Abstract

A compact and robust instrument for measuring the alignment of ocular surfaces has been designed and used in living eyes. It is based on recording Purkinje images (reflections of light at the ocular surfaces) at nine different angular fixations. A complete analysis on the causes of misalignments of Purkinje images and its relations with those physical variables to be measured (global eye tilt, lens decentration and lens tilt) is presented. A research prototype based on these ideas was built and tested in normal and pseudophakic eyes (after cataract surgery). The new analysis techniques, together with the semicircular extended source and multiple fixation tests that we used, are significant improvements towards a robust approach to measuring the misalignments of the ocular surfaces in vivo. This instrument will be of use in both basic studies of the eye’s optics and clinical ophthalmology.

© 2006 Optical Society of America

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References

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  1. J. Liang, B. Grimm, S. Goelz, and J.F. Bille. "Objective measurement of wave-aberration of the human eye with the use of a Hartmann Shack wave-front sensor," J. Opt. Soc. Am. A. 11, 1949-1957 (1994).
    [CrossRef]
  2. P.M. Prieto F. Vargas-Martín, S. Goelz, and P. Artal, "Analysis of the performance of the Hartmann-Shack sensor in the human eye," J. Opt. Soc. Am. A. 17, 1388-1398 (2000).
    [CrossRef]
  3. A. Guirao, and P. Artal, "Corneal wave aberration from videokeratography: accuracy and limitations of the procedure," J. Opt. Soc. Am. A. 17, 955-965 (2000).
    [CrossRef]
  4. P. Artal, A. Guirao, E. Berrio, and D.R. Williams, "Compensation of corneal aberrations by internal optics in the human eye," J. Vis. 1, 1-8 (2001).
    [CrossRef]
  5. P. Artal, E. Berrio, A. Guirao, and P. Piers, "Contribution of the cornea and internal surfaces to the change of ocular aberrations with age," J. Opt. Soc. Am. A. 19, 137-143 (2002).
    [CrossRef]
  6. Y . Le Grand, and S. G. El Hage, Physiological Optics (Springer Verlag, Berlin 1980).
  7. P. Phillips, J. Pérez-Emmanueilli, H.D. Rosskothen, and C.J. Koester "Measurement of intraocular lens decentration and tilt in vivo," J. Cataract. Refract. Surg. 14, 129-35 (1988).
    [PubMed]
  8. J.D. Auran, C.J. Koester, and A. Donn, "In vivo measurement of posterior chamber intraocular lens decentration and tilt," Arch. Ophthalmol. 108, 75-79 (1990).
    [CrossRef] [PubMed]
  9. J.C. Barry, K. Branman, and M.C.M. Dunne, "Catoptric properties of eyes with misaligned surfaces studied by exact ray tracing," Invest. Ophthalmol. Vis. Sci. 38, 1476-1484 (1997).
    [PubMed]
  10. J.C. Barry, M.C.M. Dunne, and T. Kirschkamp, "Phakometric measurement of ocular surface radius of curvature and alignment: evaluation of method with physical model eyes," Ophthalmic Physiol. Opt. 21,450-460 (2001).
    [CrossRef] [PubMed]
  11. T. Kirschkamp, M.C.M. Dunne, and J.C. Barry, "Phakometric measurement of ocular surface radii of curvature axial separation and alignment in relaxed and accommodated human eyes," Ophthalmic Physiol. Opt. 24, 65-73 (2004).
    [CrossRef] [PubMed]
  12. P. Rosales, and S. Marcos, "Phakometry and lens tilt and decentration using a custom-developed Purkinje imaging apparatus: validation and measurements," J. Opt. Soc. Am. A 23, 509-520 (2006).
    [CrossRef]
  13. D.L. Guyton, H. Uozato, and H.J. Wisnicki, "Rapid determination of intraocular lens tilt and decentration through the undilated pupil," Ophthalmology 97,1259-1264 (1990).
    [PubMed]
  14. R. Navarro, M. Ferro, P. Artal, and I. Miranda, "Modulation transfer-functions of eyes implanted with intraocular lenses," Appl. Opt. 32,6359-6367 (1993).
    [CrossRef] [PubMed]
  15. J. Tabernero, P. Piers, A. Benito, M. Redondo, and P. Artal, "Predicting the optical performance of eyes implanted with IOLs correcting spherical aberration," Inv.Ophthalmol.Vis.Sci. 47, 4651-4658 (2006).
    [CrossRef]
  16. J.E. Kelly, T. Mihashi, and H.C. Howland, "Compensation of corneal horizontal/vertical astigmatism, lateral coma, and spherical aberration by internal optics of the eye," J. Vis. 4,262-271 (2005).
  17. P. Artal, A. Benito, and J. Tabernero, "The human eye is an example of robust optical design," J. Vis. 6,1-7 (2006).
    [CrossRef] [PubMed]

