Abstract

A procedure to calculate the wave aberration of the human cornea from its surface shape measured by videokeratography is presented. The wave aberration was calculated as the difference in optical path between the marginal rays and the chief ray refracted at the surface, for both on- and off-axis objects. The corneal shape elevation map was obtained from videokeratography and fitted to a Zernike polynomial expansion through a Gram–Schmidt orthogonalization. The wave aberration was obtained also as a Zernike polynomial representation. The accuracy of the procedure was analyzed. For calibrated reference surface elevations, a root-mean-square error (RMSE) of 1 to 2 μm for an aperture 4–6 mm in diameter was obtained, and the RMSE associated with the experimental errors and with the fitting method was 0.2 μm. The procedure permits estimation of the corneal wave aberration from videokeratoscopic data with an accuracy of 0.05–0.2 μm for a pupil 4–6 mm in diameter, rendering the method adequate for many applications.

© 2000 Optical Society of America

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References

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    [CrossRef]
  2. H. L. Liou, N. Brennan, “Anatomically accurate, finite model eye for optical modeling,” J. Opt. Soc. Am. A 14, 1684–1695 (1997).
    [CrossRef]
  3. J. Schwiegerling, J. E. Greivenkamp, “Keratoconus detection based on videokeratoscopic height data,” Optom. Vision Sci. 73, 721–728 (1996).
    [CrossRef]
  4. J. Schwiegerling, “Cone dimensions in keratoconus using Zernike polynomials,” Optom. Vision Sci. 74, 963–969 (1997).
    [CrossRef]
  5. R. D. Applegate, G. Hilmantel, H. C. Howland, “Corneal aberrations increase with the magnitude of radial keratotomy refractive correction,” Optom. Vision Sci. 73, 585–589 (1996).
    [CrossRef]
  6. K. M. Oliver, G. P. Hemenger, M. C. Corbett, D. P. S. Obrant, S. Verna, J. Marshall, A. Tomlinson, “Corneal optical aberrations induced by photorefractive keratoctomy,” J. Refrac. Surg. 13, 246–254 (1997).
  7. S. G. El Hage, F. Berny, “Contribution of the crystalline lens to the spherical aberration of the eye,” J. Opt. Soc. Am. 63, 205–211 (1973).
    [CrossRef] [PubMed]
  8. A. Tomlinson, R. P. Hemenger, R. Garriott, “Method for estimating the spherical aberration of the human crystalline lens in vivo,” Invest. Ophthalmol. Visual Sci. 34, 621–629 (1993).
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    [CrossRef]
  10. T. C. A. Jenkins, “Aberrations of the eye and their effects on vision: part 1,” Br. J. Physiol. Opt. 20, 59–91 (1963).
    [PubMed]
  11. R. P. Hemenger, A. Tomlinson, K. Oliver, “Optical consequences of asymmetries in normal corneas,” Ophthalmic Physiol. Opt. 16, 124–129 (1996).
    [CrossRef] [PubMed]
  12. T. W. Raasch, “Corneal topography and irregular astigmatism,” Optom. Vision Sci. 72, 809–815 (1995).
    [CrossRef]
  13. R. A. Applegate, R. Nuñez, J. Buettner, H. C. Howland, “How accurately can videokeratographic systems measure surface elevation?” Optom. Vision Sci. 72, 785–792 (1995).
    [CrossRef]
  14. J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Comparison of three videokeratoscopes in measurements of toric tests surfaces,” J. Refract. Surg. 12, 229–239 (1996).
    [PubMed]
  15. R. L. Schultze, “Accuracy of corneal elevation with four corneal topography systems,” J. Refract. Surg. 14, 100–104 (1998).
    [PubMed]
  16. J. Schwiegerling, J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vision Sci. 74, 906–916 (1997).
    [CrossRef]
  17. V. N. Mahajan, Aberration Theory Made Simple (SPIE Optical Engineering Press, Bellingham, Wash., 1991).
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    [CrossRef]
  23. C-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987), Vol. X, Chap. 4. (Note that the ordering of the polynomials in the list is not universally accepted; here we used a different ordering.)
  24. D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
    [CrossRef]
  25. S. N. Bezdid’ko, “Calculation of the Strehl coefficient and determination of the best-focus plane in the case of polychromatic light,” Sov. J. Opt. Technol. 42, 514–516 (1975).
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  27. R. K. Tyson, “Conversion of Zernike aberration coefficients to Seidel and higher-order power-series aberration coefficients,” Opt. Lett. 7, 262–264 (1982).
    [CrossRef] [PubMed]
  28. C. Campbell, “Reconstruction of the corneal shape with the MasterVue corneal topography system,” Optom. Vision Sci. 74, 899–905 (1997).
    [CrossRef]
  29. By comparing Eqs. (9) and (5), we can calculate the curvature and asphericity from the Zernike coefficients as R=r022(23a4-65a9-307a22+⋯ ),K2=8R3r04 (65a9-307a22+⋯ ), where r0 is the maximum radial extent of the surface.
  30. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  31. R. P. Hemenger, A. Tomlinson, K. Oliver, “Corneal optics from videokeratographs,” Ophthalmic Physiol. Opt. 15, 63–68 (1995).
    [CrossRef] [PubMed]
  32. R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, S309 (1995).
  33. I. Iglesias, E. Berrio, P. Artal, “Estimates of the ocular wave aberration from pairs of double-pass retinal images,” J. Opt. Soc. Am. A 15, 2466–2476 (1998).
    [CrossRef]

