Abstract

We try to improve the accuracy of eccentric photorefraction by taking more information into account than just the size and tilt of the crescent. Based on Gaussian optics and the assumption of an isotropic scattering retina, a theoretical analysis of the light-intensity distribution in the pupils of astigmatic eyes is presented. The method is applied to different photorefractor setups (point light source, long linear light source, knife-edge aperture, and circular aperture). In the case of a knife-edge aperture the crescent structure can be formulated analytically. In the case of a circular aperture an analytic description is possible only for spherical refractive errors, but astigmatic refractive errors can be determined from crescent parameters with neural networks.

© 1998 Optical Society of America

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References

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  1. W. R. Bobier, O. J. Braddick, “Eccentric photorefraction: optical analysis and empirical measures,” Am. J. Optom. Physiol. Opt. 62, 614–620 (1985).
    [CrossRef] [PubMed]
  2. K. A. Kaakinen, H. O. Kaseva, H. H. Teir, “Two-flash photorefraction in screening of amblyogenic refractive errors,” Ophthalmology 94, 1036–1042 (1987).
    [CrossRef] [PubMed]
  3. M. R. Angi, A. Cocchiglia, “Il videorefrattometro binoculare infrarosso: uno strumento per lo screening dei difetti ambliogenici e lo studio dinamico della capacita accomodativa,” Bol. Ocul. 69, 305–320 (1990).
  4. R. D. Hamer, A. M. Norcia, S. H. Day, G. Haegerstrom-Portnoy, D. Lewis, C. Hsu-Winges, “Comparison of on- and off-axis photorefraction with cycloplegic retinoscopy in infants,” J. Pediatr. Ophthalmol. Strabismus 29, 232–239 (1992).
    [PubMed]
  5. H. L. Freedman, K. L. Preston, “Polaroid photoscreening for amblyogenic factors: an improved methodology,” Ophthalmology 99, 1785–1795 (1992).
    [CrossRef] [PubMed]
  6. F. Schaeffel, A. Glasser, H. C. Howland, “Accommodation, refractive error and eye growth in chickens,” Vision Res. 28, 639–657 (1988).
    [CrossRef] [PubMed]
  7. F. Schaeffel, H. C. Howland, “Properties of visual feedback loops controlling eye growth and refractive state in the chicken,” Vision Res. 31, 717–734 (1991).
    [CrossRef]
  8. G. Ueberschaar, “Die Durchführung der Skiaskopie unter Verwendung einer stationären Lichtquelle,” in Proceedings of the Annual Conference of the Wissenschaftliche Vereinigung für Augenoptik (Wissenschaftliche Vereinigung für Augenoptik und Optometrie, Mainz, Germany, 1955), Vol. 4, pp. 73–77.
  9. K. Kaakinen, “A simple method for screening of children with strabismus, anisometropia or ametropia by simultaneous photography of the corneal and the fundus reflex,” Acta Ophthalmol. 57, 161–171 (1978).
    [CrossRef]
  10. K. Kaakinen, “Simultaneous two flash static photosciascopy,” Acta Ophthalmol. 59, 378–386 (1981).
    [CrossRef]
  11. S. H. Day, A. M. Norcia, “Photographic detection of amblyogenic factors,” Ophthalmology 93, 25–28 (1986).
    [CrossRef] [PubMed]
  12. F. Schaeffel, L. Farkas, H. C. Howland, “Infrared photoretinoscope,” Appl. Opt. 26, 1505–1509 (1987).
    [CrossRef] [PubMed]
  13. H. C. Howland, “Optics of photoretinoscopy: results from ray tracing,” Am. J. Optom. Physiol. Opt. 62, 621–625 (1985).
    [CrossRef] [PubMed]
  14. W. Wesemann, A. M. Norcia, D. Allen, “Theory of eccentric photorefraction (photoretinoscopy): astigmatic eyes,” J. Opt. Soc. Am. A 8, 2038–2047 (1991).
    [CrossRef] [PubMed]
  15. F. Schaeffel, H. Wilhelm, E. Zrenner, “Inter-individual variability in the dynamics of natural accommodation in humans: relation to age and refractive errors,” J. Physiol. (London) 461, 301–320 (1993).
  16. A. Roorda, M. C. W. Campbell, W. R. Bobier, “Geometrical theory to predict eccentric photorefraction intensity profiles in the human eye,” J. Opt. Soc. Am. A 12, 1647–1656 (1995).
    [CrossRef]
  17. M. C. W. Campbell, W. R. Bobier, A. Roorda, “Effect of monochromatic aberrations on photorefractive patterns,” J. Opt. Soc. Am. A 12, 1637–1646 (1995).
    [CrossRef]
  18. A. Roorda, M. C. W. Campbell, “Slope-based eccentric photorefraction: theoretical analysis of different light source configurations and effects of ocular aberrations,” J. Opt. Soc. Am. A 14, 2547–2556 (1997).
    [CrossRef]
  19. R. Kusel, “Models describing eccentric photorefraction crescents,” in Vision Science and Its Applications, Vol. 1 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 117–120.
  20. The symmetry of the circular camera aperture is especially important for computer-assisted video setups, because several LED’s mounted at different angles can be turned on automatically, and several images can be taken in rapid succession.
  21. As aberrations are not considered, rays entering the eye at the border of the pupil form the border of the blurred retinal intensity distribution. Thus if xˆp2+yˆp2=rp2, where rp is the radius of the pupil, from Eq. 4 an ellipse [(xr-xrc)/ar]2+[(yr-yrc)/br]2=1 with half axes ar=rpp(Rx-1/p-1/l) and br=rpp(Ry-1/p-1/l) centered around (xrc, yrc)=-p/l(xl, yl) can be derived.
  22. U. Oechsner, R. Kusel, “Meridional refraction: dependence of the measurement accuracy on the number of meridians refracted,” Optom. Vis. Sci. 74, 425–433 (1997).
    [CrossRef] [PubMed]
  23. This approach was presented in part as a poster at the Association for Research in Vision and Ophthalmology, Inc. 1997: R. Kusel, U. Oechsner, W. Wesemann, J. Wattam-Bell, “A neural network for computing refractive errors from eccentric photorefraction crescents,” Invest. Ophthalmol. Visual Sci. 38, S977 (1997).
  24. A. Zell, N. Mache, R. Huebner, “SNNS (Stuttgart Neural Network Simulator),” in Neural Network Simulation Environments, J. Skrzypek, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1994), pp. 165–186.
  25. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]

