Abstract

Contrast optimization, also known as image sharpening, is a method that can be used to estimate phase errors in coherent images. However, the contrast measure of a coherent image is a random variable because of the speckle present in coherent images. The variance of this measure puts a limit on the ability of contrast optimization to focus an image. We derive the probability distribution function of the most common contrast measure, the sum of the pixel intensities raised to a power. These statistics are then verified by a number of speckle simulations and compared with measured statistics from synthetic aperture sonar images. The developed statistics can be used as a tool to understand and improve the method of contrast optimization as well as assess its performance for a given imaging system. They can also be used to predict the effect of certain image processing operations on the contrast.

© 2004 Optical Society of America

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References

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  1. R. A. Muller, A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 64, 1200–1210 (1974).
    [CrossRef]
  2. L. Nock, G. Trahey, S. Smith, “Phase aberration correction in medical ultrasound using speckle brightness as a quality factor,” J. Acoust. Soc. Am. 85, 1819–1833 (1989).
    [CrossRef] [PubMed]
  3. E. H. Attia, B. Steinberg, “Self-cohering large antenna arrays using the spatial correlation properties of radar clutter,” IEEE Trans. Antennas Propag. 37, 30–38 (1989).
    [CrossRef]
  4. J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1996).
  5. P. T. Gough, D. W. Hawkins, “Imaging algorithms for a strip-map synthetic aperture sonar: minimizing the effects of aperture errors and aperture undersampling,” IEEE J. Ocean. Eng. 22, 27–39 (1997).
    [CrossRef]
  6. R. G. Paxman, J. C. Marron, “Aberration correction of speckled imagery with an image-sharpness criterion,” in Statistical Optics, G. M. Morris, ed., Proc. SPIE976, 37–47 (1988).
    [CrossRef]
  7. S. A. Fortune, M. P. Hayes, P. T. Gough, “Contrast optimization of coherent images,” in Proceedings of Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 2622–2628.
  8. D. Blacknell, A. P. Blake, C. J. Oliver, R. G. White, “A comparison of SAR multilook registration and contrast optimization autofocus algorithms applied to real SAR data,” in Radar 92. International Conference (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 363–366.
  9. F. Berizzi, G. Corsini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst. 32, 1185–1191 (1996).
    [CrossRef]
  10. F. Berizzi, G. Corsini, M. Diani, M. Veltroni, “Autofocus of wide azimuth angle SAR images by contrast optimization,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1230–1232.
  11. L. Xi, L. Guosui, J. Ni, “Autofocusing of ISAR images based on entropy minimization,” IEEE Trans. Aerosp. Electron. Syst. 35, 1240–1252 (1999).
    [CrossRef]
  12. P. T. Gough, R. G. Lane, “Autofocussing SAR and SAS images using a conjugate gradient search algorithm,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 2, pp. 621–623.
  13. J. R. Fienup, “Synthetic-aperture radar autofocus by maximizing sharpness,” Opt. Lett. 25, 221–223 (2000).
    [CrossRef]
  14. J. R. Fienup, J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A 20, 609–620 (2003).
    [CrossRef]
  15. T. J. Sutton, S. A. Chapman, H. D. Griffiths, “Robustness and effectiveness of autofocus algorithms applied to diverse seabed environments,” in Proceedings of the Fifth European Conference on Underwater Acoustics ECUA 2000, M. E. Zakharia, ed. (European Communities, Luxembourg, 2000), Vol. 1, pp. 407–412.
  16. S. A. Fortune, M. P. Hayes, P. T. Gough, “Statistical autofocus of synthetic aperture sonar images using image contrast optimization,” in Proceedings of Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2001), Vol. 1, pp. 163–169.
  17. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related PhenomenaJ. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
  18. S. Lowenthal, H. Arsenault, “Image formation for coherent diffuse objects: statistical properties,” J. Opt. Soc. Am. 60, 1478–1483 (1970).
    [CrossRef]
  19. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IEEE Trans. Inf. Theory 8, 194–195 (1962).
    [CrossRef]
  20. J. Marron, G. M. Morris, “Image recognition in the presence of laser speckle,” J. Opt. Soc. Am. A 3, 964–971 (1986).
    [CrossRef]
  21. A. J. Hunter, M. P. Hayes, P. T. Gough, “Simulation of multiple-receiver, broadband interferometric SAS imagery,” in Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 2629–2643.
  22. H. White, Asymptotic Theory for Econometricians (Academic, Orlando, Fla., 1984).

