Abstract

We measure the center-to-center spacings and disorder in spacings between all pairs of cones in a strip of primate retina extending from the foveal center to approximately 5.75 deg of retinal eccentricity along the temporal horizontal meridian. The strip is partitioned into windows, and the positions of the cone centers in each lattice window are digitized for analysis of lattice structure and quality. We find a nearly monotonic increase in cone spacing with eccentricity. The cone mosaic is a high-quality hexagonal lattice near the foveal center, and cone positional disorder (jitter) relative to averaging spacing increases beyond about 1.5 deg. We estimate human acuity measured through the optics of the eye over a retinal region comparable with our lattice strip by pooling the results of previous investigators. When the monkey lattice is scaled to human foveal resolution, application of the sampling theorem to average cone spacing predicts these pooled visual-acuity data from the foveal center to about 1.5 deg and overestimates visual acuity more eccentrically. Orientation reversal, a new technique developed by Coletta and Williams [ J. Opt. Soc. Am. 4, 1503 ( 1987)] for estimating the Nyquist limit, estimates Nyquist frequencies from the foveal edge to beyond 5 deg of retinal eccentricity that agree with the cutoff frequencies predicted on the basis of our average spacing measurements. We conclude that the sampling theorem based on average spacing alone predicts the Nyquist limit from the foveal center to about 5 deg when that limit is measured by using the new aliasing technique. The sampling theorem based on average spacing overestimates the pooled estimate of visual acuity from the foveal edge to about 5 deg, probably because of sampling noise caused. by orientation and spacing disorder combined with demodulation as a result of the optics of the eye.

