Abstract

Results are presented from a series of experiments in which the U.S. Air Force Maui Optical Station’s 1.6-m telescope and a bare CCD speckle camera system were used to image satellites at distances of up to 1000 km. A brief overview of the image reconstruction algorithms is presented. The choice of the experiment site and various imaging parameters are described. Power spectra and power spectral signal-to-noise ratio curves that result from imaging several point stars are compared with theory. Reconstructed images of several binary stars are shown as a base-line assessment of our technique. High-quality image reconstructions of an Earth-satellite, the Hubble Space Telescope, are presented. The results confirm that speckle imaging techniques can be used with a bare CCD imaging system to provide a powerful and flexible method for imaging objects of moderate magnitude.

© 1992 Optical Society of America

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References

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  5. T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bispectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).
  6. T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, E. M. Johansson, “Extended-image reconstruction through horizontal path turbulence using bispectral speckle interferometry,” Opt. Eng. 31, 627–636 (1992).
    [CrossRef]
  7. T. Lawrence, P. Fitch, D. Goodman, “Image reconstruction using the bispectrum,” in Conference Record Twenty-Second Asilomar Conference on Signals, Systems and Computers, R. Chen, ed. (Maple, San Jose, Calif., 1988), pp. 58–62.
    [CrossRef]
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    [CrossRef]
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  12. D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. Soc. Photo-Opt.1351, 546–560 (1990).
  13. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]
  14. M. G. Miller, “Noise considerations in stellar speckle interferometry,” J. Opt. Soc. Am. 67, 1176–1184 (1977).
    [CrossRef]
  15. G. M. Cochran, T. J. B. Stanley, D. L. Fried, “White light speckle considerations,” internal document BC-436 (Optical Sciences Company, Placentia, Calif., 1987).
  16. J. G. Walker, “Optimum exposure time and filter bandwidth in speckle interferometry,” presented at International Astronomers Union Colloquium 50 (University of Maryland, College Park, Md., 1979).
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1992 (1)

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, E. M. Johansson, “Extended-image reconstruction through horizontal path turbulence using bispectral speckle interferometry,” Opt. Eng. 31, 627–636 (1992).
[CrossRef]

1990 (1)

1988 (2)

1984 (1)

1983 (1)

1980 (1)

1978 (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

1977 (1)

1973 (1)

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).

1966 (1)

Aitken, G. J. M.

Bartelt, H.

Christou, J. C.

Cobb, M. L.

Cochran, G. M.

G. M. Cochran, T. J. B. Stanley, D. L. Fried, “White light speckle considerations,” internal document BC-436 (Optical Sciences Company, Placentia, Calif., 1987).

Dainty, J. C.

Fitch, J. P.

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, E. M. Johansson, “Extended-image reconstruction through horizontal path turbulence using bispectral speckle interferometry,” Opt. Eng. 31, 627–636 (1992).
[CrossRef]

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bispectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. Soc. Photo-Opt.1351, 546–560 (1990).

Fitch, P.

T. Lawrence, P. Fitch, D. Goodman, “Image reconstruction using the bispectrum,” in Conference Record Twenty-Second Asilomar Conference on Signals, Systems and Computers, R. Chen, ed. (Maple, San Jose, Calif., 1988), pp. 58–62.
[CrossRef]

Freeman, J. D.

Fried, D. L.

D. L. Fried, “Limiting resolution looking down through the atmosphere,” J. Opt. Soc. Am. 56, 1380–1384 (1966).
[CrossRef]

G. M. Cochran, T. J. B. Stanley, D. L. Fried, “White light speckle considerations,” internal document BC-436 (Optical Sciences Company, Placentia, Calif., 1987).

Goodman, D.

T. Lawrence, P. Fitch, D. Goodman, “Image reconstruction using the bispectrum,” in Conference Record Twenty-Second Asilomar Conference on Signals, Systems and Computers, R. Chen, ed. (Maple, San Jose, Calif., 1988), pp. 58–62.
[CrossRef]

Goodman, D. M.