1997 (3)

U. Oechsner, R. Kusel, “Meridional refraction: dependence of the measurement accuracy on the number of meridians refracted,” Optom. Vis. Sci. 74, 425–433 (1997).
[CrossRef] [PubMed]

This approach was presented in part as a poster at the Association for Research in Vision and Ophthalmology, Inc. 1997: R. Kusel, U. Oechsner, W. Wesemann, J. Wattam-Bell, “A neural network for computing refractive errors from eccentric photorefraction crescents,” Invest. Ophthalmol. Visual Sci. 38, S977 (1997).

A. Roorda, M. C. W. Campbell, “Slope-based eccentric photorefraction: theoretical analysis of different light source configurations and effects of ocular aberrations,” J. Opt. Soc. Am. A 14, 2547–2556 (1997).
[CrossRef]

1995 (2)

1993 (1)

F. Schaeffel, H. Wilhelm, E. Zrenner, “Inter-individual variability in the dynamics of natural accommodation in humans: relation to age and refractive errors,” J. Physiol. (London) 461, 301–320 (1993).

1992 (2)

R. D. Hamer, A. M. Norcia, S. H. Day, G. Haegerstrom-Portnoy, D. Lewis, C. Hsu-Winges, “Comparison of on- and off-axis photorefraction with cycloplegic retinoscopy in infants,” J. Pediatr. Ophthalmol. Strabismus 29, 232–239 (1992).
[PubMed]

H. L. Freedman, K. L. Preston, “Polaroid photoscreening for amblyogenic factors: an improved methodology,” Ophthalmology 99, 1785–1795 (1992).
[CrossRef] [PubMed]

1991 (2)

F. Schaeffel, H. C. Howland, “Properties of visual feedback loops controlling eye growth and refractive state in the chicken,” Vision Res. 31, 717–734 (1991).
[CrossRef]

W. Wesemann, A. M. Norcia, D. Allen, “Theory of eccentric photorefraction (photoretinoscopy): astigmatic eyes,” J. Opt. Soc. Am. A 8, 2038–2047 (1991).
[CrossRef] [PubMed]

1990 (1)

M. R. Angi, A. Cocchiglia, “Il videorefrattometro binoculare infrarosso: uno strumento per lo screening dei difetti ambliogenici e lo studio dinamico della capacita accomodativa,” Bol. Ocul. 69, 305–320 (1990).