2006

P. Rosales, and S. Marcos, "Phakometry and lens tilt and decentration using a custom-developed Purkinje imaging apparatus: validation and measurements," J. Opt. Soc. Am. A 23, 509-520 (2006).
[CrossRef]

J. Tabernero, P. Piers, A. Benito, M. Redondo, and P. Artal, "Predicting the optical performance of eyes implanted with IOLs correcting spherical aberration," Inv.Ophthalmol.Vis.Sci. 47, 4651-4658 (2006).
[CrossRef]

P. Artal, A. Benito, and J. Tabernero, "The human eye is an example of robust optical design," J. Vis. 6,1-7 (2006).
[CrossRef] [PubMed]

2005

J.E. Kelly, T. Mihashi, and H.C. Howland, "Compensation of corneal horizontal/vertical astigmatism, lateral coma, and spherical aberration by internal optics of the eye," J. Vis. 4,262-271 (2005).

2004

T. Kirschkamp, M.C.M. Dunne, and J.C. Barry, "Phakometric measurement of ocular surface radii of curvature axial separation and alignment in relaxed and accommodated human eyes," Ophthalmic Physiol. Opt. 24, 65-73 (2004).
[CrossRef] [PubMed]

2002

P. Artal, E. Berrio, A. Guirao, and P. Piers, "Contribution of the cornea and internal surfaces to the change of ocular aberrations with age," J. Opt. Soc. Am. A. 19, 137-143 (2002).
[CrossRef]

2001

P. Artal, A. Guirao, E. Berrio, and D.R. Williams, "Compensation of corneal aberrations by internal optics in the human eye," J. Vis. 1, 1-8 (2001).
[CrossRef]

J.C. Barry, M.C.M. Dunne, and T. Kirschkamp, "Phakometric measurement of ocular surface radius of curvature and alignment: evaluation of method with physical model eyes," Ophthalmic Physiol. Opt. 21,450-460 (2001).
[CrossRef] [PubMed]

2000

P.M. Prieto F. Vargas-Martín, S. Goelz, and P. Artal, "Analysis of the performance of the Hartmann-Shack sensor in the human eye," J. Opt. Soc. Am. A. 17, 1388-1398 (2000).
[CrossRef]

A. Guirao, and P. Artal, "Corneal wave aberration from videokeratography: accuracy and limitations of the procedure," J. Opt. Soc. Am. A. 17, 955-965 (2000).
[CrossRef]

1997

J.C. Barry, K. Branman, and M.C.M. Dunne, "Catoptric properties of eyes with misaligned surfaces studied by exact ray tracing," Invest. Ophthalmol. Vis. Sci. 38, 1476-1484 (1997).
[PubMed]

1994

J. Liang, B. Grimm, S. Goelz, and J.F. Bille. "Objective measurement of wave-aberration of the human eye with the use of a Hartmann Shack wave-front sensor," J. Opt. Soc. Am. A. 11, 1949-1957 (1994).
[CrossRef]

