Abstract

Phase diversity allows one to use multiple images with known phase changes such as defocus to learn the optical characteristics of an imaging medium and to estimate the unknown object. It is shown that the method can be extended to the case in which the optical system is nonisoplanatic; thus it can recover both the extended object and the angle-dependent, aberrating phase of the medium. The technique is a timely extension of the conventional phase-diversity concept and could be used to solve the isoplanatic patch problem of adaptive optics, to determine a spatially varying point-spread function in image restoration, and to image through a single-fiber optic.

© 1994 Optical Society of America

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References

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  1. R. Gonsalves, R. Chidlaw, Proc. Soc. Photo-Opt. Instrum. Eng. 207, 32 (1979).
  2. R. Gonsalves, Proc. Soc. Photo-Opt. Instrum. Eng. 249, 153 (1980).
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    [CrossRef]
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    [CrossRef]
  6. R. G. Paxman, T. J. Schulz, J. R. Fienup, J. Opt. Soc. Am. A 9, 1072 (1992).
    [CrossRef]
  7. B. C. McCallum, J. M. Rodenbury, J. Opt. Soc. Am. A 10, 231 (1993).
    [CrossRef]
  8. R. G. Paxman, J. H. Seldin, “Fine-resolution imaging of solar features using phase-diverse speckle imaging,” presented at the 13th Sacramento Peak Summer Workshop, Sunspot, N.M., September 1992.
  9. R. A. Gonsalves, “Nonisoplanatic imaging of solar data by phase diversity,” presented at the 13th Sacramento Peak Summer Workshop, Sunspot, N.M., September 1992.
  10. R. Kendrick, “A phase wavefront sensor for adaptive optics applications,” presented at the 13th Sacramento Peak Summer Workshop, Sunspot, N.M., September 1992.
  11. J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, Nature (London) 348, 221 (1990).
    [CrossRef]
  12. V. Yu. Ivanov, V. P. Sivokon, M. A. Vorontsov, J. Opt. Soc. Am. A 9, 1515 (1992).
    [CrossRef]
  13. D. L. Fried, J. Opt. Soc. Am. 72, 52 (1982).
    [CrossRef]

1993 (1)

1992 (2)

1990 (2)

C. Roddier, F. Roddier, J. Opt. Soc. Am. A 7, 1824 (1990).
[CrossRef]

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, Nature (London) 348, 221 (1990).
[CrossRef]

1988 (1)

1982 (2)

R. Gonsalves, Opt. Eng. 21, 829 (1982).

D. L. Fried, J. Opt. Soc. Am. 72, 52 (1982).
[CrossRef]

1980 (1)

R. Gonsalves, Proc. Soc. Photo-Opt. Instrum. Eng. 249, 153 (1980).

1979 (1)

R. Gonsalves, R. Chidlaw, Proc. Soc. Photo-Opt. Instrum. Eng. 207, 32 (1979).

Angel, J. R. P.

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, Nature (London) 348, 221 (1990).
[CrossRef]

Chidlaw, R.

R. Gonsalves, R. Chidlaw, Proc. Soc. Photo-Opt. Instrum. Eng. 207, 32 (1979).

Fienup, J. R.

Fried, D. L.

Gonsalves, R.

R. Gonsalves, Opt. Eng. 21, 829 (1982).

R. Gonsalves, Proc. Soc. Photo-Opt. Instrum. Eng. 249, 153 (1980).

R. Gonsalves, R. Chidlaw, Proc. Soc. Photo-Opt. Instrum. Eng. 207, 32 (1979).

Gonsalves, R. A.

R. A. Gonsalves, “Nonisoplanatic imaging of solar data by phase diversity,” presented at the 13th Sacramento Peak Summer Workshop, Sunspot, N.M., September 1992.

Ivanov, V. Yu.

Kendrick, R.

R. Kendrick, “A phase wavefront sensor for adaptive optics applications,” presented at the 13th Sacramento Peak Summer Workshop, Sunspot, N.M., September 1992.

Lloyd-Hart, M.

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, Nature (London) 348, 221 (1990).
[CrossRef]

McCallum, B. C.

Paxman, R. G.