© 1987 Optical Society of America

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  1. R. M. Bracewell, "Strip integration in radio astronomy," Aust. J. Phys. 9, 198–217 (1956).
    [Crossref]
  2. 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]
  3. A. S. French, A. W. Snyder, and D. G. Stavenga, "Image degradation by an irregular retinal mosaic," Biol. Cybernet. 27, 229–233 (1977).
    [Crossref]
  4. T. R. J. Bossomaier, A. W. Snyder, and A. Hughes, "Irregularity and aliasing: solution?" Vision Res. 25, 145–147 (1985).
    [Crossref] [PubMed]
  5. J. L. Yen, "On non-uniform sampling of bandwidth-limited signals," IRE Trans. Circuit Theory 3, 251–257 (1956).
    [Crossref]
  6. Identification of the orientation of the retinal strip was made by location of a small fissure running medial to lateral through the exact center of the fovea. This fissure was located with respect to the optic nerve head in the section. The retinal sample is from a right eye of a 5-year-old male Macaca fascicularis. Histological methods are presented by Miller.7
  7. W. H. Miller, "Ocular optical filtering," in Handbook of Sensory Physiology, H. Autrum, ed. (Springer-Verlag, Berlin, 1979), Vol. VII/6A, pp. 70–143.
  8. W. H. Miller and G. Bernard, "Averaging over the foveal receptor aperture curtails aliasing," Vision Res. 23, 1365–1369 (1983).
    [Crossref] [PubMed]
  9. Identification of the coordinates of each photoreceptor center contributes a source of measurement error. This error is assessed by three determinations each of six lattice windows (numbers 2, 4, 6, 10,15, and 24). Positional variance for each element along x and y dimensions is pooled over multiple determinations, and the results are shown in the table below. The pooled measurement error ranges from approximately 0.10 to 0.30 µm. A second estimate of measurement error is made by determination of the coordinates of two sizes of a perfect hexagonal pattern. Results of these studies show a 3–5% measurement error, which is consistent with the above test-retest error determination method for comparably sized lattices. Pictorial representation of measurement error are shown in Fig. 2. These four windows are chosen because they are the worst-case examples, and they demonstrate that measurement error is not significant.
  10. Magnification is determined by using a stage micrometer. Negatives of the pictures of the micrometer and negatives of the pictures of the sections are placed emulsion side to emulsion side in the enlarger and printed simultaneously in order to superimpose the picture of the scale upon the picture of the retina.
  11. The conversion from micrometers to degrees of visual angle for the Macaca fascicularis is based on a 20-mm schematic eye with a focal length of 13.9 mm. The angular spacing between photoreceptors in degrees, Δø, is given by the center-to-center spacing of photoreceptors dcc over focal length F times the conversion factor from radians to degrees: [Equation]
  12. When the packing structure is hexagonal, the center-to-center spacing dcc in micrometers is determined from density counts by the following: [Equation] where n is the number of receptors per square millimeter. This method yields an estimate of average nearest-neighbor spacing with no error on the estimate. In this study we employ an alternative distribution-of-distances method to determine dcc. All distances between pairs of photoreceptors in a lattice array are distributed (see Fig. 4) and define the set of ring 1 (nearest-neighbor) distances. This method yields both the average nearest-neighbor spacing for the array, d, and the distance error, σ d: dccd¯ ± σ d• The distribution-of-distances method is assumption free and yields errors on all estimates.
  13. J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–356 (1984).
    [Crossref] [PubMed]
  14. B. D. Ripley, Spatial Statistics (Wiley, New York, 1981), p. 148.
  15. H. V. 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]
  16. T. Wertheim, "Über die indirekte Sehscharfe," Z. Psychol. Physiol. Sinnesorg. 7, 172–189 (1894).
  17. L. I. Dunsky, "Peripheral visual acuity," Am. J. Optom. Physiol. Opt. 57, 915–924 (1980) (English translation of Ref. 16).
    [Crossref]
  18. D. R. Williams, "Aliasing in human foveal vision," Vision Res. 25, 195–205 (1985).
    [Crossref] [PubMed]
  19. A. W. Snyder, and W. H. Miller, "Telephoto lens system of falconiform eye," Nature 275, 127–129 (1978).
    [Crossref] [PubMed]
  20. The effective angular separation between nearest neighbors, Δø, for a human is 0.0103 deg (0.62 min), and the focal length is approximately 16.67 mm; dcc represents the separation between nearest neighbors in micrometers: [Equation] See Eq. (1) from Note 11.
  21. F. W. Campbell, and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–583 (1965).
    [PubMed]
  22. G. Østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 1–103 (1935).
  23. A. W. Snyder, T. R. J. Bossomaier, and A. Hughes, "Optical image quality and the cone mosaic," Science 231, 499–500 (1986).
    [Crossref] [PubMed]
  24. D. R. Williams, "Seeing through the photoreceptor mosaic," Trends Neuro Sci. 9,193–198 (1986).
    [Crossref]
  25. S. L. Polyak, The Retina (U. Chicago Press, Chicago, III., 1941).
  26. B. Borwein, D. Borwein, J. Mederios, 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]
  27. D. R. Williams and N. J. Coletta, "Cone spacing and the visual resolution limit," J. Opt. Soc. Am. A 4, 1514–1523 (1987).
