Abstract

An ideal detector is derived for the discrimination of arbitrary stimuli in the two-alternative forced-choice paradigm. The ideal detector’s performance is assumed to be limited only by quantal fluctuations, the optics of the eye, and the size and spacing of the receptors in the retinal mosaic. Detailed predictions are presented for two-point acuity and hyperacuity tasks. The ideal detector’s two-point resolution, over a wide range of luminances, is approximately 10 times worse than its two-point vernier acuity or separation discrimination. Furthermore, two-point resolution is shown to vary in proportion to the −¼ power of spot intensity, but vernier acuity and separation discrimination vary in proportion to the −½ power of spot intensity. It is shown that this ideal detector can be implemented by the use of appropriately shaped receptive fields. The derivation provides a simple way to determine the shapes of these optimal receptive fields for arbitrary stimuli. The sensitivities of real (human) and ideal detectors are compared.

© 1984 Optical Society of America

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  1. G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. 18, 893–912 (1979).
  2. T. Y. Baker, G. B. Bryan, “Errors of observation,” in Proceedings of the Optical Convention (Hodder and Stroughton, London, 1912).
  3. R. N. Berry, “Quantitative relations among vernier, real depth, and stereoscopic depth acuities,” J. Exp. Psychol. 38, 708–721 (1948).
    [CrossRef] [PubMed]
  4. G. Westheimer, S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 941–947 (1977).
    [CrossRef]
  5. H. J. Howard, “A test for the judgment of distance,” Am. J. Opthalmol 2, 656–675 (1919).
  6. G. Westheimer, S. P. McKee, “Stereoscopic acuity with defocused and spatially filtered retinal images,” J. Opt. Soc. Am. 70, 772–778 (1980).
    [CrossRef]
  7. G. Westheimer, “Visual hyperacuity,” in Progress In Sensory Physiology, Vol. 1, H. Autrum, D. Ottoson, E. R. Perl, R. F. Schmidt, eds. (Springer-Verlag, New York, 1981), pp. 1–30.
  8. J. J. Gibson, The Senses Considered As Perceptual Systems (Houghton-Mifflin, Boston, 1966).
  9. C. H. Longuet-Higgins, K. Prazdny, “The interpretation of a moving retinal image,” Proc. R. Soc. London Ser. B 208, 385–397 (1980).
    [CrossRef]
  10. W. F. Clocksin, “Perception of surface slant and edge labels from optical flow: a computational approach,” Perception 9, 252–267 (1980).
    [CrossRef]
  11. D. T. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London Ser. B 204, 301–328 (1979).
    [CrossRef]
  12. J. Mayhew, “The interpretation of stereo-disparity information: the computation of surface orientation and depth,” Perception 11, 387–403 (1982).
    [CrossRef] [PubMed]
  13. C. H. Longuet-Higgins, “The role of the vertical dimension in stereoscopic vision,” Perception 11, 377–386 (1982).
    [CrossRef] [PubMed]
  14. F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).
  15. W. H. Miller, “Interocular filters,” in Handbook of Sensory Physiology, H. Autrum, R. Jung, W. R. Loewenstein, H. L. Tueber, D. M. Mackay, eds. (Springer-Verlag, Berlin, 1979), Vol. VII/6a, pp. 70–135.
  16. G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1967).
  17. D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966; Krieger, Melbourne, Fla., 1974).
  18. G. Westheimer, S. P. McKee, “Integration regions for visual hyperacuity,” Vision Res. 17, 89–93 (1977).
    [CrossRef] [PubMed]
  19. W. P. Tanner, R. C. Clark-Jones, “The ideal sensor system as approached through statistical decision theory and the theory of signal detectability,” in Vision Research Problems, A. Morrow, E. P. Horne, eds., NAS–NRC Publ. No. 712 (National Academy of Sciences, Washington, D.C., 1960).
  20. A. Rose, “The relative sensitivities of television pick-up tubes, photographic film, and the human eye,” Proc. IRE 30, 293–300 (1942).
    [CrossRef]
  21. H. de Vries, “The quantum character of light and its bearing upon the threshold of vision, the differential sensitivity and acuity of the eye,” Physica 10, 553–564 (1943).
    [CrossRef]
  22. J. Yellot, “Spectral consequences of photoreceptor sampling in the rhesus retina,” Science 22, 382–385 (1983).
    [CrossRef]
  23. S. Shlaer, “The relation between visual acuity and illumination,” J. Gen. Physiol. 21, 165–188 (1937).
    [CrossRef] [PubMed]
  24. F. L. Van Nes, M. A. Bouman, “Spatial modulation transfer in the human eye,” J. Opt. Soc. Am. 57, 401–406 (1967).
    [CrossRef]
  25. L. A. Riggs, “Visual acuity,” in Vision and Visual Perception, C. H. Graham, ed. (Wiley, New York, 1966), pp. 321–349.
  26. F. W. Campbell, C. E. Legge, “Displacement detection in human vision,” Vision Res. 21, 205–214 (1981).
    [CrossRef] [PubMed]
  27. H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London) 136, 469–488 (1957).
  28. M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, W. J. McGill, “Multiplication noise in the human visual system at threshold. 1. Quantum fluctuations and minimum detectable energy,” J. Opt. Soc. Am. 72, 419–431 (1982).
    [CrossRef] [PubMed]
  29. H. B. Barlow, “Retinal and central factors in human vision limited by noise,” in Photoreception in Vertebrates, H. B. Barlow, P. Fatt, eds. (Academic, New York, 1977), Chap. 19, pp. 337–358.
  30. W. S. Geisler, “Effects of bleaches and backgrounds on the flash response of the cone system,” J. Physiol. (London) 312, 413–434 (1981).
  31. W. S. Geisler, “Mechanisms of visual sensitivity: backgrounds and early dark adaptation,” Vision Res. 23, 1423–1432.
    [PubMed]
  32. H. B. Barlow, “Reconstructing the visual image in space and time,” Nature 79, 189–190 (1979).
    [CrossRef]
  33. H. B. Barlow, “Critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1–34 (1981).
    [CrossRef]
  34. R. J. Watt, M. J. Morgan, “Mechanisms responsible for the assessment of visual location: theory and evidence,” Vision Res. 23, 97–109 (1983).
    [CrossRef] [PubMed]
  35. Figures 5 and 7 show, for the present ideal detector, that two-point resolution follows an inverse fourth-root law and that two-point separation discrimination and vernier acuity follow an inverse square-root law. It will be proved in a subsequent paper that these relations are not true just for point sources but hold for essentially arbitrarily shaped stimuli. Thus, for any stimulus shape, the measurement of resolution and some hyperacuity as a function of luminance will provide a strong, parameter-free test of the quantum-fluctuations hypothesis.

