Abstract

The aim of this work was to study, using numerical simulations, the attainable level of accuracy to reconstruct the wavefront aberrations from tangential refractive power data measured with dynamic skiascopy. Two mathematical methods have been implemented. The first one is based on curve integration of the curvature data, previously interpolated with cubic splines. The second one reconstructs the three-dimensional wavefront surface, represented by a Zernike polynomial expansion, using a two-step least-squares method. The different factors affecting the reconstruction—noise, sampling, and wavefront patterns—were quantified. The results provide useful information to design more efficient experimental setups based on spatial dynamic skiascopy.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. J. Benjamin and I. M. Borish, Borish’s Clinical Refraction (Butterworth Heinemann/Elsevier, 2006).
  2. M. Fujieda, “Ophthalmic measurement apparatus having plural pairs of photoreceiving elements,” U.S. patent 5,907,388 (25May1999).
  3. S. MacRae and M. Fujieda, “Slit skiascopic-guided ablation using the Nidek laser,” J. Refract. Surg. 16, S576–S580(2000).
  4. D. A. Atchison, “The skew ray issue in ocular aberration measurement,” Optom. Vis. Sci. 83, 396–398 (2006).
    [CrossRef]
  5. M. Fujieda and B. Yukinobu, “Ophthalmic measurement apparatus,” U.S. patent 7,296,896 (20November2007).
  6. O. Hieda and S. Kinoshita, “Measuring of ocular wavefront aberration in large pupils using OPD-scan,” Semin. Ophthalmol. 18, 35–40 (2003).
    [CrossRef]
  7. J. Nam, L. N. Thibos, and D. R. Iskander, “Zernike radial slope polynomials for wavefront reconstruction and refraction,” J. Opt. Soc. Am. A 26, 1035–1048 (2009).
    [CrossRef]
  8. J. J. Stoker, Differential Geometry (Wiley-Interscience, 1969).
  9. A. D. Polianin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (CRC, 1995).
  10. C. Elster, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik 113, 154–158 (2002).
    [CrossRef]
  11. C. Elster, “High-accuracy reconstruction of a function f(x) when only d/dx f(x) or d2/dx2f(x) is known at discrete measurement points,” Proc. SPIE 4782, 12–20 (2002).
  12. P. Lancaster and K. Salkauskas, Curve and Surface Fitting: an Introduction (Academic, 1986).
  13. G.-M. Dai, “Modal wave-front reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 13, 1218–1225 (1996).
    [CrossRef]
  14. E. Acosta, S. Bara, and S. Rios, “Modal projectors for linear operators in Optics,” Opt. Commun. 162, 211–214 (1999).
    [CrossRef]
  15. J. Nam and J. Rubinstein, “Weighted Zernike expansion with applications to the optical aberration of the human eye,” J. Opt. Soc. Am. A 22, 1709–1716 (2005).
    [CrossRef]
  16. C. E. Campbell, “A test eye for wavefront eye refractors,” J. Refract. Surg. 21, 127–140 (2005).
  17. J. J. Rozema, D. E. M. Van Dyck, and M. J. Tassignon, “Clinical comparison of 6 aberrometers. Part 1: technical specifications,” J. Cataract Refract. Surg. 31, 1114–1127 (2005).
    [CrossRef]
  18. A. Z. Burakgazi, B. Tinio, A. Bababyan, K. K. Niksarli, and P. Asbell, “Higher order aberrations in normal eyes measured with three different aberrometers,” J. Refract. Surg. 22, 898–903 (2006).
  19. A. Cerviño, S. L. Hosking, and R. Montes-Mico, “Comparison of higher order aberrations measured by NIDEK OPD-Scan dynamic skiascopy and Zeiss WASCA Hartmann–Shack aberrometers,” J. Refract. Surg. 24, 790–796 (2008).
  20. D. U. Bartsch, K. Bessho, L. Gomez, and W. R. Freeman, “Comparison of laser ray-tracing and skiascopic ocular wavefront-sensing devices,” Eye 22, 1384–1390 (2007).
    [CrossRef]
  21. D. Zadok, Y. Levy, O. Segal, Y. Barkana, Y. Morad, and I. Avni, “Ocular higher-order aberrations in myopia and skiascopic wavefront repeatability,” J. Cataract Refract. Surg. 31, 1128–1132 (2005).
    [CrossRef]

2009 (1)

2008 (1)

A. Cerviño, S. L. Hosking, and R. Montes-Mico, “Comparison of higher order aberrations measured by NIDEK OPD-Scan dynamic skiascopy and Zeiss WASCA Hartmann–Shack aberrometers,” J. Refract. Surg. 24, 790–796 (2008).