    [Crossref] [PubMed]
  28. E. Ludvigh, "Extrafoveal visual acuity as measured by Snellen test letters," Am. J. Ophthalmol. 24, 303–310 (1941).
  29. F. Weymouth, "Visual sensory units and the minimal angle of resolution," Am. J. Ophthalmol. 46, 102–113 (1958).
    [PubMed]
  30. L. Weiskrantz and A. Cowey, "Striate cortex lesion and visual acuity of the rhesus monkey," J. Comp. Physiol. Psychol. 56, 225–231 (1963).
    [Crossref]
  31. G. Westheimer, "The spatial grain of the perifoveal visual field," Vision Res. 22, 157–162 (1982).
    [Crossref] [PubMed]
  32. Weymouth averaged the acuity data over both nasal and temporal meridia since "no meridional differences were observed." The other investigators measured acuity along the temporal horizontal meridian, which is the same meridian as the lattice strip in this study. Similar data from Levi et al.33 are not included, since acuity measurements were made in the lower visual field, where acuity tends to be lower than for other meridia.16,17 Interferometry data from Enoch and Hope34 and Green35 are also not included, since the measures are based on presentations of the stimuli that bypass the optics of the eye. Nonetheless, these data generally fall within the scatter of the data shown in Fig. 7. The best-fit line through the data points is fitted to the data between the foveal center and 3.0 deg of eccentricity, and the dashed line segment extends the line to the more peripheral regions. This fit was chosen in order to take advantage of the region where acuity measures a re most precise, and it yields the best-fit line y = 21x + 35. However, the fit to the entire data set yields a similar line of y = 25x 32, and a fit to the more limited retinal region from the foveal center to 2.0 deg is y = 22x + 34. All these fits are similar, and we have chosen our line on the basis of the largest area of highest reliability. The intercepts for all fits (32 to 35 arcsec) are in reasonable agreement with the anatomical and aliasing estimates of foveal center-to-center spacing.
  33. D. M. Levi, S. A. Klein, and A. P. Aitsebaomo, "Vernier acuity, crowding, and cortical magnification," Vision Res. 25, 963–978 (1985).
    [Crossref]
  34. J. M. Enoch and G. M. Hope, "Interferometric resolution determinations in the fovea and parafovea," Doc. Ophthalmol. 34, 143–156 (1973).
    [Crossref] [PubMed]
  35. D. Green, "Regional variation in the visual acuity for interference fringes on the retina. J. Physiol. 207, 351–356 (1970).
    [PubMed]
  36. N. J. Coletta and D. R. Williams, "Psychophysical estimate of extrafoveal cone spacing," J. Opt. Soc. Am. A 4, 1503–1513 (1987).
    [Crossref] [PubMed]
  37. A. Hughes,-"The topography of vision in mammals of contrasting life style: comparative optics and retinal organization," in Handbook of Sensory Physiology, F. Crescitelli, ed. (Springer, New York, 1977), Vol. VII/5, pp. 613–756.
    [Crossref]
  38. J. I. Yellott, Jr., "Spectral analysis of spatial sampling by photoreceptors: topological disorder prevents aliasing," Vision Res. 22, 1205–1210 (1982).
    [Crossref] [PubMed]
  39. J. I. Yellott, Jr., "Spectral consequences of photoreceptor sampling in the rhesus retina," Science 221, 382–385 (1983).
    [Crossref] [PubMed]
  40. D. R. Williams and R. Collier, "Consequences of spatial sampling by a human photoreceptor mosaic," Science 221, 385–387 (1983).
    [Crossref] [PubMed]
  41. D. R. Williams, N. Coletta, and R. Korte, "Extrafoveal grating resolution and sampling theory," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986).
  42. A. J. Ahumada, Jr., and J. I. Yellott, Jr., "A model for foveal photoreceptor placing," Invest. Ophthalmol. Suppl. 26, 10 (1985).
  43. A. J. Ahumada, Jr., and A. Poirson, "Modelling the irregularity of the foveolar mosaic," Invest. Ophthalmol. Suppl. 27, 94 (1986).
  44. A. Poirson and A. Ahumada, Jr., "Model for space-varying cone placement in the fovea," J. Opt. Soc. Am. A 3, (13), p. 70 (1986).
  45. A. J. Ahumada, Jr., and A. Poirson, "Cone sampling array models," J. Opt. Soc. Am. A 4, 1493–1502 (1987).
    [Crossref] [PubMed]
  46. J. M. Ziman, Models of Disorder (Cambridge U. Press, Cambridge, 1982), pp. 1–32.
  47. This same observation was made by Hirsch and Hylton for a similar analysis on a small central foveal window.13 The long-range parallel variance corresponding to jitter in distance was consistently lower than the long-range perpendicular variance corresponding to jitter in rotation. Both the distance disorder and the rotational disorder described here are types of short-range (nearest-neighbor) disorder. Other types of short- and long-range disorder and models of cone displacement are currently under consideration by other investigators.40–45
  48. R. Drougard, "Optical transfer properties of fiber bundles," J. Opt. Soc. Am. 54, 907–914 (1964).
    [Crossref]
  49. The unit of one standard deviation is the minimum reasonable unit of positional uncertainty because all ring 1 distance distributions are fitted by a normal distribution. Since this correction is sufficient to reconcile the acuity and sampling functions, it is applied here.
  50. This conclusion assumes that there is no convergence of sampled information as in a one-to-one correspondence between the cones and ganglion cells. This seems probable, since the cone and ganglion cell count curves do not cross in primates until about 16 deg on the temporal meridian.14
  51. Y. Le Grand, Light, Color and Vision (Wiley, New York, 1957).

