Abstract

It is commonly assumed that the visual resolution limit must be equal to or less than the Nyquist frequency of the cone mosaic. However, under some conditions, observers can see fine patterns at the correct orientation when viewing interference fringes with spatial frequencies that are as much as about 1.5 times higher than the nominal Nyquist frequency of the underlying cone mosaic. The existence of this visual ability demands a closer scrutiny of the sampling effects of the cone mosaic and the information that is sufficient for an observer to resolve a sinusoidal grating. The Nyquist frequency specifies which images can be reconstructed without aliasing by an imaging system that samples discretely. However, it is not a theoretical upper bound for psychophysical measures of visual resolution because the observer’s criteria for resolving sinusoidal gratings are less stringent than the criteria specified by the sampling theorem for perfect, alias-free image reconstruction.

© 1987 Optical Society of America

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  1. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1975).
  2. J. I. Yellott, B. A. Wandell, and T. N. Cornsweet, "The beginnings of visual perception: the retinal image and its initial encoding," in Handbook of Physiology, the Nervous System, S. R. Geiger, ed. (American Physiological Society, Bethesda, Md., 1984), Vol. Ill, Chap. 7, pp. 257–316.
  3. F. W. Campbell and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–593 (1965).
    [PubMed]
  4. D. G. Green, "Regional variations in the visual acuity for interference fringes on the retina," J. Physiol. 207, 351–356 (1970).
    [PubMed]
  5. J. M. Enoch and G. M. Hope, "Interferometric resolution determinations in the fovea and parafovea," Doc. Ophthalmol. 34, 143–156 (1973).
    [Crossref] [PubMed]
  6. N. J. Coletta and D. R. Williams, "Psychophysical estimate of extrafoveal cone spacing," J. Opt. Soc. Am. A 4, 1503–1513.(1987).
    [Crossref] [PubMed]
  7. D. R. Williams, "Aliasing in human foveal vision," Vision Res. 25, 195–205 (1985).
    [Crossref] [PubMed]
  8. D. R. Williams, "Topography of the foveal cone mosaic in the living human eye," submitted to Vision Res.
  9. G. M. Byram, "The physical and photochemical basis of visual resolving power. Part II. Visual acuity and the photochemistry of the retina," J. Opt. Soc. Am. 34, 718–738 (1944).
    [Crossref]
  10. D. R. Williams and R. J. Collier, "Consequences of spatial sampling by a human photoreceptor mosaic," Science 221, 385–387 (1983).
    [Crossref] [PubMed]
  11. D. R. Williams, R. J. Collier, and B. J. Thompson, "Spatial resolution of the short-wavelength mechanism," in Colour Vision, Physiology and Psychophysics, J. D. Mollon and L. T. Sharpe, eds. (Academic, London, 1983).
  12. D. R. Williams, "Seeing through the photoreceptor mosaic," Trends Neurosci. 9, 193–198 (1986).
    [Crossref]
  13. R. A. Smith and P. Cass, "Aliasing with incoherent light stimuli," J. Opt. Soc. Am. A 3(13), P93 (1986).
  14. L. N. Thibos and D. J. Walsh, "Detection of high frequency gratings in the periphery," J. Opt. Soc. Am. A 2(13), P64 (1985).
  15. A. B. Watson and D. G. Pelli, "QUEST: a Bayesian adaptive psychometric method," Percept. Psychophys. 33, 113–120 (1983).
    [Crossref] [PubMed]
  16. D. R. Williams, "Visibility of interference fringes near the resolution limit," J. Opt. Soc. Am. A 2, 1087–1093 (1985).
    [Crossref] [PubMed]
  17. The experiment could have presented more orientations than just horizontal and vertical in a single run to encourage identification instead of discrimination. However, pilot experiments revealed a large oblique effect that reduced resolution from 34 to 20 cycles/deg for one observer (NJC, by the method of adjustment). Because our goal was to push visual resolution to the theoretical limits of the mosaic, we abandoned the use of oblique gratings whose resolution is presumably limited by oriented mechanisms somewhere in the cortex.
  18. H. B. Barlow, "Visual resolution and the diffraction limit," Science 149, 533–555 (1965).
    [Crossref]
  19. F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson, "Visibility of aperiodic patterns compared with that of sinusoidal gratings," J. Physiol. 204, 283–298 (1969).
    [PubMed]
  20. Even when performances for horizontal and vertical fringes are considered separately, the orientation identification limit allways exceeds the nominal Nyquist limit. The nominal Nyquist frequencies obtained by the orientation-reversal technique are somewhat different for horizontal and vertical gratings. The values are 20, 21, and 20 cycles/deg for vertical gratings and 26, 23, and 21 cycles/deg for horizontal gratings for DRW, NJC, and RMK, respectively. The orientation-identification limit also depends on fringe orientation. These values were 37,30, and 31 cycles/deg for vertical gratings and 40, 38, and 26 cycles/deg for horizontal gratings for DRW, NJC, and RMK, respectively.
