Abstract

The transfer function associated with the Knox–Thompson speckle imaging technique is investigated. Numerical model transfer functions using log-normal statistics for perturbations of the complex wave front, the near-field approximation, and a Kolmogorov spectrum for atmospheric turbulence statistics are presented. Simple approximations for the transfer function are discussed. As with the transfer function of Labeyrie’s speckle interferometry technique, the portion beyond the seeing limit can be represented as the transfer function of an unaberrated telescope times a seeing-dependent constant. An additional factor depends on the frequency shift of the Knox – Thompson cross spectra. The influence of the frequency shift on the reconstructed phase error is discussed for simple reconstruction problems.

© 1988 Optical Society of America

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  1. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” As-trophys. J. 193, L45–L48 (1974).
    [CrossRef]
  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).
  3. G. Weigelt, B. Wirnitzer, “Image reconstruction by the speckle-masking method,” Opt. Lett. 8, 389–391 (1983).
    [CrossRef] [PubMed]
  4. P. Nisenson, R. V. Stachnik, M. Karovska, R. W. Noyes, “A new optical source associated with T Tauri,” Astrophys. J. 297, L17–L20 (1985).
    [CrossRef]
  5. R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
    [CrossRef]
  6. D. L. Fried, “Angular dependence of the atmospheric turbulence effect in speckle interferometry,” Opt. Acta 26, 597–613 (1979).
    [CrossRef]
  7. O. von der Lühe, “High resolution speckle imaging of solar small scale structure: the influence of anisoplanatism,” in High Resolution in Solar Physics, Vol. 233 of Lecture Notes in Physics, R. Muller, ed. (Springer-Verlag, Berlin, 1985), pp. 96–102.
    [CrossRef]
  8. C. Leinert, M. Haas, “Infrared speckle interferometry on Calar Alto,” in High Resolution Interferometric Imaging from the Ground, Proceedings of the Joint European Southern Observatory–National Optical Astronomy Observatories Conference (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 233–236.
  9. 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]
  10. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very short and very long exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  11. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in OpticsE. Wolf, ed. (Elsevier, New York, 1981), Vol. XIX.
    [CrossRef]
  12. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (Israel Program for Scientific Translations, Jerusalem, 1971).
  13. R. Barakat, P. Nisenson, “Influence of the wave-front correlation function and deterministic wave-front aberrations on the speckle image reconstruction problem,” J. Opt. Soc. Am. 71, 1390–1402 (1981).
  14. J. B. Breckinridge, “Measurement of the amplitude of phase excursions in the earth’s atmosphere,” J. Opt. Soc. Am. 66, 143–144 (1976).
    [CrossRef]
  15. C. Roddier, “Measurements of the atmospheric attenuation of the spectral components of astonomical images,” J. Opt. Soc. Am. 66, 478–482 (1976).
    [CrossRef]
  16. M. Bertolotti, M. Carnevale, A. Consortini, L. Ronchi, “Optical propagation: problems and trends,” Opt. Acta 26, 507–529 (1979).
    [CrossRef]
  17. D. P. Karo, A. M. Schneidermann, “Transfer functions, correlation scales, and phase retrieval in speckle interferometry,” J. Opt. Soc. Am. 67, 1583–1587 (1977).
    [CrossRef]
  18. R. Deron, J. C. Fontanella, “Restauration d’images dégradées par la turbulence atmosphérique selon la méthode de Knox et Thompson,” J. Opt. (Paris) 15, 15–23 (1984).
    [CrossRef]
  19. O. von der Lühe, R. B. Dunn, “Solar granulation power spectra from speckle interferometry,” Astron. Astrophys. 177, 265–276 (1987).
  20. J. D. Freeman, J. C. Christou, F. Roddier, D. W. McCarthy, M. C. Cobb, “Application of triple correlation to one-dimensional infrared speckle data,” in High Resolution Interferometric Imaging from the Ground, Proceedings of the Joint European Southern Observatory–National Optical Astronomy Observatories Conference (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 47–50.

