Abstract

A comparative study is made of the Knox–Thompson and triple-correlation techniques as applied to image restoration. Both photon-noise-degraded imaging and atmospheric-turbulence-degraded imaging are considered. The signal-to-noise ratios of the methods are studied analytically and with the aid of computer simulations. The ability to retain diffraction-limited information on imaging through turbulence is considered in terms of phase-closure relationships. On the basis of this work it is found that the two image-restoration techniques are effectively equivalent.

© 1988 Optical Society of America

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References

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  1. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
    [CrossRef]
  2. 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]
  3. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  4. J. C. Dainty, “Diffraction limited imaging of stellar objects using telescopes of low optical quality,” Opt. Commun. 7, 129–134 (1973).
    [CrossRef]
  5. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.
  6. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” Astron. J. 193, L45–L48 (1974).
    [CrossRef]
  7. K. T. Knox, “Image retrieval from astronomical speckle patterns,”J. Opt. Soc. Am. 66, 1236–1239 (1976).
    [CrossRef]
  8. G. Weigelt, “Modified speckle interferometry: speckle masking,” Opt. Commun. 21, 55 (1977).
    [CrossRef]
  9. A. W. Lohmann, G. P. Weigelt, B. Wirnitzer, “Speckle masking in astronomy—triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  10. G. P. Weigelt, B. Wirnitzer, “Image reconstruction by the speckle-masking method,” Opt. Lett. 8, 389–391 (1983).
    [CrossRef] [PubMed]
  11. R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution of small angular extent,” Mon. Not. R. Astron. Soc. 118, 276–284 (1958).
  12. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  13. B. J. Brames, J. C. Dainty, “A method of determining object intensity distributions in stellar speckle interferometry,”J. Opt. Soc. Am. 71, 1542–1545 (1981).
    [CrossRef]
  14. H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23, 3121–3129 (1984).
    [CrossRef] [PubMed]
  15. E. K. Hege, N. E. Hubbard, P. A. Strittmatterr, S. P. Worden, “Speckle interferometry observations of the triple QSO PG 1115+08,” Astrophys. J. 248, L1–L3 (1981).
    [CrossRef]
  16. F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
    [CrossRef]
  17. J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high resolution optical imaging,” Nature 320, 595–597 (1986).
    [CrossRef]
  18. J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” Publ. no. RADC-TR-76-50, March1976; “Precompensation and postcompensation of photon limited degraded images,” Publ. no. RADC-TR-76-382, December1976; “Photon limitations in imaging and restoration,” Publ. no. RADC-TR-77-175, May, 1977 (Rome Air Development Center, Griffiss Air Force Base, New York, 13441).
  19. J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 69, 786–790 (1979).
    [CrossRef]
  20. L. Koechlin, “Observational speckle interferometry,” Proc. Int. Astron. Union Colloq. 50, 24-1–24-4 (1978).
  21. C. Papaliolios, L. Mertz, “A new two-dimensional photon camera,” in Instrumentation in Astronomy IV, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.331, 360–365 (1982).
    [CrossRef]
  22. M. J. Northcott, G. R. Ayers, J. C. Dainty, “Algorithms for image reconstruction from photon-limited data using the triple correlation,” J. Opt. Soc. Am. A 5, 986–992 (1988).
    [CrossRef]
  23. 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]
  24. B. Wirnitzer, “Bispectral analysis at low light levels and astronomical speckle masking,” J. Opt. Soc. Am. A 2, 14–21 (1985).
    [CrossRef]
  25. J. C. Fontanella, A. Seve, “Reconstruction of turbulence-degraded images using the Knox–Thompson algorithm,” J. Opt. Soc. Am. A 4, 438–448 (1987).
    [CrossRef]
  26. K.-H. Hofmann, G. Weigelt, “Speckle masking observation of the central object in the giant H II region NGC3603*,” Astron. Astrophys. 167, L15–L16 (1986).
  27. G. Aitken, R. Houtman, R. Johnson, J.-M. Pochet, “Direct phase gradient measurement for speckle image reconstruction,” Appl. Opt. 24, 2926–2930 (1986).
    [CrossRef]
  28. G. J. M. Aitken, R. Houtman, R. Johnson, “Phase-gradient speckle image processing: digital implementation and noise bias terms,” Appl. Opt. 25, 1031 (1986).
    [CrossRef] [PubMed]
  29. J. C. Dainty, M. J. Northcott, “Imaging a randomly translating object at low light levels using the triple correlation,” Opt. Commun. 58, 11–14 (1986).
    [CrossRef]
  30. G. R. Ayers, J. C. Dainty, M. J. Northcott, “Photon limited imaging through turbulence,” in Inverse Problems in Optics, E. R. Pike, ed. Proc. Soc. Photo-Opt. Instrum. Eng.808, 19–25 (1987).
    [CrossRef]
  31. I. S. Reed, “On a moment theorem for complex Gaussian processes,”IRE Trans. Inf. Theory IT-8, 194–195 (1962).
    [CrossRef]
  32. A. Chelli, “Comparison between image plane phase reconstruction methods in optical interferometry,” in Interferometric Imaging in Astronomy, J. W. Goad, ed., Proceedings of the Joint Workshop on High Resolution Imaging from the Ground Using Interferometric Techniques (National Optical Astronomy Observatories, Tucson, Ariz., 1987).