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, E. M. Johansson, “Extended-image reconstruction through horizontal path turbulence using bispectral speckle interferometry,” Opt. Eng. 31, 627–636 (1992).
[CrossRef]

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. Soc. Photo-Opt.1351, 546–560 (1990).

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bispectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Hege, E. K.

Johansson, E. M.

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, E. M. Johansson, “Extended-image reconstruction through horizontal path turbulence using bispectral speckle interferometry,” Opt. Eng. 31, 627–636 (1992).
[CrossRef]

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. Soc. Photo-Opt.1351, 546–560 (1990).

Korif, D.

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).

Lawrence, T.

T. Lawrence, P. Fitch, D. Goodman, “Image reconstruction using the bispectrum,” in Conference Record Twenty-Second Asilomar Conference on Signals, Systems and Computers, R. Chen, ed. (Maple, San Jose, Calif., 1988), pp. 58–62.
[CrossRef]

Lawrence, T. W.

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, E. M. Johansson, “Extended-image reconstruction through horizontal path turbulence using bispectral speckle interferometry,” Opt. Eng. 31, 627–636 (1992).
[CrossRef]

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. Soc. Photo-Opt.1351, 546–560 (1990).

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bispectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).

Lohmann, A. W.

Massie, N. A.

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, E. M. Johansson, “Extended-image reconstruction through horizontal path turbulence using bispectral speckle interferometry,” Opt. Eng. 31, 627–636 (1992).
[CrossRef]

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bispectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).

McCarthy, D. W.

Meng, J.

Miller, M. G.

Morgan, J. S.

O’Donnell, K. A.

Roddier, F.

Sherwood, R. J.

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, E. M. Johansson, “Extended-image reconstruction through horizontal path turbulence using bispectral speckle interferometry,” Opt. Eng. 31, 627–636 (1992).
[CrossRef]

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bispectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).

Stanley, T. J. B.

G. M. Cochran, T. J. B. Stanley, D. L. Fried, “White light speckle considerations,” internal document BC-436 (Optical Sciences Company, Placentia, Calif., 1987).

Walker, J. G.

J. G. Walker, “Optimum exposure time and filter bandwidth in speckle interferometry,” presented at International Astronomers Union Colloquium 50 (University of Maryland, College Park, Md., 1979).

Weigelt, G.

Wirnitzer, B.

Appl. Opt. (2)

Astron. Astrophys. (1)

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

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, E. M. Johansson, “Extended-image reconstruction through horizontal path turbulence using bispectral speckle interferometry,” Opt. Eng. 31, 627–636 (1992).
[CrossRef]

Phys. Rp. (1)

F. Roddier, “Interferometric imaging in astronomy,” Phys. Rp. 170, 97–166 (1988).
[CrossRef]

Proc. IEEE (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Other (5)

G. M. Cochran, T. J. B. Stanley, D. L. Fried, “White light speckle considerations,” internal document BC-436 (Optical Sciences Company, Placentia, Calif., 1987).

J. G. Walker, “Optimum exposure time and filter bandwidth in speckle interferometry,” presented at International Astronomers Union Colloquium 50 (University of Maryland, College Park, Md., 1979).

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. Soc. Photo-Opt.1351, 546–560 (1990).

T. Lawrence, P. Fitch, D. Goodman, “Image reconstruction using the bispectrum,” in Conference Record Twenty-Second Asilomar Conference on Signals, Systems and Computers, R. Chen, ed. (Maple, San Jose, Calif., 1988), pp. 58–62.
[CrossRef]

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bispectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).

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Figures (13)

Fig. 1
Fig. 1

Illustration of the domain (shaded) of the shift vectors (u) used in the bispectral integration, for a given position vector, r.

Fig. 2
Fig. 2

AMOS 1.6-m telescope with the Lawrence Livermore National Laboratory speckle camera.

