Abstract

Turbulence-degraded images can be restored by using the Knox–Thompson algorithm. This algorithm is briefly described, and its photon-noise limitations are discussed. Peculiar difficulties related to the phase dislocations in the Fourier space are pointed out, and a solution to this problem is proposed. This algorithm does not require more photons than the speckle-interferometry technique and could provide much more information on complex astronomical objects. To illustrate these phase dislocations and signal-to-noise ratio problems, different restorations are presented that result from degraded images obtained with a turbulence-simulation facility and from astronomical data.

© 1987 Optical Society of America

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References

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  1. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in stars images,” Astron. Astrophys. 6, 85–87 (1970).
  2. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astron. J. 193, L45–L48 (1974).
    [Crossref]
  3. G. P. Weigelt, “Modified astronomical speckle interferometry. Speckle masking,” Opt. Commun. 21, 55–59 (1977).
    [Crossref]
  4. S. F. Gull, G. J. Daniell, “Image reconstruction from incomplete and noisy data,” Nature 272, 686–690 (1978).
    [Crossref]
  5. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik 61, 257–262 (1982).
  6. K. T. Knox, “Image retrieval from astronomical speckle patterns,”J. Opt. Soc. Am. 66, 1236–1239 (1976).
    [Crossref]
  7. R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, “Speckle image reconstruction of solar features,” Nature 266, 149–151 (1977).
    [Crossref]
  8. R. Deron, J. C. Fontanella, “Restauration d’images dégradées par la turbulence atmosphérique selon la méthode de Knox et Thompson. Etude théorique et expérimentale,”J. Opt. (Paris) 15, 15–23 (1984).
    [Crossref]
  9. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,”J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [Crossref]
  10. C. Roddier, F. Roddier, “Correlation of waves perturbed by turbulence, J. Opt. Soc. Am. 63, 661–663 (1973).
    [Crossref]
  11. C. Aime, “Measurements of averaged squared modulus of atmospheric-lens modulation transfer function,”J. Opt. Soc. Am. 64, 1129–1132 (1974).
    [Crossref]
  12. R. H. Hudgin, “Wave front reconstruction for compensated imaging,”J. Opt. Soc. Am. 67, 375–378 (1977).
    [Crossref]
  13. M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreductibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
    [Crossref] [PubMed]
  14. M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
    [Crossref]
  15. N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovitch, “Wave front dislocations: topological limitations for adaptative systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
    [Crossref]
  16. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,”J. Opt. Soc. Am. 69, 393–399 (1979).
    [Crossref]
  17. R. L. Frost, C. K. Rushforth, B. S. Baxter, “Fast FFT-based algorithm for phase estimation in speckle imaging,” Appl. Opt. 18, 2056–2061 (1979).
    [Crossref] [PubMed]
  18. B. J. Brames, J. C. Dainty, “Method for determining object intensity distributions in stellar speckle interferometry,”J. Opt. Soc. Am. 71, 1542–1545 (1981).
    [Crossref]
  19. J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
    [Crossref]
  20. J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 69, 786–790 (1979).
    [Crossref]
  21. P. Nisenson, C. Papaliolios, “Effects of photon noise on speckle image reconstruction with the Knox and Thompson algorithm,” Opt. Commun. 47, 91–96 (1983).
    [Crossref]
  22. F. Roddier, “Atmospheric limitations to high angular resolution imaging,” presented at the European Southern Observatories Conference on the Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, Garching, Federal Republic of Germany, March 24–27, 1981.
  23. M. Billard, G. Fertin, J. C. Fontanella, “Atmospheric turbulence simulation cell for optical propagation experiment,” presented at the 4th International Symposium on Gas Flow and Chemical Lasers, Stresa, Italy, September 13–17, 1982.
  24. D. Bonneau, CERGA, Observatoire de Calern, Caussols, 06460 St. Vallier de Thiey, France.
  25. A. Blazit, R. Foy, “A photon counting detector system for the E.S.O.N.T.T. spectrograph,” Proposal to European Southern Observatories (unpublished).
  26. J. C. Fontanella, “Wavefront sensing, adaptive optics and deconvolutions,” presented at the CFHT Workshop on High-Resolution Imaging in Astronomy, Kamuela, Hawaii, October 25–26, 1984).