1987 (1)

O. von der Lühe, R. B. Dunn, “Solar granulation power spectra from speckle interferometry,” Astron. Astrophys. 177, 265–276 (1987).

1985 (1)

P. Nisenson, R. V. Stachnik, M. Karovska, R. W. Noyes, “A new optical source associated with T Tauri,” Astrophys. J. 297, L17–L20 (1985).
[CrossRef]

1984 (1)

R. Deron, J. C. Fontanella, “Restauration d’images dégradées par la turbulence atmosphérique selon la méthode de Knox et Thompson,” J. Opt. (Paris) 15, 15–23 (1984).
[CrossRef]

1983 (2)

G. Weigelt, B. Wirnitzer, “Image reconstruction by the speckle-masking method,” Opt. Lett. 8, 389–391 (1983).
[CrossRef] [PubMed]

R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
[CrossRef]

1981 (1)

1979 (2)

D. L. Fried, “Angular dependence of the atmospheric turbulence effect in speckle interferometry,” Opt. Acta 26, 597–613 (1979).
[CrossRef]

M. Bertolotti, M. Carnevale, A. Consortini, L. Ronchi, “Optical propagation: problems and trends,” Opt. Acta 26, 507–529 (1979).
[CrossRef]

1977 (1)

1976 (2)

1974 (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” As-trophys. J. 193, L45–L48 (1974).
[CrossRef]

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)

Barakat, R.

Bertolotti, M.

M. Bertolotti, M. Carnevale, A. Consortini, L. Ronchi, “Optical propagation: problems and trends,” Opt. Acta 26, 507–529 (1979).
[CrossRef]

Breckinridge, J. B.

Carnevale, M.

M. Bertolotti, M. Carnevale, A. Consortini, L. Ronchi, “Optical propagation: problems and trends,” Opt. Acta 26, 507–529 (1979).
[CrossRef]

Christou, J. C.

J. D. Freeman, J. C. Christou, F. Roddier, D. W. McCarthy, M. C. Cobb, “Application of triple correlation to one-dimensional infrared speckle data,” in High Resolution Interferometric Imaging from the Ground, Proceedings of the Joint European Southern Observatory–National Optical Astronomy Observatories Conference (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 47–50.

Cobb, M. C.

J. D. Freeman, J. C. Christou, F. Roddier, D. W. McCarthy, M. C. Cobb, “Application of triple correlation to one-dimensional infrared speckle data,” in High Resolution Interferometric Imaging from the Ground, Proceedings of the Joint European Southern Observatory–National Optical Astronomy Observatories Conference (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 47–50.

Consortini, A.

M. Bertolotti, M. Carnevale, A. Consortini, L. Ronchi, “Optical propagation: problems and trends,” Opt. Acta 26, 507–529 (1979).
[CrossRef]

Deron, R.

R. Deron, J. C. Fontanella, “Restauration d’images dégradées par la turbulence atmosphérique selon la méthode de Knox et Thompson,” J. Opt. (Paris) 15, 15–23 (1984).
[CrossRef]

Dunn, R. B.

O. von der Lühe, R. B. Dunn, “Solar granulation power spectra from speckle interferometry,” Astron. Astrophys. 177, 265–276 (1987).

Fontanella, J. C.

R. Deron, J. C. Fontanella, “Restauration d’images dégradées par la turbulence atmosphérique selon la méthode de Knox et Thompson,” J. Opt. (Paris) 15, 15–23 (1984).
[CrossRef]

Freeman, J. D.

J. D. Freeman, J. C. Christou, F. Roddier, D. W. McCarthy, M. C. Cobb, “Application of triple correlation to one-dimensional infrared speckle data,” in High Resolution Interferometric Imaging from the Ground, Proceedings of the Joint European Southern Observatory–National Optical Astronomy Observatories Conference (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 47–50.

Fried, D. L.

D. L. Fried, “Angular dependence of the atmospheric turbulence effect in speckle interferometry,” Opt. Acta 26, 597–613 (1979).
[CrossRef]

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very short and very long exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
[CrossRef]

Haas, M.