1987 (3)

1986 (5)

A. J. Ahumada, Jr., and A. Poirson, "Modelling the irregularity of the foveolar mosaic," Invest. Ophthalmol. Suppl. 27, 94 (1986).

A. Poirson and A. Ahumada, Jr., "Model for space-varying cone placement in the fovea," J. Opt. Soc. Am. A 3, (13), p. 70 (1986).

D. R. Williams, N. Coletta, and R. Korte, "Extrafoveal grating resolution and sampling theory," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986).

A. W. Snyder, T. R. J. Bossomaier, and A. Hughes, "Optical image quality and the cone mosaic," Science 231, 499–500 (1986).
[Crossref] [PubMed]

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

1985 (5)

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

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

H. V. 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]

A. J. Ahumada, Jr., and J. I. Yellott, Jr., "A model for foveal photoreceptor placing," Invest. Ophthalmol. Suppl. 26, 10 (1985).

D. M. Levi, S. A. Klein, and A. P. Aitsebaomo, "Vernier acuity, crowding, and cortical magnification," Vision Res. 25, 963–978 (1985).
[Crossref]

1984 (1)

J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–356 (1984).
[Crossref] [PubMed]

1983 (3)

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

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

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

1980 (2)

L. I. Dunsky, "Peripheral visual acuity," Am. J. Optom. Physiol. Opt. 57, 915–924 (1980) (English translation of Ref. 16).
[Crossref]

B. Borwein, D. Borwein, J. Mederios, 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]

1978 (1)

A. W. Snyder, and W. H. Miller, "Telephoto lens system of falconiform eye," Nature 275, 127–129 (1978).
[Crossref] [PubMed]

1977 (2)

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. Cybernet. 27, 229–233 (1977).
[Crossref]

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

1965 (1)

F. W. Campbell, and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–583 (1965).
[PubMed]

1964 (1)

1963 (1)

L. Weiskrantz and A. Cowey, "Striate cortex lesion and visual acuity of the rhesus monkey," J. Comp. Physiol. Psychol. 56, 225–231 (1963).
[Crossref]

1958 (1)

F. Weymouth, "Visual sensory units and the minimal angle of resolution," Am. J. Ophthalmol. 46, 102–113 (1958).
[PubMed]

1956 (2)

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

R. M. Bracewell, "Strip integration in radio astronomy," Aust. J. Phys. 9, 198–217 (1956).
[Crossref]

1941 (1)

E. Ludvigh, "Extrafoveal visual acuity as measured by Snellen test letters," Am. J. Ophthalmol. 24, 303–310 (1941).

1935 (1)

G. Østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 1–103 (1935).

1894 (1)

T. Wertheim, "Über die indirekte Sehscharfe," Z. Psychol. Physiol. Sinnesorg. 7, 172–189 (1894).

Ahumada, Jr., A.

A. Poirson and A. Ahumada, Jr., "Model for space-varying cone placement in the fovea," J. Opt. Soc. Am. A 3, (13), p. 70 (1986).

Ahumada, Jr., A. J.

A. J. Ahumada, Jr., and A. Poirson, "Cone sampling array models," J. Opt. Soc. Am. A 4, 1493–1502 (1987).
[Crossref] [PubMed]

A. J. Ahumada, Jr., and A. Poirson, "Modelling the irregularity of the foveolar mosaic," Invest. Ophthalmol. Suppl. 27, 94 (1986).

A. J. Ahumada, Jr., and J. I. Yellott, Jr., "A model for foveal photoreceptor placing," Invest. Ophthalmol. Suppl. 26, 10 (1985).

Aitsebaomo, A. P.

D. M. Levi, S. A. Klein, and A. P. Aitsebaomo, "Vernier acuity, crowding, and cortical magnification," Vision Res. 25, 963–978 (1985).
[Crossref]

Bernard, G.

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

Borwein, B.

B. Borwein, D. Borwein, J. Mederios, 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, D.

B. Borwein, D. Borwein, J. Mederios, 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, 499–500 (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. M.

R. M. Bracewell, "Strip integration in radio astronomy," Aust. J. Phys. 9, 198–217 (1956).
[Crossref]

Campbell, F. W.

F. W. Campbell, and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–583 (1965).
[PubMed]

Coletta, N.

D. R. Williams, N. Coletta, and R. Korte, "Extrafoveal grating resolution and sampling theory," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986).

Coletta, N. J.

Collier, R.

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

Cowey, A.

H. V. 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]

L. Weiskrantz and A. Cowey, "Striate cortex lesion and visual acuity of the rhesus monkey," J. Comp. Physiol. Psychol. 56, 225–231 (1963).
[Crossref]

Drougard, R.

Dunsky, L. I.