  21. G. Østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 11–103 (1935).
  22. C. A. Curcio, K. R. Sloan, Jr., O. Packer, A. E. Hendrickson, and R. E. Kalina, "Distribution of cone in human and monkey retina: individual variability and radial asymmetry," Science 236, 579–582 (1987).
    [Crossref] [PubMed]
  23. The nominal Nyquist frequency, ƒN, was estimated from the data of Østerberg21 and Curcio et al.22 by calculating the Nyquist limit of a mosaic that had the same density of cones but whose packing was triangular: ƒN = (½)(0.291)(2D/√3)½, where D is the cone density in cones per square millimeter and 0.291 converts millimeters on the retina to degrees of angular subtense. This estimate is negligibly different from one based on direct measurements of the modal frequency of the disordered extrafoveal mosaic,6 with the Nyquist frequency defined as half of the modal frequency.
  24. W. H. Miller, "Ocular optical filtering," in Handbook of Sensory Physiology, H. Autrum, ed. (Springer-Verlag, Berlin, 1979), Vol. 7, Chap. 6, pp. 70–143.
  25. V. Virsu and J. Rovamo, "Visual resolution, contrast sensitivity, and the cortical magnification factor," Exp. Brain Res. 37, 475–494 (1979).
    [Crossref] [PubMed]
  26. G. Westheimer, "The spatial grain of the perifoveal visual field," Vision Res. 22, 157–162 (1982).
    [Crossref] [PubMed]
  27. D. C. Nagel, "Spatial sampling in the retina," Invest. Ophthalmol. Vis. Sci. Suppl. 20, 123 (1981).
  28. J. I. Yellott, Jr., "Spectral analysis of spatial sampling by photoreceptors: topological disorder prevents aliasing," Vision Res. 22, 1205–1210 (1982).
    [Crossref] [PubMed]
  29. J. I. Yellott, Jr., "Spectral consequences of photoreceptor sampling in the rhesus retina," Science 221, 382–385 (1983).
    [Crossref] [PubMed]
  30. For simplicity we will treat the cones here as though they sampled at infinitely small points, ignoring the effect of the cone aperture. The cone aperture is a low-pass spatial filter that demodulates the contrast of high-spatial-frequency gratings.31,32 However, it has little effect on the contrast of interference fringes until frequencies well beyond the nominal Nyquist frequency. The point has already been made for the tightly packed foveal cones.16,31,32 The effect of the cone aperture must become even less significant with increasing eccentricity because cone spacing grows more rapidly than the cone aperture.
  31. A. W. Snyder and W. H. Miller, "Photoreceptor diameter and spacing for highest resolving power," J. Opt. Soc. Am. 67, 696–698 (1977).
    [Crossref] [PubMed]
  32. W. H. Miller and G. D. Bernard, "Averaging over the foveal receptor aperture curtails aliasing," Vision Res. 23, 1365–1369 (1983).
    [Crossref] [PubMed]
  33. A. B. Watson, A. J. Ahumada, Jr., and J. E. Farrell, "Window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays," J. Opt. Soc. Am. A 3, 300–307 (1986).
    [Crossref]
  34. H. Wassle and H. J. Riemann, "The mosaic of nerve cells in the mammalian retina," Proc. R. Soc. Lond. Ser. B 200, 441–461 (1978).
    [Crossref]
  35. B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).
    [Crossref] [PubMed]
  36. V. H. Perry and A. Cowey, "The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors," Vision Res. 25, 1795–1810 (1985).
    [Crossref]
  37. J. Hirsch and W. H. Miller, "Does cone positional disorder limit resolution?" J. Opt. Soc. Am. A 4, 1481–1492 (1987).
    [Crossref] [PubMed]
  38. J. I. Yellott, Jr., "Image sampling properties of photoreceptors: a reply to Miller and Bernard," Vision Res. 24, 281–282 (1984).
    [Crossref] [PubMed]
  39. T. R. J. Bossomaier, A. W. Snyder, and A. Hughes, "Irregularity and aliasing: solution?" Vision Res. 25, 145–147 (1985).
    [Crossref] [PubMed]
  40. A. W. Snyder, T. R. J. Bossomaier, and A. Hughes, "Optical image quality and the cone mosaic," Science, 231, 499–501 (1986).
    [Crossref] [PubMed]
  41. J. L. Yen, "On non-uniform sampling of bandwidth-limited signals," IRE Trans. Circuit Theory 3, 251–257 (1956).
    [Crossref]
  42. A. S. French, A. W. Snyder, and D. G. Stavenga, "Image degradation by an irregular retinal mosaic," Biol. Cybernetics 27, 229–233 (1977).
    [Crossref]
  43. J. I. Yellott, Jr., "Consequences of spatially irregular sampling for reconstruction of photon noisy images," Invest. Ophthalmol. Visual Sci. Suppl. 28, 137 (1987).
  44. N. J. Coletta and D. R. Williams, "Undersampling by cones reverses perceived direction of motion," Invest. Ophthalmol. Vis. Sci. Suppl. 28, 232 (1987).
  45. W. S. Geisler, "Physical limits of acuity and hyperacuity," J. Opt. Soc. Am. A 1, 775–782 (1984).
    [Crossref] [PubMed]
  46. G. T. di Francia, "Resolving power and information," J. Opt. Soc. Am. 45, 497–501 (1955).
    [Crossref]
  47. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1987 (5)