1998 (3)

1997 (5)

H. L. Liou, N. Brennan, “Anatomically accurate, finite model eye for optical modeling,” J. Opt. Soc. Am. A 14, 1684–1695 (1997).
[CrossRef]

C. Campbell, “Reconstruction of the corneal shape with the MasterVue corneal topography system,” Optom. Vision Sci. 74, 899–905 (1997).
[CrossRef]

J. Schwiegerling, “Cone dimensions in keratoconus using Zernike polynomials,” Optom. Vision Sci. 74, 963–969 (1997).
[CrossRef]

K. M. Oliver, G. P. Hemenger, M. C. Corbett, D. P. S. Obrant, S. Verna, J. Marshall, A. Tomlinson, “Corneal optical aberrations induced by photorefractive keratoctomy,” J. Refrac. Surg. 13, 246–254 (1997).

J. Schwiegerling, J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vision Sci. 74, 906–916 (1997).
[CrossRef]

1996 (4)

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Comparison of three videokeratoscopes in measurements of toric tests surfaces,” J. Refract. Surg. 12, 229–239 (1996).
[PubMed]

J. Schwiegerling, J. E. Greivenkamp, “Keratoconus detection based on videokeratoscopic height data,” Optom. Vision Sci. 73, 721–728 (1996).
[CrossRef]

R. P. Hemenger, A. Tomlinson, K. Oliver, “Optical consequences of asymmetries in normal corneas,” Ophthalmic Physiol. Opt. 16, 124–129 (1996).
[CrossRef] [PubMed]

R. D. Applegate, G. Hilmantel, H. C. Howland, “Corneal aberrations increase with the magnitude of radial keratotomy refractive correction,” Optom. Vision Sci. 73, 585–589 (1996).
[CrossRef]

1995 (6)

R. P. Hemenger, A. Tomlinson, K. Oliver, “Corneal optics from videokeratographs,” Ophthalmic Physiol. Opt. 15, 63–68 (1995).
[CrossRef] [PubMed]

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, S309 (1995).

T. W. Raasch, “Corneal topography and irregular astigmatism,” Optom. Vision Sci. 72, 809–815 (1995).
[CrossRef]

R. A. Applegate, R. Nuñez, J. Buettner, H. C. Howland, “How accurately can videokeratographic systems measure surface elevation?” Optom. Vision Sci. 72, 785–792 (1995).
[CrossRef]

T. O. Salmon, D. G. Horner, “Comparison of elevation, curvature, and power descriptors for corneal topographic mapping,” Optom. Vision Sci. 72, 800–808 (1995).
[CrossRef]

J. Schwiegerling, J. E. Greivenkamp, J. M. Miller, “Representation of videokeratoscopic height data with Zernike polynomials,” J. Opt. Soc. Am. A 12, 2105–2113 (1995).
[CrossRef]

1993 (1)

A. Tomlinson, R. P. Hemenger, R. Garriott, “Method for estimating the spherical aberration of the human crystalline lens in vivo,” Invest. Ophthalmol. Visual Sci. 34, 621–629 (1993).