1984 (2)

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

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[Crossref]

1983 (3)

1982 (1)

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik 61, 257–262 (1982).

1981 (1)

1979 (3)

1978 (1)

S. F. Gull, G. J. Daniell, “Image reconstruction from incomplete and noisy data,” Nature 272, 686–690 (1978).
[Crossref]

1977 (3)

G. P. Weigelt, “Modified astronomical speckle interferometry. Speckle masking,” Opt. Commun. 21, 55–59 (1977).
[Crossref]

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, “Speckle image reconstruction of solar features,” Nature 266, 149–151 (1977).
[Crossref]

R. H. Hudgin, “Wave front reconstruction for compensated imaging,”J. Opt. Soc. Am. 67, 375–378 (1977).
[Crossref]

1976 (1)

1974 (2)

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

C. Aime, “Measurements of averaged squared modulus of atmospheric-lens modulation transfer function,”J. Opt. Soc. Am. 64, 1129–1132 (1974).
[Crossref]

1973 (1)

1970 (1)

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

1966 (1)

Aime, C.

Baranova, N. B.

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik 61, 257–262 (1982).

Baxter, B. S.

Belsher, J. F.

J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
[Crossref]

Billard, M.

M. Billard, G. Fertin, J. C. Fontanella, “Atmospheric turbulence simulation cell for optical propagation experiment,” presented at the 4th International Symposium on Gas Flow and Chemical Lasers, Stresa, Italy, September 13–17, 1982.

Blazit, A.

A. Blazit, R. Foy, “A photon counting detector system for the E.S.O.N.T.T. spectrograph,” Proposal to European Southern Observatories (unpublished).

Bonneau, D.

D. Bonneau, CERGA, Observatoire de Calern, Caussols, 06460 St. Vallier de Thiey, France.

Brames, B. J.

Dainty, J. C.

Daniell, G. J.

S. F. Gull, G. J. Daniell, “Image reconstruction from incomplete and noisy data,” Nature 272, 686–690 (1978).
[Crossref]

Deron, R.

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

Ehn, D. C.

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, “Speckle image reconstruction of solar features,” Nature 266, 149–151 (1977).
[Crossref]

Fertin, G.

M. Billard, G. Fertin, J. C. Fontanella, “Atmospheric turbulence simulation cell for optical propagation experiment,” presented at the 4th International Symposium on Gas Flow and Chemical Lasers, Stresa, Italy, September 13–17, 1982.

Fiddy, M. A.

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[Crossref]

M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreductibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
[Crossref] [PubMed]

Fontanella, J. C.

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

M. Billard, G. Fertin, J. C. Fontanella, “Atmospheric turbulence simulation cell for optical propagation experiment,” presented at the 4th International Symposium on Gas Flow and Chemical Lasers, Stresa, Italy, September 13–17, 1982.

J. C. Fontanella, “Wavefront sensing, adaptive optics and deconvolutions,” presented at the CFHT Workshop on High-Resolution Imaging in Astronomy, Kamuela, Hawaii, October 25–26, 1984).

Foy, R.

A. Blazit, R. Foy, “A photon counting detector system for the E.S.O.N.T.T. spectrograph,” Proposal to European Southern Observatories (unpublished).

Fried, D. L.

Frost, R. L.

Goodman, J. W.

J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
[Crossref]

Greenaway, A. H.

Gull, S. F.

S. F. Gull, G. J. Daniell, “Image reconstruction from incomplete and noisy data,” Nature 272, 686–690 (1978).
[Crossref]

Hall, T. J.

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[Crossref]

Hudgin, R. H.

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, “Speckle image reconstruction of solar features,” Nature 266, 149–151 (1977).
[Crossref]

R. H. Hudgin, “Wave front reconstruction for compensated imaging,”J. Opt. Soc. Am. 67, 375–378 (1977).
[Crossref]

Hunt, B. R.

