Abstract

Properties of speckle imaging lens-atmosphere transfer functions that impact phase retrieval and subsequent image reconstruction have been measured under realistic conditions using bright stellar sources. These measurements generally support assumptions made by those who have proposed phase retrieval algorithms and tested them on simulated data.

© 1977 Optical Society of America

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References

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  1. J. Texereau, “Refiguring the 82-inch McDonald reflector’s optics,” Sky Telescope 28, 345–348 (1964).
  2. S. P. Worden, C. R. Lynds, and J. W. Harvey, “Reconstructed images of Alpha Orionis using stellar speckle interferometry,” J. Opt. Soc. Am. 66, 1234–1246 (1976).
    [Crossref]
  3. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images, ” Astron. Astrophys. 6, 85–87 (1970).
  4. R. H. T. Bates and P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319–321 (1973).
  5. B. L. McGlamery, “Restoration of turbulence-degraded images,” J. Opt. Soc. Am. 57, 293–297 (1967).
    [Crossref]
  6. B. L. McGlamery, NASA Tech. Report No. SP-256, 167 (1971).
  7. K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–48 (1974).
    [Crossref]
  8. K. T. Knox, “Image retrieval from astronomical speckle patterns,” J. Opt. Soc. Am. 66, 1236–39 (1976).
    [Crossref]
  9. M. Miller (private communication), see Ref. 17 for details.
  10. J. W. Sherman, “A posteriori restoration of atmospherically degraded images using multiframe imagery,” in Image Processing, Proceedings of the S.P.I.E.74, 249–258 (1976).
  11. D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,” J. Opt. Soc. Am. 63, 971–980 (1973).
    [Crossref]
  12. D. P. Karo and A. M. Schneiderman, “Speckle interferometry lens-atmosphere MTF measurements,” J. Opt. Soc. Am. 66, 1252–1256 (1976).
    [Crossref]
  13. Using C(Δ)=exp[−12D(Δ)], where D(Δ) is the wave structure function given by D(Δ) = 6.88(Δ/r0)5/3 and this approach to defining a correlation scale produces the familiar incident wave correlation scale r0. Although in common use, this C(Δ) can not be correct since it fails to have the correct (parabolic) form for small Δ.
  14. The MTF corrected for the known low-frequency telescope MTF does follow the Fried theory out to nearly the seeing-limited spatial frequency. We weight the lowest-frequency points most strongly in determining the D/r0 value which best fits the short-exposure theory to the data.
  15. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–79 (1966).
    [Crossref]
  16. D. Korff, G. Dryden, and M. G. Miller, “Information retrieval from atmospheric induced speckle patterns,” Optics Commun. 5, 187–192 (1972).
    [Crossref]
  17. D. P. Karo and A. M. Schneiderman, “Image reconstruction in speckle interferometry,” Imaging in Astronomy Technical Digest, ThC4–1, 3 OSA Topical Meeting, June 18–21, 1975, Cambridge, Massachusetts.

1976 (3)

1974 (1)

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–48 (1974).
[Crossref]

1973 (2)

D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,” J. Opt. Soc. Am. 63, 971–980 (1973).
[Crossref]

R. H. T. Bates and P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319–321 (1973).

1972 (1)

D. Korff, G. Dryden, and M. G. Miller, “Information retrieval from atmospheric induced speckle patterns,” Optics Commun. 5, 187–192 (1972).
[Crossref]

1970 (1)

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

1967 (1)

1966 (1)

1964 (1)

J. Texereau, “Refiguring the 82-inch McDonald reflector’s optics,” Sky Telescope 28, 345–348 (1964).

Bates, R. H. T.

R. H. T. Bates and P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319–321 (1973).

Dryden, G.

D. Korff, G. Dryden, and M. G. Miller, “Information retrieval from atmospheric induced speckle patterns,” Optics Commun. 5, 187–192 (1972).
[Crossref]

Fried, D. L.

Gough, P. T.

R. H. T. Bates and P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319–321 (1973).

Harvey, J. W.

S. P. Worden, C. R. Lynds, and J. W. Harvey, “Reconstructed images of Alpha Orionis using stellar speckle interferometry,” J. Opt. Soc. Am. 66, 1234–1246 (1976).
[Crossref]

Karo, D. P.

D. P. Karo and A. M. Schneiderman, “Speckle interferometry lens-atmosphere MTF measurements,” J. Opt. Soc. Am. 66, 1252–1256 (1976).
[Crossref]

D. P. Karo and A. M. Schneiderman, “Image reconstruction in speckle interferometry,” Imaging in Astronomy Technical Digest, ThC4–1, 3 OSA Topical Meeting, June 18–21, 1975, Cambridge, Massachusetts.

Knox, K. T.

K. T. Knox, “Image retrieval from astronomical speckle patterns,” J. Opt. Soc. Am. 66, 1236–39 (1976).
[Crossref]

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–48 (1974).
[Crossref]

Korff, D.

D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,” J. Opt. Soc. Am. 63, 971–980 (1973).
[Crossref]

D. Korff, G. Dryden, and M. G. Miller, “Information retrieval from atmospheric induced speckle patterns,” Optics Commun. 5, 187–192 (1972).
[Crossref]

Labeyrie, A.

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

Lynds, C. R.