L. I. Dunsky, "Peripheral visual acuity," Am. J. Optom. Physiol. Opt. 57, 915–924 (1980) (English translation of Ref. 16).
[Crossref]

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]

French, A. S.

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

Grand, Y. Le

Y. Le Grand, Light, Color and Vision (Wiley, New York, 1957).

Green, D.

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

Green, D. G.

F. W. Campbell, and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–583 (1965).
[PubMed]

Hirsch, J.

J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–356 (1984).
[Crossref] [PubMed]

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, 499–500 (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. Hughes,-"The topography of vision in mammals of contrasting life style: comparative optics and retinal organization," in Handbook of Sensory Physiology, F. Crescitelli, ed. (Springer, New York, 1977), Vol. VII/5, pp. 613–756.
[Crossref]

Hylton, R.

J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–356 (1984).
[Crossref] [PubMed]

Klein, S. A.

D. M. Levi, S. A. Klein, and A. P. Aitsebaomo, "Vernier acuity, crowding, and cortical magnification," Vision Res. 25, 963–978 (1985).
[Crossref]

Korte, R.

D. R. Williams, N. Coletta, and R. Korte, "Extrafoveal grating resolution and sampling theory," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986).

Levi, D. M.

D. M. Levi, S. A. Klein, and A. P. Aitsebaomo, "Vernier acuity, crowding, and cortical magnification," Vision Res. 25, 963–978 (1985).
[Crossref]

Ludvigh, E.

E. Ludvigh, "Extrafoveal visual acuity as measured by Snellen test letters," Am. J. Ophthalmol. 24, 303–310 (1941).

McGowan, J. W.

B. Borwein, D. Borwein, J. Mederios, 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]

Mederios, J.

B. Borwein, D. Borwein, J. Mederios, 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.

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

A. W. Snyder, and W. H. Miller, "Telephoto lens system of falconiform eye," Nature 275, 127–129 (1978).
[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. VII/6A, pp. 70–143.

Østerberg, G.

G. Østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 1–103 (1935).

Perry, H. V.

H. V. 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]

Poirson, A.

A. J. Ahumada, Jr., and A. Poirson, "Cone sampling array models," J. Opt. Soc. Am. A 4, 1493–1502 (1987).
[Crossref] [PubMed]

A. J. Ahumada, Jr., and A. Poirson, "Modelling the irregularity of the foveolar mosaic," Invest. Ophthalmol. Suppl. 27, 94 (1986).

A. Poirson and A. Ahumada, Jr., "Model for space-varying cone placement in the fovea," J. Opt. Soc. Am. A 3, (13), p. 70 (1986).

Polyak, S. L.

S. L. Polyak, The Retina (U. Chicago Press, Chicago, III., 1941).

Ripley, B. D.

B. D. Ripley, Spatial Statistics (Wiley, New York, 1981), p. 148.

Snyder, A. W.

A. W. Snyder, T. R. J. Bossomaier, and A. Hughes, "Optical image quality and the cone mosaic," Science 231, 499–500 (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, "Telephoto lens system of falconiform eye," Nature 275, 127–129 (1978).
[Crossref] [PubMed]

A. S. French, A. W. Snyder, and D. G. Stavenga, "Image degradation by an irregular retinal mosaic," Biol. Cybernet. 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]

Stavenga, D. G.

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

Weiskrantz, L.

L. Weiskrantz and A. Cowey, "Striate cortex lesion and visual acuity of the rhesus monkey," J. Comp. Physiol. Psychol. 56, 225–231 (1963).
[Crossref]

Wertheim, T.

T. Wertheim, "Über die indirekte Sehscharfe," Z. Psychol. Physiol. Sinnesorg. 7, 172–189 (1894).

Westheimer, G.

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

Weymouth, F.

F. Weymouth, "Visual sensory units and the minimal angle of resolution," Am. J. Ophthalmol. 46, 102–113 (1958).
[PubMed]

Williams, D. R.

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

D. R. Williams and N. J. Coletta, "Cone spacing and the visual resolution limit," J. Opt. Soc. Am. A 4, 1514–1523 (1987).
[Crossref] [PubMed]

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

D. R. Williams, N. Coletta, and R. Korte, "Extrafoveal grating resolution and sampling theory," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986).

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

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

Yellott, Jr., J. I.

