Abstract

The method of deconvolution from wave-front sensing (DWFS), which is a method for improving the quality of astronomical images measured through atmospheric turbulence, uses simultaneous short-exposure measurements of both an image and the output of a wave-front sensor exposed to an image of the telescope pupil. The wave-front sensor measurements are used to reconstruct an estimate of the instantaneous generalized pupil function of the telescope, which is used to compute an estimate of the instantaneous optical transfer function (OTF). This estimate of the OTF is then used in a deconvolution procedure. We point out the existence and origin of an unnoticed bias in the estimator for the DWFS method. This bias leads to nonrandom errors in the estimated object spectrum beyond those expected to arise by virtue of low-pass filtering and noise, including the possibility of an overall system transfer function greater than unity at some spatial frequencies. An alternative measurement and postprocessing scheme to overcome this source of error is suggested.

© 1994 Optical Society of America

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References

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  1. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
    [Crossref]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, pp. 101–136.
  3. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8, pp. 361–461.
  4. 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]
  5. D. L. Fried, “Post detection wavefront compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 127–133 (1987).
  6. J. D. Gonglewski, D. G. Voelz, J. S. Fender, D. C. Dayton, B. K. Spielbusch, R. E. Pierson, “First astronomical application of postdetection turbulence compensation: images of α Aurigae, ν Ursae Majoris, and α Geminorium using self-referenced speckle holography,” Appl. Opt. 29, 4527–4529 (1990).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  9. M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electr. Eng. 18, 451–466 (1992).
    [Crossref]
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    [Crossref]
  12. R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991), Chap. 8, pp. 257–260.
  13. M. C. Roggemann, C. L. Matson, “Power spectrum and Fourier phase spectrum estimation by using fully and partially compensating adaptive optics and bispectrum post processing,” J. Opt. Soc. Am. A 9, 1525–1535 (1992).
    [Crossref]
  14. G. Cochran, “Phase screen generation,” Tech. Rep. 663 (Optical Sciences Company, Placentia, Calif., 1985).
  15. R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993), Chap. 5, pp. 279–282.
  16. P. S. Idell, A. Webster, “Resolution limits for coherent optical imaging: signal-to-noise analysis in the spatial-frequency domain,” J. Opt. Soc. Am. A 9, 43–56 (1992).
    [Crossref]

1993 (1)

1992 (3)

1990 (2)

1989 (1)

1983 (1)

1976 (1)

Cochran, G.

G. Cochran, “Phase screen generation,” Tech. Rep. 663 (Optical Sciences Company, Placentia, Calif., 1985).

Dayton, D. C.

Fender, J. S.

Fontanella, J. C.

Fried, D. L.

D. L. Fried, “Post detection wavefront compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 127–133 (1987).

Gardner, C. S.

Gonglewski, J. D.

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993), Chap. 5, pp. 279–282.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, pp. 101–136.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8, pp. 361–461.

Idell, P. S.

Matson, C. L.

Noll, R. J.

Pierson, R. E.

Primot, J.

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[Crossref]

Roggemann, M. C.

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electr. Eng. 18, 451–466 (1992).
[Crossref]

M. C. Roggemann, C. L. Matson, “Power spectrum and Fourier phase spectrum estimation by using fully and partially compensating adaptive optics and bispectrum post processing,” J. Opt. Soc. Am. A 9, 1525–1535 (1992).
[Crossref]

Rousset, G.

Spielbusch, B. K.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991), Chap. 8, pp. 257–260.

Voelz, D. G.

Von Niederhausern, R. N.

Wallner, E. P.

Webster, A.

Welsh, B. M.

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993), Chap. 5, pp. 279–282.

Appl. Opt. (2)

Comput. Electr. Eng. (1)

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electr. Eng. 18, 451–466 (1992).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Other (7)

D. L. Fried, “Post detection wavefront compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 127–133 (1987).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[Crossref]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, pp. 101–136.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8, pp. 361–461.

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991), Chap. 8, pp. 257–260.

G. Cochran, “Phase screen generation,” Tech. Rep. 663 (Optical Sciences Company, Placentia, Calif., 1985).

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993), Chap. 5, pp. 279–282.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the hardware and the computer system used in DWFS.

Fig. 2
Fig. 2

Structure functions of the input phase Dθ and of the estimated phase D θ ˜ for the case in which Dpr/r0 = 2 and SNRW = ∞, 5, 3, and 2.

Fig. 3
Fig. 3

Structure functions of the input phase Dθ and of the estimated phase D θ ˜ for the case in which Dpr/r0 = 10 and SNRW = ∞, 5, 3, and 2.

Fig. 4
Fig. 4

Fractional error E as a function of separation and SNRW ∞, 5, 3, and 2 for Dpr/r0 = 2. str. fnc., structure function.

Fig. 5
Fig. 5

Fractional error E as a function of separation and SNRW = ∞, 5, 3, and 2 for Dpr/r0 = 10.

Fig. 6
Fig. 6

Fractional error in structure function for SNRW = ∞ and for the number of Zernike modes used to reconstruct the phase: 50, 40, 30, 20, and 10.

Fig. 7
Fig. 7

Fractional error in structure function for Dpr/r0 = 2, SNRW = 2, and the number of Zernike modes used to reconstruct the phase: 61, 50, 40, 30, 20, and 10.

