Abstract

Measuring the dependence of visual sensitivity on parameters of the visual stimulus is a mainstay of vision science. However, it is not widely appreciated that visual sensitivity is a product of two factors that are each invariant with respect to many properties of the stimulus and task. By estimating these two factors, one can isolate visual processes more easily than by using sensitivity measures alone. The underlying idea is that noise limits all forms of communication, including vision. As an empirical matter, it is often useful to measure the human observer’s threshold with and without a noise background added to the display, to disentangle the observer’s ability from the observer’s intrinsic noise. And when we know how much noise there is, it is often useful to calculate ideal performance of the task at hand, as a benchmark for human performance. This strips away the intrinsic difficulty of the task to reveal a pure measure of human ability. Here we show how to do the factoring of sensitivity into efficiency and equivalent noise, and we document the invariances of the two factors.

© 1999 Optical Society of America

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  1. For examples, see T. N. Cornsweet, Visual Perception (Academic, New York, 1970).
  2. We are using the terms “factor” and “product” loosely, referring to both multipliers and divisors.
  3. A. Rose, “The sensitivity performance of the human eye on an absolute scale,” J. Opt. Soc. Am. 38, 196–208 (1948).
    [CrossRef] [PubMed]
  4. W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
    [CrossRef]
  5. A. van Meeteren, H. B. Barlow, “The statistical efficiency for detecting sinusoidal modulation of average dot density in random figures,” Vision Res. 21, 765–777 (1981).
    [CrossRef] [PubMed]
  6. 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]
  7. D. G. Pelli, “Effects of visual noise,” Ph.D. dissertation (Cambridge University, Cambridge, UK, 1981).
  8. D. G. Pelli, “The quantum efficiency of vision,” in Vision: Coding and Efficiency, C. Blakemore, ed. (Cambridge U. Press, Cambridge, UK, 1990), pp. 3–24.
  9. G. E. Legge, D. Kersten, A. E. Burgess, “Contrast discrimination in noise,” J. Opt. Soc. Am. A 4, 391–404 (1987).
    [CrossRef] [PubMed]
  10. M. S. Banks, W. S. Geisler, P. J. Bennett, “The physical limits of grating visibility,” Vision Res. 27, 1915–1924 (1987).
    [CrossRef] [PubMed]
  11. W. S. Geisler, “Sequential ideal-observer analysis of visual discriminations,” Psychol. Rev. 96, 267–314 (1989).
    [CrossRef] [PubMed]
  12. A. J. Ahumada, “Putting the visual system noise back in the picture,” J. Opt. Soc. Am. A 4, 2372–2378 (1987).
    [CrossRef] [PubMed]
  13. H. B. Barlow, D. G. Pelli, eds., special issue on the statistical efficiency of natural and artificial vision, J. Opt. Soc. Am. A 4(12) (1987) and J. Opt. Soc. Am. A 5(4) (1988), Pts. 1 and 2, respectively.
  14. White noise is indistinguishable, by the system under study, from noise whose samples are all stochastically independent. In practice, the noise samples (checks) are usually created independently, and it is enough to make sure the checks are too small to be resolved. Thus the power spectral density is constant over the range of frequencies that affect the system under study. Typically one achieves this by displaying a random checkerboard, each cell randomly black or white, or sampled from a truncated Gaussian distribution, with checks no bigger than one quarter of the period of the grating to be detected (Ref. 15), since detection of the grating is mediated by an octave-wide channel. When the mediating mechanism is unknown, the relevant band is still restricted by the visual optics. Checks finer than 2 per cycle of the optical cutoff frequency will produce white noise.
  15. H. Kukkonen, J. Rovamo, R. Näsänen, “Masking potency and whiteness of noise at various noise check sizes,” Invest. Ophthalmol. Visual Sci. 36, 513–518 (1995).
  16. A. E. Burgess, “The Rose model, revisited,” J. Opt. Soc. Am. A 16, 633–646 (1999).
    [CrossRef]
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    [CrossRef]
