Abstract

A method for reconstructing a diffraction-limited image from data consisting of many short-exposure turbulence degraded images is described. The results of a computer simulation of the method are presented. The method should prove useful for obtaining high-angular resolution images from large earth bound telescopes. The simulation indicates that the method may be applicable to astronomical objects as faint as ~11th magnitude.

© 1982 Optical Society of America

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References

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  1. P. Nisenson, R. V. Stachnik, in High Angular Resolution Stellar Interferometry, J. Davis, W. J. Tango, Eds., IAU Colloquium 50 (U. Sydney, Australia, 1979).
  2. S. P. Worden, Vistas Astron. 20, 301 (1977).
    [Crossref]
  3. J. C. Dainty, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), p. 255.
    [Crossref]
  4. A. Labeyrie, Ann. Rev. Astron. Astrophys. 16, 77 (1978).
    [Crossref]
  5. A. Labeyrie, Astron. Astrophys. 6, 85 (1970).
  6. C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 8, 372 (1973).
    [Crossref]
  7. R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).
  8. G. P. Weigelt, Appl. Opt. 17, 2660 (1978).
    [Crossref] [PubMed]
  9. K. T. Knox, B. J. Thompson, Astrophys. Lett. 193, L45 (1974).
    [Crossref]
  10. R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, Nature London 266, 149 (1977).
    [Crossref]
  11. C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
    [Crossref]
  12. J. R. Fienup, Opt. Lett. 3, 27 (1978).
    [Crossref] [PubMed]
  13. G. B. Feldkamp, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 84 (1980).
  14. J. R. Fienup, G. B. Feldkamp, Proc. Soc. Photo-Opt. Instrum. Eng. 243, 95 (1980).
  15. A. Walther, Opt. Acta 10, 41 (1963).
    [Crossref]
  16. Yu. M. Bruck, L. G. Sodin, Opt. Commun. 30, 304 (1979).
    [Crossref]
  17. A. M. J. Huiser, P. van Toorn, Opt. Lett. 5, 499 (1980).
    [Crossref] [PubMed]
  18. J. G. Walker, Opt. Acta 28, 1017 (1981).
    [Crossref]
  19. J. G. Walker, Opt. Acta 28, 735 (1981).
    [Crossref]
  20. R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).
  21. J. C. Dainty, A. H. Greenaway, J. Opt. Soc. Am. 69, 786 (1979).
    [Crossref]
  22. J. G. Walker, Opt. Acta 28, 885 (1981).
    [Crossref]

1981 (3)

J. G. Walker, Opt. Acta 28, 1017 (1981).
[Crossref]

J. G. Walker, Opt. Acta 28, 735 (1981).
[Crossref]

J. G. Walker, Opt. Acta 28, 885 (1981).
[Crossref]

1980 (3)

A. M. J. Huiser, P. van Toorn, Opt. Lett. 5, 499 (1980).
[Crossref] [PubMed]

G. B. Feldkamp, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 84 (1980).

J. R. Fienup, G. B. Feldkamp, Proc. Soc. Photo-Opt. Instrum. Eng. 243, 95 (1980).

1979 (2)

Yu. M. Bruck, L. G. Sodin, Opt. Commun. 30, 304 (1979).
[Crossref]

J. C. Dainty, A. H. Greenaway, J. Opt. Soc. Am. 69, 786 (1979).
[Crossref]

1978 (3)

1977 (2)

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, Nature London 266, 149 (1977).
[Crossref]

S. P. Worden, Vistas Astron. 20, 301 (1977).
[Crossref]

1976 (1)

C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
[Crossref]

1974 (1)

K. T. Knox, B. J. Thompson, Astrophys. Lett. 193, L45 (1974).
[Crossref]

1973 (2)

C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 8, 372 (1973).
[Crossref]

R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).

1972 (1)

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

1970 (1)

A. Labeyrie, Astron. Astrophys. 6, 85 (1970).

1963 (1)

A. Walther, Opt. Acta 10, 41 (1963).
[Crossref]

Bates, R. H. T.

R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).

Bruck, Yu. M.

Yu. M. Bruck, L. G. Sodin, Opt. Commun. 30, 304 (1979).
[Crossref]

Dainty, J. C.

J. C. Dainty, A. H. Greenaway, J. Opt. Soc. Am. 69, 786 (1979).
[Crossref]

J. C. Dainty, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), p. 255.
[Crossref]

Ehn, D. C.

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, Nature London 266, 149 (1977).
[Crossref]

Feldkamp, G. B.

G. B. Feldkamp, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 84 (1980).

J. R. Fienup, G. B. Feldkamp, Proc. Soc. Photo-Opt. Instrum. Eng. 243, 95 (1980).

Fienup, J. R.

