “The effect of notched noise on flicker detection and discrimination,” J. Vis. 9(5):21, 1–18 (2009).
[CrossRef]

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

G. B. Henning and F. A. Wichmann, “Some observations on the pedestal effect,” J. Vis. 7(1):3, 1–15 (2007).
[CrossRef]

C. M. Bird, G. B. Henning, and F. A. Wichmann, “Contrast discrimination with sinusoidal gratings of different spatial frequency,” J. Opt. Soc. Am. A 19, 1267–1273 (2002).
[CrossRef]

Hess, R. F.

E. R. Howell and R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vis. Res. 18, 369–374 (1978).
[CrossRef]

Hofer, H.

D. Koenig and H. Hofer, “The absolute threshold of cone vision,” J. Vis. 11(1):21, 1–24 (2011).
[CrossRef]

Howarth, C. I.

C. I. Howarth and M. G. Bulmer, “Non-random sequences in visual threshold experiments,” Q. J. Exp. Psychol. 8, 163–171 (1956).
[CrossRef]

Howell, E. R.

E. R. Howell and R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vis. Res. 18, 369–374 (1978).
[CrossRef]

Hubel, D. H.

D. H. Hubel and T. N. Wiesel, “Functional architecture of macaque monkey visual cortex,” Proc. R. Soc. Lond. B 198, 1–59 (1977).
[CrossRef]

Itti, L.

Kelly, D. H.

Keynes, J. M.

J. M. Keynes, The General Theory of Employment, Interest and Money (Macmillan, 1936).

Koch, C.

Kocher, E. C.

Koenig, D.

D. Koenig and H. Hofer, “The absolute threshold of cone vision,” J. Vis. 11(1):21, 1–24 (2011).
[CrossRef]

Kulikowski, J. J.

J. J. Kulikowski, “Effective contrast constancy and linearity of contrast sensation,” Vis. Res. 16, 1419–1431 (1976).

J. J. Kulikowski, “Limiting conditions of visual perception,” Prace Inst. Automat. PAN (Warsaw) 77, 1–133 (1969). (English translation)

F. W. Campbell and J. J. Kulikowski, “Orientational selectivity of the human visual system,” J. Physiol. 187, 437–445 (1966).

Kullback, S.

S. Kullback, Information Theory and Statistics (Wiley, 1959).

Laming, D.

D. Laming, “Fechner’s law: where does the log transform come from?” Seeing Perceiving 23, 155–171 (2010).

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

D. Laming, “Spatial frequency channels,” in Vision and Visual Dysfunction, Vol 5: Limits of Visual Perception, J. J. Kulikowski, V. Walsh, and I. J. Murray, eds. (Macmillan, 1991), pp. 97–105.

D. Laming, Human Judgment: The Eye of the Beholder(Thomson Learning, 2004), p. 179.

D. Laming, Mathematical Psychology (Academic, 1973).

Legge, G. E.

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

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

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

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

G. B. Weatherill and H. Levitt, “Sequential estimation of points on a psychometric function,” Brit. J. Math. Statist. Psychol. 18, 1–9 (1965).
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H. Lotze, Metaphysik; drei Bücher der Ontologie, Kosmologie und Psychologie (Hirzel, 1879), translated B. Bosanquet (Clarendon, 1884), p. 455.

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J. Rovamo, O. Luntinen, and R. Naesaenen, “Modelling the dependence of contrast sensitivity on grating area and spatial frequency,” Vis. Res. 33, 2773–2788 (1993).
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H. E. Smithson, G. B. Henning, D. I. A. MacLeod, and A. Stockman, “The effect of notched noise on flicker detection and discrimination,” J. Vis. 9(5):21, 1–18 (2009).
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J. Rovamo, J. Mustonen, and R. Naesaenen, “Modelling contrast sensitivity as a function of retinal illuminance and grating area,” Vis. Res. 34, 1301–1314 (1994).
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J. Nachmias, “On the psychometric function for contrast detection,” Vis. Res. 21, 215–223 (1981).
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J. Nachmias and R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vis. Res. 14, 1039–1042 (1974).
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M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
[CrossRef]

