Abstract

We propose to use weighted Zernike functions to represent the aberrations of an eye. Methods for computing the phase of an aberrated ophthalmic wavefront in terms of weighted Zernike functions are discussed. In particular, we consider several options for integrating the phase out of its measured slopes. The weighted functions involve a free parameter. Clinical data on subjective refraction and aberration maps of individual subjects are used to determine an estimate for this parameter.

© 2005 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
    [CrossRef]
  2. L. N. Thibos, R. A. Applegate, J. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 232–244.
  3. W. S. Stiles, B. H. Crawford, “The luminous efficiency of rays entering the eye pupil at different points,” Proc. R. Soc., London, Ser. B 112, 428–450 (1933).
    [CrossRef]
  4. R. A. Applegate, V. Lakshminarayanan, “Parametric representation of Stiles–Crawford functions: normal variation of peak location and directionality,” J. Opt. Soc. Am. A 10, 1611–1623 (1993).
    [CrossRef] [PubMed]
  5. X. Zhang, M. Ye, A. Bradley, L. N. Thibos, “Apodization by the Stiles–Crawford effect moderates the visual impact of retinal defocus,” J. Opt. Soc. Am. A 16, 812–820 (1999).
    [CrossRef]
  6. V. N. Mahajan, Optical Imaging and Aberrations I (SPIE Press, Bellingham, Wash., 1998).
    [CrossRef]
  7. J. Liang, B. Grimm, S. Goelz, J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
    [CrossRef]
  8. J. Liang, D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
    [CrossRef]
  9. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1008 (1980).
    [CrossRef]
  10. J. Rubinstein, G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
    [CrossRef]
  11. D. Malacara, J. M. Capriovladez, J. J. Sanchezmondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius disc,” Opt. Eng. (Bellingham) 29, 672–675 (1990).
  12. L. N. Thibos, X. Hong, A. Bradley, R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4, 329–351 (2004).
    [CrossRef] [PubMed]
  13. C. Coe, L. Thibos, “Objective estimates of subjective refraction from wavefront aberrometry maps,” Invest. Ophthalmol. Visual Sci. 45, 2760 (2004) Suppl. 1.
  14. L. N. Thibos, W. Wheeler, D. G. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367–375 (1997).
    [CrossRef]
  15. J. Schwiegerling, “Gauss weighting of ocular wave-front measurements,” J. Opt. Soc. Am. A 21, 2065–2072 (2004).
    [CrossRef]

2004

L. N. Thibos, X. Hong, A. Bradley, R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4, 329–351 (2004).
[CrossRef] [PubMed]

C. Coe, L. Thibos, “Objective estimates of subjective refraction from wavefront aberrometry maps,” Invest. Ophthalmol. Visual Sci. 45, 2760 (2004) Suppl. 1.

J. Schwiegerling, “Gauss weighting of ocular wave-front measurements,” J. Opt. Soc. Am. A 21, 2065–2072 (2004).
[CrossRef]

2001

J. Rubinstein, G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[CrossRef]

1999

1997

J. Liang, D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
[CrossRef]

L. N. Thibos, W. Wheeler, D. G. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367–375 (1997).
[CrossRef]

1994

1993

1990

D. Malacara, J. M. Capriovladez, J. J. Sanchezmondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius disc,” Opt. Eng. (Bellingham) 29, 672–675 (1990).

1980

1933

W. S. Stiles, B. H. Crawford, “The luminous efficiency of rays entering the eye pupil at different points,” Proc. R. Soc., London, Ser. B 112, 428–450 (1933).
[CrossRef]

Applegate, R. A.

L. N. Thibos, X. Hong, A. Bradley, R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4, 329–351 (2004).
[CrossRef] [PubMed]

R. A. Applegate, V. Lakshminarayanan, “Parametric representation of Stiles–Crawford functions: normal variation of peak location and directionality,” J. Opt. Soc. Am. A 10, 1611–1623 (1993).
[CrossRef] [PubMed]

L. N. Thibos, R. A. Applegate, J. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

Bille, J. F.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

Bradley, A.

L. N. Thibos, X. Hong, A. Bradley, R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4, 329–351 (2004).
[CrossRef] [PubMed]

X. Zhang, M. Ye, A. Bradley, L. N. Thibos, “Apodization by the Stiles–Crawford effect moderates the visual impact of retinal defocus,” J. Opt. Soc. Am. A 16, 812–820 (1999).
[CrossRef]

Capriovladez, J. M.

D. Malacara, J. M. Capriovladez, J. J. Sanchezmondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius disc,” Opt. Eng. (Bellingham) 29, 672–675 (1990).