G. B. Feldkamp, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 84 (1980).

J. R. Fienup, G. B. Feldkamp, Proc. Soc. Photo-Opt. Instrum. Eng. 243, 95 (1980).

J. R. Fienup, Opt. Lett. 3, 27 (1978).
[Crossref] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

Gough, P. T.

R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).

Greenaway, A. H.

Harvey, J. W.

C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
[Crossref]

Hudgin, R. H.

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, Nature London 266, 149 (1977).
[Crossref]

Huiser, A. M. J.

Knox, K. T.

K. T. Knox, B. J. Thompson, Astrophys. Lett. 193, L45 (1974).
[Crossref]

Labeyrie, A.

A. Labeyrie, Ann. Rev. Astron. Astrophys. 16, 77 (1978).
[Crossref]

A. Labeyrie, Astron. Astrophys. 6, 85 (1970).

Liu, C. Y. C.

C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 8, 372 (1973).
[Crossref]

Lohmann, A. W.

C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 8, 372 (1973).
[Crossref]

Lynds, C. R.

C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
[Crossref]

Napier, P. J.

R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).

Nisenson, P.

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, Nature London 266, 149 (1977).
[Crossref]

P. Nisenson, R. V. Stachnik, in High Angular Resolution Stellar Interferometry, J. Davis, W. J. Tango, Eds., IAU Colloquium 50 (U. Sydney, Australia, 1979).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

Schirf, V. E.

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, Nature London 266, 149 (1977).
[Crossref]

Sodin, L. G.

Yu. M. Bruck, L. G. Sodin, Opt. Commun. 30, 304 (1979).
[Crossref]

Stachnik, R. V.

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, Nature London 266, 149 (1977).
[Crossref]

P. Nisenson, R. V. Stachnik, in High Angular Resolution Stellar Interferometry, J. Davis, W. J. Tango, Eds., IAU Colloquium 50 (U. Sydney, Australia, 1979).

Thompson, B. J.

K. T. Knox, B. J. Thompson, Astrophys. Lett. 193, L45 (1974).
[Crossref]

van Toorn, P.

Walker, J. G.

J. G. Walker, Opt. Acta 28, 885 (1981).
[Crossref]

J. G. Walker, Opt. Acta 28, 1017 (1981).
[Crossref]

J. G. Walker, Opt. Acta 28, 735 (1981).
[Crossref]

Walther, A.

A. Walther, Opt. Acta 10, 41 (1963).
[Crossref]

Weigelt, G. P.

Worden, S. P.

S. P. Worden, Vistas Astron. 20, 301 (1977).
[Crossref]

C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
[Crossref]

Ann. Rev. Astron. Astrophys. (1)

A. Labeyrie, Ann. Rev. Astron. Astrophys. 16, 77 (1978).
[Crossref]

Appl. Opt. (1)

Astron. Astrophys. (2)

R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).

A. Labeyrie, Astron. Astrophys. 6, 85 (1970).

Astrophys. J. (1)

C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
[Crossref]

Astrophys. Lett. (1)

K. T. Knox, B. J. Thompson, Astrophys. Lett. 193, L45 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

Nature London (1)

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, Nature London 266, 149 (1977).
[Crossref]

Opt. Acta (4)

A. Walther, Opt. Acta 10, 41 (1963).
[Crossref]

J. G. Walker, Opt. Acta 28, 1017 (1981).
[Crossref]

J. G. Walker, Opt. Acta 28, 735 (1981).
[Crossref]

J. G. Walker, Opt. Acta 28, 885 (1981).
[Crossref]

Opt. Commun. (2)

Yu. M. Bruck, L. G. Sodin, Opt. Commun. 30, 304 (1979).
[Crossref]

C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 8, 372 (1973).
[Crossref]

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

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

G. B. Feldkamp, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 84 (1980).

J. R. Fienup, G. B. Feldkamp, Proc. Soc. Photo-Opt. Instrum. Eng. 243, 95 (1980).

Vistas Astron. (1)

S. P. Worden, Vistas Astron. 20, 301 (1977).
[Crossref]

Other (2)

J. C. Dainty, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), p. 255.
[Crossref]

P. Nisenson, R. V. Stachnik, in High Angular Resolution Stellar Interferometry, J. Davis, W. J. Tango, Eds., IAU Colloquium 50 (U. Sydney, Australia, 1979).

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Figures (3)

Fig. 1
Fig. 1

Block diagram of one of a number of possible iterative schemes for retrieving an object O(x,y) from measurements of |O(u,v)| and |O′(u,v)|.