Knox, K. T.

K. T. Knox, “Image retrieval from astronomical speckle patterns,”J. Opt. Soc. Am. 66, 1236–1239 (1976).
[Crossref]

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

Labeyrie, A.

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

Mamaev, A. V.

Nisenson, P.

P. Nisenson, C. Papaliolios, “Effects of photon noise on speckle image reconstruction with the Knox and Thompson algorithm,” Opt. Commun. 47, 91–96 (1983).
[Crossref]

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, “Speckle image reconstruction of solar features,” Nature 266, 149–151 (1977).
[Crossref]

Papaliolios, C.

P. Nisenson, C. Papaliolios, “Effects of photon noise on speckle image reconstruction with the Knox and Thompson algorithm,” Opt. Commun. 47, 91–96 (1983).
[Crossref]

Pilipetsky, N. F.

Roddier, C.

Roddier, F.

C. Roddier, F. Roddier, “Correlation of waves perturbed by turbulence, J. Opt. Soc. Am. 63, 661–663 (1973).
[Crossref]

F. Roddier, “Atmospheric limitations to high angular resolution imaging,” presented at the European Southern Observatories Conference on the Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, Garching, Federal Republic of Germany, March 24–27, 1981.

Rushforth, C. K.

Schirf, V. E.

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, “Speckle image reconstruction of solar features,” Nature 266, 149–151 (1977).
[Crossref]

Scivier, M. S.

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[Crossref]

Shkunov, V. V.

Stachnik, R. V.

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, “Speckle image reconstruction of solar features,” Nature 266, 149–151 (1977).
[Crossref]

Thompson, B. J.

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

Weigelt, G. P.

G. P. Weigelt, “Modified astronomical speckle interferometry. Speckle masking,” Opt. Commun. 21, 55–59 (1977).
[Crossref]

Zel’dovitch, B. Ya.

Appl. Opt. (1)

Astron. Astrophys. (1)

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

Astron. J. (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astron. J. 193, L45–L48 (1974).
[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. Etude théorique et expérimentale,”J. Opt. (Paris) 15, 15–23 (1984).
[Crossref]

J. Opt. Soc. Am. (9)

Nature (2)

R. V. Stachnik, P. Nisenson, D. C. Ehn, R. H. Hudgin, V. E. Schirf, “Speckle image reconstruction of solar features,” Nature 266, 149–151 (1977).
[Crossref]

S. F. Gull, G. J. Daniell, “Image reconstruction from incomplete and noisy data,” Nature 272, 686–690 (1978).
[Crossref]

Opt. Acta (1)

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[Crossref]

Opt. Commun. (2)

G. P. Weigelt, “Modified astronomical speckle interferometry. Speckle masking,” Opt. Commun. 21, 55–59 (1977).
[Crossref]

P. Nisenson, C. Papaliolios, “Effects of photon noise on speckle image reconstruction with the Knox and Thompson algorithm,” Opt. Commun. 47, 91–96 (1983).
[Crossref]

Opt. Lett. (1)

Optik (1)

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik 61, 257–262 (1982).

Other (6)

F. Roddier, “Atmospheric limitations to high angular resolution imaging,” presented at the European Southern Observatories Conference on the Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, Garching, Federal Republic of Germany, March 24–27, 1981.

M. Billard, G. Fertin, J. C. Fontanella, “Atmospheric turbulence simulation cell for optical propagation experiment,” presented at the 4th International Symposium on Gas Flow and Chemical Lasers, Stresa, Italy, September 13–17, 1982.

D. Bonneau, CERGA, Observatoire de Calern, Caussols, 06460 St. Vallier de Thiey, France.

A. Blazit, R. Foy, “A photon counting detector system for the E.S.O.N.T.T. spectrograph,” Proposal to European Southern Observatories (unpublished).

J. C. Fontanella, “Wavefront sensing, adaptive optics and deconvolutions,” presented at the CFHT Workshop on High-Resolution Imaging in Astronomy, Kamuela, Hawaii, October 25–26, 1984).

