Abstract

In the double-pass technique used to measure the optical performance of the eye, the double-pass image is the cross correlation of the input spread function with the output spread function [J. Opt. Soc. Am. A 12, 195 (1995)]. When entrance and exit pupil sizes are equal, the information on the point-spread function is lost from the double-pass image, although the modulation transfer function of the eye is obtained. A modification of the double-pass technique that uses unequal-sized entrance and exit pupils allows a low-resolution version of the ocular point-spread function to be recorded [J. Opt. Soc. Am. A 12, 2358 (1995)]. We propose the combined use of these two double-pass measurements as input in a phase-retrieval procedure to reconstruct the ocular point-spread function. We use an adapted version of the iterative Fourier-transform algorithm consisting of two steps. In the first step, error-reduction iterations with expanding weighting functions in the Fourier domain yield an estimation of the phase that serves as an initial guess for the second step, which consists of cycles of hybrid input–output iterations. We tested the robustness and limitations of the retrieval algorithm by using simulated data with and without noise. We then applied the procedure to reconstruct the point-spread function from actual measurements of double-pass retinal images in the living eye.

© 1998 Optical Society of America

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References

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  1. W. N. Charman, “Optics of the eye,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Chap. 24.
  2. M. F. Flamant, “Etude de la repartition de lumière dans l’image rétinienne d’une fente,” Rev. Opt. 34, 433–459 (1955).
  3. F. W. Campbell, R. W. Gubisch, “Optical image quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).
  4. J. Santamarı́a, P. Artal, J. Bescós, “Determination of the point-spread function of the human eye using a hybrid optical-digital method,” J. Opt. Soc. Am. A 4, 1109–1114 (1987).
    [CrossRef]
  5. P. Artal, R. Navarro, “Monochromatic modulation transfer function of the human eye for different pupil diameters: an analytical expression,” J. Opt. Soc. Am. A 11, 246–249 (1994).
    [CrossRef]
  6. R. Navarro, P. Artal, D. R. Williams, “Modulation transfer of the human eye as a function of retinal eccentricity,” J. Opt. Soc. Am. A 10, 201–212 (1993).
    [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. 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.
  10. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  11. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A 7, 450–458 (1990).
    [CrossRef]
  14. H. C. Howland, B. Howland, “A subjective method for the measurement of monochromatic aberrations of the eye,” J. Opt. Soc. Am. A 67, 1508–1518 (1977).
    [CrossRef]
  15. 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]
  16. J. Liang, B. Grimm, S. Goelz, J. Bille, “Objective measurement of the wave aberrations of the human eye using a Hartmann–Shack wavefront sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
    [CrossRef]
  17. 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]
  18. J. R. Fienup, T. R. Crimminis, W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,” J. Opt. Soc. Am. 72, 610–624 (1982).
    [CrossRef]
  19. T. R. Crimminis, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3–13 (1990).
    [CrossRef]
  20. B. C. McCallum, R. H. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensity,” J. Mod. Opt. 36, 619–648 (1989).
    [CrossRef]
  21. K. Konstantinides, J. R. Rasure, “The khoros software development environment for image and signal processing,” IEEE Trans. Image Process. 3, 243–252 (1994).
    [CrossRef]
  22. W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).
  23. P. Artal, J. Santamarı́a, J. Bescós, “Retrieval of the wave aberration of human eyes from actual point-spread function data,” J. Opt. Soc. Am. A 5, 1201–1206 (1988).
    [CrossRef] [PubMed]
  24. I. Iglesias, P. Artal, “Method to estimate the ocular wavefront aberration from a pair of double pass retinal images,” in Vision Science and Its Applications, Vol. 1 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 24–27.

1996

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

1994

1993

1990

1989

B. C. McCallum, R. H. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensity,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

1988

1987

1984

1982

1978

1977

H. C. Howland, B. Howland, “A subjective method for the measurement of monochromatic aberrations of the eye,” J. Opt. Soc. Am. A 67, 1508–1518 (1977).
[CrossRef]

1972

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

1966

F. W. Campbell, R. W. Gubisch, “Optical image quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

1955

M. F. Flamant, “Etude de la repartition de lumière dans l’image rétinienne d’une fente,” Rev. Opt. 34, 433–459 (1955).

Artal, P.

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]

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]

P. Artal, R. Navarro, “Monochromatic modulation transfer function of the human eye for different pupil diameters: an analytical expression,” J. Opt. Soc. Am. A 11, 246–249 (1994).
[CrossRef]

R. Navarro, P. Artal, D. R. Williams, “Modulation transfer of the human eye as a function of retinal eccentricity,” J. Opt. Soc. Am. A 10, 201–212 (1993).
[CrossRef] [PubMed]

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

J. Santamarı́a, P. Artal, J. Bescós, “Determination of the point-spread function of the human eye using a hybrid optical-digital method,” J. Opt. Soc. Am. A 4, 1109–1114 (1987).
[CrossRef]

I. Iglesias, P. Artal, “Method to estimate the ocular wavefront aberration from a pair of double pass retinal images,” in Vision Science and Its Applications, Vol. 1 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 24–27.