Fig. 3
Fig. 3

Illustration of the flight path of the Hubble Space Telescope during an orbital pass over AMOS on 31 July 1990 at 14:45 UT. The stars selected as references for the Fourier magnitude estimation are highlighted and their YBSC numbers, their magnitude corrected to 700 nm, and their angular separation from the flight path are listed

Fig. 4
Fig. 4

Power spectra of the five reference stars, ordered by star magnitude, compared with a theoretical curve for D/r0 = 17.

Fig. 5
Fig. 5

Power spectra of the five reference stars, ordered by star elevation, compared with a theoretical curve for D/r0 = 17.

Fig. 6
Fig. 6

Power spectral SNR’s of the five reference stars, ordered by star magnitude.

Fig. 7
Fig. 7

Power spectral SNR’s of the five reference stars, ordered by star elevation.

Fig. 8
Fig. 8

Power spectra of the star α-Boötis for exposure times of 1, 2, 20, 30, and 900 ms, compared with a theoretical curve for D/r0 = 13.

Fig. 9
Fig. 9

ηSκ as a function of exposure time. ηSκ is normalized by its saturation value and τ is normalized by τ0, the value of τ at the knee of the curve.

Fig. 10
Fig. 10

SNR as a function of exposure time for a photon-counting detector, with ηSκ = 10−2. The curve is normalized by its maximum value.

Fig. 11
Fig. 11

SNR as a function of exposure time for the CCD camera used in the AMOS experiment. The curve is normalized by its maximum value, which occurs at τ/τ0 ≈ 10 or τ ≈ 20 ms.

Fig. 12
Fig. 12

Comparison of the results of imaging four binary stars. For each star a sample speckle image is shown along with a reconstructed image, a one-dimensional slice through the reconstruction, and comparisons of the known magnitude ratio ΔMv and separation Δθ with estimates obtained from the reconstruction.

Fig. 13
Fig. 13

Comparison of short-exposure speckle images with reconstructions of the Hubble Space Telescope at three different points during its pass over AMOS on 31 July 1990. The range R (in km), the elevation ϕelev, the Sun angle ϕSun, and the number of frames used in each reconstruction are listed below each set. The field of view of each image is 7.4″ × 7.4″.

Tables (1)

Tables Icon

Table 1 Five Stars Chosen as References for the Orbital Pass of the Hubble Space Telescope (HST) over AMOS, 31 July 1990a

Equations (74)