J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1

Three-axis representation of the F.T. of a one-dimensional turbulence-degraded image.

Fig. 2
Fig. 2

(a) Phase dislocation. Near a zero there are two phase representations, depending on the path. (b) The vector represents the behavior of the image F.T. around a zero.

Fig. 3
Fig. 3

(a) Phase-value and phase-difference notation. (b) Dislocation compensation.

Fig. 4
Fig. 4

Typical power spectrum of a set of short-exposure images.

Fig. 5
Fig. 5

Schematic representation of the image’s cross-spectrum components.

Fig. 6
Fig. 6

Optical setup for producing turbulence-degraded images.

Fig. 7
Fig. 7

(a) Algorithm reconstruction without turbulence if dislocations are considered. (b) A short-exposure image. (c) Algorithm rconstruction with turbulence if dislocations are considered. D/r0 ~ 10 and 300 photons/speckle. (d) Reconstruction without turbulence if no dislocation is assumed. © The Walt Disney Company.

Fig. 8
Fig. 8

(a) Original image. (b) A short-exposure image; D/r0 ~ 10. (c) The long-exposure image. (d) Reconstruction with 170 photons/speckle. (e) Reconstruction with 51 photons/speckle. (f) Reconstruction with 17 photons/speckle. © The Walt Disney Company.

Fig. 9
Fig. 9

The binary star 62 UMa. (a) A typical short exposure. (b) The long-exposure image. (c) Autocorrelation. (d) Reconstruction with 1800 frames. Binary separation 0.048 arc sec (λ = 7000 Å).

Fig. 10
Fig. 10

HR4789 and its companion. (a) Autocorrelation with 3000 frames. (b) Reconstruction with 3000 frames. Binary separation 0.3 arc sec (λ = 7000 Å).

Equations (53)