1988 (1)

F. Schaeffel, A. Glasser, H. C. Howland, “Accommodation, refractive error and eye growth in chickens,” Vision Res. 28, 639–657 (1988).
[CrossRef] [PubMed]

1987 (2)

K. A. Kaakinen, H. O. Kaseva, H. H. Teir, “Two-flash photorefraction in screening of amblyogenic refractive errors,” Ophthalmology 94, 1036–1042 (1987).
[CrossRef] [PubMed]

F. Schaeffel, L. Farkas, H. C. Howland, “Infrared photoretinoscope,” Appl. Opt. 26, 1505–1509 (1987).
[CrossRef] [PubMed]

1986 (1)

S. H. Day, A. M. Norcia, “Photographic detection of amblyogenic factors,” Ophthalmology 93, 25–28 (1986).
[CrossRef] [PubMed]

1985 (2)

H. C. Howland, “Optics of photoretinoscopy: results from ray tracing,” Am. J. Optom. Physiol. Opt. 62, 621–625 (1985).
[CrossRef] [PubMed]

W. R. Bobier, O. J. Braddick, “Eccentric photorefraction: optical analysis and empirical measures,” Am. J. Optom. Physiol. Opt. 62, 614–620 (1985).
[CrossRef] [PubMed]

1981 (1)

K. Kaakinen, “Simultaneous two flash static photosciascopy,” Acta Ophthalmol. 59, 378–386 (1981).
[CrossRef]

1980 (1)

1978 (1)

K. Kaakinen, “A simple method for screening of children with strabismus, anisometropia or ametropia by simultaneous photography of the corneal and the fundus reflex,” Acta Ophthalmol. 57, 161–171 (1978).
[CrossRef]

Allen, D.

Angi, M. R.

M. R. Angi, A. Cocchiglia, “Il videorefrattometro binoculare infrarosso: uno strumento per lo screening dei difetti ambliogenici e lo studio dinamico della capacita accomodativa,” Bol. Ocul. 69, 305–320 (1990).

Bobier, W. R.

Braddick, O. J.

W. R. Bobier, O. J. Braddick, “Eccentric photorefraction: optical analysis and empirical measures,” Am. J. Optom. Physiol. Opt. 62, 614–620 (1985).
[CrossRef] [PubMed]

Campbell, M. C. W.

Cocchiglia, A.

M. R. Angi, A. Cocchiglia, “Il videorefrattometro binoculare infrarosso: uno strumento per lo screening dei difetti ambliogenici e lo studio dinamico della capacita accomodativa,” Bol. Ocul. 69, 305–320 (1990).

Day, S. H.

R. D. Hamer, A. M. Norcia, S. H. Day, G. Haegerstrom-Portnoy, D. Lewis, C. Hsu-Winges, “Comparison of on- and off-axis photorefraction with cycloplegic retinoscopy in infants,” J. Pediatr. Ophthalmol. Strabismus 29, 232–239 (1992).
[PubMed]

S. H. Day, A. M. Norcia, “Photographic detection of amblyogenic factors,” Ophthalmology 93, 25–28 (1986).
[CrossRef] [PubMed]

Farkas, L.

Freedman, H. L.

H. L. Freedman, K. L. Preston, “Polaroid photoscreening for amblyogenic factors: an improved methodology,” Ophthalmology 99, 1785–1795 (1992).
[CrossRef] [PubMed]

Glasser, A.

F. Schaeffel, A. Glasser, H. C. Howland, “Accommodation, refractive error and eye growth in chickens,” Vision Res. 28, 639–657 (1988).
[CrossRef] [PubMed]

Haegerstrom-Portnoy, G.

R. D. Hamer, A. M. Norcia, S. H. Day, G. Haegerstrom-Portnoy, D. Lewis, C. Hsu-Winges, “Comparison of on- and off-axis photorefraction with cycloplegic retinoscopy in infants,” J. Pediatr. Ophthalmol. Strabismus 29, 232–239 (1992).
[PubMed]

Hamer, R. D.

R. D. Hamer, A. M. Norcia, S. H. Day, G. Haegerstrom-Portnoy, D. Lewis, C. Hsu-Winges, “Comparison of on- and off-axis photorefraction with cycloplegic retinoscopy in infants,” J. Pediatr. Ophthalmol. Strabismus 29, 232–239 (1992).
[PubMed]

Howland, H. C.