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i n ( x ) = τ ˜ n ( x ) * o ( x ) ,
I n ( u ) = τ n ( u ) O ( u ) ,
I n ( u ) 2 n = τ n ( u ) 2 n O ( u ) 2 .
O ( u ) est = [ I n ( u ) 2 n τ n ( u ) 2 n ] 1 / 2 .
arg { O ( u + v ) } = arg { O ( u ) } + arg { O ( v ) } - arg { I B , n ( u , v ) n } ,
I B , n ( u , v ) I n ( u ) I n ( v ) I n ( - u - v ) .
O ^ ( u ) = O ( u ) O ( u ) = exp [ i arg { O ( u ) } ] ,
I B ( u , v ) = I B , n ( u , v ) n .
O ^ ( r ) = O ^ ( u ) O ^ ( r - u ) I ^ B * ( u , r - u ) .
I B ( u , r - u ) = I B ( r - u , u ) ,
O ^ ( r ) est = u 2 = 0 r 2 u 1 = 0 , u 0 r 1 / 2 O ^ ( u ) O ^ ( r - u ) I ^ B * ( u , r - u ) | u 2 = 0 r 2 u 1 = 0 , u 0 r 1 / 2 O ^ ( u ) O ^ ( r - u ) I ^ B * ( u , r - u ) | ,
O ^ ( 0 , 0 ) = 1
O ^ ( 1 , 0 ) = O ^ ( 0 , 1 ) = 1
O ( r ) est = O ( r ) est O ^ ( r ) est = [ I n ( u ) 2 n τ n ( u ) 2 n ] 1 / 2 × u 2 = 0 r 2 u 1 = 0 , u 0 r 1 / 2 O ^ ( u ) O ^ ( r - u ) I ^ B * ( u , r - u ) | u 2 = 0 r 2 u 1 = 0 , u 0 r 1 / 2 O ^ ( u ) O ^ ( r - u ) I ^ B * ( u , r - u ) | ,
I n ° ( r ) 2 = I ˜ n ( r ) 2 - I ˜ n ( 0 ) - N pix σ CCD 2 ,
I B , n ° ( u , r - u ) = I ˜ B , n ( u , r - u ) - I ˜ n ( u ) 2 - I ˜ n ( r - u ) 2 - I ˜ n ( r ) 2 + 2 I ˜ n ( 0 ) + 3 N pix σ CCD 2 ,
τ ( u ) 2 τ DL 2 ( u ) × exp [ - 6.88 ( D r 0 u ) 5 / 3 ( 1 - u 1 / 3 ) ] + τ DL ( u ) ( r 0 D ) 2 × [ 0.435 + 0.278 ( D r 0 u ) - 1 / 3 ] ,
SNR = [ m ( τ ) ] 1 / 2 1 + 2 η s κ ( τ , Δ λ ) [ 1 + N pix σ CCD 2 n T ( τ , Δ λ ) ] .
η s κ ( τ , Δ λ ) 2 n T ( τ , Δ λ ) | τ ( u = 1 2 ) | 2 | O ( u = 1 2 ) | 2
η s κ ( τ , Δ λ ) 2 = n T ( τ , Δ λ ) | τ ( u = 1 2 ) | 2 .
τ ( u ) 2 τ DL ( u ) ( r 0 D ) 2 [ 0.435 + 0.278 ( r 0 D u ) 1 / 3 ]
τ DL ( u = 1 2 ) = 2 π { arccos ( 1 2 ) - 1 2 [ 1 - ( 1 2 ) 2 ] 1 / 2 } = 0.391 ,
| τ ( u = 1 2 ) | 2 0.435 ( r 0 D ) 2 × ( 0.391 ) [ 1 + 0.639 ( 2 r 0 D ) 1 / 3 ] , 0.435 2 ( r 0 D ) 2
η s κ ( τ , Δ λ ) = n T ( τ , Δ λ ) 1 0.435 ( D / r 0 ) 2 ,
m ( τ ) = T obs τ ,
m ( τ ) = T obs τ + τ read ,
D / r 0 ~ 15 , τ 0 ~ 2 ms , η s κ ~ 14 , N pix = ( 190 ) 2 , σ CCD ~ 14.
τ 1 / 3 τ read
Δ λ 1.7 λ ( r 0 D ) 5 / 6 ,
i P 2 ( x ) = [ i ° ( x ) ] 2 + i ° ( x ) .