1990 (1)

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

1983 (1)

1982 (2)

1980 (1)

1975 (1)

S. N. Bezdid’ko, “Calculation of the Strehl coefficient and determination of the best-focus plane in the case of polychromatic light,” Sov. J. Opt. Technol. 42, 514–516 (1975).

1973 (1)

1963 (1)

T. C. A. Jenkins, “Aberrations of the eye and their effects on vision: part 1,” Br. J. Physiol. Opt. 20, 59–91 (1963).
[PubMed]

Applegate, R. A.

R. A. Applegate, R. Nuñez, J. Buettner, H. C. Howland, “How accurately can videokeratographic systems measure surface elevation?” Optom. Vision Sci. 72, 785–792 (1995).
[CrossRef]

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, S309 (1995).

Applegate, R. D.

R. D. Applegate, G. Hilmantel, H. C. Howland, “Corneal aberrations increase with the magnitude of radial keratotomy refractive correction,” Optom. Vision Sci. 73, 585–589 (1996).
[CrossRef]

Artal, P.

Berny, F.

Berrio, E.

Bezdid’ko, S. N.

S. N. Bezdid’ko, “Calculation of the Strehl coefficient and determination of the best-focus plane in the case of polychromatic light,” Sov. J. Opt. Technol. 42, 514–516 (1975).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1985).

Brennan, N.

Buettner, J.

R. A. Applegate, R. Nuñez, J. Buettner, H. C. Howland, “How accurately can videokeratographic systems measure surface elevation?” Optom. Vision Sci. 72, 785–792 (1995).
[CrossRef]

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, S309 (1995).

Campbell, C.

C. Campbell, “Reconstruction of the corneal shape with the MasterVue corneal topography system,” Optom. Vision Sci. 74, 899–905 (1997).
[CrossRef]

Carney, L. G.

P. M. Kiely, G. Smith, L. G. Carney, “The mean shape of the human cornea,” Opt. Acta 29, 1027–1040 (1982).
[CrossRef]

Carpio-Valadéz, J. M.

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Conforti, G.

Corbett, M. C.

K. M. Oliver, G. P. Hemenger, M. C. Corbett, D. P. S. Obrant, S. Verna, J. Marshall, A. Tomlinson, “Corneal optical aberrations induced by photorefractive keratoctomy,” J. Refrac. Surg. 13, 246–254 (1997).

Cottingham, A. J.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, S309 (1995).

El Hage, S. G.

Garriott, R.

A. Tomlinson, R. P. Hemenger, R. Garriott, “Method for estimating the spherical aberration of the human crystalline lens in vivo,” Invest. Ophthalmol. Visual Sci. 34, 621–629 (1993).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Greivenkamp, J. E.

J. Schwiegerling, J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vision Sci. 74, 906–916 (1997).
[CrossRef]

J. Schwiegerling, J. E. Greivenkamp, “Keratoconus detection based on videokeratoscopic height data,” Optom. Vision Sci. 73, 721–728 (1996).
[CrossRef]

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Comparison of three videokeratoscopes in measurements of toric tests surfaces,” J. Refract. Surg. 12, 229–239 (1996).
[PubMed]

J. Schwiegerling, J. E. Greivenkamp, J. M. Miller, “Representation of videokeratoscopic height data with Zernike polynomials,” J. Opt. Soc. Am. A 12, 2105–2113 (1995).
[CrossRef]

Guirao, A.

Hemenger, G. P.

K. M. Oliver, G. P. Hemenger, M. C. Corbett, D. P. S. Obrant, S. Verna, J. Marshall, A. Tomlinson, “Corneal optical aberrations induced by photorefractive keratoctomy,” J. Refrac. Surg. 13, 246–254 (1997).

Hemenger, R. P.

R. P. Hemenger, A. Tomlinson, K. Oliver, “Optical consequences of asymmetries in normal corneas,” Ophthalmic Physiol. Opt. 16, 124–129 (1996).
[CrossRef] [PubMed]

R. P. Hemenger, A. Tomlinson, K. Oliver, “Corneal optics from videokeratographs,” Ophthalmic Physiol. Opt. 15, 63–68 (1995).
[CrossRef] [PubMed]

A. Tomlinson, R. P. Hemenger, R. Garriott, “Method for estimating the spherical aberration of the human crystalline lens in vivo,” Invest. Ophthalmol. Visual Sci. 34, 621–629 (1993).