1983

J. Yellot, “Spectral consequences of photoreceptor sampling in the rhesus retina,” Science 22, 382–385 (1983).
[CrossRef]

R. J. Watt, M. J. Morgan, “Mechanisms responsible for the assessment of visual location: theory and evidence,” Vision Res. 23, 97–109 (1983).
[CrossRef] [PubMed]

1982

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, W. J. McGill, “Multiplication noise in the human visual system at threshold. 1. Quantum fluctuations and minimum detectable energy,” J. Opt. Soc. Am. 72, 419–431 (1982).
[CrossRef] [PubMed]

J. Mayhew, “The interpretation of stereo-disparity information: the computation of surface orientation and depth,” Perception 11, 387–403 (1982).
[CrossRef] [PubMed]

C. H. Longuet-Higgins, “The role of the vertical dimension in stereoscopic vision,” Perception 11, 377–386 (1982).
[CrossRef] [PubMed]

1981

W. S. Geisler, “Effects of bleaches and backgrounds on the flash response of the cone system,” J. Physiol. (London) 312, 413–434 (1981).

H. B. Barlow, “Critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1–34 (1981).
[CrossRef]

F. W. Campbell, C. E. Legge, “Displacement detection in human vision,” Vision Res. 21, 205–214 (1981).
[CrossRef] [PubMed]

1980

G. Westheimer, S. P. McKee, “Stereoscopic acuity with defocused and spatially filtered retinal images,” J. Opt. Soc. Am. 70, 772–778 (1980).
[CrossRef]

C. H. Longuet-Higgins, K. Prazdny, “The interpretation of a moving retinal image,” Proc. R. Soc. London Ser. B 208, 385–397 (1980).
[CrossRef]

W. F. Clocksin, “Perception of surface slant and edge labels from optical flow: a computational approach,” Perception 9, 252–267 (1980).
[CrossRef]

1979

D. T. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London Ser. B 204, 301–328 (1979).
[CrossRef]

H. B. Barlow, “Reconstructing the visual image in space and time,” Nature 79, 189–190 (1979).
[CrossRef]

G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. 18, 893–912 (1979).