C. Leinert, M. Haas, “Infrared speckle interferometry on Calar Alto,” in High Resolution Interferometric Imaging from the Ground, Proceedings of the Joint European Southern Observatory–National Optical Astronomy Observatories Conference (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 233–236.

Karo, D. P.

Karovska, M.

P. Nisenson, R. V. Stachnik, M. Karovska, R. W. Noyes, “A new optical source associated with T Tauri,” Astrophys. J. 297, L17–L20 (1985).
[CrossRef]

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” As-trophys. J. 193, L45–L48 (1974).
[CrossRef]

Korff, 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).

Leinert, C.

C. Leinert, M. Haas, “Infrared speckle interferometry on Calar Alto,” in High Resolution Interferometric Imaging from the Ground, Proceedings of the Joint European Southern Observatory–National Optical Astronomy Observatories Conference (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 233–236.

McCarthy, D. W.

J. D. Freeman, J. C. Christou, F. Roddier, D. W. McCarthy, M. C. Cobb, “Application of triple correlation to one-dimensional infrared speckle data,” in High Resolution Interferometric Imaging from the Ground, Proceedings of the Joint European Southern Observatory–National Optical Astronomy Observatories Conference (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 47–50.

Nisenson, P.

P. Nisenson, R. V. Stachnik, M. Karovska, R. W. Noyes, “A new optical source associated with T Tauri,” Astrophys. J. 297, L17–L20 (1985).
[CrossRef]

R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
[CrossRef]

R. Barakat, P. Nisenson, “Influence of the wave-front correlation function and deterministic wave-front aberrations on the speckle image reconstruction problem,” J. Opt. Soc. Am. 71, 1390–1402 (1981).

Noyes, R. W.

P. Nisenson, R. V. Stachnik, M. Karovska, R. W. Noyes, “A new optical source associated with T Tauri,” Astrophys. J. 297, L17–L20 (1985).
[CrossRef]

R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
[CrossRef]

Roddier, C.

Roddier, F.

J. D. Freeman, J. C. Christou, F. Roddier, D. W. McCarthy, M. C. Cobb, “Application of triple correlation to one-dimensional infrared speckle data,” in High Resolution Interferometric Imaging from the Ground, Proceedings of the Joint European Southern Observatory–National Optical Astronomy Observatories Conference (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 47–50.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in OpticsE. Wolf, ed. (Elsevier, New York, 1981), Vol. XIX.
[CrossRef]

Ronchi, L.

M. Bertolotti, M. Carnevale, A. Consortini, L. Ronchi, “Optical propagation: problems and trends,” Opt. Acta 26, 507–529 (1979).
[CrossRef]

Schneidermann, A. M.

Stachnik, R. V.

P. Nisenson, R. V. Stachnik, M. Karovska, R. W. Noyes, “A new optical source associated with T Tauri,” Astrophys. J. 297, L17–L20 (1985).
[CrossRef]

R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (Israel Program for Scientific Translations, Jerusalem, 1971).

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” As-trophys. J. 193, L45–L48 (1974).
[CrossRef]

von der Lühe, O.

O. von der Lühe, R. B. Dunn, “Solar granulation power spectra from speckle interferometry,” Astron. Astrophys. 177, 265–276 (1987).

O. von der Lühe, “High resolution speckle imaging of solar small scale structure: the influence of anisoplanatism,” in High Resolution in Solar Physics, Vol. 233 of Lecture Notes in Physics, R. Muller, ed. (Springer-Verlag, Berlin, 1985), pp. 96–102.
[CrossRef]

Weigelt, G.

Wirnitzer, B.

As-trophys. J. (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” As-trophys. J. 193, L45–L48 (1974).
[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).

O. von der Lühe, R. B. Dunn, “Solar granulation power spectra from speckle interferometry,” Astron. Astrophys. 177, 265–276 (1987).

