Abstract

A model is described for positioning cones in the retina. Each cone has a circular disk of influence, and the disks are tightly packed outward from the center. This model has three parameters that can vary with eccentricity: the mean radius of the cone disk, the standard deviation of the cone disk radius, and the standard deviation of postpacking jitter. Estimates for these parameters out to 1.6 deg are found by using measurements reported by Hirsch and Hylton [ Vision Res. 24, 347 ( 1985)] and Hirsch and Miller [ J. Opt. Soc. Am. A 4, 1481 ( 1987)] of the positions of the cone inner segments of an adult macaque. The estimation is based on fitting measures of variation in local intercone distances, and the fit to these measures is good.

© 1987 Optical Society of America

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  1. We will use the term hexagonal packing or hexagonal array to describe the array of objects or object centers resulting from a tessalation (tiling) of the plane with identical hexagons or tightly packed circles. There is an alternative convention of naming the packing scheme by the objects whose vertices form the desired array of sampling points, according to which this scheme is called triangular packing.
  2. J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–256 (1984).
    [CrossRef] [PubMed]
  3. J. Hirsch and W. H. Miller, "Does cone position disorder limit resolution?" J. Opt. Soc. Am. A 4, 1481–1492 (1987); "Irregularity of foveal cone lattice increases with eccentricity," Invest. Ophthalmol. Vis. Sci. Suppl. 26, 10 (1985).
    [CrossRef] [PubMed]
  4. In the constant-parameter version of the program (written in C), a possible disk was allowed to overlap a placed disk (other than the two it is placed against) by up to 10% of the mean radius. In the variable-parameter version (written in PASCAL), this constant was made a parameter for the sake of compatibility, but it was set to 0% for the variable-parameter simulations. The selection of the closest possible disk keeps the set of already-packed cones mostly convex, so that possible overlap with a third disk is not the usual situation. The source code is available from Albert J. Ahumada.
  5. This formula is the sum of the squares of two independent standard normal variates when the parameters are correct. The first is simply the standardized form of S1, and the second is the standardized form of the part of S2 that is uncorrelated with S1.6
  6. T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1958).
  7. At this point in the model development we also changed the rule for generating a random radius. Previously we had generated a new radius for each pair of disks for which we were trying to find a possible next neighbor. Instead, we now generate a new radius when we begin looking for the new posssible disks after the addition of a new disk to the retina and keep trying that same radius until a possible position is found for it. Although this change does save random-number generations, the change was made to avoid a rare problem. A crevice in the edge of the retina would not be filled because a large disk would be tried for the possible positions inside the crevice and a small disk would be tried for the position that closed the mouth of the crevice. The few holes thus generated would not have affected our measures appreciably, but in the interest of computational efficiency we also modified the program to search only through the neighbors of neighbors of neighbors to find possible sites and to look for overlaps. The holes permitted overlapping positions that the search could not find because they were too many generations of neighbors away. We have found only an algorithm that has worked for us with a relatively small range of parameters. We do not think that we have found a packing scheme that will always work. For example, the scheme might not work for disk sizes as disparate as rods and cones.
  8. W. S. Geisler and K. D. Davila, "Ideal discriminators in spatial vision: two-point stimuli," J. Opt. Soc. Am. A 2, 1483–1497 (1985).
    [CrossRef] [PubMed]
  9. K. R. K. Nielsen, A. B. Watson, and A. J. Ahumada, Jr., "Application of a computable model of human spatial vision to phase discrimination," J. Opt. Soc. Am. A 2, 1600–1606 (1985).
    [CrossRef] [PubMed]
  10. J. I. Yellott, Jr., "Spectral analysis of spatial sampling by photoreceptors: topological disorder presents aliasing," Vision Res. 22, 1205–1210 (1982).
    [CrossRef]
  11. J. I. Yellott, Jr., "Spectral consequences of photoreceptor sampling in the rhesus retina," Science 221, 382–385 (1983).
    [CrossRef] [PubMed]
  12. A. W. Snyder, T. R. J. Bossomaier, and A. Hughes, "Optical image quality and the cone mosaic," Science 231, 499–501 (1986).
    [CrossRef] [PubMed]
  13. D. R. Williams and N. Coletta, "Cone spacing and the visual resolution limit," J. Opt. Soc. Am. A 4, 1514–1523 (1987); "Extrafoveal grating resolution and sampling theory," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986).
    [CrossRef] [PubMed]
  14. J. I. Yellott, Jr., "Nonhomogeneous Poisson disks model the photoreceptor mosaic," Invest. Ophthalmol. Vis. Sci. Suppl. 24, 147 (1983).
  15. F. M. de Monasterio, E. P. McCrane, J. K. Newlander, and S. J. Schein, "Density profile of vlue-sensitive cones along the horizontal meridian of macaque retina," Invest. Ophthalmol. Vis. Sci. 26, 289–302 (1985).
    [PubMed]
  16. M. B. Shapiro, S. J. Schein, and F. M. de Monasterio, "Regularity and structure of the spatial pattern of blue cones of macaque retina," J. Am. Stat. Assoc. 80, 803–812 (1985).
    [CrossRef]
  17. D. R. Williams, "Aliasing in human foveal vision," Vision Res. 25, 195–205 (1985).
    [CrossRef] [PubMed]
  18. D. R. Williams, "Topography of the foveal cone mosaic in the living human eye," submitted to Vision Res.; Invest. Ophthalmol. Vis. Sci. Suppl. 26, 10 (1986).
  19. N. J. Coletta and D. R. Williams, "Psychophysical estimate of extrafoveal cone spacing," J. Opt. Soc. Am. A 4, 1503–1513 (1987); J. Opt. Soc. Am. A 3(13), P92 (1986).
    [CrossRef] [PubMed]
  20. B. Borwein, D. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey fovea photoreceptors, with special reference to the structure, shape, size and spacing of foveal cones," Am. J. Anat. 159, 125–146 (1980).
    [CrossRef] [PubMed]
  21. 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]
  22. C. A. Curcio, O. Packer, and R. E. Kalina, "A whole mount method for sequential analysis of photoreceptor and ganglion cell topography," Vision Res. 27, 9–15 (1987).
    [CrossRef]
  23. C. A. Curcio, K. R. Sloan, Jr., O. Packer, A. E. Hendrickson, and R. E. Kalina, "Distribution of cones in human and monkey retina: individual variability and radial asymmetry," Science 236, 579–582 (1987).
    [CrossRef] [PubMed]
  24. A. J. Ahumada,. Jr., and J. I. Yellott, Jr., "A model for foveal photoreceptor placing," Invest. Ophthalmol. Vis. Sci. Suppl. 26, 11 (1985); A. J. Ahumada, Jr., and A. Poirson, "Modeling the irregularity of the foveolar mosaic," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986); "Model for space-varying cone placement in the fovea," J. Opt. Soc. Am. A 3(13), P70 (1986).