Hilmantel, G.

R. D. Applegate, G. Hilmantel, H. C. Howland, “Corneal aberrations increase with the magnitude of radial keratotomy refractive correction,” Optom. Vision Sci. 73, 585–589 (1996).
[CrossRef]

Horner, D. G.

T. O. Salmon, D. G. Horner, “Comparison of elevation, curvature, and power descriptors for corneal topographic mapping,” Optom. Vision Sci. 72, 800–808 (1995).
[CrossRef]

Howland, H. C.

R. D. Applegate, G. Hilmantel, H. C. Howland, “Corneal aberrations increase with the magnitude of radial keratotomy refractive correction,” Optom. Vision Sci. 73, 585–589 (1996).
[CrossRef]

R. A. Applegate, R. Nuñez, J. Buettner, H. C. Howland, “How accurately can videokeratographic systems measure surface elevation?” Optom. Vision Sci. 72, 785–792 (1995).
[CrossRef]

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, S309 (1995).

Iglesias, I.

Jenkins, T. C. A.

T. C. A. Jenkins, “Aberrations of the eye and their effects on vision: part 1,” Br. J. Physiol. Opt. 20, 59–91 (1963).
[PubMed]

Kiely, P. M.

P. M. Kiely, G. Smith, L. G. Carney, “The mean shape of the human cornea,” Opt. Acta 29, 1027–1040 (1982).
[CrossRef]

Kim, C-J.

C-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987), Vol. X, Chap. 4. (Note that the ordering of the polynomials in the list is not universally accepted; here we used a different ordering.)

Liou, H. L.

Lowman, A. E.

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Comparison of three videokeratoscopes in measurements of toric tests surfaces,” J. Refract. Surg. 12, 229–239 (1996).
[PubMed]

Mahajan, V. N.

V. N. Mahajan, Aberration Theory Made Simple (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

Malacara, D.

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Marshall, J.

K. M. Oliver, G. P. Hemenger, M. C. Corbett, D. P. S. Obrant, S. Verna, J. Marshall, A. Tomlinson, “Corneal optical aberrations induced by photorefractive keratoctomy,” J. Refrac. Surg. 13, 246–254 (1997).

Mellinger, M. D.

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Comparison of three videokeratoscopes in measurements of toric tests surfaces,” J. Refract. Surg. 12, 229–239 (1996).
[PubMed]

Miller, J. M.

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Comparison of three videokeratoscopes in measurements of toric tests surfaces,” J. Refract. Surg. 12, 229–239 (1996).
[PubMed]

J. Schwiegerling, J. E. Greivenkamp, J. M. Miller, “Representation of videokeratoscopic height data with Zernike polynomials,” J. Opt. Soc. Am. A 12, 2105–2113 (1995).
[CrossRef]

Nuñez, R.

R. A. Applegate, R. Nuñez, J. Buettner, H. C. Howland, “How accurately can videokeratographic systems measure surface elevation?” Optom. Vision Sci. 72, 785–792 (1995).
[CrossRef]

Obrant, D. P. S.

K. M. Oliver, G. P. Hemenger, M. C. Corbett, D. P. S. Obrant, S. Verna, J. Marshall, A. Tomlinson, “Corneal optical aberrations induced by photorefractive keratoctomy,” J. Refrac. Surg. 13, 246–254 (1997).

Oliver, K.

R. P. Hemenger, A. Tomlinson, K. Oliver, “Optical consequences of asymmetries in normal corneas,” Ophthalmic Physiol. Opt. 16, 124–129 (1996).
[CrossRef] [PubMed]

R. P. Hemenger, A. Tomlinson, K. Oliver, “Corneal optics from videokeratographs,” Ophthalmic Physiol. Opt. 15, 63–68 (1995).
[CrossRef] [PubMed]

Oliver, K. M.

K. M. Oliver, G. P. Hemenger, M. C. Corbett, D. P. S. Obrant, S. Verna, J. Marshall, A. Tomlinson, “Corneal optical aberrations induced by photorefractive keratoctomy,” J. Refrac. Surg. 13, 246–254 (1997).

Raasch, T. W.

T. W. Raasch, “Corneal topography and irregular astigmatism,” Optom. Vision Sci. 72, 809–815 (1995).
[CrossRef]

Salmon, T. O.