2003 (1)

2000 (1)

1999 (1)

L. Xi, L. Guosui, J. Ni, “Autofocusing of ISAR images based on entropy minimization,” IEEE Trans. Aerosp. Electron. Syst. 35, 1240–1252 (1999).
[CrossRef]

1997 (1)

P. T. Gough, D. W. Hawkins, “Imaging algorithms for a strip-map synthetic aperture sonar: minimizing the effects of aperture errors and aperture undersampling,” IEEE J. Ocean. Eng. 22, 27–39 (1997).
[CrossRef]

1996 (1)

F. Berizzi, G. Corsini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst. 32, 1185–1191 (1996).
[CrossRef]

1989 (2)

L. Nock, G. Trahey, S. Smith, “Phase aberration correction in medical ultrasound using speckle brightness as a quality factor,” J. Acoust. Soc. Am. 85, 1819–1833 (1989).
[CrossRef] [PubMed]

E. H. Attia, B. Steinberg, “Self-cohering large antenna arrays using the spatial correlation properties of radar clutter,” IEEE Trans. Antennas Propag. 37, 30–38 (1989).
[CrossRef]

1986 (1)

1974 (1)

1970 (1)

1962 (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IEEE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

Arsenault, H.

Attia, E. H.

E. H. Attia, B. Steinberg, “Self-cohering large antenna arrays using the spatial correlation properties of radar clutter,” IEEE Trans. Antennas Propag. 37, 30–38 (1989).
[CrossRef]

Berizzi, F.

F. Berizzi, G. Corsini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst. 32, 1185–1191 (1996).
[CrossRef]

F. Berizzi, G. Corsini, M. Diani, M. Veltroni, “Autofocus of wide azimuth angle SAR images by contrast optimization,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1230–1232.

Blacknell, D.

D. Blacknell, A. P. Blake, C. J. Oliver, R. G. White, “A comparison of SAR multilook registration and contrast optimization autofocus algorithms applied to real SAR data,” in Radar 92. International Conference (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 363–366.

Blake, A. P.

D. Blacknell, A. P. Blake, C. J. Oliver, R. G. White, “A comparison of SAR multilook registration and contrast optimization autofocus algorithms applied to real SAR data,” in Radar 92. International Conference (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 363–366.

Buffington, A.

Chapman, S. A.

T. J. Sutton, S. A. Chapman, H. D. Griffiths, “Robustness and effectiveness of autofocus algorithms applied to diverse seabed environments,” in Proceedings of the Fifth European Conference on Underwater Acoustics ECUA 2000, M. E. Zakharia, ed. (European Communities, Luxembourg, 2000), Vol. 1, pp. 407–412.

Corsini, G.

F. Berizzi, G. Corsini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst. 32, 1185–1191 (1996).
[CrossRef]

F. Berizzi, G. Corsini, M. Diani, M. Veltroni, “Autofocus of wide azimuth angle SAR images by contrast optimization,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1230–1232.

Curlander, J. C.

J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1996).

Diani, M.

F. Berizzi, G. Corsini, M. Diani, M. Veltroni, “Autofocus of wide azimuth angle SAR images by contrast optimization,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1230–1232.

Fienup, J. R.

Fortune, S. A.

S. A. Fortune, M. P. Hayes, P. T. Gough, “Statistical autofocus of synthetic aperture sonar images using image contrast optimization,” in Proceedings of Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2001), Vol. 1, pp. 163–169.

S. A. Fortune, M. P. Hayes, P. T. Gough, “Contrast optimization of coherent images,” in Proceedings of Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 2622–2628.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related PhenomenaJ. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.

Gough, P. T.

P. T. Gough, D. W. Hawkins, “Imaging algorithms for a strip-map synthetic aperture sonar: minimizing the effects of aperture errors and aperture undersampling,” IEEE J. Ocean. Eng. 22, 27–39 (1997).
[CrossRef]

P. T. Gough, R. G. Lane, “Autofocussing SAR and SAS images using a conjugate gradient search algorithm,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 2, pp. 621–623.

