Abstract

The weighted-shift-and-add speckle imaging technique is analyzed using simple assumptions. The end product is shown to be a convolution of the object with a typical point-spread function (psf) that is similar in shape to the telescope psf and depends marginally on the speckle psf. A filter can be applied to each data frame before locating the maxima, either to identify the speckle locations (matched filter) or to estimate the instantaneous atmospheric psf (Wiener filter). Preliminary results show the power of the technique when applied to photon-limited data and to extended objects.

© 1986 Optical Society of America

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  1. J. C. Christou, E. K. Hege, J. D. Freeman, E. Ribak, “A selfcalibrating shift-and-add technique for speckle imaging,” J. Opt. Soc. Am. A 3, 204–209 (1986); “Images from astronomical speckle data: weighted shift-and-add analysis,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 255–262 (1985).
    [CrossRef]
  2. G. P. Weigelt, B. Wirnitzer, “Image reconstruction by the speckle masking method,” Opt. Lett. 8, 389–391 (1983).
    [CrossRef] [PubMed]
  3. R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
    [CrossRef]
  4. R. H. T. Bates, F. W. Cady, “Towards true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365–369 (1980).
    [CrossRef]
  5. C. R. Lynds, S. P. Worden, J. W. Harvey, “Digital image reconstruction applied to Alpha Ononis,” Astrophys. J. 207, 174–179 (1976).
    [CrossRef]
  6. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376; J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1983).
    [CrossRef]
  7. J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” RADC-TR-76-50, March1976; ARPA order #2646 (Rome Air Development Center, Griffiss AFB, N.Y.); “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Seminar on Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
    [CrossRef]
  8. J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,” J. Opt. Soc. Am. 69, 786–790 (1979).
    [CrossRef]
  9. B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978), Sec. 5.2.9.
    [CrossRef]
  10. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979).
  11. E. Ribak, E. K. Hege, J. C. Christou, “Identification of speckles by matched filtering,” Bull. Am. Astron. Soc. 16, 885 (1984); “Use of matched filtering to identify speckle locations,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 196–201 (1985).
    [CrossRef]
  12. A. M. Sinton, R. A. Minard, R. H. T. Bates, “Generalization of shift-and-add imaging,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 263–269 (1985).
    [CrossRef]
  13. B. R. Hunt, W. R. Fright, R. H. T. Bates, “Analysis of the shift and add method for imaging through turbulent media,” J. Opt. Soc. Am. 73, 456–465 (1983).
    [CrossRef]
  14. J. D. Freeman, E. Ribak, J. C. Christou, E. K. Hege, “Statistical analysis of the weighted shift-and-add image reconstruction technique,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 279–283 (1985).
    [CrossRef]
  15. B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1983).
    [CrossRef]
  16. J. C. Christou, A. Y. S. Cheng, E. K. Hege, C. Roddier, “Seeing calibration of optical astronomical speckle interferometric data,” Astron. J. 90, 2644–2651 (1985).
    [CrossRef]
  17. E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, “The Steward Observatory speckle interferometry system,” Opt. Acta 29, 701–715 (1982).
    [CrossRef]

1986 (1)

1985 (1)

J. C. Christou, A. Y. S. Cheng, E. K. Hege, C. Roddier, “Seeing calibration of optical astronomical speckle interferometric data,” Astron. J. 90, 2644–2651 (1985).
[CrossRef]

1984 (1)

E. Ribak, E. K. Hege, J. C. Christou, “Identification of speckles by matched filtering,” Bull. Am. Astron. Soc. 16, 885 (1984); “Use of matched filtering to identify speckle locations,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 196–201 (1985).
[CrossRef]

1983 (2)

1982 (2)

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, “The Steward Observatory speckle interferometry system,” Opt. Acta 29, 701–715 (1982).
[CrossRef]

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

1980 (1)

R. H. T. Bates, F. W. Cady, “Towards true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365–369 (1980).
[CrossRef]

1979 (1)

1976 (1)

C. R. Lynds, S. P. Worden, J. W. Harvey, “Digital image reconstruction applied to Alpha Ononis,” Astrophys. J. 207, 174–179 (1976).
[CrossRef]

Bates, R. H. T.