1993

1990

D.L. Guyton, H. Uozato, and H.J. Wisnicki, "Rapid determination of intraocular lens tilt and decentration through the undilated pupil," Ophthalmology 97,1259-1264 (1990).
[PubMed]

J.D. Auran, C.J. Koester, and A. Donn, "In vivo measurement of posterior chamber intraocular lens decentration and tilt," Arch. Ophthalmol. 108, 75-79 (1990).
[CrossRef] [PubMed]

1988

P. Phillips, J. Pérez-Emmanueilli, H.D. Rosskothen, and C.J. Koester "Measurement of intraocular lens decentration and tilt in vivo," J. Cataract. Refract. Surg. 14, 129-35 (1988).
[PubMed]

Artal, P.

J. Tabernero, P. Piers, A. Benito, M. Redondo, and P. Artal, "Predicting the optical performance of eyes implanted with IOLs correcting spherical aberration," Inv.Ophthalmol.Vis.Sci. 47, 4651-4658 (2006).
[CrossRef]

P. Artal, A. Benito, and J. Tabernero, "The human eye is an example of robust optical design," J. Vis. 6,1-7 (2006).
[CrossRef] [PubMed]

P. Artal, E. Berrio, A. Guirao, and P. Piers, "Contribution of the cornea and internal surfaces to the change of ocular aberrations with age," J. Opt. Soc. Am. A. 19, 137-143 (2002).
[CrossRef]

P. Artal, A. Guirao, E. Berrio, and D.R. Williams, "Compensation of corneal aberrations by internal optics in the human eye," J. Vis. 1, 1-8 (2001).
[CrossRef]

A. Guirao, and P. Artal, "Corneal wave aberration from videokeratography: accuracy and limitations of the procedure," J. Opt. Soc. Am. A. 17, 955-965 (2000).
[CrossRef]

R. Navarro, M. Ferro, P. Artal, and I. Miranda, "Modulation transfer-functions of eyes implanted with intraocular lenses," Appl. Opt. 32,6359-6367 (1993).
[CrossRef] [PubMed]

Auran, J.D.

J.D. Auran, C.J. Koester, and A. Donn, "In vivo measurement of posterior chamber intraocular lens decentration and tilt," Arch. Ophthalmol. 108, 75-79 (1990).
[CrossRef] [PubMed]

Barry, J.C.

T. Kirschkamp, M.C.M. Dunne, and J.C. Barry, "Phakometric measurement of ocular surface radii of curvature axial separation and alignment in relaxed and accommodated human eyes," Ophthalmic Physiol. Opt. 24, 65-73 (2004).
[CrossRef] [PubMed]

J.C. Barry, M.C.M. Dunne, and T. Kirschkamp, "Phakometric measurement of ocular surface radius of curvature and alignment: evaluation of method with physical model eyes," Ophthalmic Physiol. Opt. 21,450-460 (2001).
[CrossRef] [PubMed]

J.C. Barry, K. Branman, and M.C.M. Dunne, "Catoptric properties of eyes with misaligned surfaces studied by exact ray tracing," Invest. Ophthalmol. Vis. Sci. 38, 1476-1484 (1997).
[PubMed]

Benito, A.

J. Tabernero, P. Piers, A. Benito, M. Redondo, and P. Artal, "Predicting the optical performance of eyes implanted with IOLs correcting spherical aberration," Inv.Ophthalmol.Vis.Sci. 47, 4651-4658 (2006).
[CrossRef]

P. Artal, A. Benito, and J. Tabernero, "The human eye is an example of robust optical design," J. Vis. 6,1-7 (2006).
[CrossRef] [PubMed]

Berrio, E.