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J. Nachmias and R. M. Steinman, “Brightness and discriminability of light flashes,” Vis. Res. 5, 545–557 (1965).
[CrossRef]

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J. Rovamo, J. Mustonen, and R. Naesaenen, “Modelling contrast sensitivity as a function of retinal illuminance and grating area,” Vis. Res. 34, 1301–1314 (1994).
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J. Rovamo, O. Luntinen, and R. Naesaenen, “Modelling the dependence of contrast sensitivity on grating area and spatial frequency,” Vis. Res. 33, 2773–2788 (1993).
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J. Neyman and E. S. Pearson, “On the problem of the most efficient tests of statistical hypotheses,” Philos. Trans. R. Soc. Lond. A 231, 289–337 (1933).
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S. M. Pfafflin and M. V. Mathews, “Energy-detection model for monaural auditory detection,” J. Acoust. Soc. Am. 34, 1842–1853 (1962).
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B. Leshowitz, H. B. Taub, and D. H. Raab, “Visual detection of signals in the presence of continuous and pulsed backgrounds,” Percept. Psychophys. 4, 207–213 (1968).
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J. Rovamo, J. Mustonen, and R. Naesaenen, “Modelling contrast sensitivity as a function of retinal illuminance and grating area,” Vis. Res. 34, 1301–1314 (1994).
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J. Rovamo, O. Luntinen, and R. Naesaenen, “Modelling the dependence of contrast sensitivity on grating area and spatial frequency,” Vis. Res. 33, 2773–2788 (1993).
[CrossRef]

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J. Nachmias and R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vis. Res. 14, 1039–1042 (1974).
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H. E. Smithson, G. B. Henning, D. I. A. MacLeod, and A. Stockman, “The effect of notched noise on flicker detection and discrimination,” J. Vis. 9(5):21, 1–18 (2009).
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H. D. Speed and J. Ross, “Spatial frequency tuning of facilitation by masks,” Vis. Res. 32, 1143–1148 (1992).
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J. A. Swets, W. P. Tanner, and T. G. Birdsall, “Decision processes in perception,” Psychol. Rev. 68, 301–340 (1961).
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B. Leshowitz, H. B. Taub, and D. H. Raab, “Visual detection of signals in the presence of continuous and pulsed backgrounds,” Percept. Psychophys. 4, 207–213 (1968).
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H. E. Smithson, G. B. Henning, D. I. A. MacLeod, and A. Stockman, “The effect of notched noise on flicker detection and discrimination,” J. Vis. 9(5):21, 1–18 (2009).
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J. Nachmias and R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vis. Res. 14, 1039–1042 (1974).
[CrossRef]

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

H. D. Speed and J. Ross, “Spatial frequency tuning of facilitation by masks,” Vis. Res. 32, 1143–1148 (1992).
[CrossRef]

J. Yang and W. Makous, “Modeling pedestal experiments with amplitude instead of contrast,” Vis. Res. 35, 1979–1989 (1995).
[CrossRef]

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

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

J. Nachmias and R. M. Steinman, “Brightness and discriminability of light flashes,” Vis. Res. 5, 545–557 (1965).
[CrossRef]

E. R. Howell and R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vis. Res. 18, 369–374 (1978).
[CrossRef]

J. Rovamo, O. Luntinen, and R. Naesaenen, “Modelling the dependence of contrast sensitivity on grating area and spatial frequency,” Vis. Res. 33, 2773–2788 (1993).
[CrossRef]

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[CrossRef]

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[CrossRef]

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

Other (14)

N. v. S. Graham, Visual Pattern Analyzers (Oxford, 1989).

Detection of a grating in a dark surround under continuous inspection needs further comment. The only cue to detection of sinusoidal modulation in such a case is the variation in input to individual units as the eye moves laterally with respect to the bars of the grating. Temporal modulations generate a square-law perturbation [27, Fig. 7.4, p. 112], in this case over the face of the grating. Moreover, temporal sensitivity is maintained in the peripheral retina, so that the extreme extent of the grating in Fig. 11 does not matter. Ordinarily, comparison with a matched surround provides the more sensitive cue. However, thresholds for (0.1  c/deg) gratings with and without a surround both reach a lower limit at a grating height of 3.2λ, whereafter they do not noticeably differ. This suggests that the same limit to summation applies to both (as one should expect) and, at that limit, spatio-temporal modulations across the boundary with a matched surround are no more informative than modulations within the grating field.