S. A. Fortune, M. P. Hayes, P. T. Gough, “Contrast optimization of coherent images,” in Proceedings of Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 2622–2628.

S. A. Fortune, M. P. Hayes, P. T. Gough, “Statistical autofocus of synthetic aperture sonar images using image contrast optimization,” in Proceedings of Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2001), Vol. 1, pp. 163–169.

A. J. Hunter, M. P. Hayes, P. T. Gough, “Simulation of multiple-receiver, broadband interferometric SAS imagery,” in Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 2629–2643.

Griffiths, H. D.

T. J. Sutton, S. A. Chapman, H. D. Griffiths, “Robustness and effectiveness of autofocus algorithms applied to diverse seabed environments,” in Proceedings of the Fifth European Conference on Underwater Acoustics ECUA 2000, M. E. Zakharia, ed. (European Communities, Luxembourg, 2000), Vol. 1, pp. 407–412.

Guosui, L.

L. Xi, L. Guosui, J. Ni, “Autofocusing of ISAR images based on entropy minimization,” IEEE Trans. Aerosp. Electron. Syst. 35, 1240–1252 (1999).
[CrossRef]

Hawkins, D. W.

P. T. Gough, D. W. Hawkins, “Imaging algorithms for a strip-map synthetic aperture sonar: minimizing the effects of aperture errors and aperture undersampling,” IEEE J. Ocean. Eng. 22, 27–39 (1997).
[CrossRef]

Hayes, M. P.

S. A. Fortune, M. P. Hayes, P. T. Gough, “Contrast optimization of coherent images,” in Proceedings of Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 2622–2628.

A. J. Hunter, M. P. Hayes, P. T. Gough, “Simulation of multiple-receiver, broadband interferometric SAS imagery,” in Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 2629–2643.

S. A. Fortune, M. P. Hayes, P. T. Gough, “Statistical autofocus of synthetic aperture sonar images using image contrast optimization,” in Proceedings of Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2001), Vol. 1, pp. 163–169.

Hunter, A. J.

A. J. Hunter, M. P. Hayes, P. T. Gough, “Simulation of multiple-receiver, broadband interferometric SAS imagery,” in Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 2629–2643.

Lane, R. G.

P. T. Gough, R. G. Lane, “Autofocussing SAR and SAS images using a conjugate gradient search algorithm,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 2, pp. 621–623.

Lowenthal, S.

Marron, J.

Marron, J. C.

R. G. Paxman, J. C. Marron, “Aberration correction of speckled imagery with an image-sharpness criterion,” in Statistical Optics, G. M. Morris, ed., Proc. SPIE976, 37–47 (1988).
[CrossRef]

McDonough, R. N.

J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1996).

Miller, J. J.

Morris, G. M.

Muller, R. A.

Ni, J.

L. Xi, L. Guosui, J. Ni, “Autofocusing of ISAR images based on entropy minimization,” IEEE Trans. Aerosp. Electron. Syst. 35, 1240–1252 (1999).
[CrossRef]

Nock, L.

L. Nock, G. Trahey, S. Smith, “Phase aberration correction in medical ultrasound using speckle brightness as a quality factor,” J. Acoust. Soc. Am. 85, 1819–1833 (1989).
[CrossRef] [PubMed]

Oliver, C. J.

D. Blacknell, A. P. Blake, C. J. Oliver, R. G. White, “A comparison of SAR multilook registration and contrast optimization autofocus algorithms applied to real SAR data,” in Radar 92. International Conference (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 363–366.

Paxman, R. G.

R. G. Paxman, J. C. Marron, “Aberration correction of speckled imagery with an image-sharpness criterion,” in Statistical Optics, G. M. Morris, ed., Proc. SPIE976, 37–47 (1988).
[CrossRef]

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IEEE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

Smith, S.

L. Nock, G. Trahey, S. Smith, “Phase aberration correction in medical ultrasound using speckle brightness as a quality factor,” J. Acoust. Soc. Am. 85, 1819–1833 (1989).
[CrossRef] [PubMed]

Steinberg, B.