F. Schaeffel, H. C. Howland, “Properties of visual feedback loops controlling eye growth and refractive state in the chicken,” Vision Res. 31, 717–734 (1991).
[CrossRef]

F. Schaeffel, A. Glasser, H. C. Howland, “Accommodation, refractive error and eye growth in chickens,” Vision Res. 28, 639–657 (1988).
[CrossRef] [PubMed]

F. Schaeffel, L. Farkas, H. C. Howland, “Infrared photoretinoscope,” Appl. Opt. 26, 1505–1509 (1987).
[CrossRef] [PubMed]

H. C. Howland, “Optics of photoretinoscopy: results from ray tracing,” Am. J. Optom. Physiol. Opt. 62, 621–625 (1985).
[CrossRef] [PubMed]

Hsu-Winges, C.

R. D. Hamer, A. M. Norcia, S. H. Day, G. Haegerstrom-Portnoy, D. Lewis, C. Hsu-Winges, “Comparison of on- and off-axis photorefraction with cycloplegic retinoscopy in infants,” J. Pediatr. Ophthalmol. Strabismus 29, 232–239 (1992).
[PubMed]

Huebner, R.

A. Zell, N. Mache, R. Huebner, “SNNS (Stuttgart Neural Network Simulator),” in Neural Network Simulation Environments, J. Skrzypek, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1994), pp. 165–186.

Kaakinen, K.

K. Kaakinen, “Simultaneous two flash static photosciascopy,” Acta Ophthalmol. 59, 378–386 (1981).
[CrossRef]

K. Kaakinen, “A simple method for screening of children with strabismus, anisometropia or ametropia by simultaneous photography of the corneal and the fundus reflex,” Acta Ophthalmol. 57, 161–171 (1978).
[CrossRef]

Kaakinen, K. A.

K. A. Kaakinen, H. O. Kaseva, H. H. Teir, “Two-flash photorefraction in screening of amblyogenic refractive errors,” Ophthalmology 94, 1036–1042 (1987).
[CrossRef] [PubMed]

Kaseva, H. O.

K. A. Kaakinen, H. O. Kaseva, H. H. Teir, “Two-flash photorefraction in screening of amblyogenic refractive errors,” Ophthalmology 94, 1036–1042 (1987).
[CrossRef] [PubMed]

Kusel, R.

U. Oechsner, R. Kusel, “Meridional refraction: dependence of the measurement accuracy on the number of meridians refracted,” Optom. Vis. Sci. 74, 425–433 (1997).
[CrossRef] [PubMed]

This approach was presented in part as a poster at the Association for Research in Vision and Ophthalmology, Inc. 1997: R. Kusel, U. Oechsner, W. Wesemann, J. Wattam-Bell, “A neural network for computing refractive errors from eccentric photorefraction crescents,” Invest. Ophthalmol. Visual Sci. 38, S977 (1997).

R. Kusel, “Models describing eccentric photorefraction crescents,” in Vision Science and Its Applications, Vol. 1 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 117–120.

Lewis, D.

R. D. Hamer, A. M. Norcia, S. H. Day, G. Haegerstrom-Portnoy, D. Lewis, C. Hsu-Winges, “Comparison of on- and off-axis photorefraction with cycloplegic retinoscopy in infants,” J. Pediatr. Ophthalmol. Strabismus 29, 232–239 (1992).
[PubMed]

Mache, N.

A. Zell, N. Mache, R. Huebner, “SNNS (Stuttgart Neural Network Simulator),” in Neural Network Simulation Environments, J. Skrzypek, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1994), pp. 165–186.

Norcia, A. M.

R. D. Hamer, A. M. Norcia, S. H. Day, G. Haegerstrom-Portnoy, D. Lewis, C. Hsu-Winges, “Comparison of on- and off-axis photorefraction with cycloplegic retinoscopy in infants,” J. Pediatr. Ophthalmol. Strabismus 29, 232–239 (1992).
[PubMed]

W. Wesemann, A. M. Norcia, D. Allen, “Theory of eccentric photorefraction (photoretinoscopy): astigmatic eyes,” J. Opt. Soc. Am. A 8, 2038–2047 (1991).
[CrossRef] [PubMed]

S. H. Day, A. M. Norcia, “Photographic detection of amblyogenic factors,” Ophthalmology 93, 25–28 (1986).
[CrossRef] [PubMed]

Oechsner, U.

U. Oechsner, R. Kusel, “Meridional refraction: dependence of the measurement accuracy on the number of meridians refracted,” Optom. Vis. Sci. 74, 425–433 (1997).
[CrossRef] [PubMed]

This approach was presented in part as a poster at the Association for Research in Vision and Ophthalmology, Inc. 1997: R. Kusel, U. Oechsner, W. Wesemann, J. Wattam-Bell, “A neural network for computing refractive errors from eccentric photorefraction crescents,” Invest. Ophthalmol. Visual Sci. 38, S977 (1997).