1988 (1)

1987 (1)

1986 (6)

K.-H. Hofmann, G. Weigelt, “Speckle masking observation of the central object in the giant H II region NGC3603*,” Astron. Astrophys. 167, L15–L16 (1986).

G. Aitken, R. Houtman, R. Johnson, J.-M. Pochet, “Direct phase gradient measurement for speckle image reconstruction,” Appl. Opt. 24, 2926–2930 (1986).
[CrossRef]

G. J. M. Aitken, R. Houtman, R. Johnson, “Phase-gradient speckle image processing: digital implementation and noise bias terms,” Appl. Opt. 25, 1031 (1986).
[CrossRef] [PubMed]

J. C. Dainty, M. J. Northcott, “Imaging a randomly translating object at low light levels using the triple correlation,” Opt. Commun. 58, 11–14 (1986).
[CrossRef]

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
[CrossRef]

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

1985 (1)

1984 (1)

1983 (3)

1981 (2)

E. K. Hege, N. E. Hubbard, P. A. Strittmatterr, S. P. Worden, “Speckle interferometry observations of the triple QSO PG 1115+08,” Astrophys. J. 248, L1–L3 (1981).
[CrossRef]

B. J. Brames, J. C. Dainty, “A method of determining object intensity distributions in stellar speckle interferometry,”J. Opt. Soc. Am. 71, 1542–1545 (1981).
[CrossRef]

1979 (1)

1978 (2)

L. Koechlin, “Observational speckle interferometry,” Proc. Int. Astron. Union Colloq. 50, 24-1–24-4 (1978).

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

1977 (1)

G. Weigelt, “Modified speckle interferometry: speckle masking,” Opt. Commun. 21, 55 (1977).
[CrossRef]

1976 (1)

1974 (1)

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

1973 (1)

J. C. Dainty, “Diffraction limited imaging of stellar objects using telescopes of low optical quality,” Opt. Commun. 7, 129–134 (1973).
[CrossRef]

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)

1962 (1)

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

1958 (1)

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution of small angular extent,” Mon. Not. R. Astron. Soc. 118, 276–284 (1958).

Aitken, G.

Aitken, G. J. M.

Ayers, G. R.

M. J. Northcott, G. R. Ayers, J. C. Dainty, “Algorithms for image reconstruction from photon-limited data using the triple correlation,” J. Opt. Soc. Am. A 5, 986–992 (1988).
[CrossRef]

G. R. Ayers, J. C. Dainty, M. J. Northcott, “Photon limited imaging through turbulence,” in Inverse Problems in Optics, E. R. Pike, ed. Proc. Soc. Photo-Opt. Instrum. Eng.808, 19–25 (1987).
[CrossRef]

Baldwin, J. E.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Bartelt, H.

Belsher, J. F.

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” Publ. no. RADC-TR-76-50, March1976; “Precompensation and postcompensation of photon limited degraded images,” Publ. no. RADC-TR-76-382, December1976; “Photon limitations in imaging and restoration,” Publ. no. RADC-TR-77-175, May, 1977 (Rome Air Development Center, Griffiss Air Force Base, New York, 13441).

Brames, B. J.

Chelli, A.

A. Chelli, “Comparison between image plane phase reconstruction methods in optical interferometry,” in Interferometric Imaging in Astronomy, J. W. Goad, ed., Proceedings of the Joint Workshop on High Resolution Imaging from the Ground Using Interferometric Techniques (National Optical Astronomy Observatories, Tucson, Ariz., 1987).

Dainty, J. C.