i P 3 ( x ) = [ i ° ( x ) ] 3 + 3 [ i ° ( x ) ] 2 + i ° ( x ) .
i ˜ ( x ) = i P ( x ) + g ( x ) .
i 2 ( x ) = ξ i ( ξ ) i ( x + ξ ) ,
i 2 ( x ) DFT I ( u ) 2 .
i 3 ( x , y ) = ξ i ( ξ ) i ( x + ξ ) i ( y + ξ ) .
i 3 ( x , y ) DFT I B ( u , v ) ,
I B ( u , v ) = I ( u ) I ( v ) I ( - u - v ) = I ( u ) I ( v ) I * ( u + v ) ,
ι ˜ 2 ( x ) = ξ ι ˜ ( ξ ) ι ˜ ( x + ξ ) .
ι ˜ 2 ( x ) = ξ [ i P ( ξ ) i P ( x + ξ ) + g ( ξ ) i P ( x + ξ ) + i P ( ξ ) g ( x + ξ ) + g ( ξ ) g ( x + ξ ) ] .
i P ( ξ ) i P ( x + ξ ) = { i P ( ξ ) i P ( x + ξ ) = i ° ( ξ ) i ° ( x + ξ ) x 0 i P 2 ( ξ ) = i ° ( ξ ) + [ i ° ( ξ ) ] 2 x = 0 = i ° ( ξ ) i ° ( x + ξ ) + i ( ξ ) δ ( x ) ,
g ( ξ ) i P ( x + ξ ) = g ( ξ ) i p ( x + ξ ) = 0 ,
i P ( ξ ) g ( x + ξ ) = i P ( ξ ) g ( x + ξ ) = 0 ,
g ( ξ ) g ( x + ξ ) = { g ( ξ ) g ( x + ξ ) = 0 x 0 g 2 ( ξ )             = σ CCD 2 x = 0 = σ CCD 2 δ ( x ) ,
ι ˜ 2 ( x ) = i 2 ° ( x ) + δ ( x ) ξ ( i ° ( ξ ) + σ CCD 2 ) .
ξ σ CCD 2 = N pix σ CCD 2 ,
ι ˜ 2 ( x ) = i 2 ° ( x ) + δ ( x ) [ I ° ( 0 ) + N pix σ CCD 2 ] .
I ˜ ( u ) 2 = I ° ( u ) 2 + I ° ( 0 ) + N pix σ CCD 2 .
I n ° ( u ) 2 = I ˜ n ( u ) 2 - I ˜ n ( 0 ) - N pix σ CCD 2 ,
ι ˜ 3 ( x , y ) = ξ ι ˜ ( ξ ) ι ˜ ( x + ξ ) ι ˜ ( y + ξ ) .
i ˜ 3 ( x , y ) = ξ [ i P ( ξ ) i P ( x + ξ ) i P ( y + ξ ) + g ( ξ ) i P ( x + ξ ) i P ( y + ξ ) + i P ( ξ ) g ( x + ξ ) i P ( y + ξ ) + i P ( ξ ) i P ( x + ξ ) g ( y + ξ ) + g ( ξ ) g ( x + ξ ) i P ( y + ξ ) + g ( ξ ) i P ( x + ξ ) g ( y + ξ ) + i P ( ξ ) g ( x + ξ ) g ( y + ξ ) + g ( ξ ) g ( x + ξ ) g ( y + ξ ) ] .
i P ( ξ ) i P ( x + ξ ) i P ( y + ξ ) = { i P ( ξ ) i P ( x + ξ ) i P ( y + ξ ) = i ° ( ξ ) i ° ( x + ξ ) i ° ( y + ξ ) x 0 , y 0 , x y i P ( ξ ) i P 2 ( x + ξ ) = i ° ( ξ ) { [ i ° ( x + ξ ) ] 2 + i ° ( x + ξ ) } x 0 , y 0 , x = y i P 2 ( ξ ) i P ( x + ξ ) = { [ i ° ( ξ ) ] 2 + i ° ( ξ ) } i ° ( y + ξ ) x 0 , y = 0 i P 2 ( ξ ) i P ( y + ξ ) = { [ i ° ( ξ ) ] 2 + i ° ( ξ ) } i ° ( x + ξ ) x = 0 , y 0 i P 3 ( ξ ) = [ i ° ( ξ ) ] 3 + 3 [ i ° ( ξ ) ] 2 + i ° ( ξ ) x = 0 , y = 0 = i ° ( ξ ) i ° ( x + ξ ) i ° ( y + ξ ) + i ° ( ξ ) i ° ( y + ξ ) δ ( x ) + i ° ( ξ ) i ° ( x + ξ ) δ ( y ) + i ° ( ξ ) i ° ( x + ξ ) δ ( x - y ) + i ° ( ξ ) δ ( x ) δ ( y ) ,