A. J. Ahumada, Jr., and J. I. Yellott, Jr., "A model for foveal photoreceptor placing," Invest. Ophthalmol. Suppl. 26, 10 (1985).

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

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

Yen, J. L.

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

Ziman, J. M.

J. M. Ziman, Models of Disorder (Cambridge U. Press, Cambridge, 1982), pp. 1–32.

Acta Ophthalmol. Suppl. (1)

G. Østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 1–103 (1935).

Am. J. Anat. (1)

B. Borwein, D. Borwein, J. Mederios, 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]

Am. J. Ophthalmol. (2)

E. Ludvigh, "Extrafoveal visual acuity as measured by Snellen test letters," Am. J. Ophthalmol. 24, 303–310 (1941).

F. Weymouth, "Visual sensory units and the minimal angle of resolution," Am. J. Ophthalmol. 46, 102–113 (1958).
[PubMed]

Am. J. Optom. Physiol. Opt. (1)

L. I. Dunsky, "Peripheral visual acuity," Am. J. Optom. Physiol. Opt. 57, 915–924 (1980) (English translation of Ref. 16).
[Crossref]

Aust. J. Phys. (1)

R. M. Bracewell, "Strip integration in radio astronomy," Aust. J. Phys. 9, 198–217 (1956).
[Crossref]

Biol. Cybernet. (1)

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

Doc. Ophthalmol. (1)

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

Invest. Ophthalmol. Suppl. (2)

A. J. Ahumada, Jr., and J. I. Yellott, Jr., "A model for foveal photoreceptor placing," Invest. Ophthalmol. Suppl. 26, 10 (1985).

A. J. Ahumada, Jr., and A. Poirson, "Modelling the irregularity of the foveolar mosaic," Invest. Ophthalmol. Suppl. 27, 94 (1986).

Invest. Ophthalmol. Vis. Sci. Suppl. (1)

D. R. Williams, N. Coletta, and R. Korte, "Extrafoveal grating resolution and sampling theory," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986).

IRE Trans. Circuit Theory (1)

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

J. Comp. Physiol. Psychol. (1)

L. Weiskrantz and A. Cowey, "Striate cortex lesion and visual acuity of the rhesus monkey," J. Comp. Physiol. Psychol. 56, 225–231 (1963).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

J. Physiol. (2)

D. Green, "Regional variation 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, 576–583 (1965).
[PubMed]

Nature (1)

A. W. Snyder, and W. H. Miller, "Telephoto lens system of falconiform eye," Nature 275, 127–129 (1978).
[Crossref] [PubMed]

Science (3)

A. W. Snyder, T. R. J. Bossomaier, and A. Hughes, "Optical image quality and the cone mosaic," Science 231, 499–500 (1986).
[Crossref] [PubMed]

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

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

Trends Neuro Sci. (1)

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

Vision Res. (8)

D. M. Levi, S. A. Klein, and A. P. Aitsebaomo, "Vernier acuity, crowding, and cortical magnification," Vision Res. 25, 963–978 (1985).
[Crossref]

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

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

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

J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–356 (1984).
[Crossref] [PubMed]

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

H. V. 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]

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

Z. Psychol. Physiol. Sinnesorg. (1)

T. Wertheim, "Über die indirekte Sehscharfe," Z. Psychol. Physiol. Sinnesorg. 7, 172–189 (1894).

Other (16)

B. D. Ripley, Spatial Statistics (Wiley, New York, 1981), p. 148.

Identification of the orientation of the retinal strip was made by location of a small fissure running medial to lateral through the exact center of the fovea. This fissure was located with respect to the optic nerve head in the section. The retinal sample is from a right eye of a 5-year-old male Macaca fascicularis. Histological methods are presented by Miller.7

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

Identification of the coordinates of each photoreceptor center contributes a source of measurement error. This error is assessed by three determinations each of six lattice windows (numbers 2, 4, 6, 10,15, and 24). Positional variance for each element along x and y dimensions is pooled over multiple determinations, and the results are shown in the table below. The pooled measurement error ranges from approximately 0.10 to 0.30 µm. A second estimate of measurement error is made by determination of the coordinates of two sizes of a perfect hexagonal pattern. Results of these studies show a 3–5% measurement error, which is consistent with the above test-retest error determination method for comparably sized lattices. Pictorial representation of measurement error are shown in Fig. 2. These four windows are chosen because they are the worst-case examples, and they demonstrate that measurement error is not significant.