P. Artal, E. Berrio, A. Guirao, and P. Piers, "Contribution of the cornea and internal surfaces to the change of ocular aberrations with age," J. Opt. Soc. Am. A. 19, 137-143 (2002).
[CrossRef]

P. Artal, A. Guirao, E. Berrio, and D.R. Williams, "Compensation of corneal aberrations by internal optics in the human eye," J. Vis. 1, 1-8 (2001).
[CrossRef]

Bille, J.F.

J. Liang, B. Grimm, S. Goelz, and J.F. Bille. "Objective measurement of wave-aberration of the human eye with the use of a Hartmann Shack wave-front sensor," J. Opt. Soc. Am. A. 11, 1949-1957 (1994).
[CrossRef]

Branman, K.

J.C. Barry, K. Branman, and M.C.M. Dunne, "Catoptric properties of eyes with misaligned surfaces studied by exact ray tracing," Invest. Ophthalmol. Vis. Sci. 38, 1476-1484 (1997).
[PubMed]

Donn, A.

J.D. Auran, C.J. Koester, and A. Donn, "In vivo measurement of posterior chamber intraocular lens decentration and tilt," Arch. Ophthalmol. 108, 75-79 (1990).
[CrossRef] [PubMed]

Dunne, M.C.M.

T. Kirschkamp, M.C.M. Dunne, and J.C. Barry, "Phakometric measurement of ocular surface radii of curvature axial separation and alignment in relaxed and accommodated human eyes," Ophthalmic Physiol. Opt. 24, 65-73 (2004).
[CrossRef] [PubMed]

J.C. Barry, M.C.M. Dunne, and T. Kirschkamp, "Phakometric measurement of ocular surface radius of curvature and alignment: evaluation of method with physical model eyes," Ophthalmic Physiol. Opt. 21,450-460 (2001).
[CrossRef] [PubMed]

J.C. Barry, K. Branman, and M.C.M. Dunne, "Catoptric properties of eyes with misaligned surfaces studied by exact ray tracing," Invest. Ophthalmol. Vis. Sci. 38, 1476-1484 (1997).
[PubMed]

Ferro, M.

Goelz, S.

J. Liang, B. Grimm, S. Goelz, and J.F. Bille. "Objective measurement of wave-aberration of the human eye with the use of a Hartmann Shack wave-front sensor," J. Opt. Soc. Am. A. 11, 1949-1957 (1994).
[CrossRef]

Grimm, B.

J. Liang, B. Grimm, S. Goelz, and J.F. Bille. "Objective measurement of wave-aberration of the human eye with the use of a Hartmann Shack wave-front sensor," J. Opt. Soc. Am. A. 11, 1949-1957 (1994).
[CrossRef]

Guirao, A.

P. Artal, E. Berrio, A. Guirao, and P. Piers, "Contribution of the cornea and internal surfaces to the change of ocular aberrations with age," J. Opt. Soc. Am. A. 19, 137-143 (2002).
[CrossRef]

P. Artal, A. Guirao, E. Berrio, and D.R. Williams, "Compensation of corneal aberrations by internal optics in the human eye," J. Vis. 1, 1-8 (2001).
[CrossRef]

A. Guirao, and P. Artal, "Corneal wave aberration from videokeratography: accuracy and limitations of the procedure," J. Opt. Soc. Am. A. 17, 955-965 (2000).
[CrossRef]

Guyton, D.L.

D.L. Guyton, H. Uozato, and H.J. Wisnicki, "Rapid determination of intraocular lens tilt and decentration through the undilated pupil," Ophthalmology 97,1259-1264 (1990).
[PubMed]

Howland, H.C.

J.E. Kelly, T. Mihashi, and H.C. Howland, "Compensation of corneal horizontal/vertical astigmatism, lateral coma, and spherical aberration by internal optics of the eye," J. Vis. 4,262-271 (2005).

Kelly, J.E.

J.E. Kelly, T. Mihashi, and H.C. Howland, "Compensation of corneal horizontal/vertical astigmatism, lateral coma, and spherical aberration by internal optics of the eye," J. Vis. 4,262-271 (2005).