Preston, K. L.

H. L. Freedman, K. L. Preston, “Polaroid photoscreening for amblyogenic factors: an improved methodology,” Ophthalmology 99, 1785–1795 (1992).
[CrossRef] [PubMed]

Roorda, A.

Schaeffel, F.

F. Schaeffel, H. Wilhelm, E. Zrenner, “Inter-individual variability in the dynamics of natural accommodation in humans: relation to age and refractive errors,” J. Physiol. (London) 461, 301–320 (1993).

F. Schaeffel, H. C. Howland, “Properties of visual feedback loops controlling eye growth and refractive state in the chicken,” Vision Res. 31, 717–734 (1991).
[CrossRef]

F. Schaeffel, A. Glasser, H. C. Howland, “Accommodation, refractive error and eye growth in chickens,” Vision Res. 28, 639–657 (1988).
[CrossRef] [PubMed]

F. Schaeffel, L. Farkas, H. C. Howland, “Infrared photoretinoscope,” Appl. Opt. 26, 1505–1509 (1987).
[CrossRef] [PubMed]

Silva, D. E.

Teir, H. H.

K. A. Kaakinen, H. O. Kaseva, H. H. Teir, “Two-flash photorefraction in screening of amblyogenic refractive errors,” Ophthalmology 94, 1036–1042 (1987).
[CrossRef] [PubMed]

Ueberschaar, G.

G. Ueberschaar, “Die Durchführung der Skiaskopie unter Verwendung einer stationären Lichtquelle,” in Proceedings of the Annual Conference of the Wissenschaftliche Vereinigung für Augenoptik (Wissenschaftliche Vereinigung für Augenoptik und Optometrie, Mainz, Germany, 1955), Vol. 4, pp. 73–77.

Wang, J. Y.

Wattam-Bell, J.

This approach was presented in part as a poster at the Association for Research in Vision and Ophthalmology, Inc. 1997: R. Kusel, U. Oechsner, W. Wesemann, J. Wattam-Bell, “A neural network for computing refractive errors from eccentric photorefraction crescents,” Invest. Ophthalmol. Visual Sci. 38, S977 (1997).

Wesemann, W.

This approach was presented in part as a poster at the Association for Research in Vision and Ophthalmology, Inc. 1997: R. Kusel, U. Oechsner, W. Wesemann, J. Wattam-Bell, “A neural network for computing refractive errors from eccentric photorefraction crescents,” Invest. Ophthalmol. Visual Sci. 38, S977 (1997).

W. Wesemann, A. M. Norcia, D. Allen, “Theory of eccentric photorefraction (photoretinoscopy): astigmatic eyes,” J. Opt. Soc. Am. A 8, 2038–2047 (1991).
[CrossRef] [PubMed]

Wilhelm, H.

F. Schaeffel, H. Wilhelm, E. Zrenner, “Inter-individual variability in the dynamics of natural accommodation in humans: relation to age and refractive errors,” J. Physiol. (London) 461, 301–320 (1993).

Zell, A.

A. Zell, N. Mache, R. Huebner, “SNNS (Stuttgart Neural Network Simulator),” in Neural Network Simulation Environments, J. Skrzypek, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1994), pp. 165–186.

Zrenner, E.

F. Schaeffel, H. Wilhelm, E. Zrenner, “Inter-individual variability in the dynamics of natural accommodation in humans: relation to age and refractive errors,” J. Physiol. (London) 461, 301–320 (1993).

Acta Ophthalmol. (2)

K. Kaakinen, “A simple method for screening of children with strabismus, anisometropia or ametropia by simultaneous photography of the corneal and the fundus reflex,” Acta Ophthalmol. 57, 161–171 (1978).
[CrossRef]

K. Kaakinen, “Simultaneous two flash static photosciascopy,” Acta Ophthalmol. 59, 378–386 (1981).
[CrossRef]

Am. J. Optom. Physiol. Opt. (2)

W. R. Bobier, O. J. Braddick, “Eccentric photorefraction: optical analysis and empirical measures,” Am. J. Optom. Physiol. Opt. 62, 614–620 (1985).
[CrossRef] [PubMed]

H. C. Howland, “Optics of photoretinoscopy: results from ray tracing,” Am. J. Optom. Physiol. Opt. 62, 621–625 (1985).
[CrossRef] [PubMed]

Appl. Opt. (2)

Bol. Ocul. (1)

M. R. Angi, A. Cocchiglia, “Il videorefrattometro binoculare infrarosso: uno strumento per lo screening dei difetti ambliogenici e lo studio dinamico della capacita accomodativa,” Bol. Ocul. 69, 305–320 (1990).