Fig. 2
Fig. 2

(a) Test object with a 20-pixel diam; (b) one simulated short exposure of a point object; the diameter of the truncated Gaussian seeing disk to the 1/e2 points is 10 pixels; (c) one simulated short exposure of the test object computed by convolving the test object with point spread functions such as in (b); (d)–(g) reconstructions of the test object obtained using finite sets of data containing, respectively, 10, 25, 50, and 100 simulated object exposures and equal numbers of point object exposures. The power spectrum mode of initial data processing [Eqs. (2) and (4)] was used, and a circular top hat function with a 20-pixel diam was used as the input to the iteration. Thirty cycles of the iteration were used to obtain the reconstructions shown. The fine lines and dots within each pixel are an artifact of the microplotter and are of no real significance.

Fig. 3
Fig. 3

Reconstructions of the test object obtained using data with additive noise to simulate the effect of very low light levels. The noise levels used simulate 35-min observations using a 4-m telescope, a sample time of 10 msec, a filter bandwidth of 100 Å, a transmission through optics and filter of 50%, a 10% quantum efficiency, and objects of magnitude of 8.6, 9.3, 10.3, and 11.1, respectively. A circular top hat function with a 20-pixel diam was used as input to the iteration. Thirty cycles of the iteration were used to obtain the reconstructions shown. Fine lines and dots within each pixel are an artifact of the microplotter and are of no real significance.

Equations (18)

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i ( x , y ) = o ( x , y ) p ( x , y ) ,
W i ( u , v ) = FT [ i ( x , y ) ] 2 = FT o ( x , y ) p ( x , y ) ] 2 = O ( u , v ) 2 P ( u , v ) 2 ,
W b ( u , v ) = FT [ b ( x , y ) ] 2 = B ( u , v ) 2 = P ( u , v ) 2 ,
W i g ( u , v ) = FT [ i ( x , y ) · g ( x , y ) ] 2 = FT { [ o ( x , y ) p ( x , y ) ] · g ( x , y ) } 2 = FT { [ o ( x , y ) · g ( x , y ) ] [ p ( x , y ) · g ( x , y ) ] } 2 = O ( u , v ) G ( u , v ) 2 P ( u , v ) G ( u , v ) 2 ,
W b g ( u , v ) = FT [ b ( x , y ) · g ( x , y ) ] 2 = B ( u , v ) G ( u , v ) 2 = P ( u , v ) G ( u , v ) 2 ,
[ a ( x , y ) b ( x , y ) ] · g ( x , y ) = [ a ( x , y ) · g ( x , y ) ] [ b ( x , y ) · g ( x , y ) ] ,
O ( u , v ) = [ W i ( u , v ) / W b ( u , v ) ] 1 / 2 ,
O ( u , v ) = O ( u , v ) G ( u , v ) = [ W i g ( u , v ) / W b g ( u , v ) ] 1 / 2 .
C i ( x , y ) = i ( x , y ) * i ( x , y ) = [ o ( x , y ) p ( x , y ) ] * [ o ( x , y ) p ( x , y ) ] = [ o ( x , y ) * o ( x , y ) ] p ( x , y ) * p ( x , y ) ,
C b ( x , y ) = b ( x , y ) * b ( x , y ) = p ( x , y ) * p ( x , y ) ,
C i g ( x , y ) = [ i ( x , y ) · g ( x , y ) * [ i ( x , y ) · g ( x , y ) ] ] = { [ o ( x , y ) p ( x , y ) ] · g ( x , y ) } * { [ o ( x , y ) p ( x , y ) ] · g ( x , y ) } = { [ o ( x , y ) · g ( x , y ) ] [ p ( x , y ) · g ( x , y ) ] } * { [ o ( x , y ) · g ( x , y ) ] [ p ( x , y ) · g ( x , y ) ] } , = { [ o ( x , y ) · g ( x , y ) ] * [ o ( x , y ) · g ( x , y ) ] } [ p ( x , y ) · g ( x , y ) ] * [ p ( x , y ) · g ( x , y ) ] ,
C b g ( x , y ) = b ( x , y ) · g ( x , y ) ] * [ b ( x , y ) · g ( x , y ) ] = [ p ( x , y ) · g ( x , y ) ] * [ p ( x , y ) · g ( x , y ) ] ,
SNR = M 1 / 2 n Q ( u , v ) T ( u , v ) ,
σ = 0.5 M - 1 / 2 n - 1 .
σ 2 × 10 - 7 × 2.5 m ,
[ a ( x , y ) b ( x , y ) ] · g ( x , y ) = [ a ( x , y ) · g ( x , y ) ] [ b ( x , y ) · g ( x , y ) ] .
- a ( x , y ) b ( x - x , y - y ) g ( x , y ) d x d y = - a ( x y ) g ( x , y ) b ( x - x , y - y ) × g ( x - x , y - y ) d x d y ,
g ( x , y ) = g ( x - x , y - y ) · g ( x , y ) ,

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