1987 (5)

1986 (1)

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

1985 (7)

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]

A. J. Ahumada,. Jr., and J. I. Yellott, Jr., "A model for foveal photoreceptor placing," Invest. Ophthalmol. Vis. Sci. Suppl. 26, 11 (1985); A. J. Ahumada, Jr., and A. Poirson, "Modeling the irregularity of the foveolar mosaic," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986); "Model for space-varying cone placement in the fovea," J. Opt. Soc. Am. A 3(13), P70 (1986).

F. M. de Monasterio, E. P. McCrane, J. K. Newlander, and S. J. Schein, "Density profile of vlue-sensitive cones along the horizontal meridian of macaque retina," Invest. Ophthalmol. Vis. Sci. 26, 289–302 (1985).
[PubMed]

M. B. Shapiro, S. J. Schein, and F. M. de Monasterio, "Regularity and structure of the spatial pattern of blue cones of macaque retina," J. Am. Stat. Assoc. 80, 803–812 (1985).
[CrossRef]

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

W. S. Geisler and K. D. Davila, "Ideal discriminators in spatial vision: two-point stimuli," J. Opt. Soc. Am. A 2, 1483–1497 (1985).
[CrossRef] [PubMed]

K. R. K. Nielsen, A. B. Watson, and A. J. Ahumada, Jr., "Application of a computable model of human spatial vision to phase discrimination," J. Opt. Soc. Am. A 2, 1600–1606 (1985).
[CrossRef] [PubMed]

1984 (1)

J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–256 (1984).
[CrossRef] [PubMed]

1983 (2)

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

J. I. Yellott, Jr., "Nonhomogeneous Poisson disks model the photoreceptor mosaic," Invest. Ophthalmol. Vis. Sci. Suppl. 24, 147 (1983).