E. H. Attia, B. Steinberg, “Self-cohering large antenna arrays using the spatial correlation properties of radar clutter,” IEEE Trans. Antennas Propag. 37, 30–38 (1989).
[CrossRef]

Sutton, T. J.

T. J. Sutton, S. A. Chapman, H. D. Griffiths, “Robustness and effectiveness of autofocus algorithms applied to diverse seabed environments,” in Proceedings of the Fifth European Conference on Underwater Acoustics ECUA 2000, M. E. Zakharia, ed. (European Communities, Luxembourg, 2000), Vol. 1, pp. 407–412.

Trahey, G.

L. Nock, G. Trahey, S. Smith, “Phase aberration correction in medical ultrasound using speckle brightness as a quality factor,” J. Acoust. Soc. Am. 85, 1819–1833 (1989).
[CrossRef] [PubMed]

Veltroni, M.

F. Berizzi, G. Corsini, M. Diani, M. Veltroni, “Autofocus of wide azimuth angle SAR images by contrast optimization,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1230–1232.

White, H.

H. White, Asymptotic Theory for Econometricians (Academic, Orlando, Fla., 1984).

White, R. G.

D. Blacknell, A. P. Blake, C. J. Oliver, R. G. White, “A comparison of SAR multilook registration and contrast optimization autofocus algorithms applied to real SAR data,” in Radar 92. International Conference (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 363–366.

Xi, L.

L. Xi, L. Guosui, J. Ni, “Autofocusing of ISAR images based on entropy minimization,” IEEE Trans. Aerosp. Electron. Syst. 35, 1240–1252 (1999).
[CrossRef]

IEEE J. Ocean. Eng. (1)

P. T. Gough, D. W. Hawkins, “Imaging algorithms for a strip-map synthetic aperture sonar: minimizing the effects of aperture errors and aperture undersampling,” IEEE J. Ocean. Eng. 22, 27–39 (1997).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (2)

F. Berizzi, G. Corsini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst. 32, 1185–1191 (1996).
[CrossRef]

L. Xi, L. Guosui, J. Ni, “Autofocusing of ISAR images based on entropy minimization,” IEEE Trans. Aerosp. Electron. Syst. 35, 1240–1252 (1999).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

E. H. Attia, B. Steinberg, “Self-cohering large antenna arrays using the spatial correlation properties of radar clutter,” IEEE Trans. Antennas Propag. 37, 30–38 (1989).
[CrossRef]

IEEE Trans. Inf. Theory (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IEEE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

J. Acoust. Soc. Am. (1)

L. Nock, G. Trahey, S. Smith, “Phase aberration correction in medical ultrasound using speckle brightness as a quality factor,” J. Acoust. Soc. Am. 85, 1819–1833 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

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

Opt. Lett. (1)

Other (11)

A. J. Hunter, M. P. Hayes, P. T. Gough, “Simulation of multiple-receiver, broadband interferometric SAS imagery,” in Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 2629–2643.

H. White, Asymptotic Theory for Econometricians (Academic, Orlando, Fla., 1984).

T. J. Sutton, S. A. Chapman, H. D. Griffiths, “Robustness and effectiveness of autofocus algorithms applied to diverse seabed environments,” in Proceedings of the Fifth European Conference on Underwater Acoustics ECUA 2000, M. E. Zakharia, ed. (European Communities, Luxembourg, 2000), Vol. 1, pp. 407–412.

S. A. Fortune, M. P. Hayes, P. T. Gough, “Statistical autofocus of synthetic aperture sonar images using image contrast optimization,” in Proceedings of Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2001), Vol. 1, pp. 163–169.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related PhenomenaJ. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.

P. T. Gough, R. G. Lane, “Autofocussing SAR and SAS images using a conjugate gradient search algorithm,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 2, pp. 621–623.

F. Berizzi, G. Corsini, M. Diani, M. Veltroni, “Autofocus of wide azimuth angle SAR images by contrast optimization,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1230–1232.

J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1996).

R. G. Paxman, J. C. Marron, “Aberration correction of speckled imagery with an image-sharpness criterion,” in Statistical Optics, G. M. Morris, ed., Proc. SPIE976, 37–47 (1988).
[CrossRef]

S. A. Fortune, M. P. Hayes, P. T. Gough, “Contrast optimization of coherent images,” in Proceedings of Oceans 2003, Marine Technology and Ocean Science Conference (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 2622–2628.

