Abstract

The wave aberration of human eyes is retrieved from actual point-spread-function (PSF) data and the modulus of the pupil function. The PSF had been obtained previously by application of a hybrid optical–digital method developed recently. The retrieval is done by using a bidimensional Gerchberg–Saxton phase-retrieval algorithm joined to an iterative phase-unwrapping algorithm. To obtain an adequate convergence, the initial wave aberration for starting the retrieval–unwrapping algorithm is estimated with a nonlinear least-squares algorithm. The resulting wave aberrations for several subjects show irregular aberrations superimposed upon the regular wave-aberration components, with astigmatism being the most important asymmetric aberration.

© 1988 Optical Society of America

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References

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1987 (3)

1986 (2)

M. Nieto-Vesperinas, “A study of the performance of nonlinear least-square optimization methods in problems of phase retrieval,” Opt. Acta 33, 713–722 (1986).
[CrossRef]

R. W. Gerchberg, “The lock problem in the Gerschberg–Saxton algorithm for phase retrieval,” Optik 74, 91–93 (1986).

1985 (1)

1984 (2)

1981 (2)

1980 (1)

1978 (1)

1977 (3)

1976 (1)

1974 (1)

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

1972 (2)

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).

A. Van Meeteren, “Calculations on the optical modulation transfer function of the human eye for white light,” Opt. Acta 21, 395–412 (1972).
[CrossRef]

1961 (1)

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 6, 776–794 (1961).

Artal, P.

Barakat, R.

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. Home Dickson, ed. (Oriel, London, 1969), pp. 375–385.

Bescos, J.

Bescós, J.

Bezdid’ko, S. N.

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970).

Charman, W. N.

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg, “The lock problem in the Gerschberg–Saxton algorithm for phase retrieval,” Optik 74, 91–93 (1986).

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).

Ghiglia, D. C.

Gonsalves, R. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, London, 1950).

Howland, B.

Howland, H. C.

Jacoby, S. L. S.

S. L. S. Jacoby, Iterative Methods for Non-Linear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

Jennings, J. A. M.

J. A. M. Jennings, W. N. Charman, “Off-axis quality in the human eye,” Vision Res. 21, 445–455 (1981).
[CrossRef]

Maeda, J.

Mastin, G. A.

Murata, K.

Nahrstedt, D. A.

Navarro, R.

Newsam, G.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, “A study of the performance of nonlinear least-square optimization methods in problems of phase retrieval,” Opt. Acta 33, 713–722 (1986).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Romero, L. A.

Santamaría, 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).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

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. Home Dickson, ed. (Oriel, London, 1969), pp. 375–385.

Smirnov, M. S.

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 6, 776–794 (1961).

Southwell, W. H.

Tribolet, J. M.

J. M. Tribolet, “A new phase unwrapping algorithm,”IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
[CrossRef]

Van Meeteren, A.

A. Van Meeteren, “Calculations on the optical modulation transfer function of the human eye for white light,” Opt. Acta 21, 395–412 (1972).
[CrossRef]

Walsh, G.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970).

Appl. Opt. (2)

Biophysics (1)

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 6, 776–794 (1961).

IEEE Trans. Acoust. Speech Signal Process. (1)

J. M. Tribolet, “A new phase unwrapping algorithm,”IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (5)

Opt. Acta (2)

M. Nieto-Vesperinas, “A study of the performance of nonlinear least-square optimization methods in problems of phase retrieval,” Opt. Acta 33, 713–722 (1986).
[CrossRef]

A. Van Meeteren, “Calculations on the optical modulation transfer function of the human eye for white light,” Opt. Acta 21, 395–412 (1972).
[CrossRef]

Opt. Lett. (1)

Optik (2)

R. W. Gerchberg, “The lock problem in the Gerschberg–Saxton algorithm for phase retrieval,” Optik 74, 91–93 (1986).

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).

Sov. J. Opt. Technol. (1)

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Vision Res. (1)

J. A. M. Jennings, W. N. Charman, “Off-axis quality in the human eye,” Vision Res. 21, 445–455 (1981).
[CrossRef]

Other (6)

S. L. S. Jacoby, Iterative Methods for Non-Linear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

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. Home Dickson, ed. (Oriel, London, 1969), pp. 375–385.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970).

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, London, 1950).

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Figures (5)

Fig. 1
Fig. 1

Isometric plots of PSF data for subjects (a) PA, (b) MA, and (c) JS.

Fig. 2
Fig. 2

Block diagram of the iterative wave-aberration-retrieval–unwrapping algorithm. FT, Fourier transform; MSE, mean-square error; ASF, amplitude-spread function.

Fig. 3
Fig. 3

Contour plots at λ/2 intervals of the retrieved Wave aberration for subjects (a) PA, (b) MA, and (c) JS.

Fig. 4
Fig. 4

Comparisons of a section of the actual PSF (line with open circles) and the resulting PSF (plain solid line) obtained with the retrieved wave aberration for subjects (a) PA, (b) MA, and (c) JS.

Fig. 5
Fig. 5

Isometric plot of the evolution of sections of the PSF (y) distributed along the visual axis (z) for subject MA.

Tables (1)

Tables Icon

Table 1 Zernike Coefficients of the Regular Aberrations of the Retrieved Wave Aberrations

Equations (5)

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p ( α , β ) = exp [ ( i 2 π / λ ) W ( α , β ) ] , α 2 + β 2 < 1 = 0 , α 2 + β 2 1 ,
P s ( x , y ) = | α 2 + β 2 1 exp [ ( i 2 π / λ ) W ( α , β ) ] × exp [ i 2 π λ f ( α x + β y ) ] d α d β | 2 ,
S ( a i ) = | P s ( x , y ) P s ( x , y ) | 2 d x d y | P s ( x , y ) | 2 d x d y
W ( α , β , a i ) = a 1 5 ( 6 r 4 6 r 2 + 1 ) + a 2 3 ( 2 r 2 1 ) + a 3 6 ( α 2 β 2 ) + a 4 2 6 α β + a 5 8 ( α 2 3 β 2 ) α + a 6 8 ( β 2 3 α 2 ) β + a 7 8 ( 3 r 2 2 ) α + a 8 8 ( 3 r 2 2 ) β ,
= | P s ( x , y ) P s ( x , y ) | 2 d x d y | P s ( x , y ) | 2 d x d y .

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