  19. A. E. Burgess, “Prewhitening revisited,” in Medical Imaging, 1998: Visual Perception, H. L. Kundel, ed., Proc. SPIE3340, 55–64 (1998).
    [CrossRef]
  20. Note a subtle difference in notation. Tanner and Birdsall (Ref. 4) used one-sided power spectral density N0, whereas we use the two-sided N=2-kN0, where k is the dimensionality of the noise (e.g., k=2 for two-dimensional space), which simplifies the equations (see Ref. 8).
  21. H. B. Barlow, “Retinal and central factors in human vision limited by noise, in Vertebrate Photoreception, B. Barlow, P. Fatt, eds. (Academic, New York, 1977).
  22. P. B. Elliott, “Appendix 1—Tables of d′,” in Signal Detection and Recognition by Human Observers, J. S. Swets, ed. (Wiley, New York, 1964), pp. 651–684.
  23. D. G. Pelli, C. W. Burns, B. Farell, D. C. Moore, “Identifying letters,” Vision Res. (to be published).
  24. K. R. Gegenfurtner, D. C. Kiper, “Contrast detection in luminance and chromatic noise,” J. Opt. Soc. Am. A 9, 1880–1888 (1992).
    [CrossRef] [PubMed]
  25. A. Burgess, H. B. Barlow, “The precision of numerosity discrimination in arrays of random dots,” Vision Res. 23, 811–820 (1983).
    [CrossRef] [PubMed]
  26. J. M. Harris, A. J. Parker, “Efficiency of stereopsis in random-dot stereograms,” J. Opt. Soc. Am. A 9, 14–24 (1992).
    [CrossRef] [PubMed]
  27. J. A. Solomon, D. G. Pelli, “The visual filter mediating letter identification,” Nature (London) 369, 395–397 (1994).
    [CrossRef]
  28. N. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, “The role of spatial frequency channels in letter identification,” Vision Res. (to be published).
  29. M. Raghavan, “Sources of visual noise,” Ph.D. dissertation (Syracuse University, Syracuse, New York, 1995).
  30. The fivefold deviation from constant efficiency is only 5-fold in contrast because efficiency, like energy, is proportional to squared contrast. Parish and Sperling [D. H. Parish, G. Sperling, “Object spatial frequencies, retinal spatial frequencies, noise, and the efficiencies of letter discrimination,” Vision Res. 31, 1399–1416 (1991)] found a hint of this nonzero slope over the 32:1 range of size that they tested.
    [CrossRef]
  31. J. Rovamo, V. Virsu, R. Näsänen, “Cortical magnification factor predicts the photopic contrast sensitivity of peripheral vision,” Nature (London) 271, 54–56 (1978).
    [CrossRef]
  32. A. Burgess, “Image quality, the ideal observer, and human performance of radiologic decision tasks,” Acad. Radiol. 2, 522–526 (1995).
    [CrossRef] [PubMed]
  33. J. G. Robson, “Spatial and temporal contrast-sensitivity functions of the visual system,” J. Opt. Soc. Am. 56, 1141–1142 (1966).
    [CrossRef]
  34. H. B. Barlow, “A method of determining the overall quantum efficiency of visual discriminations,” J. Physiol. (London) 160, 155–168 (1962).
  35. H. B. Barlow, “Measurements of the quantum efficiency of discrimination in human scotopic vision,” J. Physiol. (London) 160, 169–188 (1962).
  36. W. P. Tanner, J. A. Swets, “A decision-making theory of visual detection,” Psychol. Rev. 61, 401–409 (1954); “The human use of information. I. Signal detection for the case of the signal known exactly,” Trans IRE PGIT-4, 213–221 (1954).
    [CrossRef] [PubMed]
  37. A. Rose, “Quantum effects in human vision,” Adv. Biol. Med. Phys. 5, 211–242 (1957).
    [CrossRef] [PubMed]
  38. N. S. Nagaraja, “Effect of luminance noise on contrast thresholds,” J. Opt. Soc. Am. 54, 950–955 (1964).
    [CrossRef]
  39. A. van Meeteren, J. Boogaard, “Visual contrast sensitivity with ideal image intensifiers,” Optik (Stuttgart) 37, 179–191 (1973).
  40. R. W. Engstrom, “Quantum efficiency of the eye determined by comparison with a TV camera,” J. Opt. Soc. Am. 64, 1706–1710 (1974).
    [CrossRef] [PubMed]
  41. A. Burgess, “Visual signal detection. III. On Bayesian use of prior knowledge and cross correlation,” J. Opt. Soc. Am. A 2, 1498–1507 (1985).
    [CrossRef] [PubMed]
  42. B. S. Tjan, W. L. Braje, G. E. Legge, D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053–3069 (1995).
    [CrossRef] [PubMed]

