Abstract

We apply a computational technique to retrieve the wave aberration of the eye from the point-spread function obtained from pairs of double-pass retinal images. The method consists of an adapted pyramidal version of a nonlinear least-squares fitting procedure to a wave aberration expressed as an expansion in Zernike polynomials. Although the procedure provides accurate estimates of the wave aberration, it presents several drawbacks that are discussed in detail. In particular, since a great deal of computational time is necessary to retrieve a single wave aberration, this technique is not useful for real-time applications. We present results of wave aberrations in five normal subjects in the fovea for a 4-mm-pupil diameter. In every case there is a clear presence of comalike aberrations, while the third-order spherical aberration is usually smaller than previous estimates. The root-mean-square error in the retrieved wave aberration, when defocus and astigmatism were corrected, ranges from 0.24 to 0.5 wavelength. The particular values of the aberration coefficients present a large intersubject variability.

© 1998 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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1998 (2)

1997 (4)

1996 (1)

P. Artal, S. Marcos, I. Iglesias, D. G. Green, “Optical modulation transfer and contrast sensitivity with decentered small pupils in the human eye,” Vision Res. 36, 3575–3586 (1996).
[CrossRef] [PubMed]

1995 (2)

1994 (2)

K. Konstantinides, J. R. Rasure, “The khoros software development environment for image and signal processing,” IEEE Trans. Image Process. 3, 243–252 (1994).
[CrossRef]

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

1991 (1)

W. N. Charman, “Wavefront aberration of the eye: a review,” Optom. Vision Sci. 68, 574–583 (1991).
[CrossRef]

1988 (2)

1984 (1)

1977 (2)

1976 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1963 (1)

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” IMA J. Appl. Math. 11, 431–441 (1963).

1949 (1)

1947 (1)

A. Ivanoff, “Les aberrations de chromatisme et de sphericite de l’oeil. Leur role en vison nocturne,” Rev. Opt. Theor. Instrum. 26, 145–171 (1947).

Artal, P.

Berny, F.

F. Berny, S. Slansky, “Wavefront determination resulting from Foucault test as applied to the human eye and visual instruments,” in Optical Instruments and Techniques, J. H. Dickenson, ed. (Oriel, Newcastle, UK, 1969), pp. 375–386.

Bescós, J.

Bille, J. F.

Born, M.

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

Burns, S. A.

Charman, W. N.

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 231–273.

Elsner, A. E.

Fienup, J. R.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 231–273.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1992).

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goelz, S.

Goodman, J. W.

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

Green, D. G.

P. Artal, S. Marcos, I. Iglesias, D. G. Green, “Optical modulation transfer and contrast sensitivity with decentered small pupils in the human eye,” Vision Res. 36, 3575–3586 (1996).
[CrossRef] [PubMed]

P. Artal, I. Iglesias, N. López-Gil, D. G. Green, “Double-pass measurements of the retinal-image quality with unequal entrance and exit pupil sizes and the reversibility of the eye’s optical system,” J. Opt. Soc. Am. A 12, 2358–2366 (1995).
[CrossRef]

Grimm, B.

He, J. C.

Howland, B.

Howland, H. C.

Iglesias, I.

I. Iglesias, N. López-Gil, P. Artal, “Reconstruction of the point-spread function of the human eye from two double-pass retinal images by phase retrieval algorithms,” J. Opt. Soc. Am. A 15, 326–339 (1998).
[CrossRef]

F. Vargas, I. Iglesias, P. Artal, “Images of the human fovea after correction of the ocular aberrations with a liquid crystal spatial light modulator,” Invest. Ophthalmol. Visual Sci. Suppl. 13, 513 (1997).

P. Artal, S. Marcos, I. Iglesias, D. G. Green, “Optical modulation transfer and contrast sensitivity with decentered small pupils in the human eye,” Vision Res. 36, 3575–3586 (1996).
[CrossRef] [PubMed]

P. Artal, I. Iglesias, N. López-Gil, D. G. Green, “Double-pass measurements of the retinal-image quality with unequal entrance and exit pupil sizes and the reversibility of the eye’s optical system,” J. Opt. Soc. Am. A 12, 2358–2366 (1995).
[CrossRef]

Ivanoff, A.

A. Ivanoff, “Les aberrations de chromatisme et de sphericite de l’oeil. Leur role en vison nocturne,” Rev. Opt. Theor. Instrum. 26, 145–171 (1947).

Konstantinides, K.

K. Konstantinides, J. R. Rasure, “The khoros software development environment for image and signal processing,” IEEE Trans. Image Process. 3, 243–252 (1994).
[CrossRef]

Koomen, M.

Liang, J.

López-Gil, N.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).