Coe, C.

C. Coe, L. Thibos, “Objective estimates of subjective refraction from wavefront aberrometry maps,” Invest. Ophthalmol. Visual Sci. 45, 2760 (2004) Suppl. 1.

Crawford, B. H.

W. S. Stiles, B. H. Crawford, “The luminous efficiency of rays entering the eye pupil at different points,” Proc. R. Soc., London, Ser. B 112, 428–450 (1933).
[CrossRef]

Goelz, S.

Grimm, B.

Hong, X.

L. N. Thibos, X. Hong, A. Bradley, R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4, 329–351 (2004).
[CrossRef] [PubMed]

Horner, D. G.

L. N. Thibos, W. Wheeler, D. G. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367–375 (1997).
[CrossRef]

Lakshminarayanan, V.

Liang, J.

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations I (SPIE Press, Bellingham, Wash., 1998).
[CrossRef]

Malacara, D.

D. Malacara, J. M. Capriovladez, J. J. Sanchezmondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius disc,” Opt. Eng. (Bellingham) 29, 672–675 (1990).

Rubinstein, J.

J. Rubinstein, G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[CrossRef]

Sanchezmondragon, J. J.

D. Malacara, J. M. Capriovladez, J. J. Sanchezmondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius disc,” Opt. Eng. (Bellingham) 29, 672–675 (1990).

Schwiegerling, J.

J. Schwiegerling, “Gauss weighting of ocular wave-front measurements,” J. Opt. Soc. Am. A 21, 2065–2072 (2004).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

Southwell, W. H.

Stiles, W. S.

W. S. Stiles, B. H. Crawford, “The luminous efficiency of rays entering the eye pupil at different points,” Proc. R. Soc., London, Ser. B 112, 428–450 (1933).
[CrossRef]

Thibos, L.

C. Coe, L. Thibos, “Objective estimates of subjective refraction from wavefront aberrometry maps,” Invest. Ophthalmol. Visual Sci. 45, 2760 (2004) Suppl. 1.

Thibos, L. N.

L. N. Thibos, X. Hong, A. Bradley, R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4, 329–351 (2004).
[CrossRef] [PubMed]

X. Zhang, M. Ye, A. Bradley, L. N. Thibos, “Apodization by the Stiles–Crawford effect moderates the visual impact of retinal defocus,” J. Opt. Soc. Am. A 16, 812–820 (1999).
[CrossRef]

L. N. Thibos, W. Wheeler, D. G. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367–375 (1997).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

Webb, R.

L. N. Thibos, R. A. Applegate, J. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

Wheeler, W.

L. N. Thibos, W. Wheeler, D. G. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367–375 (1997).
[CrossRef]

Williams, D. R.

Wolansky, G.

J. Rubinstein, G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

Ye, M.

Zhang, X.

Invest. Ophthalmol. Visual Sci.

C. Coe, L. Thibos, “Objective estimates of subjective refraction from wavefront aberrometry maps,” Invest. Ophthalmol. Visual Sci. 45, 2760 (2004) Suppl. 1.

J. Vis.

L. N. Thibos, X. Hong, A. Bradley, R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4, 329–351 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng. (Bellingham)

D. Malacara, J. M. Capriovladez, J. J. Sanchezmondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius disc,” Opt. Eng. (Bellingham) 29, 672–675 (1990).

Opt. Rev.

J. Rubinstein, G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[CrossRef]

Optom. Vision Sci.

L. N. Thibos, W. Wheeler, D. G. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367–375 (1997).
[CrossRef]

Proc. R. Soc., London, Ser. B

W. S. Stiles, B. H. Crawford, “The luminous efficiency of rays entering the eye pupil at different points,” Proc. R. Soc., London, Ser. B 112, 428–450 (1933).
[CrossRef]

Other

V. N. Mahajan, Optical Imaging and Aberrations I (SPIE Press, Bellingham, Wash., 1998).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

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Figures (2)

Fig. 1
Fig. 1

Error propagation in the integration by the many-orbits method. The horizontal axis indicates the number of shells taken into account, and the vertical axis provides the difference between the exact and the computed solutions in the L 2 norm. Since these are both artificial data over the unit circle, no units are needed. (a) Ψ = x 4 y 4 . (b) Ψ = cos 2 ( x 2 + y 2 ) .

Fig. 2
Fig. 2

Difference (in diopters) between the subjective refraction P s and the objective refraction P o as a function of γ. (a) The average difference. (b) The standard deviation for the sample.

