Abstract

At photopic luminance levels, the cone-cell variation of packing density across the retina provides a natural limit to the effective size of wide-field stimulus patterns. In some experiments, this eliminates the need for small test spots, which produce band-broadening effects in the spatial-frequency domain. Calculations of these effects are given, to aid in the design of such experiments.

© 1974 Optical Society of America

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References

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  1. These rules can be found in almost any book on Fourier methods; e.g., R. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill, New York, 1965); A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill, New York, 1962); or I. N. Sneddon, Fourier Transforms (McGraw–Hill, New York, 1951). A brief summary is given by D. H. Kelly, Appl. Opt. 4, 435 (1965).
    [Crossref]
  2. M. Aguilar and W. S. Stiles, Opt. Acta 1, 59 (1954).
    [Crossref]
  3. K. T. Brown and M. Murakami, Vision Res. 8, 1145 (1968), found that rod-receptor potentials in cats and monkeys were suppressed by cone activity, probably via horizontal-cell pathways. Related human psychophysical results have now been reported (e.g., at the Association for Research in Vision and Ophthalmology meeting in Sarasota, Fla., 24–28 April 1972). See W. Makous and R. Boothe, Vision Res. 14, 285 (1974).
    [Crossref] [PubMed]
  4. G. Østerberg, Acta Ophthalmol. Suppl. No. 6 (1935). Although limited to a single retina (and the techniques of 40 years ago), this painstaking study is still the standard reference for receptor-cell distributions.
  5. Among those of clinical importance are contrast sensitivity, flicker sensitivity, color discrimination, pupil response, visual acuity, local adaptation, spatial and temporal integration. See E. Aulhorn and H. Harms, in Handbook of Sensory Physiology Vol. VII/4, Visual Psychophysics, edited by D. Jameson and L. M. Hurvich (Springer, Berlin, 1972), p. 142.
  6. H. Davson, The Physiology of the Eye (Blakiston, Philadelphia, 1949).
  7. The author has been unable to find any previous calculation of this important frequency spectrum.
  8. The reader should also recall the well-known effects of ocular scattering, geometrical aberrations and diffraction, not discussed here. Scattering is essentially independent of spatial frequency; the other optical factors are readily evaluated from available data, e.g., F. W. Campbell and R. W. Gubisch, J. Physiol. Lond. 186, 558 (1966). (With a sinusoidal grating, optical attenuation is negligible when f0 is relatively low, as in determining receptive-field properties.)
  9. D. H. Kelly and R. E. Savoie, Percept. Psychophys. 14, 313 (1973); D. H. Kelly, J. Physiol. Lond. 228, 55 (1973), Vision Res. 12, 89 (1972), and J. Opt. Soc. Am. 64, 983 (1974).
    [Crossref]

1973 (1)

D. H. Kelly and R. E. Savoie, Percept. Psychophys. 14, 313 (1973); D. H. Kelly, J. Physiol. Lond. 228, 55 (1973), Vision Res. 12, 89 (1972), and J. Opt. Soc. Am. 64, 983 (1974).
[Crossref]

1968 (1)

K. T. Brown and M. Murakami, Vision Res. 8, 1145 (1968), found that rod-receptor potentials in cats and monkeys were suppressed by cone activity, probably via horizontal-cell pathways. Related human psychophysical results have now been reported (e.g., at the Association for Research in Vision and Ophthalmology meeting in Sarasota, Fla., 24–28 April 1972). See W. Makous and R. Boothe, Vision Res. 14, 285 (1974).
[Crossref] [PubMed]

1966 (1)

The reader should also recall the well-known effects of ocular scattering, geometrical aberrations and diffraction, not discussed here. Scattering is essentially independent of spatial frequency; the other optical factors are readily evaluated from available data, e.g., F. W. Campbell and R. W. Gubisch, J. Physiol. Lond. 186, 558 (1966). (With a sinusoidal grating, optical attenuation is negligible when f0 is relatively low, as in determining receptive-field properties.)