1999

1997

1995

B. S. Tjan, W. L. Braje, G. E. Legge, D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053–3069 (1995).
[CrossRef] [PubMed]

H. Kukkonen, J. Rovamo, R. Näsänen, “Masking potency and whiteness of noise at various noise check sizes,” Invest. Ophthalmol. Visual Sci. 36, 513–518 (1995).

A. Burgess, “Image quality, the ideal observer, and human performance of radiologic decision tasks,” Acad. Radiol. 2, 522–526 (1995).
[CrossRef] [PubMed]

1994

J. A. Solomon, D. G. Pelli, “The visual filter mediating letter identification,” Nature (London) 369, 395–397 (1994).
[CrossRef]

1992

1991

The fivefold deviation from constant efficiency is only 5-fold in contrast because efficiency, like energy, is proportional to squared contrast. Parish and Sperling [D. H. Parish, G. Sperling, “Object spatial frequencies, retinal spatial frequencies, noise, and the efficiencies of letter discrimination,” Vision Res. 31, 1399–1416 (1991)] found a hint of this nonzero slope over the 32:1 range of size that they tested.
[CrossRef]

1989

W. S. Geisler, “Sequential ideal-observer analysis of visual discriminations,” Psychol. Rev. 96, 267–314 (1989).
[CrossRef] [PubMed]

1987

H. B. Barlow, D. G. Pelli, eds., special issue on the statistical efficiency of natural and artificial vision, J. Opt. Soc. Am. A 4(12) (1987) and J. Opt. Soc. Am. A 5(4) (1988), Pts. 1 and 2, respectively.

M. S. Banks, W. S. Geisler, P. J. Bennett, “The physical limits of grating visibility,” Vision Res. 27, 1915–1924 (1987).
[CrossRef] [PubMed]

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

A. J. Ahumada, “Putting the visual system noise back in the picture,” J. Opt. Soc. Am. A 4, 2372–2378 (1987).
[CrossRef] [PubMed]

1985

1983

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

1981

A. van Meeteren, H. B. Barlow, “The statistical efficiency for detecting sinusoidal modulation of average dot density in random figures,” Vision Res. 21, 765–777 (1981).
[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]

1978

J. Rovamo, V. Virsu, R. Näsänen, “Cortical magnification factor predicts the photopic contrast sensitivity of peripheral vision,” Nature (London) 271, 54–56 (1978).
[CrossRef]

1974

1973

A. van Meeteren, J. Boogaard, “Visual contrast sensitivity with ideal image intensifiers,” Optik (Stuttgart) 37, 179–191 (1973).

1966

1964

1962

H. B. Barlow, “A method of determining the overall quantum efficiency of visual discriminations,” J. Physiol. (London) 160, 155–168 (1962).

H. B. Barlow, “Measurements of the quantum efficiency of discrimination in human scotopic vision,” J. Physiol. (London) 160, 169–188 (1962).

1958

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

H. B. Barlow, “Temporal and spatial summation in human vision at different background intensities,” J. Physiol. (London) 141, 337–350 (1958).