Tables (2)

Tables Icon

Table 1 Coefficients for G n m , γ = 0 , 0.5, 0 n 4 a

Tables Icon

Table 2 ( V n m , V n m ) : n = 3 , m = 3 a

Equations (31)

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Z n m ( ρ , θ ) = { N n m R n m ( ρ ) cos m θ for m 0 N n m R n m ( ρ ) sin m θ for m < 0 } ,
N n m = 2 ( n + 1 ) 1 + δ m 0 , δ i j = { 1 if i = j 0 if i j } ,
R n m ( ρ ) = s = 0 n m 2 ( 1 ) s ( n s ) ! s ! ( n + m s 2 ) ! ( n m s 2 ) ! ρ n 2 s .
( R n m ( ρ ) exp ( i m θ ) , R n m ( ρ ) exp ( i m θ ) ) = 0 2 π 0 1 R n m ( ρ ) exp ( i m θ ) R n m ( ρ ) exp ( i m θ ) ρ d ρ d θ .
I I 0 1 k 2 2 I 0 1 2 Σ W ( r ) ψ 2 ( r , θ ) r d r d θ 2 .
I I 0 1 k 2 2 I 0 1 2 n β n 2 2 .
R e = R m 2.3 γ .
V n m ( ρ , θ ) = { G n m ( ρ ) cos m θ for m 0 G n m ( ρ ) sin m θ for m < 0 } ,
( f ( ρ , θ ) , g ( ρ , θ ) ) 0 2 π 0 1 f ( ρ , θ ) g ¯ ( ρ , θ ) exp ( γ ρ 2 ) ρ d ρ d θ ,
( u ( ρ ) , v ( ρ ) ) = 0 1 u ( ρ ) v ( ρ ) exp ( γ ρ 2 ) ρ d ρ .
J k ( γ ) = 0 1 exp ( γ ρ 2 ) ρ k d ρ .
G 1 1 = 2 ρ , G 3 1 = 8.4853 ρ 3 5.6569 ρ ,
G 1 1 ( ρ ) = 2.7512 ρ , G 3 1 ( ρ ) = 11.0714 ρ 3 6.7291 ρ .
Ψ ( ρ , θ ) = n , m α n m V n m ( ρ , θ ) .
F = Ψ .
D { [ f 1 x n , m α n m V n m ( r , θ ) ] 2 + [ f 2 y n , m α n m V n m ( r , θ ) ] 2 } W ( r , θ ) r d r d θ ,
H ( Ψ ) = D Ψ F 2 d x d y
Δ Ψ = F , ( x , y ) D ,
Ψ ν = F ν , ( x , y ) D .
Ψ x = f 1 , Ψ y = f 2 ,
( x i , y j ) = ( i 1 6 , j 1 6 ) ,
J 3 ( γ ) = 0 1 exp ( γ ρ 2 ) ρ 3 d ρ = d d γ J 1 ( γ ) = 1 2 γ 2 1 2 exp ( γ ) ( γ 1 + γ 2 ) .
J 0 ( γ ) = 1 γ π 2 erf ( γ ) .
J 2 ( γ ) = 0 1 exp ( γ ρ 2 ) ρ 2 d ρ = d d γ J 0 ( γ ) = 1 2 γ 1 [ J 0 ( γ ) exp ( γ ) ] .
J 13 ( γ ) = ( 6 ) ( 5 ) ( 4 ) ( 3 ) γ 7 1 2 exp ( γ ) ( γ 1 + 6 γ 2 + 30 γ 3 + 120 γ 4 + 360 γ 5 + 720 γ 6 + 720 γ 7 ) .
J 0 ( γ ) = π 2 n = 0 ( γ ) n n ! ( 2 n + 1 ) , J 1 ( γ ) = 1 2 e γ n = 0 γ n ( n + 1 ) ! ,
J 2 ( γ ) = n = 0 ( γ ) n n ! ( 2 n + 3 ) , J 3 ( γ ) = 1 2 e γ n = 0 γ n ( n + 2 ) ! .
U ( x , y ) = n , m C n m ( γ ) V n m ( ρ , θ , γ ) ,
C n m ( γ ) = { 1 π ( U ( x , y ) , V n m ( ρ , θ , γ ) ) if m 0 1 2 π ( U ( x , y ) , V n m ( ρ , θ , γ ) ) if m = 0 } .
L ( C n m ) = 0 2 π 0 1 [ U ( x , y ) n , m C n m V n m ] 2 exp ( γ ρ 2 ) ρ d ρ d θ .
n , m C ̂ n m ( V n m , V n m ) = ( U , V n m ) .

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