J. M. Keynes, The General Theory of Employment, Interest and Money (Macmillan, 1936).

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

S. Kullback, Information Theory and Statistics (Wiley, 1959).

D. Marr, Vision (Freeman, 1982).

N. v. S. Graham, “Spatial-frequency channels in human vision: detecting edges without edge detectors,” in Visual Coding and Adaptability, C. S. Harris, ed. (Erlbaum, 1980), pp. 215–262.

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D. Laming, Mathematical Psychology (Academic, 1973).

D. Laming, Human Judgment: The Eye of the Beholder(Thomson Learning, 2004), p. 179.

H. Lotze, Metaphysik; drei Bücher der Ontologie, Kosmologie und Psychologie (Hirzel, 1879), translated B. Bosanquet (Clarendon, 1884), p. 455.

D. Laming, “Spatial frequency channels,” in Vision and Visual Dysfunction, Vol 5: Limits of Visual Perception, J. J. Kulikowski, V. Walsh, and I. J. Murray, eds. (Macmillan, 1991), pp. 97–105.

The argument by Sachs et al. ([4]; see their Fig. 4) looks compelling, except that they have assumed the psychometric function for detection of a grating to be normal with respect to contrast. They could alternatively have accommodated their experimental results by supposing the psychometric function to be normal with respect to some power of contrast (as in Fig. 1 here), with detectability depending on the summation of that power over all grating components. There would then have been no need for probability summation, nor for a fixed threshold. The evidence, as at that time, pointing to a power-law transform of small near-threshold stimuli had already been summarized by Nachmias and Kocher [6]. This article shows that the idea that Sachs et al. did not explore, that is, of detectability depending on the summation of power-law transforms over all grating components, provides a more comprehensive account of visual sensitivity.

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

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Figures (14)

Fig. 1.
Fig. 1.

Detectability data for gratings of 1.2, 3.6, and 10.8c/deg and for their compounds in sine and cosine phases. Contrast is expressed relative to the 75% threshold (at 0 dB) for each grating. The continuous curves are Weibull functions [Eq. (3)] with parameter 3.96, one of them shifted 2.8dB with respect to the other. The dashed curves are normal integrals with respect to Eq. (A9) in Appendix A. Data from [11, Fig. 7]. [Adapted from Sensory Analysis, by D. Laming (Academic Press, 1986), p. 54. Copyright Elsevier 1986. Reproduced by permission.]

Fig. 2.
Fig. 2.

Thresholds for sinusoidal gratings as a function of stimulus duration. The straight lines have gradient 1/4 with respect to the logarithmic axes, showing that threshold decreases as the inverse fourth root of duration. Data for one observer from [12]. [Adapted from Sensory Analysis, by D. Laming (Academic Press, 1986), p. 196. Copyright Elsevier 1986. Reproduced by permission.]

Fig. 3.
Fig. 3.

Proportions of correct second guesses (conditional on an incorrect first response) in the 4AFC experiment by Swets et al. [19, Fig. 10]. The proportion correct increases with d, which measures the detectability of the flash. The continuous curve represents calculations based on the normal, equal-variance signal-detection model. (The American Psychological Association, 1961. Reproduced by permission.)

Fig. 4.
Fig. 4.

Signal-detection operating characteristics and the gradient at a typical operating point. The illustrative data are from [21, Expt. 3]. [From Mathematical Psychology, by D. Laming (Academic Press, 1973), p. 75. Copyright Elsevier 1973. Reproduced by permission.]

Fig. 5.
Fig. 5.