Invest. Ophthalmol. Visual Sci. (1)

This approach was presented in part as a poster at the Association for Research in Vision and Ophthalmology, Inc. 1997: R. Kusel, U. Oechsner, W. Wesemann, J. Wattam-Bell, “A neural network for computing refractive errors from eccentric photorefraction crescents,” Invest. Ophthalmol. Visual Sci. 38, S977 (1997).

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

J. Pediatr. Ophthalmol. Strabismus (1)

R. D. Hamer, A. M. Norcia, S. H. Day, G. Haegerstrom-Portnoy, D. Lewis, C. Hsu-Winges, “Comparison of on- and off-axis photorefraction with cycloplegic retinoscopy in infants,” J. Pediatr. Ophthalmol. Strabismus 29, 232–239 (1992).
[PubMed]

J. Physiol. (London) (1)

F. Schaeffel, H. Wilhelm, E. Zrenner, “Inter-individual variability in the dynamics of natural accommodation in humans: relation to age and refractive errors,” J. Physiol. (London) 461, 301–320 (1993).

Ophthalmology (3)

H. L. Freedman, K. L. Preston, “Polaroid photoscreening for amblyogenic factors: an improved methodology,” Ophthalmology 99, 1785–1795 (1992).
[CrossRef] [PubMed]

K. A. Kaakinen, H. O. Kaseva, H. H. Teir, “Two-flash photorefraction in screening of amblyogenic refractive errors,” Ophthalmology 94, 1036–1042 (1987).
[CrossRef] [PubMed]

S. H. Day, A. M. Norcia, “Photographic detection of amblyogenic factors,” Ophthalmology 93, 25–28 (1986).
[CrossRef] [PubMed]

Optom. Vis. Sci. (1)

U. Oechsner, R. Kusel, “Meridional refraction: dependence of the measurement accuracy on the number of meridians refracted,” Optom. Vis. Sci. 74, 425–433 (1997).
[CrossRef] [PubMed]

Vision Res. (2)

F. Schaeffel, A. Glasser, H. C. Howland, “Accommodation, refractive error and eye growth in chickens,” Vision Res. 28, 639–657 (1988).
[CrossRef] [PubMed]

F. Schaeffel, H. C. Howland, “Properties of visual feedback loops controlling eye growth and refractive state in the chicken,” Vision Res. 31, 717–734 (1991).
[CrossRef]

Other (5)

G. Ueberschaar, “Die Durchführung der Skiaskopie unter Verwendung einer stationären Lichtquelle,” in Proceedings of the Annual Conference of the Wissenschaftliche Vereinigung für Augenoptik (Wissenschaftliche Vereinigung für Augenoptik und Optometrie, Mainz, Germany, 1955), Vol. 4, pp. 73–77.

R. Kusel, “Models describing eccentric photorefraction crescents,” in Vision Science and Its Applications, Vol. 1 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 117–120.

The symmetry of the circular camera aperture is especially important for computer-assisted video setups, because several LED’s mounted at different angles can be turned on automatically, and several images can be taken in rapid succession.

As aberrations are not considered, rays entering the eye at the border of the pupil form the border of the blurred retinal intensity distribution. Thus if xˆp2+yˆp2=rp2, where rp is the radius of the pupil, from Eq. 4 an ellipse [(xr-xrc)/ar]2+[(yr-yrc)/br]2=1 with half axes ar=rpp(Rx-1/p-1/l) and br=rpp(Ry-1/p-1/l) centered around (xrc, yrc)=-p/l(xl, yl) can be derived.

A. Zell, N. Mache, R. Huebner, “SNNS (Stuttgart Neural Network Simulator),” in Neural Network Simulation Environments, J. Skrzypek, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1994), pp. 165–186.

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Figures (11)

Fig. 1
Fig. 1

Geometric optical situation of an eccentric photorefractor for an eye with myopic astigmatism in the coordinate system of the principal meridians (see text).

Fig. 2
Fig. 2

(a) Illumination path with a light source centered on the z axis. (b) Observation path of a retinal point centered on the z axis. Both figures depict the xz plane.

Fig. 3
Fig. 3

(a) Knife-edge aperture with eccentric pointlike light source. (b) Knife-edge aperture with eccentric long light source mounted perpendicular to the knife edge.