C. A. Curcio, K. R. Sloan, Jr., O. Packer, A. E. Hendrickson, and R. E. Kalina, "Distribution of cone in human and monkey retina: individual variability and radial asymmetry," Science 236, 579–582 (1987).
[Crossref] [PubMed]

J. I. Yellott, Jr., "Consequences of spatially irregular sampling for reconstruction of photon noisy images," Invest. Ophthalmol. Visual Sci. Suppl. 28, 137 (1987).

N. J. Coletta and D. R. Williams, "Undersampling by cones reverses perceived direction of motion," Invest. Ophthalmol. Vis. Sci. Suppl. 28, 232 (1987).

J. Hirsch and W. H. Miller, "Does cone positional disorder limit resolution?" J. Opt. Soc. Am. A 4, 1481–1492 (1987).
[Crossref] [PubMed]

N. J. Coletta and D. R. Williams, "Psychophysical estimate of extrafoveal cone spacing," J. Opt. Soc. Am. A 4, 1503–1513.(1987).
[Crossref] [PubMed]

1986 (4)

A. B. Watson, A. J. Ahumada, Jr., and J. E. Farrell, "Window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays," J. Opt. Soc. Am. A 3, 300–307 (1986).
[Crossref]

A. W. Snyder, T. R. J. Bossomaier, and A. Hughes, "Optical image quality and the cone mosaic," Science, 231, 499–501 (1986).
[Crossref] [PubMed]

D. R. Williams, "Seeing through the photoreceptor mosaic," Trends Neurosci. 9, 193–198 (1986).
[Crossref]

R. A. Smith and P. Cass, "Aliasing with incoherent light stimuli," J. Opt. Soc. Am. A 3(13), P93 (1986).

1985 (5)

L. N. Thibos and D. J. Walsh, "Detection of high frequency gratings in the periphery," J. Opt. Soc. Am. A 2(13), P64 (1985).