1954 (1)

M. Aguilar and W. S. Stiles, Opt. Acta 1, 59 (1954).
[Crossref]

1935 (1)

G. Østerberg, Acta Ophthalmol. Suppl. No. 6 (1935). Although limited to a single retina (and the techniques of 40 years ago), this painstaking study is still the standard reference for receptor-cell distributions.

Aguilar, M.

M. Aguilar and W. S. Stiles, Opt. Acta 1, 59 (1954).
[Crossref]

Aulhorn, E.

Among those of clinical importance are contrast sensitivity, flicker sensitivity, color discrimination, pupil response, visual acuity, local adaptation, spatial and temporal integration. See E. Aulhorn and H. Harms, in Handbook of Sensory Physiology Vol. VII/4, Visual Psychophysics, edited by D. Jameson and L. M. Hurvich (Springer, Berlin, 1972), p. 142.

Bracewell, R.

These rules can be found in almost any book on Fourier methods; e.g., R. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill, New York, 1965); A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill, New York, 1962); or I. N. Sneddon, Fourier Transforms (McGraw–Hill, New York, 1951). A brief summary is given by D. H. Kelly, Appl. Opt. 4, 435 (1965).
[Crossref]

Brown, K. T.

K. T. Brown and M. Murakami, Vision Res. 8, 1145 (1968), found that rod-receptor potentials in cats and monkeys were suppressed by cone activity, probably via horizontal-cell pathways. Related human psychophysical results have now been reported (e.g., at the Association for Research in Vision and Ophthalmology meeting in Sarasota, Fla., 24–28 April 1972). See W. Makous and R. Boothe, Vision Res. 14, 285 (1974).
[Crossref] [PubMed]

Campbell, F. W.

The reader should also recall the well-known effects of ocular scattering, geometrical aberrations and diffraction, not discussed here. Scattering is essentially independent of spatial frequency; the other optical factors are readily evaluated from available data, e.g., F. W. Campbell and R. W. Gubisch, J. Physiol. Lond. 186, 558 (1966). (With a sinusoidal grating, optical attenuation is negligible when f0 is relatively low, as in determining receptive-field properties.)

Davson, H.

H. Davson, The Physiology of the Eye (Blakiston, Philadelphia, 1949).

Gubisch, R. W.

The reader should also recall the well-known effects of ocular scattering, geometrical aberrations and diffraction, not discussed here. Scattering is essentially independent of spatial frequency; the other optical factors are readily evaluated from available data, e.g., F. W. Campbell and R. W. Gubisch, J. Physiol. Lond. 186, 558 (1966). (With a sinusoidal grating, optical attenuation is negligible when f0 is relatively low, as in determining receptive-field properties.)

Harms, H.

Among those of clinical importance are contrast sensitivity, flicker sensitivity, color discrimination, pupil response, visual acuity, local adaptation, spatial and temporal integration. See E. Aulhorn and H. Harms, in Handbook of Sensory Physiology Vol. VII/4, Visual Psychophysics, edited by D. Jameson and L. M. Hurvich (Springer, Berlin, 1972), p. 142.

Kelly, D. H.

D. H. Kelly and R. E. Savoie, Percept. Psychophys. 14, 313 (1973); D. H. Kelly, J. Physiol. Lond. 228, 55 (1973), Vision Res. 12, 89 (1972), and J. Opt. Soc. Am. 64, 983 (1974).
[Crossref]

Murakami, M.

K. T. Brown and M. Murakami, Vision Res. 8, 1145 (1968), found that rod-receptor potentials in cats and monkeys were suppressed by cone activity, probably via horizontal-cell pathways. Related human psychophysical results have now been reported (e.g., at the Association for Research in Vision and Ophthalmology meeting in Sarasota, Fla., 24–28 April 1972). See W. Makous and R. Boothe, Vision Res. 14, 285 (1974).
[Crossref] [PubMed]

Østerberg, G.

G. Østerberg, Acta Ophthalmol. Suppl. No. 6 (1935). Although limited to a single retina (and the techniques of 40 years ago), this painstaking study is still the standard reference for receptor-cell distributions.

Savoie, R. E.

D. H. Kelly and R. E. Savoie, Percept. Psychophys. 14, 313 (1973); D. H. Kelly, J. Physiol. Lond. 228, 55 (1973), Vision Res. 12, 89 (1972), and J. Opt. Soc. Am. 64, 983 (1974).
[Crossref]

Stiles, W. S.