Fig. 4
Fig. 4

Lines of constant brightness and vertical relative brightness profile of the crescent through the center of the pupil that appears when a point light source mounted 2 mm below a horizontal knife edge is used (Fx=-4 D, Fy=-2 D, ϕ=45°, c=l=1 m, rp=2.5 mm).

Fig. 5
Fig. 5

(a) Determination of the crescent size from the straight border of the crescent [Eq. (14)]. If the crescent is located above the border line, then γ=γ and the eye is hypermetropic. If the eye is myopic, the crescent is located below the border line and γ=π+γ.

Fig. 6
Fig. 6

Crescent size db versus spherical refractive error F (knife-edge method). Parameter is the eccentricity |xlL| of the light source (c=l=1 m, rp=2.5 mm).

Fig. 7
Fig. 7

Lines of constant brightness and vertical relative brightness profile of the crescent through the center of the pupil that appears when a linear light source is used (see Fig. 3) (Fx=-4 D, Fy=-2 D, ϕ=45°, c=l=1 m, rp=2.5 mm). The crescent is normalized by cErp.

Fig. 8
Fig. 8

Lines of constant brightness and vertical relative brightness profile of the crescent through the center of the pupil that appears when a point light source is mounted 2 mm below a circular aperture (Fx=-4 D, Fy=-2 D, ϕ=45°, c=l=1 m, rp=2.5 mm, rc=10 mm).

Fig. 9
Fig. 9

Plateau and border of the light-intensity distribution in the pupil plane as it results from Eqs. (19) but as seen in the laboratory system (Fx=-4 D, Fy=-2 D, ϕ=45°, c=l=1 m, rp=2.5 mm, rc=10 mm, xlL=-12 mm, ylL=0 mm). The plateau with relative intensity 1 lies outside of the pupil. The border replicates the crescent margin shown in Fig. 8.

Fig. 10
Fig. 10

(a) Error of the estimated refractive error in the xL meridian of the laboratory system as a function of the refractive error in this meridian. The refractive error in the y meridian was 2.5 D smaller; the tilt of the x meridian with respect to the xL meridian was 30°. With the exception of relative refractive errors of less than 1 D, where the crescents are small, the error is smaller than ±0.25D. (b) Error of the estimated refractive error in the xL meridian of the laboratory system as a function of the angle ϕ of the tilt of the x meridian with respect to the xL meridian. The relative refractive error was 6 D in the x meridian and 3.5 D in the y meridian. While in the range around 0° the error is clearly below ±0.25 D, it increases to ±0.5 D for angles from 90° to 150°.

Fig. 11
Fig. 11

Illustration of the relation of E and γ to the principal meridional components of the relative refractive error. EPR crescents in knife-edge setups are determined mainly by the projections of the principal-axis components on the xL axis perpendicular to the knife edge. These projected components add up in the sense of the error-propagation law to the quantity E.

Equations (63)