2007 (1)

D. U. Bartsch, K. Bessho, L. Gomez, and W. R. Freeman, “Comparison of laser ray-tracing and skiascopic ocular wavefront-sensing devices,” Eye 22, 1384–1390 (2007).
[CrossRef]

2006 (2)

A. Z. Burakgazi, B. Tinio, A. Bababyan, K. K. Niksarli, and P. Asbell, “Higher order aberrations in normal eyes measured with three different aberrometers,” J. Refract. Surg. 22, 898–903 (2006).

D. A. Atchison, “The skew ray issue in ocular aberration measurement,” Optom. Vis. Sci. 83, 396–398 (2006).
[CrossRef]

2005 (4)

D. Zadok, Y. Levy, O. Segal, Y. Barkana, Y. Morad, and I. Avni, “Ocular higher-order aberrations in myopia and skiascopic wavefront repeatability,” J. Cataract Refract. Surg. 31, 1128–1132 (2005).
[CrossRef]

J. Nam and J. Rubinstein, “Weighted Zernike expansion with applications to the optical aberration of the human eye,” J. Opt. Soc. Am. A 22, 1709–1716 (2005).
[CrossRef]

C. E. Campbell, “A test eye for wavefront eye refractors,” J. Refract. Surg. 21, 127–140 (2005).

J. J. Rozema, D. E. M. Van Dyck, and M. J. Tassignon, “Clinical comparison of 6 aberrometers. Part 1: technical specifications,” J. Cataract Refract. Surg. 31, 1114–1127 (2005).
[CrossRef]

2003 (1)

O. Hieda and S. Kinoshita, “Measuring of ocular wavefront aberration in large pupils using OPD-scan,” Semin. Ophthalmol. 18, 35–40 (2003).
[CrossRef]

2002 (2)

C. Elster, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik 113, 154–158 (2002).
[CrossRef]

C. Elster, “High-accuracy reconstruction of a function f(x) when only d/dx f(x) or d2/dx2f(x) is known at discrete measurement points,” Proc. SPIE 4782, 12–20 (2002).

2000 (1)

S. MacRae and M. Fujieda, “Slit skiascopic-guided ablation using the Nidek laser,” J. Refract. Surg. 16, S576–S580(2000).

1999 (1)

E. Acosta, S. Bara, and S. Rios, “Modal projectors for linear operators in Optics,” Opt. Commun. 162, 211–214 (1999).
[CrossRef]

1996 (1)

Acosta, E.

E. Acosta, S. Bara, and S. Rios, “Modal projectors for linear operators in Optics,” Opt. Commun. 162, 211–214 (1999).
[CrossRef]

Asbell, P.

A. Z. Burakgazi, B. Tinio, A. Bababyan, K. K. Niksarli, and P. Asbell, “Higher order aberrations in normal eyes measured with three different aberrometers,” J. Refract. Surg. 22, 898–903 (2006).

Atchison, D. A.

D. A. Atchison, “The skew ray issue in ocular aberration measurement,” Optom. Vis. Sci. 83, 396–398 (2006).
[CrossRef]

Avni, I.

D. Zadok, Y. Levy, O. Segal, Y. Barkana, Y. Morad, and I. Avni, “Ocular higher-order aberrations in myopia and skiascopic wavefront repeatability,” J. Cataract Refract. Surg. 31, 1128–1132 (2005).
[CrossRef]

Bababyan, A.

A. Z. Burakgazi, B. Tinio, A. Bababyan, K. K. Niksarli, and P. Asbell, “Higher order aberrations in normal eyes measured with three different aberrometers,” J. Refract. Surg. 22, 898–903 (2006).

Bara, S.

E. Acosta, S. Bara, and S. Rios, “Modal projectors for linear operators in Optics,” Opt. Commun. 162, 211–214 (1999).
[CrossRef]

Barkana, Y.

D. Zadok, Y. Levy, O. Segal, Y. Barkana, Y. Morad, and I. Avni, “Ocular higher-order aberrations in myopia and skiascopic wavefront repeatability,” J. Cataract Refract. Surg. 31, 1128–1132 (2005).
[CrossRef]

Bartsch, D. U.

D. U. Bartsch, K. Bessho, L. Gomez, and W. R. Freeman, “Comparison of laser ray-tracing and skiascopic ocular wavefront-sensing devices,” Eye 22, 1384–1390 (2007).
[CrossRef]

Benjamin, W. J.