B. R. Hunt, W. R. Fright, R. H. T. Bates, “Analysis of the shift and add method for imaging through turbulent media,” J. Opt. Soc. Am. 73, 456–465 (1983).
[CrossRef]

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

R. H. T. Bates, F. W. Cady, “Towards true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365–369 (1980).
[CrossRef]

A. M. Sinton, R. A. Minard, R. H. T. Bates, “Generalization of shift-and-add imaging,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 263–269 (1985).
[CrossRef]

Belsher, J. F.

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” RADC-TR-76-50, March1976; ARPA order #2646 (Rome Air Development Center, Griffiss AFB, N.Y.); “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Seminar on Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
[CrossRef]

Cady, F. W.

R. H. T. Bates, F. W. Cady, “Towards true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365–369 (1980).
[CrossRef]

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979).

Cheng, A. Y. S.

J. C. Christou, A. Y. S. Cheng, E. K. Hege, C. Roddier, “Seeing calibration of optical astronomical speckle interferometric data,” Astron. J. 90, 2644–2651 (1985).
[CrossRef]

Christou, J. C.

J. C. Christou, E. K. Hege, J. D. Freeman, E. Ribak, “A selfcalibrating shift-and-add technique for speckle imaging,” J. Opt. Soc. Am. A 3, 204–209 (1986); “Images from astronomical speckle data: weighted shift-and-add analysis,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 255–262 (1985).
[CrossRef]

J. C. Christou, A. Y. S. Cheng, E. K. Hege, C. Roddier, “Seeing calibration of optical astronomical speckle interferometric data,” Astron. J. 90, 2644–2651 (1985).
[CrossRef]

E. Ribak, E. K. Hege, J. C. Christou, “Identification of speckles by matched filtering,” Bull. Am. Astron. Soc. 16, 885 (1984); “Use of matched filtering to identify speckle locations,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 196–201 (1985).
[CrossRef]

J. D. Freeman, E. Ribak, J. C. Christou, E. K. Hege, “Statistical analysis of the weighted shift-and-add image reconstruction technique,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 279–283 (1985).
[CrossRef]

Cocke, W. J.

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, “The Steward Observatory speckle interferometry system,” Opt. Acta 29, 701–715 (1982).
[CrossRef]

Dainty, J. C.

Freeman, J. D.

J. C. Christou, E. K. Hege, J. D. Freeman, E. Ribak, “A selfcalibrating shift-and-add technique for speckle imaging,” J. Opt. Soc. Am. A 3, 204–209 (1986); “Images from astronomical speckle data: weighted shift-and-add analysis,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 255–262 (1985).
[CrossRef]

J. D. Freeman, E. Ribak, J. C. Christou, E. K. Hege, “Statistical analysis of the weighted shift-and-add image reconstruction technique,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 279–283 (1985).
[CrossRef]

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1983).
[CrossRef]

Fright, W. R.

Goodman, J. W.

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” RADC-TR-76-50, March1976; ARPA order #2646 (Rome Air Development Center, Griffiss AFB, N.Y.); “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Seminar on Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
[CrossRef]

Greenaway, A. H.

Harvey, J. W.

C. R. Lynds, S. P. Worden, J. W. Harvey, “Digital image reconstruction applied to Alpha Ononis,” Astrophys. J. 207, 174–179 (1976).
[CrossRef]

Hege, E. K.