1957

A. Rose, “Quantum effects in human vision,” Adv. Biol. Med. Phys. 5, 211–242 (1957).
[CrossRef] [PubMed]

1954

W. P. Tanner, J. A. Swets, “A decision-making theory of visual detection,” Psychol. Rev. 61, 401–409 (1954); “The human use of information. I. Signal detection for the case of the signal known exactly,” Trans IRE PGIT-4, 213–221 (1954).
[CrossRef] [PubMed]

1948

Abbey, C. K.

Ahumada, A. J.

Banks, M. S.

M. S. Banks, W. S. Geisler, P. J. Bennett, “The physical limits of grating visibility,” Vision Res. 27, 1915–1924 (1987).
[CrossRef] [PubMed]

Barlow, H. B.

A. 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]

A. van Meeteren, H. B. Barlow, “The statistical efficiency for detecting sinusoidal modulation of average dot density in random figures,” Vision Res. 21, 765–777 (1981).
[CrossRef] [PubMed]

H. B. Barlow, “A method of determining the overall quantum efficiency of visual discriminations,” J. Physiol. (London) 160, 155–168 (1962).

H. B. Barlow, “Measurements of the quantum efficiency of discrimination in human scotopic vision,” J. Physiol. (London) 160, 169–188 (1962).

H. B. Barlow, “Temporal and spatial summation in human vision at different background intensities,” J. Physiol. (London) 141, 337–350 (1958).

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

Bennett, P. J.

M. S. Banks, W. S. Geisler, P. J. Bennett, “The physical limits of grating visibility,” Vision Res. 27, 1915–1924 (1987).
[CrossRef] [PubMed]

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]

Boogaard, J.

A. van Meeteren, J. Boogaard, “Visual contrast sensitivity with ideal image intensifiers,” Optik (Stuttgart) 37, 179–191 (1973).

Braje, W. L.

B. S. Tjan, W. L. Braje, G. E. Legge, D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053–3069 (1995).
[CrossRef] [PubMed]

Burgess, A.

A. Burgess, “Image quality, the ideal observer, and human performance of radiologic decision tasks,” Acad. Radiol. 2, 522–526 (1995).
[CrossRef] [PubMed]

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

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

Burgess, A. E.

Burns, C. W.

D. G. Pelli, C. W. Burns, B. Farell, D. C. Moore, “Identifying letters,” Vision Res. (to be published).

Elliott, P. B.

P. B. Elliott, “Appendix 1—Tables of d′,” in Signal Detection and Recognition by Human Observers, J. S. Swets, ed. (Wiley, New York, 1964), pp. 651–684.

Engstrom, R. W.

Farell, B.

D. G. Pelli, C. W. Burns, B. Farell, D. C. Moore, “Identifying letters,” Vision Res. (to be published).

Gegenfurtner, K. R.

Geisler, W. S.

W. S. Geisler, “Sequential ideal-observer analysis of visual discriminations,” Psychol. Rev. 96, 267–314 (1989).
[CrossRef] [PubMed]

M. S. Banks, W. S. Geisler, P. J. Bennett, “The physical limits of grating visibility,” Vision Res. 27, 1915–1924 (1987).
[CrossRef] [PubMed]

Harris, J. M.

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.

B. S. Tjan, W. L. Braje, G. E. Legge, D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053–3069 (1995).
[CrossRef] [PubMed]

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

Kiper, D. C.

Kukkonen, H.

H. Kukkonen, J. Rovamo, R. Näsänen, “Masking potency and whiteness of noise at various noise check sizes,” Invest. Ophthalmol. Visual Sci. 36, 513–518 (1995).

Kurshan, P.

N. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, “The role of spatial frequency channels in letter identification,” Vision Res. (to be published).

Legge, G. E.

B. S. Tjan, W. L. Braje, G. E. Legge, D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053–3069 (1995).
[CrossRef] [PubMed]

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

Li, X.

Majaj, N.

N. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, “The role of spatial frequency channels in letter identification,” Vision Res. (to be published).

Moore, D. C.

D. G. Pelli, C. W. Burns, B. Farell, D. C. Moore, “Identifying letters,” Vision Res. (to be published).

Nagaraja, N. S.

