Abstract

An object is iteratively reconstructed from two atmospherically degraded images, one of which is detected by shifting of a known amount of phase with respect to the other. The iterative reconstruction scheme is based on the ideas of the blind-deconvolution method and the error-reduction algorithm. An optical system to do phase shifting is proposed in which a deformable mirror is employed. Several computer simulations are conducted to exemplify the usefulness of our method.

© 1994 Optical Society of America

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References

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  1. A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  2. G. R. Ayers, J. C. Dainty, “An iterative blind deconvolution algorithm and its applications,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef] [PubMed]
  3. B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued images,” Opt. Commun. 69, 353–356 (1989).
    [CrossRef]
  4. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
  5. J. A. Högbom, “On the intensity distribution over the focal volume,” in High Spatial Resolution Solar Observations, proceedings of the Tenth Sacramento Peak Summer Workshop (Sacramento Peak Observatory, Sunspot, N. Mex., 1988).
  6. R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. 5, 914–923 (1988).
    [CrossRef]
  7. R. G. Paxman, T. J. Schultz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [CrossRef]
  8. N. Baba, E. Kenmochi, “Wavefront retrieval with use of defocused PSF data,” Optik 84, 70–72 (1990).
  9. R. K. Tyson, Principles of Adaptive Optics (Academic, New York, 1991), pp. 1–24.
  10. J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
    [CrossRef]
  11. 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).
  12. G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).
  13. R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
    [CrossRef]
  14. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  15. X. Shi, R. K. Ward, “Restoration of images degraded by atmospheric turbulence and detection noise,” J. Opt. Soc. Am. A 9, 364–370 (1992).
    [CrossRef]
  16. F. Merkle, “Adaptive optics developments at ESO,” in Proceedings of the ESO Conference on Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 639–656.

1992 (2)

1991 (1)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

1990 (3)

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

N. Baba, E. Kenmochi, “Wavefront retrieval with use of defocused PSF data,” Optik 84, 70–72 (1990).

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
[CrossRef]

1989 (1)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued images,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

1988 (2)

R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. 5, 914–923 (1988).
[CrossRef]

G. R. Ayers, J. C. Dainty, “An iterative blind deconvolution algorithm and its applications,” Opt. Lett. 13, 547–549 (1988).
[CrossRef] [PubMed]

1982 (2)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

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).

1970 (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Ayers, G. R.

Baba, N.

N. Baba, E. Kenmochi, “Wavefront retrieval with use of defocused PSF data,” Optik 84, 70–72 (1990).

Bates, R. H. T.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued images,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Boyer, C.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Dainty, J. C.

Davey, B. L. K.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued images,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Fienup, J. R.

Fontanella, J. C.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
[CrossRef]

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Fried, D. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Gaffard, J. P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

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).

Gigan, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

Högbom, J. A.

J. A. Högbom, “On the intensity distribution over the focal volume,” in High Spatial Resolution Solar Observations, proceedings of the Tenth Sacramento Peak Summer Workshop (Sacramento Peak Observatory, Sunspot, N. Mex., 1988).

Jagourel, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Kenmochi, E.

N. Baba, E. Kenmochi, “Wavefront retrieval with use of defocused PSF data,” Optik 84, 70–72 (1990).

Kern, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lane, R. G.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued images,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Lena, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Merkle, F.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

F. Merkle, “Adaptive optics developments at ESO,” in Proceedings of the ESO Conference on Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 639–656.

Paxman, R. G.

R. G. Paxman, T. J. Schultz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. 5, 914–923 (1988).
[CrossRef]

Primot, J.

Rigaut, F.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Rousset, G.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
[CrossRef]

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

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).

Schultz, T. J.

Shi, X.

Tyler, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, New York, 1991), pp. 1–24.

Ward, R. K.