J. C. Christou, E. K. Hege, J. D. Freeman, E. Ribak, “A selfcalibrating shift-and-add technique for speckle imaging,” J. Opt. Soc. Am. A 3, 204–209 (1986); “Images from astronomical speckle data: weighted shift-and-add analysis,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 255–262 (1985).
[CrossRef]

J. C. Christou, A. Y. S. Cheng, E. K. Hege, C. Roddier, “Seeing calibration of optical astronomical speckle interferometric data,” Astron. J. 90, 2644–2651 (1985).
[CrossRef]

E. Ribak, E. K. Hege, J. C. Christou, “Identification of speckles by matched filtering,” Bull. Am. Astron. Soc. 16, 885 (1984); “Use of matched filtering to identify speckle locations,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 196–201 (1985).
[CrossRef]

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, “The Steward Observatory speckle interferometry system,” Opt. Acta 29, 701–715 (1982).
[CrossRef]

J. D. Freeman, E. Ribak, J. C. Christou, E. K. Hege, “Statistical analysis of the weighted shift-and-add image reconstruction technique,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 279–283 (1985).
[CrossRef]

Hubbard, E. N.

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, “The Steward Observatory speckle interferometry system,” Opt. Acta 29, 701–715 (1982).
[CrossRef]

Hunt, B. R.

Lynds, C. R.

C. R. Lynds, S. P. Worden, J. W. Harvey, “Digital image reconstruction applied to Alpha Ononis,” Astrophys. J. 207, 174–179 (1976).
[CrossRef]

Minard, R. A.

A. M. Sinton, R. A. Minard, R. H. T. Bates, “Generalization of shift-and-add imaging,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 263–269 (1985).
[CrossRef]

Ribak, E.

J. C. Christou, E. K. Hege, J. D. Freeman, E. Ribak, “A selfcalibrating shift-and-add technique for speckle imaging,” J. Opt. Soc. Am. A 3, 204–209 (1986); “Images from astronomical speckle data: weighted shift-and-add analysis,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 255–262 (1985).
[CrossRef]

E. Ribak, E. K. Hege, J. C. Christou, “Identification of speckles by matched filtering,” Bull. Am. Astron. Soc. 16, 885 (1984); “Use of matched filtering to identify speckle locations,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 196–201 (1985).
[CrossRef]

J. D. Freeman, E. Ribak, J. C. Christou, E. K. Hege, “Statistical analysis of the weighted shift-and-add image reconstruction technique,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 279–283 (1985).
[CrossRef]

Roddier, C.

J. C. Christou, A. Y. S. Cheng, E. K. Hege, C. Roddier, “Seeing calibration of optical astronomical speckle interferometric data,” Astron. J. 90, 2644–2651 (1985).
[CrossRef]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376; J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1983).
[CrossRef]

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978), Sec. 5.2.9.
[CrossRef]

Sinton, A. M.

A. M. Sinton, R. A. Minard, R. H. T. Bates, “Generalization of shift-and-add imaging,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 263–269 (1985).
[CrossRef]

Strittmatter, P. A.

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, “The Steward Observatory speckle interferometry system,” Opt. Acta 29, 701–715 (1982).
[CrossRef]

Weigelt, G. P.

Wirnitzer, B.

Worden, S. P.

C. R. Lynds, S. P. Worden, J. W. Harvey, “Digital image reconstruction applied to Alpha Ononis,” Astrophys. J. 207, 174–179 (1976).
[CrossRef]

Astron. J. (1)

J. C. Christou, A. Y. S. Cheng, E. K. Hege, C. Roddier, “Seeing calibration of optical astronomical speckle interferometric data,” Astron. J. 90, 2644–2651 (1985).
[CrossRef]

Astrophys. J. (1)

C. R. Lynds, S. P. Worden, J. W. Harvey, “Digital image reconstruction applied to Alpha Ononis,” Astrophys. J. 207, 174–179 (1976).
[CrossRef]

Bull. Am. Astron. Soc. (1)