Bates, R. H.

B. C. McCallum, R. H. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensity,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

Bescós, J.

Bille, J.

Campbell, F. W.

F. W. Campbell, R. W. Gubisch, “Optical image quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

Charman, W. N.

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]

W. N. Charman, “Optics of the eye,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Chap. 24.

Crimminis, T. R.

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.

Fienup, J. R.

Flamant, M. F.

M. F. Flamant, “Etude de la repartition de lumière dans l’image rétinienne d’une fente,” Rev. Opt. 34, 433–459 (1955).

Gerchberg, R. W.

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

Goelz, S.

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.

Gubisch, R. W.

F. W. Campbell, R. W. Gubisch, “Optical image quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

Holsztynski, W.

Howland, B.

H. C. Howland, B. Howland, “A subjective method for the measurement of monochromatic aberrations of the eye,” J. Opt. Soc. Am. A 67, 1508–1518 (1977).
[CrossRef]

Howland, H. C.

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]

H. C. Howland, B. Howland, “A subjective method for the measurement of monochromatic aberrations of the eye,” J. Opt. Soc. Am. A 67, 1508–1518 (1977).
[CrossRef]

Iglesias, I.

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]

I. Iglesias, P. Artal, “Method to estimate the ocular wavefront aberration from a pair of double pass retinal images,” in Vision Science and Its Applications, Vol. 1 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 24–27.

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]

Kowalczyk, A. M.

Liang, J.

López-Gil, N.

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]

McCallum, B. C.

B. C. McCallum, R. H. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensity,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

Navarro, R.

Pratt, W. K.

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

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]

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 (Stuttgart) 35, 237–246 (1972).

Thelen, B. J.

Walsh, G.

Williams, D. R.

Appl. Opt.

IEEE Trans. Image Process

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. Mod. Opt.

B. C. McCallum, R. H. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensity,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

T. R. Crimminis, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3–13 (1990).
[CrossRef]

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A 7, 450–458 (1990).
[CrossRef]

H. C. Howland, B. Howland, “A subjective method for the measurement of monochromatic aberrations of the eye,” J. Opt. Soc. Am. A 67, 1508–1518 (1977).
[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. Bille, “Objective measurement of the wave aberrations of the human eye using a Hartmann–Shack wavefront sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
[CrossRef]

J. Santamarı́a, P. Artal, J. Bescós, “Determination of the point-spread function of the human eye using a hybrid optical-digital method,” J. Opt. Soc. Am. A 4, 1109–1114 (1987).
[CrossRef]

P. Artal, R. Navarro, “Monochromatic modulation transfer function of the human eye for different pupil diameters: an analytical expression,” J. Opt. Soc. Am. A 11, 246–249 (1994).
[CrossRef]

R. Navarro, P. Artal, D. R. Williams, “Modulation transfer of the human eye as a function of retinal eccentricity,” J. Opt. Soc. Am. A 10, 201–212 (1993).
[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]

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

J. Physiol. (London)

F. W. Campbell, R. W. Gubisch, “Optical image quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

Opt. Lett.

Optik (Stuttgart)

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

Rev. Opt.

M. F. Flamant, “Etude de la repartition de lumière dans l’image rétinienne d’une fente,” Rev. Opt. 34, 433–459 (1955).

Vision Res.

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

W. N. Charman, “Optics of the eye,” in Handbook of Optics, 2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Chap. 24.

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.

I. Iglesias, P. Artal, “Method to estimate the ocular wavefront aberration from a pair of double pass retinal images,” in Vision Science and Its Applications, Vol. 1 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 24–27.

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

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Figures (10)

Fig. 1
Fig. 1

(a) Flow chart of the iterative Fourier-transform algorithm of ER or IO. The restrictions imposed in the Fourier and diffraction planes are computed from the double-pass images (see text for details). (b) Flow chart of the iterative Fourier-transform algorithm for obtaining an initial estimation of the PTF for the hybrid algorithm. (c) Schematic diagram of the complete method for reconstructing the PTF. BLOCK 1 consists of ER iterations with sequential phase retrieval and forced modulus to obtain an estimation of the PTF that serves as start for the second part of the algorithm (BLOCK 2) consisting of the hybrid algorithm with the correct MTF.

Fig. 2
Fig. 2

Simulated data to test the reconstruction algorithm. (a) PSF test pD(x) computed from a pure coma aberration with a pupil diameter of 128 pixels (in a window of 256 pixels). (b) iD(x), autocorrelation of the PSF pD(x). Each image is represented in a contour line graph subtending a central region of 32×32 pixels extracted from the full 256×256 pixels. (c) MTF MD(u) represented in a contour map (256×256 pixel image). (d) Principal value of the PTF FD(u) represented in a gray-level image. The cutoff frequency uD is 128 pixels.

Fig. 3
Fig. 3

(a) PSF pd(x) for the small pupil computed from the same coma aberration as for Fig. 2(a) but with a pupil diameter of 48 pixels (in a window of 256 pixels). (b) id(x), convolution of (a) pd(x) and [Fig. 2(a)] pD(x). Each image is represented in a contour line graph subtending a central region of 32×32 pixels extracted from the full 256×256 pixels. (c) Low-spatial-frequency estimation of the PTF Fd(u) computed from id(x). The cutoff frequency uD is 48 pixels.