W. J. Benjamin and I. M. Borish, Borish’s Clinical Refraction (Butterworth Heinemann/Elsevier, 2006).

Bessho, K.

D. U. Bartsch, K. Bessho, L. Gomez, and W. R. Freeman, “Comparison of laser ray-tracing and skiascopic ocular wavefront-sensing devices,” Eye 22, 1384–1390 (2007).
[CrossRef]

Borish, I. M.

W. J. Benjamin and I. M. Borish, Borish’s Clinical Refraction (Butterworth Heinemann/Elsevier, 2006).

Burakgazi, A. Z.

A. Z. Burakgazi, B. Tinio, A. Bababyan, K. K. Niksarli, and P. Asbell, “Higher order aberrations in normal eyes measured with three different aberrometers,” J. Refract. Surg. 22, 898–903 (2006).

Campbell, C. E.

C. E. Campbell, “A test eye for wavefront eye refractors,” J. Refract. Surg. 21, 127–140 (2005).

Cerviño, A.

A. Cerviño, S. L. Hosking, and R. Montes-Mico, “Comparison of higher order aberrations measured by NIDEK OPD-Scan dynamic skiascopy and Zeiss WASCA Hartmann–Shack aberrometers,” J. Refract. Surg. 24, 790–796 (2008).

Dai, G.-M.

Elster, C.

C. Elster, “High-accuracy reconstruction of a function f(x) when only d/dx f(x) or d2/dx2f(x) is known at discrete measurement points,” Proc. SPIE 4782, 12–20 (2002).

C. Elster, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik 113, 154–158 (2002).
[CrossRef]

Freeman, W. R.

D. U. Bartsch, K. Bessho, L. Gomez, and W. R. Freeman, “Comparison of laser ray-tracing and skiascopic ocular wavefront-sensing devices,” Eye 22, 1384–1390 (2007).
[CrossRef]

Fujieda, M.

S. MacRae and M. Fujieda, “Slit skiascopic-guided ablation using the Nidek laser,” J. Refract. Surg. 16, S576–S580(2000).

M. Fujieda and B. Yukinobu, “Ophthalmic measurement apparatus,” U.S. patent 7,296,896 (20November2007).

M. Fujieda, “Ophthalmic measurement apparatus having plural pairs of photoreceiving elements,” U.S. patent 5,907,388 (25May1999).

Gerhardt, J.

C. Elster, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik 113, 154–158 (2002).
[CrossRef]

Gomez, L.

D. U. Bartsch, K. Bessho, L. Gomez, and W. R. Freeman, “Comparison of laser ray-tracing and skiascopic ocular wavefront-sensing devices,” Eye 22, 1384–1390 (2007).
[CrossRef]

Hieda, O.

O. Hieda and S. Kinoshita, “Measuring of ocular wavefront aberration in large pupils using OPD-scan,” Semin. Ophthalmol. 18, 35–40 (2003).
[CrossRef]

Hosking, S. L.

A. Cerviño, S. L. Hosking, and R. Montes-Mico, “Comparison of higher order aberrations measured by NIDEK OPD-Scan dynamic skiascopy and Zeiss WASCA Hartmann–Shack aberrometers,” J. Refract. Surg. 24, 790–796 (2008).

Iskander, D. R.

Kinoshita, S.

O. Hieda and S. Kinoshita, “Measuring of ocular wavefront aberration in large pupils using OPD-scan,” Semin. Ophthalmol. 18, 35–40 (2003).
[CrossRef]

Lancaster, P.

P. Lancaster and K. Salkauskas, Curve and Surface Fitting: an Introduction (Academic, 1986).

Levy, Y.

D. Zadok, Y. Levy, O. Segal, Y. Barkana, Y. Morad, and I. Avni, “Ocular higher-order aberrations in myopia and skiascopic wavefront repeatability,” J. Cataract Refract. Surg. 31, 1128–1132 (2005).
[CrossRef]

MacRae, S.

S. MacRae and M. Fujieda, “Slit skiascopic-guided ablation using the Nidek laser,” J. Refract. Surg. 16, S576–S580(2000).

Montes-Mico, R.

A. Cerviño, S. L. Hosking, and R. Montes-Mico, “Comparison of higher order aberrations measured by NIDEK OPD-Scan dynamic skiascopy and Zeiss WASCA Hartmann–Shack aberrometers,” J. Refract. Surg. 24, 790–796 (2008).

Morad, Y.