Astrophys. J. (2)

P. Nisenson, R. V. Stachnik, M. Karovska, R. W. Noyes, “A new optical source associated with T Tauri,” Astrophys. J. 297, L17–L20 (1985).
[CrossRef]

R. V. Stachnik, P. Nisenson, R. W. Noyes, “Speckle image reconstruction of solar features,” Astrophys. J. 271, L37–L40 (1983).
[CrossRef]

J. Opt. (Paris) (1)

R. Deron, J. C. Fontanella, “Restauration d’images dégradées par la turbulence atmosphérique selon la méthode de Knox et Thompson,” J. Opt. (Paris) 15, 15–23 (1984).
[CrossRef]

J. Opt. Soc. Am. (6)

Opt. Acta (2)

D. L. Fried, “Angular dependence of the atmospheric turbulence effect in speckle interferometry,” Opt. Acta 26, 597–613 (1979).
[CrossRef]

M. Bertolotti, M. Carnevale, A. Consortini, L. Ronchi, “Optical propagation: problems and trends,” Opt. Acta 26, 507–529 (1979).
[CrossRef]

Opt. Lett. (1)

Other (5)

O. von der Lühe, “High resolution speckle imaging of solar small scale structure: the influence of anisoplanatism,” in High Resolution in Solar Physics, Vol. 233 of Lecture Notes in Physics, R. Muller, ed. (Springer-Verlag, Berlin, 1985), pp. 96–102.
[CrossRef]

C. Leinert, M. Haas, “Infrared speckle interferometry on Calar Alto,” in High Resolution Interferometric Imaging from the Ground, Proceedings of the Joint European Southern Observatory–National Optical Astronomy Observatories Conference (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 233–236.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in OpticsE. Wolf, ed. (Elsevier, New York, 1981), Vol. XIX.
[CrossRef]

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (Israel Program for Scientific Translations, Jerusalem, 1971).

J. D. Freeman, J. C. Christou, F. Roddier, D. W. McCarthy, M. C. Cobb, “Application of triple correlation to one-dimensional infrared speckle data,” in High Resolution Interferometric Imaging from the Ground, Proceedings of the Joint European Southern Observatory–National Optical Astronomy Observatories Conference (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 47–50.

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Figures (6)

Fig. 1
Fig. 1

Geometry of the four-circle overlap. The centers of the circles are located at the arrowheads labeled A, B, C, and D. The arrows indicate the magnitude and direction of the vectors q, δ, and Δ. In the case shown here, the overlap area is determined by only the three circles A, B, and D.

Fig. 2
Fig. 2

Decay of signal in the cross-spectrum transfer as a function of |Δ|/α. Solid line, approximate result β(Δ/α), [Eq. (26)]; □ and (Image), ratios of the CTF models and the STF models for ξ = 0 and ξ = 90 deg, respectively, averaged over the spatial-frequency range from −0.8 to 0.8 and multiplied by 0.342; dotted–dashed line, measurement of Cτ,17 multiplied by 0.342; dashed line, measurement of Cτ,17 with a constant 0.1 subtracted and then multiplied by 0.342.

Fig. 3
Fig. 3

CTF models by numerical evaluation of Eq. (11), for ξ = 0 deg and α = 0.1. Dotted line, Labeyrie case (STF); solid lines, models for (in descending order of the curves) |Δ| = 0.02, 0.05, 0.1, 0.15, 0.2. For negative relative wave numbers, the curves represent CTF models for ξ = π.

Fig. 4
Fig. 4

CTF models by numerical evaluation of Eq. (11), for |Δ| = 0.05 and α = 0.1. Solid line, ξ = 0; dotted–dashed line, ξ = π/4; dashed line, ξ = π/2.

Fig. 5
Fig. 5

CTF models by numerical evaluation of Eq. (11). In all cases, Δ = 1 and ξ = 0. In descending order of the curves, α = 0.05, 0.1, 0.15, 0.2, 0.25.

Fig. 6
Fig. 6

Phase error in the reconstruction for the one-dimensional reconstruction algorithm [Eqs. (38)]. The approximate result in Eq. (26) was used for the STF/CTF ratio in Eq. (37). (a) Phase error as a function of frequency for |Δ| = 0.02 and α = 0.1 and values for the product of SNR and the frame number N of (in descending order of the curves) 100, 200, 500, and 1000. (b) Reconstructed rms phase error averaged over all spatial frequencies as a function of |Δ|, for α = 0.1. N(SNR) values: ▵, 200; +, 500; ◊, 1000.