1977

G. Westheimer, S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 941–947 (1977).
[CrossRef]

G. Westheimer, S. P. McKee, “Integration regions for visual hyperacuity,” Vision Res. 17, 89–93 (1977).
[CrossRef] [PubMed]

1967

1966

F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

1957

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London) 136, 469–488 (1957).

1948

R. N. Berry, “Quantitative relations among vernier, real depth, and stereoscopic depth acuities,” J. Exp. Psychol. 38, 708–721 (1948).
[CrossRef] [PubMed]

1943

H. de Vries, “The quantum character of light and its bearing upon the threshold of vision, the differential sensitivity and acuity of the eye,” Physica 10, 553–564 (1943).
[CrossRef]

1942

A. Rose, “The relative sensitivities of television pick-up tubes, photographic film, and the human eye,” Proc. IRE 30, 293–300 (1942).
[CrossRef]

1937

S. Shlaer, “The relation between visual acuity and illumination,” J. Gen. Physiol. 21, 165–188 (1937).
[CrossRef] [PubMed]

1919

H. J. Howard, “A test for the judgment of distance,” Am. J. Opthalmol 2, 656–675 (1919).

Baker, T. Y.

T. Y. Baker, G. B. Bryan, “Errors of observation,” in Proceedings of the Optical Convention (Hodder and Stroughton, London, 1912).

Barlow, H. B.

H. B. Barlow, “Critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1–34 (1981).
[CrossRef]

H. B. Barlow, “Reconstructing the visual image in space and time,” Nature 79, 189–190 (1979).
[CrossRef]

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London) 136, 469–488 (1957).

H. B. Barlow, “Retinal and central factors in human vision limited by noise,” in Photoreception in Vertebrates, H. B. Barlow, P. Fatt, eds. (Academic, New York, 1977), Chap. 19, pp. 337–358.

Berry, R. N.

R. N. Berry, “Quantitative relations among vernier, real depth, and stereoscopic depth acuities,” J. Exp. Psychol. 38, 708–721 (1948).
[CrossRef] [PubMed]

Bouman, M. A.

Breton, M. E.

Bryan, G. B.

T. Y. Baker, G. B. Bryan, “Errors of observation,” in Proceedings of the Optical Convention (Hodder and Stroughton, London, 1912).

Campbell, F. W.

F. W. Campbell, C. E. Legge, “Displacement detection in human vision,” Vision Res. 21, 205–214 (1981).
[CrossRef] [PubMed]

F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

Clark-Jones, R. C.

W. P. Tanner, R. C. Clark-Jones, “The ideal sensor system as approached through statistical decision theory and the theory of signal detectability,” in Vision Research Problems, A. Morrow, E. P. Horne, eds., NAS–NRC Publ. No. 712 (National Academy of Sciences, Washington, D.C., 1960).

Clocksin, W. F.

W. F. Clocksin, “Perception of surface slant and edge labels from optical flow: a computational approach,” Perception 9, 252–267 (1980).
[CrossRef]

de Vries, H.

H. de Vries, “The quantum character of light and its bearing upon the threshold of vision, the differential sensitivity and acuity of the eye,” Physica 10, 553–564 (1943).
[CrossRef]

Geisler, W. S.

W. S. Geisler, “Effects of bleaches and backgrounds on the flash response of the cone system,” J. Physiol. (London) 312, 413–434 (1981).

W. S. Geisler, “Mechanisms of visual sensitivity: backgrounds and early dark adaptation,” Vision Res. 23, 1423–1432.
[PubMed]

Gibson, J. J.

J. J. Gibson, The Senses Considered As Perceptual Systems (Houghton-Mifflin, Boston, 1966).

Green, D. M.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966; Krieger, Melbourne, Fla., 1974).

Gubisch, R. W.

F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

Howard, H. J.

H. J. Howard, “A test for the judgment of distance,” Am. J. Opthalmol 2, 656–675 (1919).

Legge, C. E.

F. W. Campbell, C. E. Legge, “Displacement detection in human vision,” Vision Res. 21, 205–214 (1981).
[CrossRef] [PubMed]

Longuet-Higgins, C. H.

C. H. Longuet-Higgins, “The role of the vertical dimension in stereoscopic vision,” Perception 11, 377–386 (1982).
[CrossRef] [PubMed]

C. H. Longuet-Higgins, K. Prazdny, “The interpretation of a moving retinal image,” Proc. R. Soc. London Ser. B 208, 385–397 (1980).
[CrossRef]

Marr, D. T.