g ( ξ ) g ( x + ξ ) i P ( y + ξ ) = { g 2 ( ξ ) i P ( y + ξ ) = σ CCD 2 i ° ( y + ξ ) x = 0 g ( ξ ) g ( x + ξ ) i P ( y + ξ ) = 0 x 0 = σ CCD 2 i ° ( y + ξ ) δ ( x ) .
g ( ξ ) i P ( x + ξ ) g ( y + ξ ) = σ CCD 2 i ° ( x + ξ ) δ ( y ) ,
i P ( ξ ) g ( x + ξ ) g ( y + ξ ) = σ CCD 2 i ° ( ξ ) δ ( x - y ) .
i ˜ 3 ( x , y ) = i 3 ° ( x , y ) + i 2 ° ( y ) δ ( x ) + i 2 ° ( x ) δ ( y ) + i 2 ° ( x ) δ ( x - y ) + I ° ( 0 ) δ ( x ) δ ( y ) + σ CCD 2 I ° ( 0 ) [ δ ( x ) + δ ( y ) + δ ( x - y ) ] .
I ˜ B ( u , v ) = I B ° ( u , v ) + I ° ( v ) 2 + I ° ( u ) 2 + I ° ( u + v ) 2 + I ° ( 0 ) + σ CCD 2 I ° ( 0 ) [ δ ( v ) + δ ( u ) + δ ( u + v ) ] .
I B ° ( u , v ) = I ˜ B ( u , v ) - I ° ( v ) 2 - I ° ( u ) 2 - I ° ( u + v ) 2 - I ° ( 0 ) .
I B , n ° ( u , v ) = I ˜ B , n ( u , v ) - I ˜ n ( v ) 2 - I ˜ n ( u ) 2 - I ˜ n ( u + v ) 2 + 2 I ˜ n ( 0 ) + 3 N pix σ CCD 2 .
i ˜ w ( x ) = w ( x ) i ˜ ( x ) .
i ˜ w ( x ) = w ( x ) [ i p ( x ) + g ( x ) ] .
i ˜ 2 w ( x ) = i 2 w ° ( x ) + δ ( x ) [ I w 2 ° ( 0 ) + N w 2 pix σ CCD 2 ] ,
I w 2 ° ( 0 ) = ξ w 2 ( ξ ) i ° ( ξ ) ,
N w 2 pix = ξ w 2 ( ξ ) .
I ˜ w ( u ) 2 = I w ° ( u ) 2 + I w 2 ° ( 0 ) + N w 2 pix σ CCD 2 .
I w n ° ( u ) 2 = I ˜ w n ( u ) 2 - I ˜ w 2 n ( 0 ) - N w 2 pix σ CCD 2 .
i ˜ 3 w ( x , y ) = i 3 w ° ( x , y ) + i 2 w 3 ° ( y ) δ ( x ) + i 2 w 3 ° ( x ) δ ( y ) + i 2 w 3 ° ( x ) δ ( x - y ) + I w 3 ° ( 0 ) δ ( x ) δ ( y ) + σ CCD 2 I w 3 ° ( 0 ) [ δ ( x ) + δ ( y ) + δ ( x - y ) ] ,
i 2 w 3 ° ( x ) = ξ w 2 ( ξ ) w ( x + ξ ) i ° ( ξ ) i ° ( x + ξ ) ,
I w 3 ° ( 0 ) = ξ w 3 ( ξ ) i ° ( ξ ) .
i 2 w 3 ° ( x ) DFT I w ° ( u ) [ I w 2 ° ( u ) ] * ,
w 2 ( x ) i ° ( x ) DFT I w 2 ° ( u ) .
I ˜ B w ( u , v ) = I B w ° ( u , v ) + I w ° ( v ) [ I w 2 ° ( v ) ] * + I w ° ( u ) [ I w 2 ° ( u ) ] * + I w ° ( u + v ) × [ I w 2 ° ( u + v ) ] * + I w 3 ° ( 0 ) + σ CCD 2 I w 3 ° ( 0 ) × [ δ ( v ) + δ ( u ) + δ ( u + v ) ] .
I w ° ( u ) [ I w 2 ° ( u ) ] * I w ° ( u ) 2 ,
I w 3 ° ( 0 ) I w 2 ° ( 0 ) .
I B w , n ° ( u , v ) = I ˜ B w , n ( u , v ) - I ˜ w n ( v ) 2 - I ˜ w n ( u ) 2 - I ˜ w n ( u + v ) 2 + 2 I ˜ w 2 n ( 0 ) + 3 N w 2 pix σ CCD 2 .

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