AC is the operating characteristic for a single-threshold unit with probability A of detection and BC shows how that characteristic is modified by probability summation applied to multiple units. The data points (for comparison) are the cumulative proportions of false positives and correct detections from the rating experiment by [19].

Fig. 6.
Fig. 6.

Difference thresholds for contrast as a function of pedestal contrast from [30]. The three panels present the data from three different observers. The curves are based on the assumption that Weber’s law applies exactly to the transform of Eq. (A8) in Appendix A. The vertical arrows mark the respective detection thresholds on the abscissa.

Fig. 7.
Fig. 7.

Normal deviate transforms of the proportions of correct detections (filled circles) and discriminations (open circles) in 2AFC detection/discrimination of contrast. The discrimination data relate to discrimination of a contrast with respect to its 75% detection threshold, and the normal deviate transforms are accordingly augmented by 0.6745. The curves fitted to these measurements are Eq. (A8) in Appendix A with a suitable scaling constant aL0. The data, for one observer, are from [32].

Fig. 8.
Fig. 8.

Inverse contrast thresholds for monocular (both left and right eyes) and binocular viewing. The sinusoidal grating was rectangular in outline, 2° by 1.3°, and was surrounded by a uniform luminance equal to its space average (80cd/m2). The ratio of monocular to binocular thresholds is shown by the filled squares at the bottom of the diagram (linear scale at right-hand side) and is consistently close to 2. The thresholds were determined by the method of adjustment and are reproduced from [39]. [Reproduced from Sensory Analysis, by D. Laming (Academic Press, 1986), p. 220. Copyright Elsevier 1986. Reproduced by permission.]

Fig. 9.
Fig. 9.

Arrangement of transfer functions relative to binocular fusion. Each rectangular box represents a component transform [Eq. (A6) in Appendix A], initially square law for the smallest contrasts, but transiting to linear for supra-threshold contrasts. The first component is specific to each eye; the second is common to both eyes. In combination, the two transforms in cascade deliver the complete transfer function of Fig. 7, initially fourth power, but transiting to linear for supra-threshold contrasts.

Fig. 10.
Fig. 10.

Monocular and dichoptic masking of contrast. Detection thresholds are shown for five different test wavenumbers, indicated by the arrows, masked by gratings of various wavenumbers and of fixed contrast 0.19. The data, averaged over the two observers, are reproduced from Legge [41]. [Adapted from Sensory Analysis, by D. Laming (Academic Press, 1986), p. 216. Copyright Elsevier 1986. Reproduced by permission.]

Fig. 11.
Fig. 11.

Monocular and dichoptic discrimination of contrast. Thresholds for four different wavenumbers have been normalized with respect to their unmasked detection thresholds, so that the data points cluster round a common empirical trend. The dashed curves are copied from Fig. 6(a) and represent the prediction of the fourth-power nonlinearity in Fig. 7. The dotted curves are an equivalent square-law nonlinearity, calculated with the same parameter values as the fourth-power transform. In the dichoptic diagram, the dotted curve has been expanded twofold vertically to take account of the hypothesis that dichoptic thresholds relate to (ΔCdich)2. The data, averaged over the same two observers as in Fig. 10, are reproduced from [41]. [Adapted from Sensory Analysis, by D. Laming (Academic Press, 1986), p. 214. Copyright Elsevier 1986. Reproduced by permission.]

Fig. 12.
Fig. 12.

Sample oscilloscope traces to illustrate the sensory analysis of a sinusoidal grating. (a) Luminance profile. (b) Positive and negative Poisson inputs to a receptive field unit; the amplitude in the negative input has been attenuated more than that in the positive. (c) Linear combination of the two traces in (b). (d) Half-wave rectified output, where z=L0aCcos2πgu/L0; the high-frequency components of the Poisson noise have been attenuated in recognition of the bandpass character of the receptive field units. [Adapted from “Spatial frequency channels,” by D. Laming, in J. J. Kulikowski, V. Walsh, and I. J. Murray, eds., Vision and Visual Dysfunction, Vol. 5: Limits of Visual Perception (Macmillan, 1991), p. 100. Copyright Macmillan, 1991. Reproduced by permission.]