M. J. Northcott, G. R. Ayers, J. C. Dainty, “Algorithms for image reconstruction from photon-limited data using the triple correlation,” J. Opt. Soc. Am. A 5, 986–992 (1988).
[CrossRef]

J. C. Dainty, M. J. Northcott, “Imaging a randomly translating object at low light levels using the triple correlation,” Opt. Commun. 58, 11–14 (1986).
[CrossRef]

B. J. Brames, J. C. Dainty, “A method of determining object intensity distributions in stellar speckle interferometry,”J. Opt. Soc. Am. 71, 1542–1545 (1981).
[CrossRef]

J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 69, 786–790 (1979).
[CrossRef]

J. C. Dainty, “Diffraction limited imaging of stellar objects using telescopes of low optical quality,” Opt. Commun. 7, 129–134 (1973).
[CrossRef]

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

G. R. Ayers, J. C. Dainty, M. J. Northcott, “Photon limited imaging through turbulence,” in Inverse Problems in Optics, E. R. Pike, ed. Proc. Soc. Photo-Opt. Instrum. Eng.808, 19–25 (1987).
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Fontanella, J. C.

Fried, D. L.

Goodman, J. W.

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” Publ. no. RADC-TR-76-50, March1976; “Precompensation and postcompensation of photon limited degraded images,” Publ. no. RADC-TR-76-382, December1976; “Photon limitations in imaging and restoration,” Publ. no. RADC-TR-77-175, May, 1977 (Rome Air Development Center, Griffiss Air Force Base, New York, 13441).

Greenaway, A. H.

Haniff, C. A.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Hege, E. K.

E. K. Hege, N. E. Hubbard, P. A. Strittmatterr, S. P. Worden, “Speckle interferometry observations of the triple QSO PG 1115+08,” Astrophys. J. 248, L1–L3 (1981).
[CrossRef]

Hofmann, K.-H.

K.-H. Hofmann, G. Weigelt, “Speckle masking observation of the central object in the giant H II region NGC3603*,” Astron. Astrophys. 167, L15–L16 (1986).

Houtman, R.

Hubbard, N. E.

E. K. Hege, N. E. Hubbard, P. A. Strittmatterr, S. P. Worden, “Speckle interferometry observations of the triple QSO PG 1115+08,” Astrophys. J. 248, L1–L3 (1981).
[CrossRef]

Jennison, R. C.

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution of small angular extent,” Mon. Not. R. Astron. Soc. 118, 276–284 (1958).

Johnson, 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]

Koechlin, L.

L. Koechlin, “Observational speckle interferometry,” Proc. Int. Astron. Union Colloq. 50, 24-1–24-4 (1978).

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).

Lohmann, A. W.

Mackay, C. D.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Mertz, L.

C. Papaliolios, L. Mertz, “A new two-dimensional photon camera,” in Instrumentation in Astronomy IV, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.331, 360–365 (1982).
[CrossRef]

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]

Northcott, M. J.

M. J. Northcott, G. R. Ayers, J. C. Dainty, “Algorithms for image reconstruction from photon-limited data using the triple correlation,” J. Opt. Soc. Am. A 5, 986–992 (1988).
[CrossRef]

J. C. Dainty, M. J. Northcott, “Imaging a randomly translating object at low light levels using the triple correlation,” Opt. Commun. 58, 11–14 (1986).
[CrossRef]

G. R. Ayers, J. C. Dainty, M. J. Northcott, “Photon limited imaging through turbulence,” in Inverse Problems in Optics, E. R. Pike, ed. Proc. Soc. Photo-Opt. Instrum. Eng.808, 19–25 (1987).
[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]

C. Papaliolios, L. Mertz, “A new two-dimensional photon camera,” in Instrumentation in Astronomy IV, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.331, 360–365 (1982).
[CrossRef]

Pochet, J.-M.

Reed, I. S.

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

Roddier, F.

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
[CrossRef]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
[CrossRef]

Seve, A.

Strittmatterr, P. A.

E. K. Hege, N. E. Hubbard, P. A. Strittmatterr, S. P. Worden, “Speckle interferometry observations of the triple QSO PG 1115+08,” Astrophys. J. 248, L1–L3 (1981).
[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]

Warner, P. J.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Weigelt, G.

K.-H. Hofmann, G. Weigelt, “Speckle masking observation of the central object in the giant H II region NGC3603*,” Astron. Astrophys. 167, L15–L16 (1986).

G. Weigelt, “Modified speckle interferometry: speckle masking,” Opt. Commun. 21, 55 (1977).
[CrossRef]

Weigelt, G. P.

Wirnitzer, B.

Worden, S. P.

E. K. Hege, N. E. Hubbard, P. A. Strittmatterr, S. P. Worden, “Speckle interferometry observations of the triple QSO PG 1115+08,” Astrophys. J. 248, L1–L3 (1981).
[CrossRef]

Appl. Opt. (4)

Astron. Astrophys. (2)

K.-H. Hofmann, G. Weigelt, “Speckle masking observation of the central object in the giant H II region NGC3603*,” Astron. Astrophys. 167, L15–L16 (1986).