D. Blacknell, A. P. Blake, C. J. Oliver, R. G. White, “A comparison of SAR multilook registration and contrast optimization autofocus algorithms applied to real SAR data,” in Radar 92. International Conference (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 363–366.

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Figures (11)

Fig. 1
Fig. 1

Variation in contrast of a field stripmap SAS image as the phase error is varied. Phase error is a coefficient times the quadratic towfish path over the whole image. Contrast is C2 defined in Eq. (14).

Fig. 2
Fig. 2

Example of speckle in SAS intensity image. The across-track bandwidth is double the along-track bandwidth.

Fig. 3
Fig. 3

Probability distribution function of pixel intensity over a patch of seafloor in a field SAS image. The data are compared with the model given by Eq. (2).

Fig. 4
Fig. 4

Mean of contrast (C2) as a function of the mean speckle intensity σU. The data are compared with the model: E[C2]=2σU2, from Eq. (25).

Fig. 5
Fig. 5

Variance of contrast (C2) of uncorrelated speckle as a function of the mean speckle intensity σU. The data are compared with the model: Var[C2]=20σU4/Np, from Eq. (28).

Fig. 6
Fig. 6

Variance of contrast C2 as a function of speckle correlation length Lc. The data are compared with the model: Var[C2]=0.45×20σU4Lc2/Np, from relation (34).

Fig. 7
Fig. 7

Variance of contrast C2 for point target strength λ plus speckle as a function of speckle correlation length Lc. The data are compared with the model from relation (36).

Fig. 8
Fig. 8

Probability distribution of contrast (C2) of simulated uncorrelated speckle. The data are compared with the normal distribution model in Eq. (39).

Fig. 9
Fig. 9

Probability distribution of contrast (C2) of simulated highly correlated speckle (correlation length Lc=20). The data are compared with the normal distribution model in Eq. (39).

Fig. 10
Fig. 10

Probability distribution of contrast of strips of SAS data compared with the model in Eq. (39). The model mean is calculated from Eq. (25) and the variance from Eq. (28).

Fig. 11
Fig. 11

Contrast gradient C2/ϕ(v) for simulated spotlight scene; rt is the target-to-speckle energy ratio. The gradient is normalized by rt. The actual phase error is quadratic.

Equations (55)