E. Ribak, E. K. Hege, J. C. Christou, “Identification of speckles by matched filtering,” Bull. Am. Astron. Soc. 16, 885 (1984); “Use of matched filtering to identify speckle locations,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 196–201 (1985).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, “The Steward Observatory speckle interferometry system,” Opt. Acta 29, 701–715 (1982).
[CrossRef]

Opt. Commun. (1)

R. H. T. Bates, F. W. Cady, “Towards true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365–369 (1980).
[CrossRef]

Opt. Lett. (1)

Phys. Rep. (1)

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

Other (7)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376; J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1983).
[CrossRef]

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” RADC-TR-76-50, March1976; ARPA order #2646 (Rome Air Development Center, Griffiss AFB, N.Y.); “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Seminar on Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
[CrossRef]

B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978), Sec. 5.2.9.
[CrossRef]

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979).

A. M. Sinton, R. A. Minard, R. H. T. Bates, “Generalization of shift-and-add imaging,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 263–269 (1985).
[CrossRef]

J. D. Freeman, E. Ribak, J. C. Christou, E. K. Hege, “Statistical analysis of the weighted shift-and-add image reconstruction technique,” in International Conference on Speckle, H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 279–283 (1985).
[CrossRef]

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1983).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

A, B, Alpha Ononis and C, D, Gamma Ononis for two apodizations applied to the matched filter. A, C, 28 pixels FWHM. B, D, 52 pixels FWHM. Frame size is 128 × 128. Observations were made with a 3.8-m telescope at 650 ± 1 nm. The image of Gamma Orionis is unresolved. The images of Alpha Orionis are different by 10% in size (50 frames for each image).

Fig. 2
Fig. 2

An image of an unresolved star (Zeta Aquaril A) with the 2.28-m telescope at 750 ± 10 nm, showing the (aberrated) telescopedetector psf. Logarithmic intensity scale. Processed from 100 frames, about 100 photons each. All figures are with North down, East to the right.

Fig. 3
Fig. 3

A, The binary star Capella (Alpha Aurigae, 3.8-m telescope, 550 ± 10 nm), with two sidelobes (at the top and bottom), created by double detections of single speckles. B, The impulse power spectrum for the same data, showing fringes at the 0.5% level produced by this effect. No apodization was applied to the matched filter (20 frames).

Fig. 4
Fig. 4

General flow chart for filtered WSA (refer to the text for explanation). The frame loop is temporarily broken and the filter updated when enough speckles have been counted. Capital letters signify Fourier space quantities, FT is a Fourier transform, s is the current frame, TH and F are the threshold and matched filters, a is the impulse frame, o is the image estimate, and B.P.F. is the bandpass filter.