Näsänen, R.

H. Kukkonen, J. Rovamo, R. Näsänen, “Masking potency and whiteness of noise at various noise check sizes,” Invest. Ophthalmol. Visual Sci. 36, 513–518 (1995).

J. Rovamo, V. Virsu, R. Näsänen, “Cortical magnification factor predicts the photopic contrast sensitivity of peripheral vision,” Nature (London) 271, 54–56 (1978).
[CrossRef]

Palomares, M.

N. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, “The role of spatial frequency channels in letter identification,” Vision Res. (to be published).

Parish, D. H.

The fivefold deviation from constant efficiency is only 5-fold in contrast because efficiency, like energy, is proportional to squared contrast. Parish and Sperling [D. H. Parish, G. Sperling, “Object spatial frequencies, retinal spatial frequencies, noise, and the efficiencies of letter discrimination,” Vision Res. 31, 1399–1416 (1991)] found a hint of this nonzero slope over the 32:1 range of size that they tested.
[CrossRef]

Parker, A. J.

Pelli, D. G.

J. A. Solomon, D. G. Pelli, “The visual filter mediating letter identification,” Nature (London) 369, 395–397 (1994).
[CrossRef]

D. G. Pelli, C. W. Burns, B. Farell, D. C. Moore, “Identifying letters,” Vision Res. (to be published).

N. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, “The role of spatial frequency channels in letter identification,” Vision Res. (to be published).

D. G. Pelli, “Effects of visual noise,” Ph.D. dissertation (Cambridge University, Cambridge, UK, 1981).

D. G. Pelli, “The quantum efficiency of vision,” in Vision: Coding and Efficiency, C. Blakemore, ed. (Cambridge U. Press, Cambridge, UK, 1990), pp. 3–24.

Raghavan, M.

M. Raghavan, “Sources of visual noise,” Ph.D. dissertation (Syracuse University, Syracuse, New York, 1995).

Robson, J. G.

Rose, A.

Rovamo, J.

H. Kukkonen, J. Rovamo, R. Näsänen, “Masking potency and whiteness of noise at various noise check sizes,” Invest. Ophthalmol. Visual Sci. 36, 513–518 (1995).

J. Rovamo, V. Virsu, R. Näsänen, “Cortical magnification factor predicts the photopic contrast sensitivity of peripheral vision,” Nature (London) 271, 54–56 (1978).
[CrossRef]

Solomon, J. A.

J. A. Solomon, D. G. Pelli, “The visual filter mediating letter identification,” Nature (London) 369, 395–397 (1994).
[CrossRef]

Sperling, G.

The fivefold deviation from constant efficiency is only 5-fold in contrast because efficiency, like energy, is proportional to squared contrast. Parish and Sperling [D. H. Parish, G. Sperling, “Object spatial frequencies, retinal spatial frequencies, noise, and the efficiencies of letter discrimination,” Vision Res. 31, 1399–1416 (1991)] found a hint of this nonzero slope over the 32:1 range of size that they tested.
[CrossRef]

Swets, J. A.

W. P. Tanner, J. A. Swets, “A decision-making theory of visual detection,” Psychol. Rev. 61, 401–409 (1954); “The human use of information. I. Signal detection for the case of the signal known exactly,” Trans IRE PGIT-4, 213–221 (1954).
[CrossRef] [PubMed]

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]

W. P. Tanner, J. A. Swets, “A decision-making theory of visual detection,” Psychol. Rev. 61, 401–409 (1954); “The human use of information. I. Signal detection for the case of the signal known exactly,” Trans IRE PGIT-4, 213–221 (1954).
[CrossRef] [PubMed]

Tjan, B. S.

B. S. Tjan, W. L. Braje, G. E. Legge, D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053–3069 (1995).
[CrossRef] [PubMed]

van Meeteren, A.

