Abstract

Multiframe blind deconvolution - the process of restoring resolution to blurred imagery when the precise form of the blurs is unknown - is discussed as an estimation-theoretic method for improving the resolving power of ground-based telescopes used for space surveillance. The imaging problem is posed in an estimation-theoretic framework whereby the object’s incoherent scattering function is estimated through the simultaneous identification and correction of the distorting effects of atmospheric turbulence. An iterative method derived via the expectation-maximization (EM) procedure is reviewed, and results obtained from telescope imagery of the Hubble Space Telescope are presented.

©1997 Optical Society of America

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References

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  1. M. C. Roggemann and B. Welsh, Imaging Through Turbulence, CRC Press, Inc. (1996).
  2. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys.,  6, 85 (1970).
  3. K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short exposure images,” Astrophys. J.,  193, L45–L48 (1974).
    [Crossref]
  4. A. W. Lohmann, G. Weigelt, and B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. ,  22, 4028–4037 (1983).
    [Crossref] [PubMed]
  5. T. J. Schulz, “Multi-frame blind deconvolution of astronomical images”, J. Opt. Soc. Am. A ,  10, 1064–1073 (1993).
    [Crossref]
  6. Strictly speaking, because of image inversion and magnification an imaging system is never spatially invariant; however, if one views the input signal as the inverted and magnified object, many imaging systems are then well-modeled as spatially invariant.
  7. D. L. Snyder, A. M. Hammoud, and R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A ,  10, 1014–1023 (1993).
    [Crossref] [PubMed]
  8. D. L. Snyder, C. W. Helstrom, A. D. Lanterman, M. Faisal, and R. L. White, “Compensation for readout noise in CCD images,” J. Opt. Soc. Am. A ,  12, 272–283 (1985).
    [Crossref]
  9. G. R. Ayers and J. C. Dainty, “Iterative blind deconvolution method and its application”, Opt. Lett. ,  13, 547–549 (1988).
    [Crossref] [PubMed]
  10. R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A ,  9, 1508–1514 (1992).
    [Crossref]
  11. S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution”, Astrophys. J.,  63, 862–874 (1993).
    [Crossref]
  12. E. Thiebaut and J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A ,  12, 485–492 (1995).
    [Crossref]
  13. T. J. Cornwell, “Where have we been, where are we now, where are we going?”, in The Restoration of HST Images and Spectra II, B. Hanisch and R. L. White, editors, (Space Telescope Science Institute, Baltimore, MD, 1993) pp. 369–372.
  14. T. J. Schulz, “Movie of processed Hubble Space Telescope imagery,” http://www.ee.mtu.edu/faculty/schulz/mpeg/hst.mpeg
  15. R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations using phase diversity,” J. Opt. Soc. Am. A ,  9, 1072–1085 (1992).
    [Crossref]

1995 (1)

1993 (3)

1992 (2)

1988 (1)

1985 (1)

1983 (1)

1974 (1)

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short exposure images,” Astrophys. J.,  193, L45–L48 (1974).
[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 (1970).

Ayers, G. R.

Christou, J. C.

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution”, Astrophys. J.,  63, 862–874 (1993).
[Crossref]

Conan, J.-M.

Cornwell, T. J.

T. J. Cornwell, “Where have we been, where are we now, where are we going?”, in The Restoration of HST Images and Spectra II, B. Hanisch and R. L. White, editors, (Space Telescope Science Institute, Baltimore, MD, 1993) pp. 369–372.

Dainty, J. C.

Faisal, M.

Fienup, J. R.

Hammoud, A. M.

Helstrom, C. W.

Jefferies, S. M.

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution”, Astrophys. J.,  63, 862–874 (1993).
[Crossref]

Knox, K. T.

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short exposure images,” Astrophys. J.,  193, L45–L48 (1974).
[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 (1970).

Lane, R. G.

Lanterman, A. D.

Lohmann, A. W.

Paxman, R. G.

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence, CRC Press, Inc. (1996).

Schulz, T. J.

Snyder, D. L.

Thiebaut, E.

Thompson, B. J.

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short exposure images,” Astrophys. J.,  193, L45–L48 (1974).
[Crossref]

Weigelt, G.

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence, CRC Press, Inc. (1996).

White, R. L.

Wirnitzer, B.