Kirschkamp, T.

T. Kirschkamp, M.C.M. Dunne, and J.C. Barry, "Phakometric measurement of ocular surface radii of curvature axial separation and alignment in relaxed and accommodated human eyes," Ophthalmic Physiol. Opt. 24, 65-73 (2004).
[CrossRef] [PubMed]

J.C. Barry, M.C.M. Dunne, and T. Kirschkamp, "Phakometric measurement of ocular surface radius of curvature and alignment: evaluation of method with physical model eyes," Ophthalmic Physiol. Opt. 21,450-460 (2001).
[CrossRef] [PubMed]

Koester, C.J.

J.D. Auran, C.J. Koester, and A. Donn, "In vivo measurement of posterior chamber intraocular lens decentration and tilt," Arch. Ophthalmol. 108, 75-79 (1990).
[CrossRef] [PubMed]

P. Phillips, J. Pérez-Emmanueilli, H.D. Rosskothen, and C.J. Koester "Measurement of intraocular lens decentration and tilt in vivo," J. Cataract. Refract. Surg. 14, 129-35 (1988).
[PubMed]

Liang, J.

J. Liang, B. Grimm, S. Goelz, and J.F. Bille. "Objective measurement of wave-aberration of the human eye with the use of a Hartmann Shack wave-front sensor," J. Opt. Soc. Am. A. 11, 1949-1957 (1994).
[CrossRef]

Marcos, S.

Mihashi, T.

J.E. Kelly, T. Mihashi, and H.C. Howland, "Compensation of corneal horizontal/vertical astigmatism, lateral coma, and spherical aberration by internal optics of the eye," J. Vis. 4,262-271 (2005).

Miranda, I.

Navarro, R.

Pérez-Emmanueilli, J.

P. Phillips, J. Pérez-Emmanueilli, H.D. Rosskothen, and C.J. Koester "Measurement of intraocular lens decentration and tilt in vivo," J. Cataract. Refract. Surg. 14, 129-35 (1988).
[PubMed]

Phillips, P.

P. Phillips, J. Pérez-Emmanueilli, H.D. Rosskothen, and C.J. Koester "Measurement of intraocular lens decentration and tilt in vivo," J. Cataract. Refract. Surg. 14, 129-35 (1988).
[PubMed]

Piers, P.

J. Tabernero, P. Piers, A. Benito, M. Redondo, and P. Artal, "Predicting the optical performance of eyes implanted with IOLs correcting spherical aberration," Inv.Ophthalmol.Vis.Sci. 47, 4651-4658 (2006).
[CrossRef]

P. Artal, E. Berrio, A. Guirao, and P. Piers, "Contribution of the cornea and internal surfaces to the change of ocular aberrations with age," J. Opt. Soc. Am. A. 19, 137-143 (2002).
[CrossRef]

Redondo, M.

J. Tabernero, P. Piers, A. Benito, M. Redondo, and P. Artal, "Predicting the optical performance of eyes implanted with IOLs correcting spherical aberration," Inv.Ophthalmol.Vis.Sci. 47, 4651-4658 (2006).
[CrossRef]

Rosales, P.

Rosskothen, H.D.

P. Phillips, J. Pérez-Emmanueilli, H.D. Rosskothen, and C.J. Koester "Measurement of intraocular lens decentration and tilt in vivo," J. Cataract. Refract. Surg. 14, 129-35 (1988).
[PubMed]

Tabernero, J.

J. Tabernero, P. Piers, A. Benito, M. Redondo, and P. Artal, "Predicting the optical performance of eyes implanted with IOLs correcting spherical aberration," Inv.Ophthalmol.Vis.Sci. 47, 4651-4658 (2006).
[CrossRef]

P. Artal, A. Benito, and J. Tabernero, "The human eye is an example of robust optical design," J. Vis. 6,1-7 (2006).
[CrossRef] [PubMed]

Uozato, H.