Magnification is determined by using a stage micrometer. Negatives of the pictures of the micrometer and negatives of the pictures of the sections are placed emulsion side to emulsion side in the enlarger and printed simultaneously in order to superimpose the picture of the scale upon the picture of the retina.

The conversion from micrometers to degrees of visual angle for the Macaca fascicularis is based on a 20-mm schematic eye with a focal length of 13.9 mm. The angular spacing between photoreceptors in degrees, Δø, is given by the center-to-center spacing of photoreceptors dcc over focal length F times the conversion factor from radians to degrees: [Equation]

When the packing structure is hexagonal, the center-to-center spacing dcc in micrometers is determined from density counts by the following: [Equation] where n is the number of receptors per square millimeter. This method yields an estimate of average nearest-neighbor spacing with no error on the estimate. In this study we employ an alternative distribution-of-distances method to determine dcc. All distances between pairs of photoreceptors in a lattice array are distributed (see Fig. 4) and define the set of ring 1 (nearest-neighbor) distances. This method yields both the average nearest-neighbor spacing for the array, d, and the distance error, σ d: dccd¯ ± σ d• The distribution-of-distances method is assumption free and yields errors on all estimates.

Weymouth averaged the acuity data over both nasal and temporal meridia since "no meridional differences were observed." The other investigators measured acuity along the temporal horizontal meridian, which is the same meridian as the lattice strip in this study. Similar data from Levi et al.33 are not included, since acuity measurements were made in the lower visual field, where acuity tends to be lower than for other meridia.16,17 Interferometry data from Enoch and Hope34 and Green35 are also not included, since the measures are based on presentations of the stimuli that bypass the optics of the eye. Nonetheless, these data generally fall within the scatter of the data shown in Fig. 7. The best-fit line through the data points is fitted to the data between the foveal center and 3.0 deg of eccentricity, and the dashed line segment extends the line to the more peripheral regions. This fit was chosen in order to take advantage of the region where acuity measures a re most precise, and it yields the best-fit line y = 21x + 35. However, the fit to the entire data set yields a similar line of y = 25x 32, and a fit to the more limited retinal region from the foveal center to 2.0 deg is y = 22x + 34. All these fits are similar, and we have chosen our line on the basis of the largest area of highest reliability. The intercepts for all fits (32 to 35 arcsec) are in reasonable agreement with the anatomical and aliasing estimates of foveal center-to-center spacing.

S. L. Polyak, The Retina (U. Chicago Press, Chicago, III., 1941).

The effective angular separation between nearest neighbors, Δø, for a human is 0.0103 deg (0.62 min), and the focal length is approximately 16.67 mm; dcc represents the separation between nearest neighbors in micrometers: [Equation] See Eq. (1) from Note 11.

The unit of one standard deviation is the minimum reasonable unit of positional uncertainty because all ring 1 distance distributions are fitted by a normal distribution. Since this correction is sufficient to reconcile the acuity and sampling functions, it is applied here.

This conclusion assumes that there is no convergence of sampled information as in a one-to-one correspondence between the cones and ganglion cells. This seems probable, since the cone and ganglion cell count curves do not cross in primates until about 16 deg on the temporal meridian.14

Y. Le Grand, Light, Color and Vision (Wiley, New York, 1957).

A. Hughes,-"The topography of vision in mammals of contrasting life style: comparative optics and retinal organization," in Handbook of Sensory Physiology, F. Crescitelli, ed. (Springer, New York, 1977), Vol. VII/5, pp. 613–756.
[Crossref]

J. M. Ziman, Models of Disorder (Cambridge U. Press, Cambridge, 1982), pp. 1–32.

This same observation was made by Hirsch and Hylton for a similar analysis on a small central foveal window.13 The long-range parallel variance corresponding to jitter in distance was consistently lower than the long-range perpendicular variance corresponding to jitter in rotation. Both the distance disorder and the rotational disorder described here are types of short-range (nearest-neighbor) disorder. Other types of short- and long-range disorder and models of cone displacement are currently under consideration by other investigators.40–45

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Figures (10)

Fig. 1
Fig. 1

Tangential sections of Macaca fascicularis retina, showing cone and rod apertures (a) near the external limiting membrane at the retinal center, (b) at approximately 1 deg of retinal eccentricity, and (c) at approximately 5 deg of retinal eccentricity. The foveal window consists only of cones. Arrowheads in (b) mark the locations of rods intruding into the cone lattice. The large cells in the 5-deg lattice (c) are sections of cones, and the small cells are rods. The scale bar in (b) applies to all three pictures and indicates 10 μm. Dot arrays of cone center positions that correspond to the lattice windows (a), (b), and (c) are shown in (d), (e), and (f), respectively.