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x=xL cos ϕ+yL sin ϕ,
y=-xL sin ϕ+yL cos ϕ,
xL=x cos ϕ-y sin ϕ,
yL=x sin ϕ+y cos ϕ,
-xrxˆp=zr-zlzl-zp xr=-zr-zlzl-zpxˆp.
Rx=1l+1zl-zp
zl=1Rx-1/l+zp.
(xr, yr)=-pRx-1p-1lxˆp,Ry-1p-1lyˆp
(xr0, yr0)=-pl(xl, yl),
(xr, yr)=-pxll+Rx-1p-1lxˆp, yll+Ry-1p-1lyˆp.
(xc, yc)=-cxrp+Rx-1p-1cxp,yrp+Ry-1p-1cyp.
(xc, yc)=cxll+Rx-1p-1lxˆp-Rx-1p-1cxp, yll+Ry-1p-1lyˆp-Ry-1p-1cyp.
Fx=1p-Rx(refractiveerrorinthexmeridian),
Fy=1p-Ry(refractiveerrorintheymeridian)
(xc, yc)=cxll-1l+Fxxˆp+1c+Fxxp, yll-1l+Fyyˆp+1c+Fyyp.
xc-clxl-c1c+Fxxpc1l+Fxrp2
+yc-clyl-c1c+Fyypc1l+Fyrp2=1.
(xcc, ycc)=cxll+1c+Fxxp, yll+1c+Fyyp,
a=c|1/l+Fx|rp,b=c|1/l+Fy|rp.
xcc-clxl(1+cFx)rp2+ycc-clyl(1+cFy)rp2=1.
I(xpL, ypL)=1π[π/2+arcsin u+u(1-u2)1/2]
u=1E1c+Fx+Fy2+Fx-Fy2cos 2ϕ xpLrp+Fx-Fy2sin 2ϕ ypLrp-|xlL|crp=xpLrpcos γ+ypLrpsin γ-|xlL|cErp,
E=[(1/c+Fx)2 cos2 ϕ+(1/c+Fy)2 sin2 ϕ]1/2.
IxpL, IypL=2πrp(1-u2)1/2(cos γ, sin γ).
cos γ=1E1c+Fx+Fy2+Fx-Fy2cos 2ϕ,
xpLrp=-ypLrptan γ+|xlL|/(cErp)-1cos γ.
x¯=|xlL|/(cE)-rpcos γ,y¯=|xlL|/(cE)-rpsin γ.
h=x¯y¯ (+)(x¯2+y¯2)1/2=sgn(cos γ)(|xlL|/(cE)-rp).
db=2rp-|xlL|/(cE).
-1c1-|xlL|2rpF-1c1+|xlL|2rp.
dd=2rp-db=|xlL|/(cE).
(ξ, η)=|xlL|/(cE)(cos γ, sin γ)
ξ0=|xlL|c(1/c+Fx)(1/c+Fy)1c+Fx+Fy2,
η0=0,
r=|xlL|c(1/c+Fx)(1/c+Fy)Fx-Fy2
I(xpL, ypL)=1/π[π/2+arcsin u+u(1-u2)1/2],
u=u0xlL+u1,
xmaxL=-1+u1u0,
I(xpL, ypL)=AπxmaxL0 π2+arcsin u+u(1-u2)1/2dxlL=AcErpu12+u1πarcsin u1+2+u123π(1-u12)1/2.
IxpL, IypL=AcEππ2+arcsin u1+u1(1-u12)1/2(cos γ, sin γ).
IcErp=Au12+u1πarcsin u1+2+u123π(1-u12)1/2,
u1=(cos γ, sin γ)xpLrpypLrp.
u1=xp/rp
(xc-xcc)2+(yc-ycc)2=r2.
(xcc, ycc)=cxll+1c+Fxp, yll+1c+Fyp.
xc2+yc2=rc2
I(xp, yp)=(A1+A2)/(πr2),
A1=rc2π2-u1(1-u12)1/2-arcsin u1,
A2=r2π2+u2(1-u22)1/2+arcsin u2,
u1=1rcrc2-r22(xcc2+ycc2)1/2+12(xcc2+ycc2)1/2,
u2=1rrc2-r22(xcc2+ycc2)1/2+12(xcc2+ycc2)1/2-xcc.
xcc2+ycc2(1+cF)2=xp+cl(1+cF)xl2+yp+cl(1+cF)yl2.
(xpc, ypc)=-xll(1/c+Fx), yll(1/c+Fy),
(xpcL, ypcL)=-xlLl(1/c+Fx)(1/c+Fy)×1c+Fx+Fy2-Fx-Fy2×cos 2ϕ,-Fx-Fy2sin 2ϕ,
δ=arctan -Fx-Fy2sin 2ϕ1c+Fx+Fy2-Fx-Fy2cos 2ϕ.
ap=rc/c+|1/l+Fx|rp|1/c+Fx|,bp=rc/c+|1/l+Fy|rp|1/c+Fy|.
xpbL=-xlLl(1/c+Fx)(1/c+Fy)×1c+Fx+Fy2-Fx-Fy2cos 2ϕ+xcc cos ϕ1+cFx-ycc sin ϕ1+cFy,
ypbL=xlLl(1/c+Fx)(1/c+Fy)Fx-Fy2sin 2ϕ+xcc sin ϕ1+cFx+ycc cos ϕ1+cFy,
xcc=rc cos ψ±(1/l+Fx)2crp cos ψ[(1/l+Fx)2 cos2 ψ+(1/l+Fy)2 sin2 ψ]1/2,
ycc=rc sin ψ±(1/l+Fy)2crp sin ψ[(1/l+Fx)2 cos2 ψ+(1/l+Fy)2 sin2 ψ]1/2,
γ=ϕ-arctan1/c+Fy1/c+Fxtan ϕ
1c+Fx+Fy2+Fx-Fy2cos 2ϕ.
Ei cos γi=1c+Fx+Fy2+Fx-Fy2cos 2(ϕ-ϕi),

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