D.L. Guyton, H. Uozato, and H.J. Wisnicki, "Rapid determination of intraocular lens tilt and decentration through the undilated pupil," Ophthalmology 97,1259-1264 (1990).
[PubMed]

Williams, D.R.

P. Artal, A. Guirao, E. Berrio, and D.R. Williams, "Compensation of corneal aberrations by internal optics in the human eye," J. Vis. 1, 1-8 (2001).
[CrossRef]

Wisnicki, H.J.

D.L. Guyton, H. Uozato, and H.J. Wisnicki, "Rapid determination of intraocular lens tilt and decentration through the undilated pupil," Ophthalmology 97,1259-1264 (1990).
[PubMed]

Appl. Opt.

Arch. Ophthalmol.

J.D. Auran, C.J. Koester, and A. Donn, "In vivo measurement of posterior chamber intraocular lens decentration and tilt," Arch. Ophthalmol. 108, 75-79 (1990).
[CrossRef] [PubMed]

Inv.Ophthalmol.Vis.Sci.

J. Tabernero, P. Piers, A. Benito, M. Redondo, and P. Artal, "Predicting the optical performance of eyes implanted with IOLs correcting spherical aberration," Inv.Ophthalmol.Vis.Sci. 47, 4651-4658 (2006).
[CrossRef]

Invest. Ophthalmol. Vis. Sci.

J.C. Barry, K. Branman, and M.C.M. Dunne, "Catoptric properties of eyes with misaligned surfaces studied by exact ray tracing," Invest. Ophthalmol. Vis. Sci. 38, 1476-1484 (1997).
[PubMed]

J. Cataract. Refract. Surg.

P. Phillips, J. Pérez-Emmanueilli, H.D. Rosskothen, and C.J. Koester "Measurement of intraocular lens decentration and tilt in vivo," J. Cataract. Refract. Surg. 14, 129-35 (1988).
[PubMed]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

P. Artal, E. Berrio, A. Guirao, and P. Piers, "Contribution of the cornea and internal surfaces to the change of ocular aberrations with age," J. Opt. Soc. Am. A. 19, 137-143 (2002).
[CrossRef]

J. Liang, B. Grimm, S. Goelz, and J.F. Bille. "Objective measurement of wave-aberration of the human eye with the use of a Hartmann Shack wave-front sensor," J. Opt. Soc. Am. A. 11, 1949-1957 (1994).
[CrossRef]

P.M. Prieto F. Vargas-Martín, S. Goelz, and P. Artal, "Analysis of the performance of the Hartmann-Shack sensor in the human eye," J. Opt. Soc. Am. A. 17, 1388-1398 (2000).
[CrossRef]

A. Guirao, and P. Artal, "Corneal wave aberration from videokeratography: accuracy and limitations of the procedure," J. Opt. Soc. Am. A. 17, 955-965 (2000).
[CrossRef]

J. Vis.

P. Artal, A. Guirao, E. Berrio, and D.R. Williams, "Compensation of corneal aberrations by internal optics in the human eye," J. Vis. 1, 1-8 (2001).
[CrossRef]

J.E. Kelly, T. Mihashi, and H.C. Howland, "Compensation of corneal horizontal/vertical astigmatism, lateral coma, and spherical aberration by internal optics of the eye," J. Vis. 4,262-271 (2005).

P. Artal, A. Benito, and J. Tabernero, "The human eye is an example of robust optical design," J. Vis. 6,1-7 (2006).
[CrossRef] [PubMed]

Ophthalmic Physiol. Opt.