T. R. J. Bossomaier, A. W. Snyder, and A. Hughes, "Irregularity and aliasing: solution?" Vision Res. 25, 145–147 (1985).
[Crossref] [PubMed]

D. R. Williams, "Aliasing in human foveal vision," Vision Res. 25, 195–205 (1985).
[Crossref] [PubMed]

V. H. Perry and A. Cowey, "The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors," Vision Res. 25, 1795–1810 (1985).
[Crossref]

D. R. Williams, "Visibility of interference fringes near the resolution limit," J. Opt. Soc. Am. A 2, 1087–1093 (1985).
[Crossref] [PubMed]

1984 (2)

W. S. Geisler, "Physical limits of acuity and hyperacuity," J. Opt. Soc. Am. A 1, 775–782 (1984).
[Crossref] [PubMed]

J. I. Yellott, Jr., "Image sampling properties of photoreceptors: a reply to Miller and Bernard," Vision Res. 24, 281–282 (1984).
[Crossref] [PubMed]

1983 (4)

W. H. Miller and G. D. Bernard, "Averaging over the foveal receptor aperture curtails aliasing," Vision Res. 23, 1365–1369 (1983).
[Crossref] [PubMed]

A. B. Watson and D. G. Pelli, "QUEST: a Bayesian adaptive psychometric method," Percept. Psychophys. 33, 113–120 (1983).
[Crossref] [PubMed]

D. R. Williams and R. J. Collier, "Consequences of spatial sampling by a human photoreceptor mosaic," Science 221, 385–387 (1983).
[Crossref] [PubMed]

J. I. Yellott, Jr., "Spectral consequences of photoreceptor sampling in the rhesus retina," Science 221, 382–385 (1983).
[Crossref] [PubMed]

1982 (2)

J. I. Yellott, Jr., "Spectral analysis of spatial sampling by photoreceptors: topological disorder prevents aliasing," Vision Res. 22, 1205–1210 (1982).
[Crossref] [PubMed]

G. Westheimer, "The spatial grain of the perifoveal visual field," Vision Res. 22, 157–162 (1982).
[Crossref] [PubMed]

1981 (1)

D. C. Nagel, "Spatial sampling in the retina," Invest. Ophthalmol. Vis. Sci. Suppl. 20, 123 (1981).

1980 (1)

B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).
[Crossref] [PubMed]

1979 (1)

V. Virsu and J. Rovamo, "Visual resolution, contrast sensitivity, and the cortical magnification factor," Exp. Brain Res. 37, 475–494 (1979).
[Crossref] [PubMed]

1978 (1)

H. Wassle and H. J. Riemann, "The mosaic of nerve cells in the mammalian retina," Proc. R. Soc. Lond. Ser. B 200, 441–461 (1978).
[Crossref]

1977 (2)

A. S. French, A. W. Snyder, and D. G. Stavenga, "Image degradation by an irregular retinal mosaic," Biol. Cybernetics 27, 229–233 (1977).
[Crossref]

A. W. Snyder and W. H. Miller, "Photoreceptor diameter and spacing for highest resolving power," J. Opt. Soc. Am. 67, 696–698 (1977).
[Crossref] [PubMed]

1973 (1)

J. M. Enoch and G. M. Hope, "Interferometric resolution determinations in the fovea and parafovea," Doc. Ophthalmol. 34, 143–156 (1973).
[Crossref] [PubMed]

1970 (1)

D. G. Green, "Regional variations in the visual acuity for interference fringes on the retina," J. Physiol. 207, 351–356 (1970).
[PubMed]

1969 (1)

F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson, "Visibility of aperiodic patterns compared with that of sinusoidal gratings," J. Physiol. 204, 283–298 (1969).
[PubMed]

1965 (2)

F. W. Campbell and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–593 (1965).
[PubMed]

H. B. Barlow, "Visual resolution and the diffraction limit," Science 149, 533–555 (1965).
[Crossref]

1956 (1)

J. L. Yen, "On non-uniform sampling of bandwidth-limited signals," IRE Trans. Circuit Theory 3, 251–257 (1956).
[Crossref]

1955 (1)

1944 (1)

1935 (1)

G. Østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 11–103 (1935).

di Francia, G. T.