Equations (48)

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I i ( x ) = I 0 ( x ) * PSF i ( x ) ,
F i ( f ) = F 0 ( f ) S i ( f ) ,
1 N i = 1 N F i ( f ) F i * ( f Δ ) = F 0 ( f ) F 0 * ( f Δ ) × 1 N i = 1 N S i ( f ) S i * ( f Δ )
S i ( f ) = W ( r + λ f 2 ) W * ( r λ f 2 ) × exp { j [ ϕ i ( r + λ f 2 ) ϕ i ( r λ f 2 ) ] } d r .
D ( ρ ) = [ ϕ ( r ) ϕ ( r ρ ) ] 2 .
D ( ρ ) = 6.88 ( | ρ | r 0 ) 5 / 3 .
S i ( q ) = W ( r + q / 2 ) W * ( r q / 2 ) × exp { j [ ϕ i ( r + q / 2 ) ϕ i ( r q / 2 ) ] } d r .
D ( ρ ) = 6.88 ( | ρ | α ) 5 / 3 .
CTF Δ ( q ) = S i ( q ) S i * ( q Δ ) .
CTF Δ ( q ) = W ( r + q / 2 ) W * ( r q / 2 ) × exp { j [ ϕ ( r + q / 2 ) ϕ ( r q / 2 ) } d r × W * [ r + ( q Δ ) / 2 ] W [ r ( q Δ ) / 2 ] × exp ( j { ϕ [ r + ( q Δ ) / 2 ] + ϕ [ r + ( q Δ ) / 2 ] } ) d r .
CTF Δ ( q ) = W ( σ + δ + q 2 ) W * ( σ + δ q 2 ) × W * ( σ δ + q Δ 2 ) W ( σ δ q + Δ 2 ) d σ × exp { ½ [ D ( q ) + D ( q Δ ) + D ( δ + Δ / 2 ) + D ( δ Δ / 2 ) D ( q Δ / 2 + δ ) D ( q Δ / 2 δ ) ] } d δ .
CTF Δ ( 0 ) = | W ( σ + δ 2 ) | 2 W * ( σ δ Δ 2 ) × W ( σ δ + Δ 2 ) d σ d δ exp [ ½ D ( Δ ) ] .
CTF Δ ( 0 ) = | W ( r ) | 2 d r W * ( r Δ / 2 ) × W ( r + Δ / 2 ) d r exp [ ½ D ( Δ ) ] .
W ( r ) = Π ( r ) { 1 if | r | < ½ ½ if | r | = ½ 0 if | r | > ½ .
CTF Δ ( 0 ) = π 8 [ cos 1 Δ Δ ( 1 Δ 2 ) 1 / 2 ] exp [ 3.44 ( Δ α ) 5 / 3 ] .
6.88 ( x + α ) 5 / 3 6.88 ( x α ) 5 / 3 { 1 + 5 6 [ | | | x | cos θ + ( | | | x | ) 2 ( 1 1 12 cos 2 θ ) + ] } ,
6.88 { ( q α ) 5 / 3 + ( | q Δ | α ) 5 / 3 + 5 6 [ | q | | δ Δ / 2 | cos θ ( | δ Δ / 2 | α ) 5 / 3 | q | | δ + Δ / 2 | cos θ ( | δ + Δ / 2 | α ) 5 / 3 ] } .
6.88 [ ( q α ) 5 / 3 + ( | q Δ | α ) 5 / 3 ] .
CTF Δ ( q ) W ( σ + δ + q 2 ) W * ( σ + δ q 2 ) × W * ( σ δ + q Δ 2 ) W ( σ δ q + Δ 2 ) d σ d δ × exp { 3.44 [ ( q α ) 5 / 3 + ( | q Δ | α ) 5 / 3 ] } .