J.C. Barry, M.C.M. Dunne, and T. Kirschkamp, "Phakometric measurement of ocular surface radius of curvature and alignment: evaluation of method with physical model eyes," Ophthalmic Physiol. Opt. 21,450-460 (2001).
[CrossRef] [PubMed]

T. Kirschkamp, M.C.M. Dunne, and J.C. Barry, "Phakometric measurement of ocular surface radii of curvature axial separation and alignment in relaxed and accommodated human eyes," Ophthalmic Physiol. Opt. 24, 65-73 (2004).
[CrossRef] [PubMed]

Ophthalmology

D.L. Guyton, H. Uozato, and H.J. Wisnicki, "Rapid determination of intraocular lens tilt and decentration through the undilated pupil," Ophthalmology 97,1259-1264 (1990).
[PubMed]

Other

Y . Le Grand, and S. G. El Hage, Physiological Optics (Springer Verlag, Berlin 1980).

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Figures (13)

Fig. 1.
Fig. 1.

Pupil edge and Purkinje images of a semicircular source (array of LEDs) in a perfectly aligned model eye.

Fig. 2.
Fig. 2.

Global eye tilt, lens tilt and lens decentration cause Purkinje images generated by a semicircular source to be misaligned.

Fig. 3.
Fig. 3.

Schematic representation of angle kappa. It is important to note that due to angle kappa the lens is systematically tilted with respect to the line of sight.

Fig. 4.
Fig. 4.

On the left column, Purkinje images III and IV generated by a point source are represented for different lens tilts (2, 4 and 8 degrees). On the right column, the same Purkinje images are represented after a global eye tilt that aligns both.

Fig. 5.
Fig. 5.

The global eye tilt required to align PIII and PIV as a function of lens tilt (on the left) and lens decentration (on the right).

Fig. 6.
Fig. 6.

Photographs of the experimental prototype.

Fig. 7.
Fig. 7.

Simulations of Purkinje images in an eye model with nominal values of tilt and decentration for different eye rotations simulating fixations at five different angular positions.

Fig. 8.
Fig. 8.

The position of each Purkinje image with respect to the pupil is plotted versus angular rotations in both horizontal and vertical directions.

Fig. 9.
Fig. 9.

Movies of the measurement sessions from the two pseudophakic eyes included in this work. The positions of the Purkinje images change with the fixation target (upper left corner in each movie). [Media 1] [Media 2]

Fig. 10.
Fig. 10.

The positions of each Purkinje image with respect to the entrance pupil center versus angular fixation for the first pseudophakic eye in both horizontal and vertical direction.

Fig. 11.
Fig. 11.

Visualizing PI, PIII and PIV for the two normal eyes included in this study. The third column is a schematic draw of the position of each Purkinje image.

Fig. 12.
Fig. 12.

The positions of each Purkinje image with respect to the entrance pupil center versus angular fixation for the first normal eye included in this study in both horizontal and vertical direction.

Fig. 13.
Fig. 13.

Uncertainties in the measurement of each component of tilt, decentration and kappa angle for the four eyes included in this study.

Equations (13)

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

β AL PIII + PIV = β Globa Rot . + β Lens Tilt + β Lens Dec .
β AL PIII + PIV = + β Lens Tilt ( LoS ) + β Lens Dec .
β AL PIII + PIV = + β Global _ Rot . + A 1 Tilt + A 2 Dec
Y = a 1 X + b 1
Y = a 2 X + b 2
X C = ( b 1 b 2 a 1 a 2 )
Y C = a 1 X C + b 1
Δ X C = X C a 1 Δ a 1 + X C a 2 Δ a 2 + X C b 1 Δ b 1 + X C b 2 Δ b 2
Δ Y C = Y C a 1 Δ a 1 + Y C b 1 Δ b 1 + Y C X C Δ X C
Δ X C = 1 a 1 a 2 [ X C ( Δ a 1 + Δ a 2 ) + ( Δ b 1 + Δ b 2 ) ]
Δ Y C = X C Δ a 1 + Δ X C a 1 + Δ b 1
X C = b a
Δ X C X C = Δ a a + Δ b b

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