T. O. Salmon, D. G. Horner, “Comparison of elevation, curvature, and power descriptors for corneal topographic mapping,” Optom. Vision Sci. 72, 800–808 (1995).
[CrossRef]

Sánchez-Mondragón, J. J.

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Schultze, R. L.

R. L. Schultze, “Accuracy of corneal elevation with four corneal topography systems,” J. Refract. Surg. 14, 100–104 (1998).
[PubMed]

Schwiegerling, J.

J. Schwiegerling, “Cone dimensions in keratoconus using Zernike polynomials,” Optom. Vision Sci. 74, 963–969 (1997).
[CrossRef]

J. Schwiegerling, J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vision Sci. 74, 906–916 (1997).
[CrossRef]

J. Schwiegerling, J. E. Greivenkamp, “Keratoconus detection based on videokeratoscopic height data,” Optom. Vision Sci. 73, 721–728 (1996).
[CrossRef]

J. Schwiegerling, J. E. Greivenkamp, J. M. Miller, “Representation of videokeratoscopic height data with Zernike polynomials,” J. Opt. Soc. Am. A 12, 2105–2113 (1995).
[CrossRef]

Schwiegerling, J. T.

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Comparison of three videokeratoscopes in measurements of toric tests surfaces,” J. Refract. Surg. 12, 229–239 (1996).
[PubMed]

Shannon, R. R.

C-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987), Vol. X, Chap. 4. (Note that the ordering of the polynomials in the list is not universally accepted; here we used a different ordering.)

Sharp, R. P.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, S309 (1995).

Silva, D. E.

Smith, G.

P. M. Kiely, G. Smith, L. G. Carney, “The mean shape of the human cornea,” Opt. Acta 29, 1027–1040 (1982).
[CrossRef]

Snyder, R. W.

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Comparison of three videokeratoscopes in measurements of toric tests surfaces,” J. Refract. Surg. 12, 229–239 (1996).
[PubMed]

Tomlinson, A.

K. M. Oliver, G. P. Hemenger, M. C. Corbett, D. P. S. Obrant, S. Verna, J. Marshall, A. Tomlinson, “Corneal optical aberrations induced by photorefractive keratoctomy,” J. Refrac. Surg. 13, 246–254 (1997).

R. P. Hemenger, A. Tomlinson, K. Oliver, “Optical consequences of asymmetries in normal corneas,” Ophthalmic Physiol. Opt. 16, 124–129 (1996).
[CrossRef] [PubMed]

R. P. Hemenger, A. Tomlinson, K. Oliver, “Corneal optics from videokeratographs,” Ophthalmic Physiol. Opt. 15, 63–68 (1995).
[CrossRef] [PubMed]

A. Tomlinson, R. P. Hemenger, R. Garriott, “Method for estimating the spherical aberration of the human crystalline lens in vivo,” Invest. Ophthalmol. Visual Sci. 34, 621–629 (1993).

Tyson, R. K.

Verna, S.

K. M. Oliver, G. P. Hemenger, M. C. Corbett, D. P. S. Obrant, S. Verna, J. Marshall, A. Tomlinson, “Corneal optical aberrations induced by photorefractive keratoctomy,” J. Refrac. Surg. 13, 246–254 (1997).

Wang, J. Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1985).

Yee, R. W.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, S309 (1995).

Appl. Opt. (1)

Br. J. Physiol. Opt. (1)

T. C. A. Jenkins, “Aberrations of the eye and their effects on vision: part 1,” Br. J. Physiol. Opt. 20, 59–91 (1963).
[PubMed]

Invest. Ophthalmol. Visual Sci. (2)

A. Tomlinson, R. P. Hemenger, R. Garriott, “Method for estimating the spherical aberration of the human crystalline lens in vivo,” Invest. Ophthalmol. Visual Sci. 34, 621–629 (1993).

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, S309 (1995).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Refrac. Surg. (1)

K. M. Oliver, G. P. Hemenger, M. C. Corbett, D. P. S. Obrant, S. Verna, J. Marshall, A. Tomlinson, “Corneal optical aberrations induced by photorefractive keratoctomy,” J. Refrac. Surg. 13, 246–254 (1997).