D. Zadok, Y. Levy, O. Segal, Y. Barkana, Y. Morad, and I. Avni, “Ocular higher-order aberrations in myopia and skiascopic wavefront repeatability,” J. Cataract Refract. Surg. 31, 1128–1132 (2005).
[CrossRef]

Nam, J.

Niksarli, K. K.

A. Z. Burakgazi, B. Tinio, A. Bababyan, K. K. Niksarli, and P. Asbell, “Higher order aberrations in normal eyes measured with three different aberrometers,” J. Refract. Surg. 22, 898–903 (2006).

Polianin, A. D.

A. D. Polianin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (CRC, 1995).

Rios, S.

E. Acosta, S. Bara, and S. Rios, “Modal projectors for linear operators in Optics,” Opt. Commun. 162, 211–214 (1999).
[CrossRef]

Rozema, J. J.

J. J. Rozema, D. E. M. Van Dyck, and M. J. Tassignon, “Clinical comparison of 6 aberrometers. Part 1: technical specifications,” J. Cataract Refract. Surg. 31, 1114–1127 (2005).
[CrossRef]

Rubinstein, J.

Salkauskas, K.

P. Lancaster and K. Salkauskas, Curve and Surface Fitting: an Introduction (Academic, 1986).

Schulz, M.

C. Elster, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik 113, 154–158 (2002).
[CrossRef]

Segal, O.

D. Zadok, Y. Levy, O. Segal, Y. Barkana, Y. Morad, and I. Avni, “Ocular higher-order aberrations in myopia and skiascopic wavefront repeatability,” J. Cataract Refract. Surg. 31, 1128–1132 (2005).
[CrossRef]

Stoker, J. J.

J. J. Stoker, Differential Geometry (Wiley-Interscience, 1969).

Tassignon, M. J.

J. J. Rozema, D. E. M. Van Dyck, and M. J. Tassignon, “Clinical comparison of 6 aberrometers. Part 1: technical specifications,” J. Cataract Refract. Surg. 31, 1114–1127 (2005).
[CrossRef]

Thibos, L. N.

Thomsen-Schmidt, P.

C. Elster, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik 113, 154–158 (2002).
[CrossRef]

Tinio, B.

A. Z. Burakgazi, B. Tinio, A. Bababyan, K. K. Niksarli, and P. Asbell, “Higher order aberrations in normal eyes measured with three different aberrometers,” J. Refract. Surg. 22, 898–903 (2006).

Van Dyck, D. E. M.

J. J. Rozema, D. E. M. Van Dyck, and M. J. Tassignon, “Clinical comparison of 6 aberrometers. Part 1: technical specifications,” J. Cataract Refract. Surg. 31, 1114–1127 (2005).
[CrossRef]

Weingartner, I.

C. Elster, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik 113, 154–158 (2002).
[CrossRef]

Yukinobu, B.

M. Fujieda and B. Yukinobu, “Ophthalmic measurement apparatus,” U.S. patent 7,296,896 (20November2007).

Zadok, D.

D. Zadok, Y. Levy, O. Segal, Y. Barkana, Y. Morad, and I. Avni, “Ocular higher-order aberrations in myopia and skiascopic wavefront repeatability,” J. Cataract Refract. Surg. 31, 1128–1132 (2005).
[CrossRef]

Zaitsev, V. F.

A. D. Polianin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (CRC, 1995).

Eye (1)

D. U. Bartsch, K. Bessho, L. Gomez, and W. R. Freeman, “Comparison of laser ray-tracing and skiascopic ocular wavefront-sensing devices,” Eye 22, 1384–1390 (2007).
[CrossRef]

J. Cataract Refract. Surg. (2)

D. Zadok, Y. Levy, O. Segal, Y. Barkana, Y. Morad, and I. Avni, “Ocular higher-order aberrations in myopia and skiascopic wavefront repeatability,” J. Cataract Refract. Surg. 31, 1128–1132 (2005).
[CrossRef]

J. J. Rozema, D. E. M. Van Dyck, and M. J. Tassignon, “Clinical comparison of 6 aberrometers. Part 1: technical specifications,” J. Cataract Refract. Surg. 31, 1114–1127 (2005).
[CrossRef]

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

J. Refract. Surg. (4)

S. MacRae and M. Fujieda, “Slit skiascopic-guided ablation using the Nidek laser,” J. Refract. Surg. 16, S576–S580(2000).

C. E. Campbell, “A test eye for wavefront eye refractors,” J. Refract. Surg. 21, 127–140 (2005).

A. Z. Burakgazi, B. Tinio, A. Bababyan, K. K. Niksarli, and P. Asbell, “Higher order aberrations in normal eyes measured with three different aberrometers,” J. Refract. Surg. 22, 898–903 (2006).