A. van Meeteren, H. B. Barlow, “The statistical efficiency for detecting sinusoidal modulation of average dot density in random figures,” Vision Res. 21, 765–777 (1981).
[CrossRef] [PubMed]

A. van Meeteren, J. Boogaard, “Visual contrast sensitivity with ideal image intensifiers,” Optik (Stuttgart) 37, 179–191 (1973).

Virsu, V.

J. Rovamo, V. Virsu, R. Näsänen, “Cortical magnification factor predicts the photopic contrast sensitivity of peripheral vision,” Nature (London) 271, 54–56 (1978).
[CrossRef]

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]

Acad. Radiol.

A. Burgess, “Image quality, the ideal observer, and human performance of radiologic decision tasks,” Acad. Radiol. 2, 522–526 (1995).
[CrossRef] [PubMed]

Adv. Biol. Med. Phys.

A. Rose, “Quantum effects in human vision,” Adv. Biol. Med. Phys. 5, 211–242 (1957).
[CrossRef] [PubMed]

Invest. Ophthalmol. Visual Sci.

H. Kukkonen, J. Rovamo, R. Näsänen, “Masking potency and whiteness of noise at various noise check sizes,” Invest. Ophthalmol. Visual Sci. 36, 513–518 (1995).

J. Acoust. Soc. Am.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Physiol. (London)

H. B. Barlow, “Temporal and spatial summation in human vision at different background intensities,” J. Physiol. (London) 141, 337–350 (1958).

H. B. Barlow, “A method of determining the overall quantum efficiency of visual discriminations,” J. Physiol. (London) 160, 155–168 (1962).

H. B. Barlow, “Measurements of the quantum efficiency of discrimination in human scotopic vision,” J. Physiol. (London) 160, 169–188 (1962).

Nature (London)

J. A. Solomon, D. G. Pelli, “The visual filter mediating letter identification,” Nature (London) 369, 395–397 (1994).
[CrossRef]

J. Rovamo, V. Virsu, R. Näsänen, “Cortical magnification factor predicts the photopic contrast sensitivity of peripheral vision,” Nature (London) 271, 54–56 (1978).
[CrossRef]

Optik (Stuttgart)

A. van Meeteren, J. Boogaard, “Visual contrast sensitivity with ideal image intensifiers,” Optik (Stuttgart) 37, 179–191 (1973).

Psychol. Rev.

W. P. Tanner, J. A. Swets, “A decision-making theory of visual detection,” Psychol. Rev. 61, 401–409 (1954); “The human use of information. I. Signal detection for the case of the signal known exactly,” Trans IRE PGIT-4, 213–221 (1954).
[CrossRef] [PubMed]

W. S. Geisler, “Sequential ideal-observer analysis of visual discriminations,” Psychol. Rev. 96, 267–314 (1989).
[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]

Vision Res.

M. S. Banks, W. S. Geisler, P. J. Bennett, “The physical limits of grating visibility,” Vision Res. 27, 1915–1924 (1987).
[CrossRef] [PubMed]

A. van Meeteren, H. B. Barlow, “The statistical efficiency for detecting sinusoidal modulation of average dot density in random figures,” Vision Res. 21, 765–777 (1981).
[CrossRef] [PubMed]

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

The fivefold deviation from constant efficiency is only 5-fold in contrast because efficiency, like energy, is proportional to squared contrast. Parish and Sperling [D. H. Parish, G. Sperling, “Object spatial frequencies, retinal spatial frequencies, noise, and the efficiencies of letter discrimination,” Vision Res. 31, 1399–1416 (1991)] found a hint of this nonzero slope over the 32:1 range of size that they tested.
[CrossRef]

B. S. Tjan, W. L. Braje, G. E. Legge, D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053–3069 (1995).
[CrossRef] [PubMed]

Other

N. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, “The role of spatial frequency channels in letter identification,” Vision Res. (to be published).