Ahumada, Jr., A. J.

Barlow, H. B.

H. B. Barlow, "Visual resolution and the diffraction limit," Science 149, 533–555 (1965).
[Crossref]

Bernard, G. D.

W. H. Miller and G. D. Bernard, "Averaging over the foveal receptor aperture curtails aliasing," Vision Res. 23, 1365–1369 (1983).
[Crossref] [PubMed]

Borwein, B.

B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).
[Crossref] [PubMed]

Borwein, C.

B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).
[Crossref] [PubMed]

Bossomaier, T. R. J.

A. W. Snyder, T. R. J. Bossomaier, and A. Hughes, "Optical image quality and the cone mosaic," Science, 231, 499–501 (1986).
[Crossref] [PubMed]

T. R. J. Bossomaier, A. W. Snyder, and A. Hughes, "Irregularity and aliasing: solution?" Vision Res. 25, 145–147 (1985).
[Crossref] [PubMed]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1975).

Byram, G. M.

Campbell, F. W.

F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson, "Visibility of aperiodic patterns compared with that of sinusoidal gratings," J. Physiol. 204, 283–298 (1969).
[PubMed]

F. W. Campbell and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–593 (1965).
[PubMed]

Carpenter, R. H. S.

F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson, "Visibility of aperiodic patterns compared with that of sinusoidal gratings," J. Physiol. 204, 283–298 (1969).
[PubMed]

Cass, P.

R. A. Smith and P. Cass, "Aliasing with incoherent light stimuli," J. Opt. Soc. Am. A 3(13), P93 (1986).

Coletta, N. J.

N. J. Coletta and D. R. Williams, "Psychophysical estimate of extrafoveal cone spacing," J. Opt. Soc. Am. A 4, 1503–1513.(1987).
[Crossref] [PubMed]

N. J. Coletta and D. R. Williams, "Undersampling by cones reverses perceived direction of motion," Invest. Ophthalmol. Vis. Sci. Suppl. 28, 232 (1987).

Collier, R. J.

D. R. Williams and R. J. Collier, "Consequences of spatial sampling by a human photoreceptor mosaic," Science 221, 385–387 (1983).
[Crossref] [PubMed]

D. R. Williams, R. J. Collier, and B. J. Thompson, "Spatial resolution of the short-wavelength mechanism," in Colour Vision, Physiology and Psychophysics, J. D. Mollon and L. T. Sharpe, eds. (Academic, London, 1983).

Cornsweet, T. N.

J. I. Yellott, B. A. Wandell, and T. N. Cornsweet, "The beginnings of visual perception: the retinal image and its initial encoding," in Handbook of Physiology, the Nervous System, S. R. Geiger, ed. (American Physiological Society, Bethesda, Md., 1984), Vol. Ill, Chap. 7, pp. 257–316.

Cowey, A.

V. H. Perry and A. Cowey, "The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors," Vision Res. 25, 1795–1810 (1985).
[Crossref]

Curcio, C. A.

C. A. Curcio, K. R. Sloan, Jr., O. Packer, A. E. Hendrickson, and R. E. Kalina, "Distribution of cone in human and monkey retina: individual variability and radial asymmetry," Science 236, 579–582 (1987).
[Crossref] [PubMed]

Enoch, J. M.

J. M. Enoch and G. M. Hope, "Interferometric resolution determinations in the fovea and parafovea," Doc. Ophthalmol. 34, 143–156 (1973).
[Crossref] [PubMed]

Farrell, J. E.

French, A. S.

A. S. French, A. W. Snyder, and D. G. Stavenga, "Image degradation by an irregular retinal mosaic," Biol. Cybernetics 27, 229–233 (1977).
[Crossref]

Geisler, W. S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Green, D. G.

D. G. Green, "Regional variations in the visual acuity for interference fringes on the retina," J. Physiol. 207, 351–356 (1970).
[PubMed]

F. W. Campbell and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–593 (1965).
[PubMed]

Hendrickson, A. E.