Wopat, L. M.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (2)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

J. Opt. Soc. Am. (1)

R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. 5, 914–923 (1988).
[CrossRef]

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

Nature (London) (1)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Opt. Commun. (1)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued images,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

Opt. Lett. (1)

Optik (2)

N. Baba, E. Kenmochi, “Wavefront retrieval with use of defocused PSF data,” Optik 84, 70–72 (1990).

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).

Other (3)

F. Merkle, “Adaptive optics developments at ESO,” in Proceedings of the ESO Conference on Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 639–656.

R. K. Tyson, Principles of Adaptive Optics (Academic, New York, 1991), pp. 1–24.

J. A. Högbom, “On the intensity distribution over the focal volume,” in High Spatial Resolution Solar Observations, proceedings of the Tenth Sacramento Peak Summer Workshop (Sacramento Peak Observatory, Sunspot, N. Mex., 1988).

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Figures (5)

Fig. 1
Fig. 1

Optical system for phase-diversity imaging. A part of the incident wave from a stellar object is analyzed by a wave-front sensor at wavelength λ3. The phase diversity is delivered by a deformable mirror so as to correct the atmospheric disturbance slightly. An ordinary speckle image is detected at D1 with wavelength λ1. A speckle image with phase diversity is detected at D2 with wavelength λ2. The phase-diversity imaging is conducted by use of two speckle images and the information of the delivered phase.

Fig. 2
Fig. 2

Iterative reconstruction scheme for phase-diversity imaging. In wave-front estimation the Gerchberg–Saxton approach is employed. The iterative blind-deconvolution technique is applied to estimate an object distribution and PSF’s, in which the nonnegativity constraint is explicitly used.

Fig. 3
Fig. 3

(a) Supposed object in computer simulations and (b) its diffraction-limited image. The object is octagonally shaped and has uniform distribution (unit intensity) except for a circular dark spot (zero intensity). These are displayed on 64 × 64 pixels, and the contour maps are drawn in 10 equal-intensity levels between the maximum and the minimum values in each map.

Fig. 4
Fig. 4

Speckle images (a) without and (b) with defocus; (c) reconstructed image obtained by use of these speckle images and the information of the phase diversity, 1.0λ defocus.

Fig. 5
Fig. 5

Speckle images (a) without and (b) with phase diversity, which is based on the wave-front sensing. The image of (b) is improved slightly compared with the one of (a). (c) The reconstructed image from these speckle images. (d) The image averaged with five reconstructed images obtained by use of five pairs of speckle images. The noise is suppressed by the averaging operation.

Equations (12)

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s 1 ( x , y ) = o ( x , y ) * p 1 ( x , y ) + n 1 ( x , y ) ,
h 1 ( x , y ) = [ p 1 ( x , y ) ] 1 / 2 exp [ i ψ ( x , y ) ] ,
H 1 ( u , υ ) = A ( u , υ ) exp [ i θ ( u , υ ) ] ,
H 2 ( u , υ ) = A ( u , υ ) exp { i [ θ ( u , υ ) + Φ ( u , υ ) ] } .
s 2 ( x , y ) = o ( x , y ) * p 2 ( x , y ) + n 2 ( x , y ) ,
p 2 ( x , y ) = | h 2 ( x , y ) | 2
h j k ( x , y ) = p j k ( x , y ) 1 / 2 exp [ i ψ j k 1 ( x , y ) ] , j = 1 , 2 ,
H j k ( u , υ ) = P j k ( u , υ ) exp [ i θ j k ( u , υ ) ] .
H j k ( u , υ ) = A ( u , υ ) exp [ i θ j k ( u , υ ) ] ,
O k ( u , υ ) = ( H 1 k + ) * S 1 + ( H 2 k + ) * S 2 | H 1 k + | 2 + | H 2 k + | 2 + σ n 2 / W o ,
P j k ( u , υ ) = ( O k ) * S j | O k | 2 + σ n / W p j ,
θ i 0 ( u , υ ) = 0 .

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