J. Refract. Surg. (2)

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Other (6)

By comparing Eqs. (9) and (5), we can calculate the curvature and asphericity from the Zernike coefficients as R=r022(23a4-65a9-307a22+⋯ ),K2=8R3r04 (65a9-307a22+⋯ ), where r0 is the maximum radial extent of the surface.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

V. N. Mahajan, Aberration Theory Made Simple (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

The Taylor expansion of a function with three variables is given by f(x)=Σk=0∞(1/k!)f(k)(x0), with x=(X, Y, Z),x0=(0, 0, 0), and f(k)(x)=(x∇)kf(x).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1985).

C-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987), Vol. X, Chap. 4. (Note that the ordering of the polynomials in the list is not universally accepted; here we used a different ordering.)

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Figures (9)

Fig. 1
Fig. 1

Imaging of an off-axis point object by a refracting surface separating media of refractive indices n and n. The exit pupil is located at the vertex of the surface.

Fig. 2
Fig. 2

(a) Elevation (zi) of the corneal surface at the sampled point (ri, θi) of the exit pupil. (b) Plane of reference and sample of points where the elevation is measured.

Fig. 3
Fig. 3

RMSE between the actual values of surface elevation and the measured values for each ring of the MasterVue system, for six calibrated surfaces. The horizontal axis represents the radial average distance from each ring to the vertex of the surface. Solid circles, RMSE from the MasterVue elevation file; open circles, from elevations calculated from the MasterVue curvature file.

Fig. 4
Fig. 4

RMSE between the measured corneal elevations and the elevations given by Eq. (9), for the fitting of the central area (5 mm in diameter) of a cornea as a function of the order of the Zernike expansion. Open circles represent an expansion with four terms (first-order plus second-order focus term) and an expansion with nine terms (second-order plus third-order cylindrical terms plus fourth-order spherical term).

Fig. 5
Fig. 5

(a) Error in the spherical aberration when calculated from the surface elevation measured with the MasterVue system, as a function of the distance to the axis, for two calibrated surfaces. Solid circles, sphere (R=7.94 mm); open circles, ellipsoid (R=7.99 mm and K=0.828). The value n=1.3375 was used to calculate the aberration. (b) Modulation transfer function for the spherical surface mentioned in (a), calculated with the actual wave aberration (solid curve) and the estimated wave aberration (dotted curve). Pupil diameter, 4 mm.

Fig. 6
Fig. 6

Zernike coefficient values corresponding to the wave aberration of a cornea, calculated up to fourth-order (white bars), fifth-order (gray bars), and sixth-order (black bars).

Fig. 7
Fig. 7

(a) Mean value and error bars (two standard deviations), from four videokeratographs, of the first 15 Zernike coefficients representing a calibrated spherical surface (R=8 mm), for a 4-mm-diameter pupil. (b) Same as (a), except for a cornea. (c) Mean value and error bars (two standard deviations) of the Zernike coefficients corresponding to the wave aberration calculated from the data of (a). (d) Same as (c), except for the wave aberration of the cornea of (b).

Fig. 8
Fig. 8

(a) MTF from the wave aberration of the sphere of Fig. 7(c), calculated with the mean values of the Zernike coefficients (solid curve), and with the mean values plus two standard deviations (dotted curve). (b) Same as (a), except for the cornea of Fig. 7(d).

Fig. 9
Fig. 9

First 15 Zernike coefficients for the corneal wave aberration calculated in a subject when the subject was instructed to blink before recording the videokeratography (white bars) and not to blink during the preceding 5 s (gray bars).

Tables (4)

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Table 1 First 15 Zernike Polynomials and Their Monomial Representation

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Table 2 Aberration Seidel Coefficients in the Function of the Fourth-Order Zernike Coefficients

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Table 3 Radius of Curvature and Asphericity of Six Calibrated Surfaces Estimated from Videokeratoscopic Data, in Comparison with the Actual Values, by Use of 15 and 20 Rings of the MasterVue System

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Table 4 Zernike Coefficient Values When a Corneal Surface Is Modeled with the 4, 6, 9, or 15 First Terms Listed in Table 1

Equations (39)