C. A. Curcio, K. R. Sloan, Jr., O. Packer, A. E. Hendrickson, and R. E. Kalina, "Distribution of cone in human and monkey retina: individual variability and radial asymmetry," Science 236, 579–582 (1987).
[Crossref] [PubMed]

Hirsch, J.

Hope, G. M.

J. M. Enoch and G. M. Hope, "Interferometric resolution determinations in the fovea and parafovea," Doc. Ophthalmol. 34, 143–156 (1973).
[Crossref] [PubMed]

Hughes, A.

A. W. Snyder, T. R. J. Bossomaier, and A. Hughes, "Optical image quality and the cone mosaic," Science, 231, 499–501 (1986).
[Crossref] [PubMed]

T. R. J. Bossomaier, A. W. Snyder, and A. Hughes, "Irregularity and aliasing: solution?" Vision Res. 25, 145–147 (1985).
[Crossref] [PubMed]

Kalina, R. E.

C. A. Curcio, K. R. Sloan, Jr., O. Packer, A. E. Hendrickson, and R. E. Kalina, "Distribution of cone in human and monkey retina: individual variability and radial asymmetry," Science 236, 579–582 (1987).
[Crossref] [PubMed]

Levinson, J. Z.

F. W. Campbell, R. H. S. Carpenter, and J. Z. Levinson, "Visibility of aperiodic patterns compared with that of sinusoidal gratings," J. Physiol. 204, 283–298 (1969).
[PubMed]

McGowan, J. W.

B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).
[Crossref] [PubMed]

Medeiros, J.

B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).
[Crossref] [PubMed]

Miller, W. H.

J. Hirsch and W. H. Miller, "Does cone positional disorder limit resolution?" J. Opt. Soc. Am. A 4, 1481–1492 (1987).
[Crossref] [PubMed]

W. H. Miller and G. D. Bernard, "Averaging over the foveal receptor aperture curtails aliasing," Vision Res. 23, 1365–1369 (1983).
[Crossref] [PubMed]

A. W. Snyder and W. H. Miller, "Photoreceptor diameter and spacing for highest resolving power," J. Opt. Soc. Am. 67, 696–698 (1977).
[Crossref] [PubMed]

W. H. Miller, "Ocular optical filtering," in Handbook of Sensory Physiology, H. Autrum, ed. (Springer-Verlag, Berlin, 1979), Vol. 7, Chap. 6, pp. 70–143.

Nagel, D. C.

D. C. Nagel, "Spatial sampling in the retina," Invest. Ophthalmol. Vis. Sci. Suppl. 20, 123 (1981).

Østerberg, G.

G. Østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 11–103 (1935).

Packer, O.

C. A. Curcio, K. R. Sloan, Jr., O. Packer, A. E. Hendrickson, and R. E. Kalina, "Distribution of cone in human and monkey retina: individual variability and radial asymmetry," Science 236, 579–582 (1987).
[Crossref] [PubMed]

Pelli, D. G.

A. B. Watson and D. G. Pelli, "QUEST: a Bayesian adaptive psychometric method," Percept. Psychophys. 33, 113–120 (1983).
[Crossref] [PubMed]

Perry, V. H.

V. H. Perry and A. Cowey, "The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors," Vision Res. 25, 1795–1810 (1985).
[Crossref]

Riemann, H. J.

H. Wassle and H. J. Riemann, "The mosaic of nerve cells in the mammalian retina," Proc. R. Soc. Lond. Ser. B 200, 441–461 (1978).
[Crossref]

Rovamo, J.

V. Virsu and J. Rovamo, "Visual resolution, contrast sensitivity, and the cortical magnification factor," Exp. Brain Res. 37, 475–494 (1979).
[Crossref] [PubMed]

Sloan, Jr., K. R.

C. A. Curcio, K. R. Sloan, Jr., O. Packer, A. E. Hendrickson, and R. E. Kalina, "Distribution of cone in human and monkey retina: individual variability and radial asymmetry," Science 236, 579–582 (1987).
[Crossref] [PubMed]

Smith, R. A.

R. A. Smith and P. Cass, "Aliasing with incoherent light stimuli," J. Opt. Soc. Am. A 3(13), P93 (1986).

Snyder, A. W.