D. T. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London Ser. B 204, 301–328 (1979).
[CrossRef]

Mayhew, J.

J. Mayhew, “The interpretation of stereo-disparity information: the computation of surface orientation and depth,” Perception 11, 387–403 (1982).
[CrossRef] [PubMed]

McGill, W. J.

McKee, S. P.

G. Westheimer, S. P. McKee, “Stereoscopic acuity with defocused and spatially filtered retinal images,” J. Opt. Soc. Am. 70, 772–778 (1980).
[CrossRef]

G. Westheimer, S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 941–947 (1977).
[CrossRef]

G. Westheimer, S. P. McKee, “Integration regions for visual hyperacuity,” Vision Res. 17, 89–93 (1977).
[CrossRef] [PubMed]

Miller, W. H.

W. H. Miller, “Interocular filters,” in Handbook of Sensory Physiology, H. Autrum, R. Jung, W. R. Loewenstein, H. L. Tueber, D. M. Mackay, eds. (Springer-Verlag, Berlin, 1979), Vol. VII/6a, pp. 70–135.

Morgan, M. J.

R. J. Watt, M. J. Morgan, “Mechanisms responsible for the assessment of visual location: theory and evidence,” Vision Res. 23, 97–109 (1983).
[CrossRef] [PubMed]

Poggio, T.

D. T. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London Ser. B 204, 301–328 (1979).
[CrossRef]

Prazdny, K.

C. H. Longuet-Higgins, K. Prazdny, “The interpretation of a moving retinal image,” Proc. R. Soc. London Ser. B 208, 385–397 (1980).
[CrossRef]

Prucnal, P. R.

Riggs, L. A.

L. A. Riggs, “Visual acuity,” in Vision and Visual Perception, C. H. Graham, ed. (Wiley, New York, 1966), pp. 321–349.

Rose, A.

A. Rose, “The relative sensitivities of television pick-up tubes, photographic film, and the human eye,” Proc. IRE 30, 293–300 (1942).
[CrossRef]

Shlaer, S.

S. Shlaer, “The relation between visual acuity and illumination,” J. Gen. Physiol. 21, 165–188 (1937).
[CrossRef] [PubMed]

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1967).

Swets, J. A.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966; Krieger, Melbourne, Fla., 1974).

Tanner, W. P.

W. P. Tanner, R. C. Clark-Jones, “The ideal sensor system as approached through statistical decision theory and the theory of signal detectability,” in Vision Research Problems, A. Morrow, E. P. Horne, eds., NAS–NRC Publ. No. 712 (National Academy of Sciences, Washington, D.C., 1960).

Teich, M. C.

Van Nes, F. L.

Vannucci, G.

Watt, R. J.

R. J. Watt, M. J. Morgan, “Mechanisms responsible for the assessment of visual location: theory and evidence,” Vision Res. 23, 97–109 (1983).
[CrossRef] [PubMed]

Westheimer, G.

G. Westheimer, S. P. McKee, “Stereoscopic acuity with defocused and spatially filtered retinal images,” J. Opt. Soc. Am. 70, 772–778 (1980).
[CrossRef]

G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. 18, 893–912 (1979).

G. Westheimer, S. P. McKee, “Integration regions for visual hyperacuity,” Vision Res. 17, 89–93 (1977).
[CrossRef] [PubMed]

G. Westheimer, S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 941–947 (1977).
[CrossRef]

G. Westheimer, “Visual hyperacuity,” in Progress In Sensory Physiology, Vol. 1, H. Autrum, D. Ottoson, E. R. Perl, R. F. Schmidt, eds. (Springer-Verlag, New York, 1981), pp. 1–30.

Wyszecki, G.

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1967).

Yellot, J.

J. Yellot, “Spectral consequences of photoreceptor sampling in the rhesus retina,” Science 22, 382–385 (1983).
[CrossRef]

Am. J. Opthalmol

H. J. Howard, “A test for the judgment of distance,” Am. J. Opthalmol 2, 656–675 (1919).

Invest. Ophthalmol.

G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. 18, 893–912 (1979).

J. Exp. Psychol.

R. N. Berry, “Quantitative relations among vernier, real depth, and stereoscopic depth acuities,” J. Exp. Psychol. 38, 708–721 (1948).
[CrossRef] [PubMed]

J. Gen. Physiol.

S. Shlaer, “The relation between visual acuity and illumination,” J. Gen. Physiol. 21, 165–188 (1937).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Physiol. (London)

F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London) 136, 469–488 (1957).