M. Raghavan, “Sources of visual noise,” Ph.D. dissertation (Syracuse University, Syracuse, New York, 1995).

White noise is indistinguishable, by the system under study, from noise whose samples are all stochastically independent. In practice, the noise samples (checks) are usually created independently, and it is enough to make sure the checks are too small to be resolved. Thus the power spectral density is constant over the range of frequencies that affect the system under study. Typically one achieves this by displaying a random checkerboard, each cell randomly black or white, or sampled from a truncated Gaussian distribution, with checks no bigger than one quarter of the period of the grating to be detected (Ref. 15), since detection of the grating is mediated by an octave-wide channel. When the mediating mechanism is unknown, the relevant band is still restricted by the visual optics. Checks finer than 2 per cycle of the optical cutoff frequency will produce white noise.

A. E. Burgess, “Prewhitening revisited,” in Medical Imaging, 1998: Visual Perception, H. L. Kundel, ed., Proc. SPIE3340, 55–64 (1998).
[CrossRef]

Note a subtle difference in notation. Tanner and Birdsall (Ref. 4) used one-sided power spectral density N0, whereas we use the two-sided N=2-kN0, where k is the dimensionality of the noise (e.g., k=2 for two-dimensional space), which simplifies the equations (see Ref. 8).

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

P. B. Elliott, “Appendix 1—Tables of d′,” in Signal Detection and Recognition by Human Observers, J. S. Swets, ed. (Wiley, New York, 1964), pp. 651–684.

D. G. Pelli, C. W. Burns, B. Farell, D. C. Moore, “Identifying letters,” Vision Res. (to be published).

D. G. Pelli, “Effects of visual noise,” Ph.D. dissertation (Cambridge University, Cambridge, UK, 1981).

D. G. Pelli, “The quantum efficiency of vision,” in Vision: Coding and Efficiency, C. Blakemore, ed. (Cambridge U. Press, Cambridge, UK, 1990), pp. 3–24.

For examples, see T. N. Cornsweet, Visual Perception (Academic, New York, 1970).

We are using the terms “factor” and “product” loosely, referring to both multipliers and divisors.

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Figures (2)

Fig. 1
Fig. 1

(a) Letters. The vertical scale indicates the contrast of the letter. The bottom horizontal scale indicates the letter size, assuming a viewing distance of 48 cm. The top horizontal scale indicates nominal spatial frequency, in cycles per degree (c/deg), assuming 3 cycles per letter.27 The actual experiment was similar to this demonstration but showed only one letter at a time, randomly chosen from the entire alphabet. (b) Letters in noise. The noise power spectral density N is 2×10-4 deg2. (c) Letters in noise. The noise is scaled with the signal. The largest letter is an “a.” (d) Threshold contrast for 64%-correct identification, as a function of letter size, with no noise (open symbols), unscaled white noise (dashed curve), and scaled white noise (filled symbols). Circles, data for observer WT; squares, data for observer DM. The horizontal line is the ideal observer’s threshold with scaled noise. The × is DM’s threshold for 82%-correct letter detection of 1° letters in noise.

Fig. 2
Fig. 2

(a) Threshold energy for identification as a function of letter size, replotted from Fig. 1(d). Energy equals squared contrast times letter area. The × is DM’s threshold for detection. (b) Equivalent input noise of the observer, as a function of letter size, computed by Eq. (6) from the data shown in (a). Note that the equivalent noise for detection (×) is about the same as that for identification (filled symbols). (c) Efficiency η* as a function of letter size, computed by Eq. (9) from the data shown in (a). The efficiency for detection (×) is one tenth that for identification (filled symbols).

Equations (17)

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E=c2(x, y)dxdy.
c(x, y)=[L(x, y)-Lb]/Lb.
D=E/N.
E*=E-E0.
E=D* (N+Neq),
D*=EN+Neq=E-E0N=E*N
Neq=E0E-E0N
η*=EidealE*=DidealD*=η N+NeqN,
E=Didealη*(N+Neq).
η*=EidealE-E0.
E0=D* Neq,
E0=Didealη*Neq.
η=EidealE.
D*=E-E0N,
Neq=E0E-E0N.
η*=DidealD*=EidealE-E0,
Neq=E0E-E0N.

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