Marcos, S.

P. Artal, S. Marcos, I. Iglesias, D. G. Green, “Optical modulation transfer and contrast sensitivity with decentered small pupils in the human eye,” Vision Res. 36, 3575–3586 (1996).
[CrossRef] [PubMed]

P. Artal, S. Marcos, R. Navarro, D. R. Williams, “Odd aberrations and double-pass measurements of retinal image quality,” J. Opt. Soc. Am. A 12, 195–201 (1995).
[CrossRef]

Marquardt, D.

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” IMA J. Appl. Math. 11, 431–441 (1963).

Miller, D. T.

Navarro, R.

Noll, R. J.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1992).

Prieto, P.

Rasure, J. R.

K. Konstantinides, J. R. Rasure, “The khoros software development environment for image and signal processing,” IEEE Trans. Image Process. 3, 243–252 (1994).
[CrossRef]

Rodier, F.

Santamari´a, J.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Scoluik, R.

Slansky, S.

F. Berny, S. Slansky, “Wavefront determination resulting from Foucault test as applied to the human eye and visual instruments,” in Optical Instruments and Techniques, J. H. Dickenson, ed. (Oriel, Newcastle, UK, 1969), pp. 375–386.

Southwell, W. H.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1992).

Tousey, R.

Vargas, F.

F. Vargas, I. Iglesias, P. Artal, “Images of the human fovea after correction of the ocular aberrations with a liquid crystal spatial light modulator,” Invest. Ophthalmol. Visual Sci. Suppl. 13, 513 (1997).

Vargas-Martin, F.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1992).

Walsh, G.

Williams, D. R.

Wolf, E.

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

Wu, S.

Appl. Opt. (1)

IEEE Trans. Image Process. (1)

K. Konstantinides, J. R. Rasure, “The khoros software development environment for image and signal processing,” IEEE Trans. Image Process. 3, 243–252 (1994).
[CrossRef]

IMA J. Appl. Math. (1)

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” IMA J. Appl. Math. 11, 431–441 (1963).

Invest. Ophthalmol. Visual Sci. Suppl. (1)

F. Vargas, I. Iglesias, P. Artal, “Images of the human fovea after correction of the ocular aberrations with a liquid crystal spatial light modulator,” Invest. Ophthalmol. Visual Sci. Suppl. 13, 513 (1997).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (10)

S. A. Burns, S. Wu, J. C. He, A. E. Elsner, “Variations in photoreceptor directionality across the central retina,” J. Opt. Soc. Am. A 14, 2033–2040 (1997).
[CrossRef]

F. Vargas-Martin, P. Prieto, P. Artal, “Correction of the aberrations in the human eye with a liquid-crystal spatial light modulator: limits to performance,” J. Opt. Soc. Am. A 15, 2552–2562 (1998).
[CrossRef]

J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
[CrossRef]

G. Walsh, W. N. Charman, H. C. Howland, “Objective technique for the determination of monochromatic aberrations of the human eye,” J. Opt. Soc. Am. A 1, 987–992 (1984).
[CrossRef] [PubMed]

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

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]

P. Artal, J. Santamarı́a, J. Bescós, “Retrieval of wave aberration of human eyes from actual point-spread-function data,” J. Opt. Soc. Am. A 5, 1201–1206 (1988).
[CrossRef] [PubMed]

P. Artal, S. Marcos, R. Navarro, D. R. Williams, “Odd aberrations and double-pass measurements of retinal image quality,” J. Opt. Soc. Am. A 12, 195–201 (1995).
[CrossRef]

P. Artal, I. Iglesias, N. López-Gil, D. G. Green, “Double-pass measurements of the retinal-image quality with unequal entrance and exit pupil sizes and the reversibility of the eye’s optical system,” J. Opt. Soc. Am. A 12, 2358–2366 (1995).
[CrossRef]

I. Iglesias, N. López-Gil, P. Artal, “Reconstruction of the point-spread function of the human eye from two double-pass retinal images by phase retrieval algorithms,” J. Opt. Soc. Am. A 15, 326–339 (1998).
[CrossRef]

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Optom. Vision Sci. (1)

W. N. Charman, “Wavefront aberration of the eye: a review,” Optom. Vision Sci. 68, 574–583 (1991).
[CrossRef]

Rev. Opt. Theor. Instrum. (1)

A. Ivanoff, “Les aberrations de chromatisme et de sphericite de l’oeil. Leur role en vison nocturne,” Rev. Opt. Theor. Instrum. 26, 145–171 (1947).