1982 (1)

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

1980 (1)

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

de Monasterio, F. M.

F. M. de Monasterio, E. P. McCrane, J. K. Newlander, and S. J. Schein, "Density profile of vlue-sensitive cones along the horizontal meridian of macaque retina," Invest. Ophthalmol. Vis. Sci. 26, 289–302 (1985).
[PubMed]

M. B. Shapiro, S. J. Schein, and F. M. de Monasterio, "Regularity and structure of the spatial pattern of blue cones of macaque retina," J. Am. Stat. Assoc. 80, 803–812 (1985).
[CrossRef]

Ahumada Jr., A. J.

A. J. Ahumada,. Jr., and J. I. Yellott, Jr., "A model for foveal photoreceptor placing," Invest. Ophthalmol. Vis. Sci. Suppl. 26, 11 (1985); A. J. Ahumada, Jr., and A. Poirson, "Modeling the irregularity of the foveolar mosaic," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986); "Model for space-varying cone placement in the fovea," J. Opt. Soc. Am. A 3(13), P70 (1986).

Ahumada, Jr., A. J.

Anderson, T. W.

T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1958).

Borwein, B.

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

Borwein, D.

B. Borwein, D. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey fovea photoreceptors, with special reference to the structure, shape, size and spacing of 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]

Coletta, N.

Coletta, N. J.

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, O. Packer, and R. E. Kalina, "A whole mount method for sequential analysis of photoreceptor and ganglion cell topography," Vision Res. 27, 9–15 (1987).
[CrossRef]

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

Davila, K. D.

Geisler, W. S.

Hendrickson, A. E.

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

Hirsch, J.

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]

Hylton, R.

J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–256 (1984).
[CrossRef] [PubMed]

Kalina, R. E.

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

C. A. Curcio, O. Packer, and R. E. Kalina, "A whole mount method for sequential analysis of photoreceptor and ganglion cell topography," Vision Res. 27, 9–15 (1987).
[CrossRef]

McCrane, E. P.

F. M. de Monasterio, E. P. McCrane, J. K. Newlander, and S. J. Schein, "Density profile of vlue-sensitive cones along the horizontal meridian of macaque retina," Invest. Ophthalmol. Vis. Sci. 26, 289–302 (1985).
[PubMed]

McGowan, J. W.

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

Medeiros, J.

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

Miller, W. H.

Newlander, J. K.

F. M. de Monasterio, E. P. McCrane, J. K. Newlander, and S. J. Schein, "Density profile of vlue-sensitive cones along the horizontal meridian of macaque retina," Invest. Ophthalmol. Vis. Sci. 26, 289–302 (1985).
[PubMed]

Nielsen, K. R. K.

Packer, O.

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

C. A. Curcio, O. Packer, and R. E. Kalina, "A whole mount method for sequential analysis of photoreceptor and ganglion cell topography," Vision Res. 27, 9–15 (1987).
[CrossRef]

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]

Schein, S. J.

F. M. de Monasterio, E. P. McCrane, J. K. Newlander, and S. J. Schein, "Density profile of vlue-sensitive cones along the horizontal meridian of macaque retina," Invest. Ophthalmol. Vis. Sci. 26, 289–302 (1985).
[PubMed]

M. B. Shapiro, S. J. Schein, and F. M. de Monasterio, "Regularity and structure of the spatial pattern of blue cones of macaque retina," J. Am. Stat. Assoc. 80, 803–812 (1985).
[CrossRef]

Shapiro, M. B.

M. B. Shapiro, S. J. Schein, and F. M. de Monasterio, "Regularity and structure of the spatial pattern of blue cones of macaque retina," J. Am. Stat. Assoc. 80, 803–812 (1985).
[CrossRef]

Sloan, Jr., K. R.

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

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]

Watson, A. B.