M. Aguilar and W. S. Stiles, Opt. Acta 1, 59 (1954).
[Crossref]

Acta Ophthalmol. Suppl. No. 6 (1)

G. Østerberg, Acta Ophthalmol. Suppl. No. 6 (1935). Although limited to a single retina (and the techniques of 40 years ago), this painstaking study is still the standard reference for receptor-cell distributions.

J. Physiol. Lond. (1)

The reader should also recall the well-known effects of ocular scattering, geometrical aberrations and diffraction, not discussed here. Scattering is essentially independent of spatial frequency; the other optical factors are readily evaluated from available data, e.g., F. W. Campbell and R. W. Gubisch, J. Physiol. Lond. 186, 558 (1966). (With a sinusoidal grating, optical attenuation is negligible when f0 is relatively low, as in determining receptive-field properties.)

Opt. Acta (1)

M. Aguilar and W. S. Stiles, Opt. Acta 1, 59 (1954).
[Crossref]

Percept. Psychophys. (1)

D. H. Kelly and R. E. Savoie, Percept. Psychophys. 14, 313 (1973); D. H. Kelly, J. Physiol. Lond. 228, 55 (1973), Vision Res. 12, 89 (1972), and J. Opt. Soc. Am. 64, 983 (1974).
[Crossref]

Vision Res. (1)

K. T. Brown and M. Murakami, Vision Res. 8, 1145 (1968), found that rod-receptor potentials in cats and monkeys were suppressed by cone activity, probably via horizontal-cell pathways. Related human psychophysical results have now been reported (e.g., at the Association for Research in Vision and Ophthalmology meeting in Sarasota, Fla., 24–28 April 1972). See W. Makous and R. Boothe, Vision Res. 14, 285 (1974).
[Crossref] [PubMed]

Other (4)

Among those of clinical importance are contrast sensitivity, flicker sensitivity, color discrimination, pupil response, visual acuity, local adaptation, spatial and temporal integration. See E. Aulhorn and H. Harms, in Handbook of Sensory Physiology Vol. VII/4, Visual Psychophysics, edited by D. Jameson and L. M. Hurvich (Springer, Berlin, 1972), p. 142.

H. Davson, The Physiology of the Eye (Blakiston, Philadelphia, 1949).

The author has been unable to find any previous calculation of this important frequency spectrum.

These rules can be found in almost any book on Fourier methods; e.g., R. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill, New York, 1965); A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill, New York, 1962); or I. N. Sneddon, Fourier Transforms (McGraw–Hill, New York, 1951). A brief summary is given by D. H. Kelly, Appl. Opt. 4, 435 (1965).
[Crossref]

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Figures (2)

FIG. 1
FIG. 1

Distribution of cone cells in the human retina, after Østerberg (Ref. 4). Each point represents the number of cells counted in an area of 0.0069 mm2. The angular scale incorporates Østerberg’s correction for the shrinkage of his preparation. The four ordinates indicate field sizes typically used in visual experiments. The continuous curve is the sum of the two exponential distributions shown as dashed and dotted lines; the filled and open arrows indicate their respective 1/e radii.

FIG. 2
FIG. 2

Two-dimensional Fourier transforms of the functions given in Fig. 1, showing the effects in the frequency domain of truncating the cone distribution in the space domain. The continuous heavy curve, a plot of Eq. (2), shows the minimum bandwidth obtainable with a Ganzfeld stimulus. The two terms of Eq. (2) are also shown separately, by the dashed and dotted curves; the arrows indicate their corner frequencies. The lighter curves show the band broadening that occurs when the cone distribution is truncated at the indicated radii.

Equations (2)

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f ( r ) = 1500 exp ( - 1.879 r ) + 350 exp ( - 0.1503 r ) ,
F ( ρ ) = 0 f ( r ) J 0 ( ρ r ) r d r = 2818.5 ( 3.5306 + ρ 2 ) - 3 2 + 52.605 ( 0.0226 + ρ 2 ) - 3 2 ,