W. S. Geisler, “Effects of bleaches and backgrounds on the flash response of the cone system,” J. Physiol. (London) 312, 413–434 (1981).

Nature

H. B. Barlow, “Reconstructing the visual image in space and time,” Nature 79, 189–190 (1979).
[CrossRef]

Perception

W. F. Clocksin, “Perception of surface slant and edge labels from optical flow: a computational approach,” Perception 9, 252–267 (1980).
[CrossRef]

J. Mayhew, “The interpretation of stereo-disparity information: the computation of surface orientation and depth,” Perception 11, 387–403 (1982).
[CrossRef] [PubMed]

C. H. Longuet-Higgins, “The role of the vertical dimension in stereoscopic vision,” Perception 11, 377–386 (1982).
[CrossRef] [PubMed]

Physica

H. de Vries, “The quantum character of light and its bearing upon the threshold of vision, the differential sensitivity and acuity of the eye,” Physica 10, 553–564 (1943).
[CrossRef]

Proc. IRE

A. Rose, “The relative sensitivities of television pick-up tubes, photographic film, and the human eye,” Proc. IRE 30, 293–300 (1942).
[CrossRef]

Proc. R. Soc. London Ser. B

C. H. Longuet-Higgins, K. Prazdny, “The interpretation of a moving retinal image,” Proc. R. Soc. London Ser. B 208, 385–397 (1980).
[CrossRef]

H. B. Barlow, “Critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1–34 (1981).
[CrossRef]

D. T. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London Ser. B 204, 301–328 (1979).
[CrossRef]

Science

J. Yellot, “Spectral consequences of photoreceptor sampling in the rhesus retina,” Science 22, 382–385 (1983).
[CrossRef]

Vision Res.

G. Westheimer, S. P. McKee, “Integration regions for visual hyperacuity,” Vision Res. 17, 89–93 (1977).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “Mechanisms responsible for the assessment of visual location: theory and evidence,” Vision Res. 23, 97–109 (1983).
[CrossRef] [PubMed]

F. W. Campbell, C. E. Legge, “Displacement detection in human vision,” Vision Res. 21, 205–214 (1981).
[CrossRef] [PubMed]

G. Westheimer, S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 941–947 (1977).
[CrossRef]

W. S. Geisler, “Mechanisms of visual sensitivity: backgrounds and early dark adaptation,” Vision Res. 23, 1423–1432.
[PubMed]

Other

T. Y. Baker, G. B. Bryan, “Errors of observation,” in Proceedings of the Optical Convention (Hodder and Stroughton, London, 1912).

G. Westheimer, “Visual hyperacuity,” in Progress In Sensory Physiology, Vol. 1, H. Autrum, D. Ottoson, E. R. Perl, R. F. Schmidt, eds. (Springer-Verlag, New York, 1981), pp. 1–30.

J. J. Gibson, The Senses Considered As Perceptual Systems (Houghton-Mifflin, Boston, 1966).

W. H. Miller, “Interocular filters,” in Handbook of Sensory Physiology, H. Autrum, R. Jung, W. R. Loewenstein, H. L. Tueber, D. M. Mackay, eds. (Springer-Verlag, Berlin, 1979), Vol. VII/6a, pp. 70–135.

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1967).

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966; Krieger, Melbourne, Fla., 1974).

H. B. Barlow, “Retinal and central factors in human vision limited by noise,” in Photoreception in Vertebrates, H. B. Barlow, P. Fatt, eds. (Academic, New York, 1977), Chap. 19, pp. 337–358.

Figures 5 and 7 show, for the present ideal detector, that two-point resolution follows an inverse fourth-root law and that two-point separation discrimination and vernier acuity follow an inverse square-root law. It will be proved in a subsequent paper that these relations are not true just for point sources but hold for essentially arbitrarily shaped stimuli. Thus, for any stimulus shape, the measurement of resolution and some hyperacuity as a function of luminance will provide a strong, parameter-free test of the quantum-fluctuations hypothesis.

W. P. Tanner, R. C. Clark-Jones, “The ideal sensor system as approached through statistical decision theory and the theory of signal detectability,” in Vision Research Problems, A. Morrow, E. P. Horne, eds., NAS–NRC Publ. No. 712 (National Academy of Sciences, Washington, D.C., 1960).

L. A. Riggs, “Visual acuity,” in Vision and Visual Perception, C. H. Graham, ed. (Wiley, New York, 1966), pp. 321–349.

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