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

Pr(X, Y)=12πσX2exp-X2+Y22σX2.
Pr(U)=1σUexp-UσUU00otherwise ,
E[Un]=n!E[U]nnpositiveintegerΓ(n+1)E[U]nn>-1 .
E[U]=σU,
Var[U]=E[U2]-E[U]2=σU2.
Ui=Xi2+Yi2.
ρijCorr[Xi, Xj]=Corr[Yi, Yj].
E[Xi2Xj2]=E[Yi2Yj2]=1+2ρij2,
E[UiUj]=σU2(1+ρij2).
Corr[Ui,Uj]=E[UiUj]-E[Ui]E[Uj]Var[Ui]Var[Uj]=ρij2.
I(m, n)=V(m, n)U(m, n),
E[I(m, n)]=μUV(m, n),
Var[I(m, n)]=σU2V2(m, n),
μU=E[U(m, n)],
σU2=Var[U(m, n)].
Cβ=1Np(m, n)Iβ(m, n),
Cβ=1Np(m, n)Vβ(m, n)Uβ(m, n).
Cβ=1Np(m, n)Vβ(m, n)Z(m, n).
Pr(Z)=Z(1/β-1)βσUexp-Z1/βσUZ00otherwise.
E[Z]=β!σUβ
Var[Z]=E[Z2]-(E[Z])2=[(2β)!-(β!)2]σU2β.
E[Xi4Xj2]=E[Xi2Xj4]=E[Yi4Yj2]=E[Yi2Yj4]=12ρij2+3,
E[Xi4Xj4]=E[Yi4Yj4]=24ρij4+72ρij2+9.
E[ZiZj]=σU4/16(2E[Xi4Xj4]+8E[Xi4Xj2]E[Xi2]+2E[Xi4]E[Xj4]+4E[Xi2Xj2]2)=4σU4(ρij4+4ρij2+1).
Corr[Zi, Zj]=E[ZiZj]-E[Zi]E[Zj]Var[Zi]Var[Zj]=ρij4+4ρij25.
E[Cβ]=E[Z] 1NpiViβ=E[Z]Cβ^,
E[Cβ]=β!σUβCβ^
Var[Cβ]=Var[Z] 1Np2iVi2β+Var[Z]Np2ijViβVjβCorr[ZiZj].
Var[Cβ]=Var[Z] 1Np2iVi2β=1NpVar[Z]Kβ^,
Kβ^1Npi=1NpVi2β.
Var[Cβ]=1Np [(2β)!-(β!)2]σU2βKβ^.
ρij=Corr[X(m, n),X(r, s)]=exp{-π[(m-r)/Lx]2}exp{-π[(n-s)/Ly]2}=exp[-π(Δx/Lx)2]exp[-π(Δy/Ly)2],
RZ(Δx, Δy)
=Corr[Z(x, y),Z(x+Δx, y+Δy)]=15exp[-4π(Δx/Lx)2]exp[-4π(Δy/Ly)2]+45exp[-2π(Δx/Lx)2]exp[-2π(Δy/Ly)2].
x=-NNexp[-π(x/Lc)2]Lc forLc>1,NLc,
Δx=-NxNxΔy=-NyNyRZ(Δx, Δy)0.45LxLy
forLx, Ly>2,Nx, NyLx, Ly.
Var[C2]=20σU4Np2m=1Nxn=1Nyr=1Nxs=1NyV2(m, n)V2(r, s)×RZ(m-r, n-s),
Var[C2]=20σU4Np2x=1Nxy=1NyΔx=1-NxNx-1Δy=1-NyNy-1V2(x, y)×V2(x+Δx, y+Δy)RZ(Δx, Δy).
Var[C2]=20σU4Np2x=1Nxy=1NyV4(x, y)×Δx=1-NxNx-1Δy=1-NyNy-1RZ(Δx, Δy)20σU4Np K2^(0.45LxLy)Lx,Ly220σU4Np K2^LxLyotherwise.
Vc(x, y)=1,
Vt(x, y)=λfor(x, y)=(Ntx, Nty)0otherwise,
V(x, y)=Vc(x, y)+Vt(x, y).
Var[C2]=20σU4Np2xyΔxΔyRZ(Δx, Δy)[Vc(x, y)+Vt(x, y)]2[Vc(x+Δx, y+Δy)+Vt(x+Δx, y+Δy)]2.
Var[C2]20σU4Np2xyΔxΔyRZ(Δx, Δy)[Vc2(x, y)+Vt2(x, y)][Vc2(x+Δx, y+Δy)+Vt2(x+Δx, y+Δy)].
Var[C2]20σU4Np2xyΔxΔyRZ(Δx, Δy)[1+Vt2(x, y)+Vt2(x+Δx, y+Δy)+Vt2(x, y)Vt2(x+Δx, y+Δy)],
20σU4Nx2Ny2xyΔxΔyRZ(Δx, Δy)+xyVt2(x, y)ΔxΔyRZ(Δx, Δy)+xyΔxΔyVt2(x+Δx, y+Δy)RZ(Δx, Δy)+xyΔxΔyVt2(x, y)Vt2(x+Δx, y+Δy)RZ(Δx, Δy)
20σU4Np2 [0.45NpLxLy+0.9λ2LxLy+λ4]Lx,Ly220σU4Np2 [Np+2λ2+λ4]otherwise.
rt=λ2/Np.
Var[C2]20σU4(0.45LxLy/Np+0.9rtLxLy/Np+rt2)Lx,Ly220σU4(1/Np+2rt/Np+rt2)otherwise.
Var[Wt]0,t.
E[|Wt-E[Wt]|2+δ]<Δ<forsomeΔ>0andt.
σ¯N2>δ>0,N.
Pr(C)=12π Var[C]exp-(C-E[C])22 Var[C].
Cβϕ(v)=2NvxIm{G(x, v)×[Fyv{βg(x, y)|g(x, y)|2(β-1)}]*},

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