Fig. 2
Fig. 2

Twenty-five continuous cone-center arrays. The upper-left-hand window includes the foveal center, and successive windows progress to the right and down. The bottom-right-hand window is the farthest from the center. Sequence number, retinal region (degrees), and number of cones are shown for each window in Table 1.

Fig. 3
Fig. 3

Representations of measurement error. Four lattice windows, numbers 2, 4, 6, and 10 [(a), (c), (b), and (d), respectively], are each coded three times. The coordinates for all repeated measures for each window are shown.

Fig. 4
Fig. 4

Example of a distance histogram from lattice window 1 (0.0–0.23 deg). All distances between lattice points are binned according to bin sizes of 0.1 μm, and the frequency of occurrence (y axis) is plotted against the distance scale (x axis) in micrometers. The boundaries between these distributions provide the definitions of nearest neighbors (ring 1), second-nearest neighbors (ring 2), and third-nearest neighbors (ring 3). For example, any distance between cone centers that falls within the first distribution is considered a nearest-neighbor (ring 1) distance.

Fig. 5
Fig. 5

Digital representations of five retinal windows each 11.49 μm × 11.49 μm arranged as two orthogonal strips. Summary statistics are shown in Table 3 for each window separately and for the combined windows in the row and column strips.

Fig. 6
Fig. 6

Average angle of row separations ±σ in seconds of arc as a function of retinal eccentricity (degrees). Data are from Table 4, with the exception of the value for the foveal point, which is from the center window in Table 3.

Fig. 7
Fig. 7

Human visual resolution (seconds of arc) as a function of retinal eccentricity (degrees). The data are from Ref. 31 (open and filled circles), Ref. 28 (diamonds), Ref. 30 (filled squares), and Ref. 29 (open squares). The solid line is the best-fit line of acuity versus retinal eccentricity from the foveal center to 3 deg, where the scatter in the data is nearly stable. The dashed line segment extends this line through the more peripheral retinal regions, where the scatter in acuity data is much larger.

Fig. 8
Fig. 8

Retinal sampling and visual reduction. The left- and right-hand ordinate axes correspond to d and 1/2d, respectively, where d is center-to-center spacing in degrees. The minimum angle of resolution (lines) and the sampling limits (dots) are plotted against retinal eccentricity. The line represents the best fit as described in Fig. 7. The dots are the primate sampling limits from Table 4 and are scaled to match the human acuity data at the foveal point.

Fig. 9
Fig. 9

Disorder among nearest neighbors in the photoreceptor lattice. Fractional spacing error [distance σ normalized by the mean of the distance distribution (filled circles)] and fractional rotation error [angle σ normalized by the mean of the angle distribution (open diamonds)] are plotted against retinal eccentricity.

Fig. 10
Fig. 10

Effective retinal sampling and visual resolution. This display is similar to that of Fig. 8. The solid and dashed line summarizes the best-fit acuity data from Fig. 7. The dots show photoreceptor sampling limits based on the corrected separation between nearest neighbors. The correction increases the average center-to-center spacing by a constant equal to the distance σ. The correspondence between measures of acuity and corrected sampling limits illustrates that the effects of sampling disorder may be operationally expressed as increased center-to-center spacing.

Tables (6)

Tables Icon

Table 1 Lattice Windows 1 through 25, Corresponding to the Digitized Samples Shown in Fig. 2a

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Table 2 Distribution of Distances by Ring

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Table 3 Summary of 11.49 μm × 11.49 μm Windows from Fig. 5

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Table 4 Minimum Distances and Retinal Sampling

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Table 5 Lattice Disordera

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Table 6 Distance Measurement Error Determined by Repeated Measurements

Equations (4)

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Δϕ=dcc(57.3)F.
dcc=1000n(23)1/2,
dccd¯±σd.
dcc=Δϕ(focallength)57.3.

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