Fig. 13.
Fig. 13.

Thresholds for the detection of sinusoidal gratings under continuous inspection. The gratings, of the wavenumbers indicated, were vertically oriented and extended exactly five cycles (eight cycles for 0.1c/deg) horizontally. The heights of the gratings varied as shown in units of their wavelengths. Open symbols represent thresholds for gratings set in a surround of equal mean luminance (100cd/m2); the broken lines have gradient 1/4 showing that in this case threshold decreases as the inverse fourth root of the height (or area) up to a limit. The filled circles represent thresholds for a grating of 0.1c/deg set in a dark surround; the continuous line has gradient 1/2 showing that in the absence of a surround luminance threshold decreases as the inverse square root of the height. Data for one observer from [53]. [Adapted from Sensory Analysis, by D. Laming (Academic Press, 1986), p. 191. Copyright Elsevier 1986. Reproduced by permission.]

Fig. 14.
Fig. 14.

Further thresholds for the detection of sinusoidal gratings from [53]. The height of the gratings, again vertically oriented, was fixed at a value giving minimum threshold in Fig. 13, and the number of cycles varied as shown. As the number of cycles increased, threshold decreased initially as (No. of cycles)1/4 (continuous lines), then as (No. of cycles)1/8 (broken lines). Data for the same observer as Fig. 13. [Adapted from Sensory Analysis, by D. Laming (Academic Press, 1986), p. 193. Copyright Elsevier 1986. Reproduced by permission.]

Tables (1)

Tables Icon

Table 1. Typical Threshold Values Paired with the Gradient of the Corresponding Psychometric Function

Equations (32)

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

pcomp=1(1p1)(1p2)(1p3)
pcomp=1exp(aC1k)exp(aC2k)exp(aC3k)=1exp(a[C1k+C2k+C3k]),
P·correct=112exp{aCk},
P(False positive)=gP(Correct detection)=p+g(1p),},
P(False positive)=α(x)P(Correct detection)=β(x)}
λ=dβ(x)/dα(x),
ΔC=ΔC0C,
amCmask+(amΔCdich)2amCmask+amΔCmon,
(ΔCdich)2am1ΔCmon.
ΔCmon/CmaskamΔCmon/c0.
ΔCmon/Cmask=Θ,
amΔCmon/c0=[ν(C)]1/2Θ.
ΔCmon=(c0/am)[ν(C)]1/2Θ=Cmask[ν(Cmask)]1/2Θ
(amΔCdich)2/c0=[ν(C)]1/2Θ,
amΔCdich=(c0[ν(C)]1/2Θ)1/2.
L(u)=L0[1+Ccos2πgu],
2Es/N0=ln1+M(ed21),
L(u)=L0[1+Ccos2πgu],
[L0(C+C)cos2πgu]/2L0aCcos2πgu,
(2πL0)1/20xexp{(xL0aCcos2πgu)2/2L0}dx=L0aCcos2πguΦ(L0aCcos2πgu/L0)+L0Φ(L0aCcos2πgu/L0),
(2πL0)1/20xexp{(xL0aCcos2πgu)2/2L0}dx=L0/2π+(L0aCcos2πgu/L0)/2+L0/2π(L0aCcos2πgu/L0)2/2+
L0[zcos2πguΦ(zcos2πgu)+Φ(zcos2πgu)].
μ(z)=[z(Φ(z)1/2)+(Φ(z)+Φ(0))]/2,
h(z)=[z(Φ(z)1/2)+(Φ(z)Φ(0))]/2(11/π).
Contrast transform=h(h(aCL0)).
P·correct=Φ{h(h(aCL0))}.
μbckgd={1+1/[8(11/π)]}/π1.
h(h(z))+μbckgd=z/[8(11/π)],
C+ΔC=(1+Θ)C,
h(h(a(C+ΔC)))+μbckgd=(1+Θ)[h(h(aC))+μbckgd],
h(h(aΔC0))=Θμbckgd,
h(h(a(C+ΔC)))=(1+Θ)h(h(aC))+h(h(aΔC0)).

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