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

n k ( x ) = i k ( x ) o ( x ) * s k ( x ) .
i k ( x ) = d ( x ) * [ o ( x ) * s k ( x ) + n k ( x ) ] .
I k ( u ) = D ( u ) [ O ( u ) S k ( u ) + N k ( u ) ]
C k ( u ) = D ( u ) [ O ( u ) S k ( u ) + N k ( u ) ] A k * ( u ) ,
R k ( u ) = A k ( u ) A k * ( u ) = | A k ( u ) | 2 .
W ( u ) C ( u ) R ( u ) = D ( u ) [ O ( u ) S ( u ) A * ( u ) + N ( u ) A * ( u ) ] | ( u ) | 2 .
W ( u ) D ( u ) O ( u ) S ( u ) A * ( u ) | A ( u ) | 2 .
s k ( x ) = | FT [ ψ k ( u ) P ( u ) ] | 2 ,
a k ( x ) = | FT [ ψ k ( u ) P ( u ) ] | 2 | F T [ P ( u ) ] | 2 ,
S ( u ) A * ( u ) = d v d v ψ ( v ) ψ * ( v + u ) ψ * ( v ) ψ ( v + u ) P ( v ) P * ( v + u ) P * ( v ) P ( v + u ) S 1 d v P ( v ) P * ( v + u ) = S 2 d u M ( u , u ) H ( u , u ) S 1 d v P ( v ) P * ( v + u ) = P s ( u ) T ( u ) ,
| A ( u ) | 2 = d v d v ψ ( v ) ψ * ( v + u ) ψ * ( v ) ψ ( v + u ) P ( v ) P * ( v + u ) P * ( v ) P ( v + u ) S 2 d v d v P ( v ) P * ( v + u ) P * ( v ) P ( v + u ) = S 2 d u M ( u , u ) H ( u , u ) S 1 d u H ( u , u ) = P s ( u ) T 2 ( u ) .
W ( u ) D ( u ) O ( u ) T ( u ) .
i k ( x ) = d ( x ) * n = 1 N k δ ( x x n ) .
a k ( x ) = l = 1 L k δ ( x x l ) + j = 1 K k δ ( x x j ) .
E n l j [ C k ( u ) ] = D ( u ) { n = 1 N k l = 1 L k E n l [ exp 2 π i u · ( x n x l ) ] + n = 1 N k j = 1 K k E n j [ exp 2 π i u · ( x n x j ) ] } .
λ k ( x ) exp 2 π i u · x d x μ k ( x ) exp 2 π i u · x d x λ k ( x ) d x μ k ( x ) d x = Λ k ( u ) M k * ( u ) Λ k ( 0 ) M k * ( 0 ) ,
E n l j [ C k ( u ) ] = D ( u ) [ K k + ( K k N k K k ) | Λ k ( u ) | 2 | Λ k ( 0 ) | 2 + L k N k Λ k ( u ) M k * ( u ) Λ h ( 0 ) M k * ( 0 ) ] .
E n l j , NL [ C k ( u ) ] = D ( u ) [ K + ( K N K ) | Λ ( u ) | 2 | Λ ( 0 ) | 2 + L N Λ ( u ) M * ( u ) Λ ( 0 ) M * ( 0 ) ] ,
E n l , NL [ R k ( u ) ] = L + L ( 2 ) | M ( u ) | 2 | M ( 0 ) | 2 + K + K ( 2 ) | Λ ( u ) | 2 | Λ ( 0 ) | 2 + K L Λ ( u ) M * ( u ) Λ ( 0 ) M * ( 0 ) ,
W ( u ) = D ( u ) L N O ( u ) P s ( u ) / T ( u ) + K ( N 1 ) | O ( u ) | 2 P s ( u ) + K L ( 2 ) P s ( u ) / T 2 ( u ) + K ( 2 ) | O ( u ) | 2 P s ( u ) + K L O ( u ) P s ( u ) / T ( u ) + K + L ,
W ( u ) ( N / L ) D ( u ) O ( u ) Q ( u ) ,
Q ( u ) = T ( u ) P s ( u ) P s ( u ) + T 2 ( u ) / L
P S ( u ) B 2 ( u ) T 2 ( u ) + T ( u ) / L ,
L = 2.3 ( D / r 0 ) 2 = S / d u B 2 ( u ) .
Q ( u ) = { T ( u ) , u r 0 / λ T ( u ) T ( u ) + 1 , u r 0 / λ .
F m ( u ) = C exp ( 2 π i u ) x O * ( u ) / P n ( u ) ,
F m ( u ) = O * ( u ) exp 2 π i u · x 0 .
A k ( u ) = F w ( u ) I k ( u ) = F w ( u ) D ( u ) [ O ( u ) S k ( u ) + N ( u ) ] .
e = d x | s ( x ) a ( x ) | 2 = d u | S ( u ) A ( u ) | 2
F w ( u ) = O ( u ) * P s ( u ) D ( u ) [ | O ( u ) | 2 P s ( u ) + P n ( u ) ] ,
P i ( u ) = | D ( u ) | 2 [ | O ( u ) | 2 P s ( u ) + P n ( u ) ]
F w ( u ) = [ D ( u ) O ( u ) ] * P s ( u ) P i ( u ) .

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