Williams, D. R.

Yellott, Jr., J. I.

A. J. Ahumada,. Jr., and J. I. Yellott, Jr., "A model for foveal photoreceptor placing," Invest. Ophthalmol. Vis. Sci. Suppl. 26, 11 (1985); A. J. Ahumada, Jr., and A. Poirson, "Modeling the irregularity of the foveolar mosaic," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986); "Model for space-varying cone placement in the fovea," J. Opt. Soc. Am. A 3(13), P70 (1986).

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

J. I. Yellott, Jr., "Nonhomogeneous Poisson disks model the photoreceptor mosaic," Invest. Ophthalmol. Vis. Sci. Suppl. 24, 147 (1983).

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

Am. J. Anat. (1)

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

Invest. Ophthalmol. Vis. Sci. (1)

F. M. de Monasterio, E. P. McCrane, J. K. Newlander, and S. J. Schein, "Density profile of vlue-sensitive cones along the horizontal meridian of macaque retina," Invest. Ophthalmol. Vis. Sci. 26, 289–302 (1985).
[PubMed]

Invest. Ophthalmol. Vis. Sci. Suppl. (2)

J. I. Yellott, Jr., "Nonhomogeneous Poisson disks model the photoreceptor mosaic," Invest. Ophthalmol. Vis. Sci. Suppl. 24, 147 (1983).

A. J. Ahumada,. Jr., and J. I. Yellott, Jr., "A model for foveal photoreceptor placing," Invest. Ophthalmol. Vis. Sci. Suppl. 26, 11 (1985); A. J. Ahumada, Jr., and A. Poirson, "Modeling the irregularity of the foveolar mosaic," Invest. Ophthalmol. Vis. Sci. Suppl. 27, 94 (1986); "Model for space-varying cone placement in the fovea," J. Opt. Soc. Am. A 3(13), P70 (1986).

J. Am. Stat. Assoc. (1)

M. B. Shapiro, S. J. Schein, and F. M. de Monasterio, "Regularity and structure of the spatial pattern of blue cones of macaque retina," J. Am. Stat. Assoc. 80, 803–812 (1985).
[CrossRef]

J. Opt. Soc. Am. A (5)

Science (3)

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

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

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

Vision Res. (5)

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

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

J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–256 (1984).
[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]

C. A. Curcio, O. Packer, and R. E. Kalina, "A whole mount method for sequential analysis of photoreceptor and ganglion cell topography," Vision Res. 27, 9–15 (1987).
[CrossRef]

Other (6)

We will use the term hexagonal packing or hexagonal array to describe the array of objects or object centers resulting from a tessalation (tiling) of the plane with identical hexagons or tightly packed circles. There is an alternative convention of naming the packing scheme by the objects whose vertices form the desired array of sampling points, according to which this scheme is called triangular packing.

In the constant-parameter version of the program (written in C), a possible disk was allowed to overlap a placed disk (other than the two it is placed against) by up to 10% of the mean radius. In the variable-parameter version (written in PASCAL), this constant was made a parameter for the sake of compatibility, but it was set to 0% for the variable-parameter simulations. The selection of the closest possible disk keeps the set of already-packed cones mostly convex, so that possible overlap with a third disk is not the usual situation. The source code is available from Albert J. Ahumada.

This formula is the sum of the squares of two independent standard normal variates when the parameters are correct. The first is simply the standardized form of S1, and the second is the standardized form of the part of S2 that is uncorrelated with S1.6

T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1958).

At this point in the model development we also changed the rule for generating a random radius. Previously we had generated a new radius for each pair of disks for which we were trying to find a possible next neighbor. Instead, we now generate a new radius when we begin looking for the new posssible disks after the addition of a new disk to the retina and keep trying that same radius until a possible position is found for it. Although this change does save random-number generations, the change was made to avoid a rare problem. A crevice in the edge of the retina would not be filled because a large disk would be tried for the possible positions inside the crevice and a small disk would be tried for the position that closed the mouth of the crevice. The few holes thus generated would not have affected our measures appreciably, but in the interest of computational efficiency we also modified the program to search only through the neighbors of neighbors of neighbors to find possible sites and to look for overlaps. The holes permitted overlapping positions that the search could not find because they were too many generations of neighbors away. We have found only an algorithm that has worked for us with a relatively small range of parameters. We do not think that we have found a packing scheme that will always work. For example, the scheme might not work for disk sizes as disparate as rods and cones.