S. P. Worden, C. R. Lynds, and J. W. Harvey, “Reconstructed images of Alpha Orionis using stellar speckle interferometry,” J. Opt. Soc. Am. 66, 1234–1246 (1976).
[Crossref]

McGlamery, B. L.

Miller, M.

M. Miller (private communication), see Ref. 17 for details.

Miller, M. G.

D. Korff, G. Dryden, and M. G. Miller, “Information retrieval from atmospheric induced speckle patterns,” Optics Commun. 5, 187–192 (1972).
[Crossref]

Schneiderman, A. M.

D. P. Karo and A. M. Schneiderman, “Speckle interferometry lens-atmosphere MTF measurements,” J. Opt. Soc. Am. 66, 1252–1256 (1976).
[Crossref]

D. P. Karo and A. M. Schneiderman, “Image reconstruction in speckle interferometry,” Imaging in Astronomy Technical Digest, ThC4–1, 3 OSA Topical Meeting, June 18–21, 1975, Cambridge, Massachusetts.

Sherman, J. W.

J. W. Sherman, “A posteriori restoration of atmospherically degraded images using multiframe imagery,” in Image Processing, Proceedings of the S.P.I.E.74, 249–258 (1976).

Texereau, J.

J. Texereau, “Refiguring the 82-inch McDonald reflector’s optics,” Sky Telescope 28, 345–348 (1964).

Thompson, B. J.

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–48 (1974).
[Crossref]

Worden, S. P.

S. P. Worden, C. R. Lynds, and J. W. Harvey, “Reconstructed images of Alpha Orionis using stellar speckle interferometry,” J. Opt. Soc. Am. 66, 1234–1246 (1976).
[Crossref]

Astron. Astrophys. (2)

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

R. H. T. Bates and P. T. Gough, “Speckle interferometry gives holograms of multiple star systems,” Astron. Astrophys. 22, 319–321 (1973).

Astrophys. J. Lett. (1)

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–48 (1974).
[Crossref]

J. Opt. Soc. Am. (6)

Optics Commun. (1)

D. Korff, G. Dryden, and M. G. Miller, “Information retrieval from atmospheric induced speckle patterns,” Optics Commun. 5, 187–192 (1972).
[Crossref]

Sky Telescope (1)

J. Texereau, “Refiguring the 82-inch McDonald reflector’s optics,” Sky Telescope 28, 345–348 (1964).

Other (6)

D. P. Karo and A. M. Schneiderman, “Image reconstruction in speckle interferometry,” Imaging in Astronomy Technical Digest, ThC4–1, 3 OSA Topical Meeting, June 18–21, 1975, Cambridge, Massachusetts.

Using C(Δ)=exp[−12D(Δ)], where D(Δ) is the wave structure function given by D(Δ) = 6.88(Δ/r0)5/3 and this approach to defining a correlation scale produces the familiar incident wave correlation scale r0. Although in common use, this C(Δ) can not be correct since it fails to have the correct (parabolic) form for small Δ.

The MTF corrected for the known low-frequency telescope MTF does follow the Fried theory out to nearly the seeing-limited spatial frequency. We weight the lowest-frequency points most strongly in determining the D/r0 value which best fits the short-exposure theory to the data.

B. L. McGlamery, NASA Tech. Report No. SP-256, 167 (1971).

M. Miller (private communication), see Ref. 17 for details.

J. W. Sherman, “A posteriori restoration of atmospherically degraded images using multiframe imagery,” in Image Processing, Proceedings of the S.P.I.E.74, 249–258 (1976).

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Figures (3)

FIG. 1
FIG. 1

Short-exposure lens-atmosphere modulation transfer function was obtained from two hundred images of α Lyra taken with 5 ms exposure times and with light from 4900 to 5100 Å in wavelength. The measured transfer function shows the presence of near-diffraction-limited spatial frequency information in the original images.

FIG. 2
FIG. 2

Average phase (●) and statistical deviations (○) of the phase of the OTF as a function of frequency. The solid lines show the ± σ levels expected for the average phase measured from two hundred samples and the expected deviation level under the assumption of phase uniformly distributed from − π to +π rad. The seeing-limited frequency is 0.07fDL.

FIG. 3
FIG. 3

The measured autocovariance functions for the OTF (Cτ), MTF (C|τ|), and PTF (Cϕ) are plotted as a function of frequency offset Δ measured in units of the telescope’s diffraction-limited frequency fDL. For the conditions of the data, the seeing-limited frequency was 0.07fDL.

Equations (2)

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π 4 r 2 ( f ) = 0 C ( Δ , f ) 2 π Δ d Δ .
C τ ( Δ , f ) = R τ ( Δ , f ) τ ( f ) τ * ( f + Δ ) σ τ ( f ) 2 = τ ( f ) τ * ( f + Δ ) σ τ ( f ) 2 ( OTF ) , C ϕ ( Δ , f ) = R ϕ ( Δ , f ) ϕ ( f ) ϕ ( f + Δ ) σ ϕ ( f ) 2 = ϕ ( f ) ϕ ( f + Δ ) σ ϕ ( f ) 2 ( PTF ) , C | τ | ( Δ , f ) = R | τ | ( Δ , f ) | τ ( f ) | | τ ( f + Δ ) | σ | τ ( f ) | 2 = | τ ( f ) | | τ ( f + Δ ) | | τ ( f ) | | τ ( f + Δ ) | σ | τ ( f ) | 2 ( MTF ) ,