A. W. Snyder, T. R. J. Bossomaier, and A. Hughes, "Optical image quality and the cone mosaic," Science, 231, 499–501 (1986).
[Crossref] [PubMed]

T. R. J. Bossomaier, A. W. Snyder, and A. Hughes, "Irregularity and aliasing: solution?" Vision Res. 25, 145–147 (1985).
[Crossref] [PubMed]

A. W. Snyder and W. H. Miller, "Photoreceptor diameter and spacing for highest resolving power," J. Opt. Soc. Am. 67, 696–698 (1977).
[Crossref] [PubMed]

A. S. French, A. W. Snyder, and D. G. Stavenga, "Image degradation by an irregular retinal mosaic," Biol. Cybernetics 27, 229–233 (1977).
[Crossref]

Stavenga, D. G.

A. S. French, A. W. Snyder, and D. G. Stavenga, "Image degradation by an irregular retinal mosaic," Biol. Cybernetics 27, 229–233 (1977).
[Crossref]

Thibos, L. N.

L. N. Thibos and D. J. Walsh, "Detection of high frequency gratings in the periphery," J. Opt. Soc. Am. A 2(13), P64 (1985).

Thompson, B. J.

D. R. Williams, R. J. Collier, and B. J. Thompson, "Spatial resolution of the short-wavelength mechanism," in Colour Vision, Physiology and Psychophysics, J. D. Mollon and L. T. Sharpe, eds. (Academic, London, 1983).

Virsu, V.

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Other (10)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

For simplicity we will treat the cones here as though they sampled at infinitely small points, ignoring the effect of the cone aperture. The cone aperture is a low-pass spatial filter that demodulates the contrast of high-spatial-frequency gratings.31,32 However, it has little effect on the contrast of interference fringes until frequencies well beyond the nominal Nyquist frequency. The point has already been made for the tightly packed foveal cones.16,31,32 The effect of the cone aperture must become even less significant with increasing eccentricity because cone spacing grows more rapidly than the cone aperture.

Even when performances for horizontal and vertical fringes are considered separately, the orientation identification limit allways exceeds the nominal Nyquist limit. The nominal Nyquist frequencies obtained by the orientation-reversal technique are somewhat different for horizontal and vertical gratings. The values are 20, 21, and 20 cycles/deg for vertical gratings and 26, 23, and 21 cycles/deg for horizontal gratings for DRW, NJC, and RMK, respectively. The orientation-identification limit also depends on fringe orientation. These values were 37,30, and 31 cycles/deg for vertical gratings and 40, 38, and 26 cycles/deg for horizontal gratings for DRW, NJC, and RMK, respectively.

The nominal Nyquist frequency, ƒN, was estimated from the data of Østerberg21 and Curcio et al.22 by calculating the Nyquist limit of a mosaic that had the same density of cones but whose packing was triangular: ƒN = (½)(0.291)(2D/√3)½, where D is the cone density in cones per square millimeter and 0.291 converts millimeters on the retina to degrees of angular subtense. This estimate is negligibly different from one based on direct measurements of the modal frequency of the disordered extrafoveal mosaic,6 with the Nyquist frequency defined as half of the modal frequency.

W. H. Miller, "Ocular optical filtering," in Handbook of Sensory Physiology, H. Autrum, ed. (Springer-Verlag, Berlin, 1979), Vol. 7, Chap. 6, pp. 70–143.

D. R. Williams, "Topography of the foveal cone mosaic in the living human eye," submitted to Vision Res.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1975).

J. I. Yellott, B. A. Wandell, and T. N. Cornsweet, "The beginnings of visual perception: the retinal image and its initial encoding," in Handbook of Physiology, the Nervous System, S. R. Geiger, ed. (American Physiological Society, Bethesda, Md., 1984), Vol. Ill, Chap. 7, pp. 257–316.

The experiment could have presented more orientations than just horizontal and vertical in a single run to encourage identification instead of discrimination. However, pilot experiments revealed a large oblique effect that reduced resolution from 34 to 20 cycles/deg for one observer (NJC, by the method of adjustment). Because our goal was to push visual resolution to the theoretical limits of the mosaic, we abandoned the use of oblique gratings whose resolution is presumably limited by oriented mechanisms somewhere in the cortex.