Vision Res. (1)

P. Artal, S. Marcos, I. Iglesias, D. G. Green, “Optical modulation transfer and contrast sensitivity with decentered small pupils in the human eye,” Vision Res. 36, 3575–3586 (1996).
[CrossRef] [PubMed]

Other (6)

F. Berny, S. Slansky, “Wavefront determination resulting from Foucault test as applied to the human eye and visual instruments,” in Optical Instruments and Techniques, J. H. Dickenson, ed. (Oriel, Newcastle, UK, 1969), pp. 375–386.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 231–273.

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

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1992).

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

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the pyramidal strategy for an optimized phase retrieval (LM pyramidal algorithm). The panels on the right are 64 × 64 pixel central sections of different subsampled versions of the PSF. The panels on the left are the associated WA maps, for pupil and normalization radius of 128, 96, and 64 pixels, respectively (from top to bottom), within an image size of 256 × 256 pixels.

Fig. 2
Fig. 2

Sample of results of the whole process of WA retrieval in subject AP. Double-pass images and PSF’s subtend 19.8 arcmin. The WA map corresponds to a 4-mm-pupil diameter. (a), (b) Experimental double-pass images recorded with symmetric (4–4 mm) and asymmetric (4–1.5-mm-) diameter pupil size configurations, respectively. (c) Reconstructed PSF from the double-pass images of (a) and (b). (d) Map of the retrieved WA modeled with the first 15 terms of the Zernike polynomial expansion. The gray scale was adjusted between the P–V values. (e) PSF associated with the retrieved WA. (f), (g) Pair of double-pass images computed from the retrieved WA, to be compared with the experimental double-pass images (a) and (b), respectively.

Fig. 3
Fig. 3

(a) Comparison between the PSF’s (horizontal sections). The solid curve corresponds to the actual data [Fig. 2(c)]; the dashed curve, to the PSF associated with the retrieved WA [Fig. 2(e)]. (b) Comparison of the averaged radial profiles of the modulation transfer functions (MTF’s). The solid curve is the MTF computed from the double-pass retinal image [Fig. 2(a)]; the dashed curve, the MTF computed from the retrieved WA.

Fig. 4
Fig. 4

Two samples of complete results in subject PA corresponding to two different pairs of double-pass images recorded under the same experimental conditions. [(a), (b)], [(h), (i)]: Pairs of experimental double-pass images registered with symmetric pupil configuration of 4–4-mm diameter and asymmetric pupil configuration of 4–1.5-mm diameter. (c), (j): PSF’s reconstructed from double-pass images: (d), (k): Maps of retrieved WA’s, represented without the coefficients of tilt as contour plots with line steps of 0.1 λ. Dashed curves represent negative values of the aberration. (e), (l): PSF’s associated with the WA estimates. [(f), (g)], [(m), (n)]: Double-pass images computed from the retrieved WA’s. These images should be compared with the experimental images of panels [(a), (b)], [(h), (i)].

Fig. 5
Fig. 5

WA results in the five subjects considered. For each subject (identified by initials on the left), the input PSF (left-hand panel), the retrieved WA (central panel), and the associated PSF (right-hand panel) are represented. The WA’s are in contour line graphs (without coefficients of tilt), with 0.1-λ separation among adjacent lines and negative values represented in dashed curves.

Fig. 6
Fig. 6

Values of the Zernike coefficient (a7a15) for the five subjects in number of λ.

Fig. 7
Fig. 7

Comparison of the radial profiles of the MTF’s in four subjects (identified by initials). The solid curve is the MTF computed from the double-pass retinal image; the dashed curve, the MTF computed from the retrieved WA.

Tables (5)

Tables Icon

Table 1 Zernike Polynomial Expansion (First 15 Terms) Used in This Study

Tables Icon

Table 2 Zernike Coefficients (in λ) Obtained from Two Pairs of Double-Pass Images in Subject PAa

Tables Icon

Table 3 Zernike Coefficients (in λ) for the Five Subjects Participating in This Studya

Tables Icon

Table 4 Values (in λ) of the Seidel Spherical and Coma Aberration for the Five Subjects, Computed from Eqs. (9) with the Zernike Coefficients of Table 3

Tables Icon

Table 5 Values of Strehl Ratio, P–V Values, and Root-Mean-Square Error in the WA for Every Subjecta

Equations (10)

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

P(x)=FTm(ν)expi2πλW(ν)2,
W(ν, a)=kakZk(r, θ),
G(x, a)=FTm(ν)expi2πλW(ν, a),
E(a)=x[|A(x)|-|G(x, a)|]2,
E(b)E(a)-kβkδk+klαklδkδl,
βk=-12E(a)ak,αkl=122E(a)akal.
lαklδl=βk.
αkl=αkl(kl),αkk=αkk(1+),
Ac=38(a72+a82),
As=65a11.

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