Fig. 4
Fig. 4

Results obtained in the noise-free simulation (see text for additional details). (a) PSF and (b) associated PTF obtained by use of only BLOCK 2 of the algorithm with a constant phase as the initial guess. (c) PSF and (d) associated PTF obtained with only BLOCK 1 of the algorithm. The PTF in (d) is used as the initial guess for BLOCK 2 in the complete reconstruction procedure. (e) Reconstructed PSF and (f) associated PTF obtained by applying the complete algorithm.

Fig. 5
Fig. 5

(a) Diffraction-limited MTF for a 4-mm pupil diameter (long-dashed curve). Radially averaged MTF’s obtained from actual double-pass images in two eyes (short-curves dashed and dotted curve). Radially averaged MTF computed from the simulated double-pass image iD(x) [Fig. 2(b)] contaminated with additive noise (solid curve). (b) Radially averaged MTF computed from the simulated double-pass image iD(x) [Fig. 2(b)] without noise (dotted curve), section of the adapted Butterworth filter (solid curve), and radially averaged MTF computed from the simulated double-pass image iD(x) [Fig. 2(b)] contaminated with additive noise (long-dashed curve). The short-dashed curve shows filtered MTF’s obtained from the noise-contaminated iD(x).

Fig. 6
Fig. 6

Input data and results obtained in the simulation with noise. (a) PTF Fd(u) computed from the noise-contaminated id(x) represented in a gray-level image (256×256 pixels). (b) MTF, MD(u) (256×256 pixels) computed from the noise-contaminated iD(x). (c) Reconstructed PSF (only the central 32×32 pixels of the full image are shown).

Fig. 7
Fig. 7

Schematic diagram of the double-pass apparatus to record simultaneously a pair of double-pass images: He–NE laser (543 nm); DF, neutral-density filter; SF, spatial filter; M, mirror; L1L3, achromatic doublets; BS, BS 1, BS 2, pellicle beam splitters. BS2 splits the output beam into two paths, one with a effective small pupil (1.5-mm diameter) and the other with a large pupil diameter (the same as in the first passage). In the full frame of the CCD camera, two images are recorded, one corresponding to the autocorrelation of the PSF iD(x), and the other to the convolution of the PSF and the near diffraction-limited pattern id(x).

Fig. 8
Fig. 8

Results for subject PA. (a) id(x) recorded with 4–1.5-mm pupil size configuration, (b) iD(x) recorded with 4–4-mm pupil size configuration. These two images are normalized to one and represented in a contour line plot. Only the central section of the full image is shown (64×64 pixels, corresponding to 14.7×14.7). (c) PTF in the interval [0, ud=48 c/deg] obtained from (a) id(x). (d) MTF (256×256 pixels, corresponding to 128 c/deg at the edge of the image) computed from iD(x). (e) Reconstructed PSF (64×64 pixels). (f) Principal value of the retrieved PTF.

Fig. 9
Fig. 9

Results for subject NL. (a) id(x) recorded with 4–1.5-mm pupil size configuration, (b) iD(x) recorded with 4–4-mm pupil size configuration. These two images are normalized to one and represented in a contour line plot. Only the central section of the full image is shown (64×64 pixels, corresponding to 14.7×14.7arcmin). (c) PTF in the interval [0, ud=48 c/deg] obtained from (a) id(x). (d) MTF (256×256 pixels, corresponding to 128 c/deg at the edge of the image) computed from iD(x). (e) Reconstructed PSF (64×64 pixels). (f) Principal value of the retrieved PTF.

Fig. 10
Fig. 10

(a) Convolution of the reconstructed PSF for subject PA with the text diffraction-limited pattern for the 1.5-mm pupil diameter (64×64 pixels). (b) Autocorrelation of the reconstructed PSF for subject PA (64×64 pixels). (c) Convolution of the reconstructed PSF for subject NL with the diffraction-limited pattern for the 1.5-mm pupil diameter (64×64 pixels). (d) Autocorrelation of the reconstructed PSF for subject NL (64×64 pixels). (e), (f) MTF’s computed from the reconstructed PSF’s for subjects PA and NL, respectively. These images should be compared with the experimental double-pass images and their associated MTF’s in Figs. 8 and 9 (see text for additional details).

Equations (10)

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

iD(x)=pD(x)pD(-x),
ID(u)=[MD(u)]2,
id(x)=pd(x)pD(-x),
Id(u)=MD(u)Ad(u)exp[-i(FD(u)],
Fd(u)=tan-1 Im[Id(u)]Re[Id(u)].
MDk(u)exp[iFDk(u)].
pDk+1(x)=pDk(x)ifxγ0ifxγ,
u|MDk(u)-MD(u)|2u|MD(u)|21/2,
pDk+1(x)=pDk(x)ifxγpDk(x)-βpDk(x)ifxγ,
x|pDt(x)-pDf(x)|2xpdt(x)21/2

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