D. R. Williams, R. J. Collier, and B. J. Thompson, "Spatial resolution of the short-wavelength mechanism," in Colour Vision, Physiology and Psychophysics, J. D. Mollon and L. T. Sharpe, eds. (Academic, London, 1983).

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Figures (5)

Fig. 1
Fig. 1

Contrast sensitivity for interference fringes as a function of spatial frequency at 3.8 deg in the nasal retina for observers DRW and NJC. Contrast sensitivity is shown for detection of gratings (■) and for orientation discrimination (□). Data are the average for vertical and horizontal gratings. Also shown is the 75% correct level (○) for orientation discrimination (observer NJC) for 100% contrast gratings. If the contrast threshold for either the horizontal or the vertical fringe could not be measured at maximum contrast, no point was plotted.

Fig. 2
Fig. 2

Psychometric functions for orientation identification at 3.8 deg of eccentricity for three observers. Gratings were vertical or horizontal interference fringes of unity contrast. Error bars represent ±1 standard error of the mean based on the variability between runs. Means are based on 4 runs (120 trials per frequency) for observers DRW and RMK and 10 runs (300 trials per frequency) for observers NJC. The orientation-identification limit was chosen to be the spatial frequency required for 75% correct responses, calculated from the smooth-curve fit to the mean data for both horizontal and vertical gratings. The nominal Nyquist frequency is calculated from Østerberg’s cone spacing data21 (filled arrows) and from psychophysical measurements of cone spacing (open arrows) obtained on the same individuals by the technique described by Coletta and Williams.6

Fig. 3
Fig. 3

Orientation-identification limit and cone Nyquist frequency as a function of retinal eccentricity. Filled symbols depict forced-choice orientation-identification limits for interference fringes for observers DRW (squares), NJC (circles), and RMK (triangles). All data are for temporal retina, except points at 3.8 deg, which are for nasal retina. Data are the mean ±1 standard error of the mean for vertical and horizontal fringes. Open symbols depict the cone Nyquist frequency obtained by the psychophysical technique described by Coletta and Williams6 for the same subjects. Solid and dashed lines are nominal Nyquist frequencies calculated from the anatomical cone spacing data of Østerberg21 and Curcio et al.,22 respectively. The × indicates the nominal Nyquist frequency calculated from the anatomical cone spacing data at the fovea from Østerberg,21 Curcio et al.,22 and Miller.24

Fig. 4
Fig. 4

Sampling properties of the primate extrafoveal cone mosaic. Effects of sampling horizontal (left) and vertical (right) gratings that exceed the nominal Nyquist frequency are shown in the spatial domain in the upper half of the figure. Dots represent locations of individual cones at 3.8 deg in the monkey parafovea. The sample is roughly 1 deg of visual angle across. The grating spatial frequency was 1.25 times the nominal Nyquist frequency. The lower half of the figure shows the effects of sampling in the two-dimensional frequency plane. The optical transform of the sampled horizontal grating is shown at the lower left; that of the vertical grating is shown at the lower right.

Fig. 5
Fig. 5

Description in the frequency plane of two possible explanations for supra-Nyquist orientation identification. Both images show the optical transform of a vertical sinusoidal grating at 1.25 times the nominal Nyquist frequency of the cone mosaic sampled by the extrafoveal primate mosaic. The dark disk at the center of each transform represents the window of visibility, which is a hypothetical spatial filter in the postreceptoral visual system. (a) The supra-Nyquist resolution hypothesis: the spatial bandwidth of the filter is sufficient to pass the delta functions corresponding to the original grating. (b) The aliasing hypothesis: the spatial bandwidth of the filter is no greater than the Nyquist frequency, and supra-Nyquist orientation identification is mediated by the aliasing noise passing predominantly through the left- and right-hand edges of the window.

Equations (2)

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f ( x ) = ( 0.5 ) exp - ( x / α ) β + 0.5 ,
fN=(1/2)(0.291)(2D/3)1/2,

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