Appl. Opt. (1)

Astron. Astrophys. (1)

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

Astrophys. J. (2)

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short exposure images,” Astrophys. J.,  193, L45–L48 (1974).
[Crossref]

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution”, Astrophys. J.,  63, 862–874 (1993).
[Crossref]

J. Opt. Soc. Am. A (6)

Opt. Lett. (1)

Other (4)

T. J. Cornwell, “Where have we been, where are we now, where are we going?”, in The Restoration of HST Images and Spectra II, B. Hanisch and R. L. White, editors, (Space Telescope Science Institute, Baltimore, MD, 1993) pp. 369–372.

T. J. Schulz, “Movie of processed Hubble Space Telescope imagery,” http://www.ee.mtu.edu/faculty/schulz/mpeg/hst.mpeg

Strictly speaking, because of image inversion and magnification an imaging system is never spatially invariant; however, if one views the input signal as the inverted and magnified object, many imaging systems are then well-modeled as spatially invariant.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence, CRC Press, Inc. (1996).

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Figures (2)

Figure 1:
Figure 1: Four short-exposure images of the Hubble Space Telescope as acquired by the 1.6 m AMOS telescope.

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Figure 2:
Figure 2: Four restored images of the Hubble Space Telescope.

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Tables (1)

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Table 1: Telescope and imaging system parameters for the AMOS imagery of the Hubble Space Telescope.

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Equations (21)

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h ( y ; θ t ) = ∣ ∫ A ( u ) exp { j [ θ t ( u ) − 2 π λ d o u · y ] } du ∣ 2 ,
i ( y ; θ t , o ) = ∫ h ( y − x ; θ t ) o ( x ) dx
= h ( y ; θ t ) * o ( y ) ,
d k [ n ] = N k [ n ] + M k [ n ] + g k [ n ] + b [ n ] ,
E ( N k [ n ] ) = I k [ n ]
= γ [ n ] ∫ y n ∫ t k t k + T i ( y ; θ t , o ) dydt ,
I k [ n ] ≃ γ [ n ] T ∣ y ∣ i k ( y n ; θ t k , o ) ,
i k ( y n ; θ t k , o ) ≃ ∑ m h ( y n − x m ; θ t k ) o ( x m ) Δ x 2 ,
h ( y n ; θ t k ) ≃ ∣ ∑ l A ( u l ) exp { j [ θ t k ( u l ) − 2 π Δ u Δ y λ d o u l · y n ] } Δ u 2 ∣ 2 ,
Pr { N k [ n ] + M k [ n ] = N ; θ t k , o } = exp { − ( I k [ n ] + I b [ n ] ) } ( I k [ n ] + I b [ n ] ) N N ! ,
I k [ n ] = γ [ n ] T ∣ y ∣ ∑ m h ( y n − x m ; θ t k ) o ( x m ) Δ x 2
= γ [ n ] ∑ m h ( y n − x m ; θ t k ) o [ m ] .
p g k [ n ] ( g ) = ( 2 π σ 2 [ n ] ) − 1 2 exp [ − g 2 ( 2 σ 2 [ n ] ) ] ,
p d k [ n ] ( d , θ t k , o ) = ∑ N = 0 ∞ p g k [ n ] ( d − b [ n ] − N ) Pr { N k [ n ] + M k [ n ] = N ; θ t k , o } .
l o θ = ∏ k = 1 K ∏ n p d k [ n ] ( d k [ n ] ; θ t k , o ) ,
𝐿 o θ = ln l o θ
= ∑ k = 1 K ∑ n ln p d k [ n ] ( d k [ n ] ; θ t k , o ) .
d ˜ k [ n ] = d k [ n ] − b [ n ] + σ 2 [ n ]
= N k [ n ] + M k [ n ] + g k [ n ] + σ 2 [ n ] ,
Pr { d ˜ k [ n ] = D ; θ t k , o } = exp { − ( I k [ n ] + I b [ n ] + σ 2 [ n ] ) } ( I k [ n ] + I b [ n ] + σ 2 [ n ] ) D D ! ,
𝐿 o θ = ∑ k = 1 K ∑ n { − ( I k [ n ] + I b [ n ] + σ 2 [ n ] ) + d ˜ k [ n ] ln ( I k [ n ] + I b [ n ] + σ 2 [ n ] ) } ,

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