D. R. Williams, "Topography of the foveal cone mosaic in the living human eye," submitted to Vision Res.; Invest. Ophthalmol. Vis. Sci. Suppl. 26, 10 (1986).

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Figures (9)

Fig. 1
Fig. 1

Disks placed by the model (solid lines) and possible disks to be placed next (dashed lines). (a) The first disk has been placed in the center, and the second has been placed to its right on the x axis. (b) The first six disks have been placed.

Fig. 2
Fig. 2

Distribution of differences between cone-center coordinates of near neighbors in the mean HH data. Only the top half of the distribution is shown, since the distribution is symmetric about the center. The circular boundaries indicate the limits defining the first ring and the second ring.

Fig. 3
Fig. 3

The region above and to the left of the continuous curve is a 95% confidence region for the model parameters σr and σj based on the χ2 method described in the text. The points show the grid of parameter values at which simulations were done to measure the joint distribution of S1 and S2. The × marks the minimum χ2 estimate of σr and σj.

Fig. 4
Fig. 4

(a) The HH mean data sample points; (b) a sample generated by the model with σr equal to 3.2% and σj equal to 5.4%.

Fig. 5
Fig. 5

(a) The HM data sample points for 0–0.23 deg; (b) a sample generated by the constant-parameter version of the model with the same parameters as Fig. 4.

Fig. 6
Fig. 6

The number of cones in each of the first seven cuts of the HM data (triangles) and the largest and smallest number obtained in 10 simulations of the model (points connected by solid lines).

Fig. 7
Fig. 7

(a) First-ring variability ratio measure R1 for the first eight cuts of the HM data (circles). The solid lines connect the minimum, the median, and the maximum values (bottom to top, respectively) of R1 for 10 model simulations out to 7 cuts. (b) Same as (a) for second-ring variability ratio measure R2. (c) Same as (a) for hexagonality variability ratio measure R3.

Fig. 8
Fig. 8

(a) HM data points (left) and sample model points (right) for 0–0.23 deg. The circles all have the same radius and hence do not represent disk size or cone size. They do make it easier to see the increase in spacing with eccentricity in cut 1. (b) Same as (a) for 0.46–0.69 deg. (c) Same as (a) for 1.38–1.61 deg.

Fig. 9
Fig. 9

The local-variability measures R1, R2, and R3 for all 25 cuts of the HM data.

Tables (4)

Tables Icon

Table 1 HH Data Values for Normalized First- and Second-Ring-Distance Standard Deviations S1 and S2 and the Subring Corrected Measure S2

Tables Icon

Table 2 Regression Coefficients for Predicting the Means and the Standard Deviations of and the Correlation between the Two Measures S1 and S2 from the Parameters σr and σja

Tables Icon

Table 3 Estimates for Model Parameters σr and σj from the Statistics S1 and S2a

Tables Icon

Table 4 Ring 1 Distance Standard Deviation S1 for the First Cut of the HM Dataa

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

μ r = ( r min + r max ) / 2 ,
σ r = ( r max - r min ) / 12 .
S 2 = ( S 2 2 - 0.134 2 ) 1 / 2 .
χ 2 = z 1 2 + [ ( z 2 - r 12 z 1 ) / ( 1 - r 12 ) ] 2 ,
z i = ( S i - μ i ) / σ i ,             i = 1 , 2.
σ j , n 2 = σ t 2 + σ e 2 / n ,
σ e 2 = 2 ( σ j , 1 2 - σ j , 2 2 ) .
μ r ( ) = 1 + 0.50 ,
R 1 = 1 - [ i ( D i , 1 / D i , 4 ) ] / N 4 ,
R 2 = 1 - [ i ( D i , 9 / D i , 12 ) ] / N 12 ,
R 3 = 1 - [ i ( D i , 3 / D i , 6 ) ] / N 6 .
c r ( ) = σ r ( ) / μ r ( )
c j ( ) = σ j ( ) / μ r ( ) ,

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