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

I ˜ i ( f ) 2 = O ˜ ( f ) 2 S ˜ i ( f ) 2 ,
S ˜ i ( f ) 2 = 0.435 ( r 0 D ) T ˜ 2 ( f ) ,
N 0 = 1 0.435 ( D r 0 ) 2 ,
S ˜ i ( f ) 2 = 1 N 0 T ˜ ( f ) .
A ( f , Δ f ) = I ˜ * i ( f ) I ˜ i ( f + Δ f ) ,
O ˜ ( f ) = O ˜ ( f ) exp [ i φ ( f ) ] ,
S ˜ i ( f ) = S ˜ i ( f ) exp [ i ψ i ( f ) ] .
A ( f , Δ f ) O ˜ ( f ) 2 exp [ i Δ φ ( f ) ] S ˜ i ( f ) 2 exp [ i Δ ψ i ( f ) ] ,
O ˜ ( f + Δ f ) O ˜ ( f ) ,
S ˜ i ( f + Δ f ) S ˜ i ( f ) .
Δ φ ( f ) = φ ( f + Δ f ) - φ ( f ) ,
Δ ψ i ( f ) = ψ i ( f + Δ f ) - ψ i ( f ) .
Δ ψ i ( f ) Δ ψ ( f )
exp [ i Δ ψ i ( f ) ] exp [ i Δ ψ ( f ) ] .
Δ φ ( f ) + Δ ψ ( f ) = arctan Im ( A ) Re ( A ) ,
φ ( f ) = Σ Δ φ ( f ) .
A i , ( f , Δ f ) = I ˜ * i ( f ) I ˜ i ) f + Δ f )
ξ 2 = i = 1 M j = 1 N = ( Δ φ i j u - φ i j + φ i - 1 , j ) 2 + ( Δ φ i j v - φ i j + φ i , j - 1 ) 2 ,
4 φ i j = φ i - 1 , j + φ i + 1 , j + φ i , j - 1 + φ i , j + 1 + Δ φ i j u - Δ φ i + 1 , j u + Δ φ i j v - Δ φ i , j + 1 v
φ i - 1 , j + Δ φ i j u - φ i , j - 1 - Δ φ i j v > π
φ i - 1 , j + Δ φ i j u - φ i , j - 1 + Δ φ i j v + K 2 π < π
σ φ 2 σ Δ φ 2 .
I i ( x ) = k = 1 n i ϕ c ( x - x i k ) ,
I ˜ i ( f ) = ϕ ˜ c ( f ) exp ( - 2 i π f x i k ) .
I ˜ 0 ( f ) = ϕ ˜ c ( f ) exp ( - 2 i π f x i k ) .
A ( f , Δ f ) = ϕ ˜ * c ( f ) ϕ ˜ c ( f + Δ f ) exp ( 2 i π f x i k ) × exp [ - 2 π i ( f + Δ f ) x i k ] .
n i 2 - n i = n 2 ,
A ( f , Δ f ) = ϕ ˜ * c ( f ) ϕ ˜ c ( f + Δ f ) × [ n I ˜ 0 ( Δ f ) ϕ ˜ ( Δ f ) + n 2 O ˜ * ( f ) O ˜ ( f + Δ f ) S ˜ * i ( f ) S ˜ i ( f + Δ f ) ] .
ϕ ˜ c ( f + Δ f ) ϕ ˜ c ( f ) ,
ϕ ˜ c ( Δ f ) ϕ ˜ c ( 0 ) .
A N = ϕ ˜ c ( f ) 2 n I ˜ ( Δ f ) ϕ ˜ c ( 0 ) .
A ( M ) ( f , Δ f ) = 1 M 1 M I ˜ * i ( f ) I ˜ i ( f + Δ f ) .
δ Δ θ Im [ A ( M ) ] - Im ( A ) Re ( A - A N ) .
δ Δ θ 2 = σ Δ θ 2 σ 2 ( Im A ( M ) - Im A ) ( A - A N ) 2 .
I ˜ i ( f ) * = P R 1 ( i ) - i P I 1 ( i ) ,
I ˜ i ( f + Δ f ) = P R 2 ( i ) - i P I 2 ( i ) .
P R 1 2 = P I 1 2 = ½ I ˜ i ( f ) 2 .
P R 1 2 P R 2 2
P I 1 2 P I 2 2 .
Im [ I ˜ * i ( f ) I ˜ i ( f + Δ f ) ] = P R 1 ( i ) P I 2 ( i ) - P R 2 ( i ) P I 1 ( i )
Im [ I ˜ * i ( f ) I ˜ i ( f + Δ f ) ] - Im A = P R 1 ( i ) P I 2 ( i ) - P R 2 ( i ) P I 1 ( i ) - P R 1 ( i ) P I 2 ( i ) + P R 2 ( i ) P I 1 ( i ) .
{ Im [ I ˜ * i ( f ) I ˜ i ( f + Δ f ) ] - Im A } 2 P R 1 ( i ) 2 P I 2 ( i ) 2 + P R 2 ( i ) 2 P I 1 ( i ) 2 .
{ Im [ I ˜ * i ( f ) I i ( f + Δ f ) ] - Im ( A ) } 2 = 2 P R 2 ( i ) 2 P I 2 ( i ) 2 = ½ [ I ˜ i ( f ) 2 ] 2 .
I ˜ i ( f ) 2 = N 2 p 2 ϕ ( f ) W ( f ) + N p + N 0 p 0 ,
( A - A N ) 2 [ N 2 p 2 ϕ ( f ) W ( f ) ] 2 .
σ Δ θ 2 1 2 M [ N 2 p 1 ϕ ( f ) W ( f ) + N p + N 0 p 0 N 2 p 2 ϕ ( f ) W ( f ) ] 2 .
σ 2 θ σ 2 Δ θ ,
1 2 M [ N 2 p 2 ϕ ( f ) W ( f ) + N p + N 0 p 0 N 2 p 2 ϕ ( f ) W ( f ) N 2 p 2 ] 2 < ( π 6 ) 2 .
N p ϕ ( f ) W ( f ) M > 6 2 π .
M p W ( f ) > 10.
M p W ( f ) > 10.
p > 100.
M > 80.

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