CTF Δ ( q ) | | q | α S t , a ( q ) exp [ 3.44 ( | q | α ) 5 / 3 ] S t , a * ( q Δ ) × exp [ 3.44 ( | q Δ | α ) 5 / 3 ] S LE ( q ) S LE ( q Δ ) .
6.88 [ ( | δ + Δ / 2 | α ) 5 / 3 + ( | δ + Δ / 2 | α ) 5 / 3 + ( q α ) 5 / 3 + ( q Δ α ) 5 / 3 2 ( q Δ / 2 α ) 5 / 3 + 5 3 δ 2 | q Δ / 2 | 2 ( 1 1 12 cos 2 θ ) ] .
6.88 [ ( | δ + Δ / 2 | α ) 5 / 3 + ( | δ + Δ / 2 | α ) 5 / 3 ] .
CTF Δ ( q ) W ( σ + q 2 ) W * ( σ q 2 ) × W * ( σ + q Δ 2 ) W ( σ q + Δ 2 ) d σ × exp { 3.44 [ ( | δ + Δ / 2 | α ) 5 / 3 + ( | δ Δ / 2 | α ) 5 / 3 ] } d δ .
W ( σ + q 2 ) W * ( σ q 2 ) W * ( σ + q Δ 2 ) × W ( σ q + Δ 2 ) d σ S 0 ( q ) ,
ρ = δ α , η = Δ 2 α , d δ = α 2 d ρ ;
exp { 3.44 [ ( | δ + Δ / 2 | α ) 5 / 3 + ( | δ Δ / 2 | α ) 5 / 3 ] } d δ = α 2 exp [ 3.44 ( | ρ + η | 5 / 3 + | ρ η | 5 / 3 ) ] d ρ .
CTF Δ ( q ) | | q | α α 2 β ( Δ α ) S 0 ( q ) .
CTF Δ ( q ) = S LE ( q ) S LE ( q Δ ) + α 2 β ( Δ α ) S 0 ( q ) .
STF ( q ) = S LE 2 ( q ) + 0.342 α 2 S 0 ( q ) .
σ ϕ i 2 = σ Im [ F i ( q ) F i * ( q Δ ) ] 2 [ | F 0 ( q ) | 2 STF ( q ) ] 2 ,
σ ϕ i 2 = σ Im [ F i ( q ) F i * ( q Δ ) ] 2 [ | F 0 ( q ) F 0 * ( q Δ ) | CTF Δ ( q ) ] 2 .
σ Im [ F i ( q ) F i * ( q Δ ) ] 2 σ N 2 | F 0 ( q ) | 2 STF ( q ) ,
σ ϕ i 2 σ N 2 | F 0 ( q ) | 2 STF ( q ) [ | F 0 ( q ) F 0 * ( q Δ ) | CTF Δ ( q ) ] 2
σ ϕ i 2 1 N σ N 2 | F 0 ( q ) | 2 STF ( q ) [ | F 0 ( q ) F 0 * ( q Δ ) | CTF Δ ( q ) ] 2
σ ϕ i 2 1 N σ N 2 | F 0 ( q ) | 2 STF ( q ) [ CTF Δ ( q ) ] 2 .
SNR ( q ) = | F 0 ( q ) | 2 STF ( q ) σ N 2 ,
σ ϕ i 2 1 N 1 SNR ( q ) [ STF ( q ) CTF Δ ( q ) ] 2 .
ψ k = ψ k 1 + ϕ k , ψ 0 = 0 or ψ k = j = 1 k ϕ j .
F o , k = F i , k F i , k 1 * CTF Δ , k F o , k 1 * , F o , 0 = 1 ,
CTF Δ ( q ) = W ( r + q / 2 ) W * ( r q / 2 ) × exp { j [ ϕ ( r + q / 2 ) ϕ ( r q / 2 ) ] } d r × W * [ r + ( q Δ ) / 2 ] W [ r ( q Δ ) / 2 ] × exp ( j { ϕ [ r + ( q Δ ) / 2 ] + ϕ [ r + ( q Δ ) / 2 ] } ) d r .