R. G. Paxman, T. J. Schulz, J. R. Fienup, J. Opt. Soc. Am. A 9, 1072 (1992).
[CrossRef]

R. G. Paxman, J. R. Fienup, J. Opt. Soc. Am. A 5, 914 (1988).
[CrossRef]

R. G. Paxman, J. H. Seldin, “Fine-resolution imaging of solar features using phase-diverse speckle imaging,” presented at the 13th Sacramento Peak Summer Workshop, Sunspot, N.M., September 1992.

Roddier, C.

Roddier, F.

Rodenbury, J. M.

Sandler, D.

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, Nature (London) 348, 221 (1990).
[CrossRef]

Schulz, T. J.

Seldin, J. H.

R. G. Paxman, J. H. Seldin, “Fine-resolution imaging of solar features using phase-diverse speckle imaging,” presented at the 13th Sacramento Peak Summer Workshop, Sunspot, N.M., September 1992.

Sivokon, V. P.

Vorontsov, M. A.

Wizinowich, P.

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, Nature (London) 348, 221 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nature (1)

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, Nature (London) 348, 221 (1990).
[CrossRef]

Opt. Eng. (1)

R. Gonsalves, Opt. Eng. 21, 829 (1982).

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

R. Gonsalves, R. Chidlaw, Proc. Soc. Photo-Opt. Instrum. Eng. 207, 32 (1979).

R. Gonsalves, Proc. Soc. Photo-Opt. Instrum. Eng. 249, 153 (1980).

Other (3)

R. G. Paxman, J. H. Seldin, “Fine-resolution imaging of solar features using phase-diverse speckle imaging,” presented at the 13th Sacramento Peak Summer Workshop, Sunspot, N.M., September 1992.

R. A. Gonsalves, “Nonisoplanatic imaging of solar data by phase diversity,” presented at the 13th Sacramento Peak Summer Workshop, Sunspot, N.M., September 1992.

R. Kendrick, “A phase wavefront sensor for adaptive optics applications,” presented at the 13th Sacramento Peak Summer Workshop, Sunspot, N.M., September 1992.

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Figures (2)

Fig. 1
Fig. 1

x-displaced point sources and their nonisoplanatic PSF’s: (a) eight x-displaced sources, (b) in-focus PSF’s, (c) eight x-displaced sources, (d) out-of-focus PSF’s.

Fig. 2
Fig. 2

Example of nonisoplanatic phase-diversity imaging: (a) original object, (b) in-focus image, (c) estimated object, (d) out-of-focus image.

Tables (1)

Tables Icon

Table 1 Actual and Estimated Parameters of the Phase in Eq. (12)

Equations (12)

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i ( x ) = o ( σ ) p ( x , σ ) d σ ,
p ( x , σ ) = h ( x , σ ) 2 = | A ( u ) exp [ j θ ( u , σ ) ] exp ( j 2 π x u ) d u | 2 .
θ ( u , σ ) = θ T ( u , σ ) + θ k ( u ) ,
I k ( u ) = O ( u ) P k ( u ) .
Z 1 ( u ) = I 1 ( u ) + N 1 ( u ) , Z 2 ( u ) = I 2 ( u ) + N 2 ( u ) ,
O ^ ( u ) = Z 1 ( u ) P ^ 1 * ( u ) + Z 2 ( u ) P ^ 2 * ( u ) P ^ 1 ( u ) P ^ 1 * ( u ) + P ^ 2 ( u ) P ^ 2 * ( u ) ,
Z ^ 1 ( u ) = O ^ ( u ) P ^ 1 ( u ) , Z ^ 2 ( u ) = O ^ ( u ) P ^ 2 ( u ) .
mse = [ Z ^ 1 ( u ) - Z 1 ( u ) 2 + Z ^ 2 ( u ) - Z 2 ( u ) 2 ] d u .
Z = [ Z 1 Z 2 ] = [ P 1 O + N 1 P 2 O + N 2 ] = PO + N ,
O ^ = ( P ^ P ^ T ) - 1 P ^ T Z ,
mse = Z - P ^ O ^ 2 ,
θ ( u , σ ) = 2 π [ ( p 1 + p 2 σ ) u + ( p 3 + p 4 σ ) ( 2 u 2 - 1 ) + ( p 5 + p 6 σ ) ( 2 u 3 - u ) ] .

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