Fig. 8
Fig. 8

Fractional error in structure function for Dpr/r0 = 10, SNRW = 2, and the number of Zernike modes used to reconstruct the phase: 61, 50, 40, 30, 20, and 10.

Fig. 9
Fig. 9

Numerator, denominator, and DWFS transfer function for the case Dpr/r0 = 2, SNRW = ∞, with 61 Zernike modes used. A u-axis slice is shown.

Fig. 10
Fig. 10

Numerator, denominator, and DWFS transfer function for the case Dpr/r0 = 10, SNRW = ∞, with 61 Zernike modes used. A u-axis slice is shown.

Equations (39)

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i ( x , y ) = o ( x , y ) h ( x , y ) .
I ( u , v ) = O ( u , v ) H ( u , v ) ,
H ˜ i ( u , v ) = P ( x , y ) P ( x - u λ f , y - v λ f ) exp { j [ θ ˜ i ( x , y ) - θ ˜ i ( x - u λ f , y - v λ f ) ] } d x d y P ( x , y ) 2 d x d y ,
O ˜ ( u , v ) = I i ( u , v ) H ˜ i * ( u , v ) H ˜ i ( u , v ) 2 ,
O ˜ ( u , v ) = O ( u , v ) H i ( u , v ) H ˜ i * ( u , v ) H ˜ i ( u , v ) 2 ,
S ( u , v ) = H i ( u , v ) H ˜ i * ( u , v ) H ˜ i ( u , v ) 2 .
O ˜ ( u , v ) = S ( u , v ) O ( u , v ) .
θ ˜ i ( x , y ) = j c j r j ( x , y ) ,
c j = n M j n s n ,
Δ = ( s - H c ) ,
h n j = s n c j .
c opt = ( H T H ) - 1 H T s .
M j n = ( H T H ) - 1 H T .
Num ( u , v ) = H i ( u , v ) H ˜ i * ( u , v )
Denom ( u , v ) = H ˜ i ( u , v ) 2 .
Num ( u ) = N - 2 P ( x ) P * ( x - ρ ) × exp { j [ θ i ( x ) - θ i ( x - ρ ) ] } d 2 x · P * ( x ) P ( x - ρ ) × exp { - j [ θ ˜ i ( x ) - θ ˜ i ( x - ρ ) ] } d 2 x ,
Num ( u ) = N - 2 P ( x ) P * ( x - ρ ) P * ( x ) P ( x - ρ ) × exp { - 1 2 [ θ i ( x ) - θ i ( x - ρ ) - θ ˜ i ( x ) + θ ˜ i ( x - ρ ) ] 2 } d 2 x d 2 x .
Denom ( u ) = N - 2 P ( x ) P * ( x - ρ ) P * ( x ) P ( x - ρ ) × exp { - 1 2 [ θ ˜ i ( x ) - θ ˜ i ( x - ρ ) - θ ˜ i ( x ) + θ ˜ i ( x - ρ ) ] 2 } d 2 x d 2 x .
{ [ θ i ( x ) - θ i ( x - ρ ) - θ ˜ i ( x ) + θ ˜ i ( x - ρ ) ] 2 = [ θ ˜ i ( x ) - θ ˜ i ( x - ρ ) - θ ˜ i ( x ) + θ ˜ i ( x - ρ ) ] 2 ,
D θ ( Δ x , Δ y ) = D θ ( r ) = [ θ i ( x , y ) - θ i ( x - Δ x , y - Δ y ) ] 2 = 6.8839 ( r r 0 ) 5 / 3 .
r = ( Δ x 2 + Δ y 2 ) 1 / 2 .
D θ ˜ ( x , y ; x , y ) = [ θ ˜ i ( x , y ) - θ ˜ i ( x , y ) ] 2 .
D θ ˜ ( x , y ; x , y ) = { j n M j n s n [ r j ( x , y ) - r j ( x , y ) ] } × { j n M j n s n [ r j ( x , y ) - r j ( x , y ) ] } .
D θ ˜ ( x , y ; x , y ) = j n j n M j n M j n S n n × [ r j ( x , y ) - r j ( x , y ) ] × [ r j ( x , y ) - r j ( x , y ) ] ,
S n n = s n s n = - ( 1 2 ) W n ( x , y ) W n ( x , y ) × D θ ( x - x , y - y ) d x d y d x d y + α n α n ,
α n α n = σ α 2 δ n n ,
σ α = 0.74 π η SNR W L ,
SNR W = K ¯ W ,
SNR W = K ¯ W ( K ¯ W + P σ r 2 ) 1 / 2 .
δ = D pr 2 N ,
E = ( D θ ˜ - D θ ) D θ ,
σ WFS 2 = 0.35 π 2 4 SNR W 2 .
σ fit 2 = 0.2944 N m - 3 / 2 ( D pr r 0 ) 5 / 3 .
R 61 ( γ ) = 211.069 γ 10 - 337.710 γ 8 + 131.332 γ 6 ,
S ( u , v ) = Num ( u , v ) Denom ( u , v ) + ,
O ˜ ( u , v ) = I i ( u , v ) H ˜ i * ( u , v ) H k ref ( u , v ) [ H ˜ k ref ( u , v ) ] * + γ ( u , v ) ,
γ ( u , v ) = 1 SNR ( u , v ) ,
SNR ( u , v ) = I i ( u , v ) H ˜ i * ( u , v ) [ var { I i ( u , v ) H ˜ i * ( u , v ) } ] 1 / 2 .
var { Z } = Z 2 - Z 2 .

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