CTF Δ ( q ) = W ( r + q / 2 ) W * ( r q / 2 ) × W * [ r + ( q Δ ) / 2 ] W [ r ( q Δ ) / 2 ] × exp ( j { ϕ ( r + q / 2 ) ϕ ( r q / 2 ) ϕ [ r + ( q Δ ) / 2 ] + ϕ [ r ( q Δ ) / 2 ] } ) d r d r .
exp ( j { ϕ ( r + q / 2 ) ϕ ( r q / 2 ) ϕ [ r + ( q Δ ) / 2 ] + ϕ [ r + ( q Δ ) / 2 ] } ) = exp ( ½ { ϕ ( r + q / 2 ) ϕ ( r q / 2 ) ϕ [ r + ( q Δ ) / 2 ] + ϕ [ r ( q Δ ) / 2 ] } 2 ) .
{ ϕ ( r + q / 2 ) ϕ ( r q / 2 ) ϕ [ r + ( q Δ ) / 2 ] + ϕ [ r ( q Δ ) / 2 ] } 2 = [ ϕ ( r + q / 2 ) ϕ ( r q / 2 ) ] 2 + { ϕ [ r + ( q + Δ ) / 2 ] ϕ [ r ( q Δ ) / 2 ] } 2 2 ϕ ( r + q / 2 ) ϕ [ r + ( q Δ ) / 2 ] 2 ϕ ( r q / 2 ) ϕ [ r + ( q Δ ) / 2 ] + 2 ϕ ( r q / 2 ) ϕ [ r + ( q Δ ) / 2 ] + 2 ϕ ( r + q / 2 ) ϕ [ r + ( q Δ ) / 2 ] .
ϕ 2 ( r + q / 2 ) + ϕ 2 [ r + ( q Δ ) / 2 ) + ϕ 2 ( r q / 2 ) + ϕ 2 [ r ( q + Δ ) / 2 ) ϕ 2 ( r + q / 2 ) ϕ 2 [ r + ( q Δ ) / 2 ) ϕ 2 ( r q / 2 ) ϕ 2 [ r ( q + Δ ) / 2 ] .
{ ϕ ( r + q / 2 ) ϕ ( r q / 2 ) ϕ [ r + ( q Δ ) / 2 ] + ϕ [ r ( q Δ ) / 2 ] } 2 = [ ϕ ( r + q / 2 ) ϕ ( r q / 2 ) ] 2 + { ϕ [ r + ( q + Δ ) / 2 ] ϕ [ r ( q Δ ) / 2 ] } 2 + [ ϕ ( r + q / 2 ) ϕ [ r + ( q Δ ) / 2 ] ] 2 + { ϕ ( r q / 2 ) ϕ [ r + ( q Δ ) / 2 ] } 2 [ ϕ ( r q / 2 ) ϕ [ r + ( q Δ ) / 2 ] ] 2 [ ϕ ( r + q / 2 ) ϕ [ r ( q Δ ) / 2 ] ] 2 = D ( q ) + D ( q Δ ) + D ( r + r + Δ / 2 ) + D ( r r Δ / 2 ) D ( q + r r + Δ / 2 ) D ( q r + r Δ / 2 ) .
CTF Δ ( q ) = W ( r + q / 2 ) W * ( r q / 2 ) × W * ( r + ( q Δ ) / 2 ) W [ r ( q Δ ) / 2 ] × exp { ½ [ D ( q ) + D ( q Δ ) + D ( r r + Δ / 2 ) + D ( r r Δ / 2 ) D ( q + r r Δ / 2 ) D ( q r + r Δ / 2 ) ] } d r d r .
σ = r + r , δ = r r , r = σ + δ 2 , r = σ δ 2 .
CTF Δ ( q ) = W ( σ + δ + q 2 ) W * ( σ + δ q 2 ) × W * ( σ δ + q Δ 2 ) W ( σ δ q + Δ 2 ) d σ × exp { ½ [ D ( q ) + D ( q Δ ) + D ( δ + Δ / 2 ) + D ( δ Δ / 2 ) D ( q Δ / 2 + δ ) D ( q Δ / 2 + δ ) ] } d δ .

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