A. Cerviño, S. L. Hosking, and R. Montes-Mico, “Comparison of higher order aberrations measured by NIDEK OPD-Scan dynamic skiascopy and Zeiss WASCA Hartmann–Shack aberrometers,” J. Refract. Surg. 24, 790–796 (2008).

Opt. Commun. (1)

E. Acosta, S. Bara, and S. Rios, “Modal projectors for linear operators in Optics,” Opt. Commun. 162, 211–214 (1999).
[CrossRef]

Optik (1)

C. Elster, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik 113, 154–158 (2002).
[CrossRef]

Optom. Vis. Sci. (1)

D. A. Atchison, “The skew ray issue in ocular aberration measurement,” Optom. Vis. Sci. 83, 396–398 (2006).
[CrossRef]

Proc. SPIE (1)

C. Elster, “High-accuracy reconstruction of a function f(x) when only d/dx f(x) or d2/dx2f(x) is known at discrete measurement points,” Proc. SPIE 4782, 12–20 (2002).

Semin. Ophthalmol. (1)

O. Hieda and S. Kinoshita, “Measuring of ocular wavefront aberration in large pupils using OPD-scan,” Semin. Ophthalmol. 18, 35–40 (2003).
[CrossRef]

Other (6)

P. Lancaster and K. Salkauskas, Curve and Surface Fitting: an Introduction (Academic, 1986).

J. J. Stoker, Differential Geometry (Wiley-Interscience, 1969).

A. D. Polianin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (CRC, 1995).

M. Fujieda and B. Yukinobu, “Ophthalmic measurement apparatus,” U.S. patent 7,296,896 (20November2007).

W. J. Benjamin and I. M. Borish, Borish’s Clinical Refraction (Butterworth Heinemann/Elsevier, 2006).

M. Fujieda, “Ophthalmic measurement apparatus having plural pairs of photoreceiving elements,” U.S. patent 5,907,388 (25May1999).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Wavefront aberration patterns used in the simulations (Campbell’s eye tests [16]): (a) test eye A4, (b) test eye A3, and (c) test eye L2. The grayscale pattern represents wave aberration heights in micrometers.

Fig. 2.
Fig. 2.

NRMSE, expressed as a percentage, as a function of curvature SNR in decibels for test eye A4. Sampling patterns S1, S2, and S3 are plotted with a solid curve, a dashed curve with circles, and a solid curve with squares, respectively. Numerical reconstruction for (a) the modal fitting and (b) the curve integration reconstruction algorithms.

Fig. 3.
Fig. 3.

NRMSE, expressed as a percentage, as a function of curvature SNR in decibels for test eye A3. Sampling patterns S1, S2, and S3 are plotted with a solid curve, a dashed curve with circles, and a solid curve with squares, respectively. Numerical reconstruction for (a) the modal fitting and (b) the curve integration reconstruction algorithms.

Fig. 4.
Fig. 4.

NRMSE, expressed as a percentage, as a function of curvature SNR in decibels for test eye L2. Sampling patterns S1, S2, and S3 are plotted with a solid curve, a dashed curve with circles, and a solid curve with squares, respectively. Numerical reconstruction for (a) the modal fitting and (b) the curve integration reconstruction algorithms.

Fig. 5.
Fig. 5.

Errors in the reconstruction of the Zernike coefficients of the reconstructed wavefronts, both experimentally (as reported in [16]) and numerically. Curvature SNR was 20 dB. Sampling pattern was S1. Test eyes: (a) A4, (b) A3, and (c) L2. Experimental data, surface fitting, and curve integration reconstructions are plotted with a solid curve, a dashed curve with circles, and a dashed curve with triangles, respectively.

Tables (1)

Tables Icon

Table 1. NRMSE (%) of the Algorithms for Different Tests (A4, A3, and L2) and Sampling Patterns (S1, S2, and S3)a

Equations (8)

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

y=Kt(1+y2)3/2,
ψ(x)=0xKt(ξ)dξ,
y(x)=ψ(x)+α1(ψ(x)+α)2,
α=y(0)1+(y(0))2.
urr(r,θ)=k=1nCk2Zk(r,θ)r2.
urr(r,θ)=Kt(r,θ)(1+ur(r,θ)2)3/2,
Kt(r,θ)=k=1nCk2Z(r,θ)r2.
MF=i(KiK˜i)2.

Metrics