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W=nd+nd-nl-nl.
l=s(1+Y2)1/2,
l=s(1+Y2)1/2,
d=s(1+X2+Y2+Z2+2Z-2AXY)1/2,
d=s(1+X2+Y2+Z2-2Z+2AXY)1/2,
W(r, θ; z)
=ns+ns-n21ns+1nssin2 ω r22-ns3+ns3 r48+n(n2-n2)2n2×r(cos β sin θ+sin β cos θ)sin3 ω+n24 1ns+1nsr2(cos 2β cos 2θ-sin 2β sin 2θ)sin2 ω+n2 1s2-1s2r3(cos β sin θ+sin β cos θ)sin ω+(n-n)1+n2n sin2 ωz+n1s+1srz(cos β sin θ+sin β cos θ)sin ω+n22 1ns+1nsz2 sin2 ω+n1s2-1s2×rz2(cos β sin θ+sin β cos θ)sin ω+ns3+ns3 r2z22+ns2-ns2 r2z2.
z(r, θ)=r22R+r48R3 K2+r616R5 K4++Δz(r, θ),
W(r,θ)=Wsphere(r,θ)-e2(n-n)8R3 r4+[(n-n)+L sin2 ω+Mr2+N(cos β sin θ+sin β cos θ)r sin ω]Δz+[T sin2 ω+Ur2+V(cos β sin θ+sin β cos θ)r sin ω]Δz2,
L=(n-n) n2n,M=12 ns2-ns2,
N=n1s+1s,T=n22 1ns+1ns,
U=12 ns3+ns3,V=n1s2-1s2.
Wsphere(r,θ)=adr2+asr4+at(cos β r sin θ+sin β r cos θ)+aa(cos 2βr2 cos 2θ-sin 2βr2 sin 2θ)+ac(cos βr3 sin θ+sin βr3 cos θ),
as=n-n8R3+M2R-U4,
ad=L2R-Tsin2 ω-nΔs2s2,
at=n(n2-n2)2n2 sin3 ω,
ac=12 NR-Vsin ω,aa=T2 sin2 ω
W(r,θ)=-nΔs2s2 r2+asP+K2(n-n)8R3r4+(n-n)Δz+Mr2Δz+U r2Δz2,
z(ρ,θ)=j=1Laj Zj(ρ,θ),
Vj=Zj+s=1j-1DjsVs,
z(ρ,θ)=j=1LbjVj(ρ,θ).
aj=bj+k=j+1LakDkj,aL=bL.
W(ρ,θ)=j=1LAj Zj(ρ,θ).
A9=(n-n)a9+r0465 asP,
A4=6523 A9+123 adr02+Nr023 [(a2-8a7)sin β+(a3-8a8)cos β]sin ω,
A2(3)=(n-n)a2(3)+at2+acr023r0 sin β(cos β)+(n-n)n2n sin2 ω+23 γ[a2(3)-8a7(8)]+Nr06 -(a5-15a12)sin β(cos β)+(a6-15a13)cos β(sin β)sin ω,
A5(6)=(n-n)a5(6)-aar026 cos 2β(sin 2β)+(n-n)×n2n sin2 ω+34 γ[a5(6)-15a12(13)]+Nr026 2(a2-8a7)sin β(cos β)-2(a3-15a8)cos β(sin β)sin ω,
A7(8)=(n-n)a7(8)+acr0338 sin β(cos β)+(n-n) 238 γ[a2(3)-8a7(8)]+Nr048 (a5-15a12)sin β(cos β)+(a6-15a13)cos β(sin β)sin ω,
A10(11)=(n-n)a10(11)+324 Nr0(a5-15a12)sin β(cos β)-(a6-15a13)cos β(sin β)sin ω,
A12(13)=(n-n)a12(13)+(n-n)×345 γ[a5(6)-15a12(13)],
A14(15)=(n-n)a14(15),
K02=8R3(n-n) asP.
WSeidel=Adr2+Asr4+Atr cos(θ-θt)+Aar2 cos(θ-θa)+Acr3 cos(θ-θc).
z(ri)=R-(R2-K2ri2)1/2K2,
RMSE=1N i=1N[zi-z(ri)]21/2.
RMSE=1N i=1Nzi-j=1LajZj(ρi, θi)21/2.
z=r22R+K02r48R3+(K2-K02)r48R3=r22R+K02r48R3+As(n-n) r4r04,
R=r022(23a4-65a9-307a22+ ),
K2=8R3r04 (65a9-307a22+ ),

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