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star 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]

Astrophys. J. (1)

E. K. Hege, N. E. Hubbard, P. A. Strittmatterr, S. P. Worden, “Speckle interferometry observations of the triple QSO PG 1115+08,” Astrophys. J. 248, L1–L3 (1981).
[CrossRef]

IRE Trans. Inf. Theory (1)

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

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

Mon. Not. R. Astron. Soc. (1)

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution of small angular extent,” Mon. Not. R. Astron. Soc. 118, 276–284 (1958).

Nature (1)

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Opt. Commun. (5)

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
[CrossRef]

J. C. Dainty, “Diffraction limited imaging of stellar objects using telescopes of low optical quality,” Opt. Commun. 7, 129–134 (1973).
[CrossRef]

G. Weigelt, “Modified speckle interferometry: speckle masking,” Opt. Commun. 21, 55 (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]

J. C. Dainty, M. J. Northcott, “Imaging a randomly translating object at low light levels using the triple correlation,” Opt. Commun. 58, 11–14 (1986).
[CrossRef]

Opt. Lett. (2)

Proc. Int. Astron. Union Colloq. (1)

L. Koechlin, “Observational speckle interferometry,” Proc. Int. Astron. Union Colloq. 50, 24-1–24-4 (1978).

Other (6)

C. Papaliolios, L. Mertz, “A new two-dimensional photon camera,” in Instrumentation in Astronomy IV, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.331, 360–365 (1982).
[CrossRef]

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” Publ. no. RADC-TR-76-50, March1976; “Precompensation and postcompensation of photon limited degraded images,” Publ. no. RADC-TR-76-382, December1976; “Photon limitations in imaging and restoration,” Publ. no. RADC-TR-77-175, May, 1977 (Rome Air Development Center, Griffiss Air Force Base, New York, 13441).

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
[CrossRef]

G. R. Ayers, J. C. Dainty, M. J. Northcott, “Photon limited imaging through turbulence,” in Inverse Problems in Optics, E. R. Pike, ed. Proc. Soc. Photo-Opt. Instrum. Eng.808, 19–25 (1987).
[CrossRef]

A. Chelli, “Comparison between image plane phase reconstruction methods in optical interferometry,” in Interferometric Imaging in Astronomy, J. W. Goad, ed., Proceedings of the Joint Workshop on High Resolution Imaging from the Ground Using Interferometric Techniques (National Optical Astronomy Observatories, Tucson, Ariz., 1987).

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Figures (14)

Fig. 1
Fig. 1

Corresponding subplanes of (a) the bispectrum (u1 = u1x, u2 = u2x) and (b) the KT (u1 = u1x, Δu = Δux) four-dimensional transfer functions of a circular-aperture lens.

Fig. 2
Fig. 2

Pictorial representation of correlation techniques in the image plane: (a) the autocorrelation; (b) the KT correlation for a single plane, Δu = Δux; (c) a single bispectrum plane, u2 = 2Δu = 2Δux, represented as a double correlation in the image plane; and (d) correlations required for the phase-gradient technique, a = ax. The asterisk indicates the correlation of the two-dimensional images, and multiplication by a function is indicated by the overlaid dashed axes and one-dimensional function. The functions are constant in the orthogonal direction. The real and imaginary parts of the correlations are shown separately.

Fig. 3
Fig. 3

(a) Plots of the theoretical error ӨE for a single image, of the phase-difference estimates resulting from using the bispectrum plane (u2x = Δux, u1y = u2y = 0) and the KT plane (1y = 0, Δuy = 0). (b) Plots of SNRm values for the same two planes. Plots (a) and (b) are on a natural logarithmic scale. (c) Asteroid-type object and its Fourier modulus.

Fig. 4
Fig. 4

(a) Comparison of the phase errors on corresponding cross sections of the bispectrum (u1 = u1x, u2 = u2x) and the KT (u1 = u1x, Δu = Δux) subplanes for u2x = Δux = 4 frequency-sampling intervals. (b) Comparison of the SNRm values of the same two cross sections of the KT and bispectrum subplanes.

Fig. 5
Fig. 5

Plots of the ratio ӨE, defined by relation (30), to 1 ( 2 SNR m ) for (a) the bispectrum plane defined by u2x = Δux, u1y = u2y = 0, and (b) the KT plane defined by u1y = 0, Δuy = 0. Departure of the ratio from unity occurs only near the axes.

Fig. 6
Fig. 6

Diagrammatic representation of systems of pupil subapertures of diameter r0, showing how phase closure is realized. The phase-closure requirement can be redefined in terms of spatial-frequency vectors forming a closed loop. (a) Vectors contributing to the Fourier spatial frequency u. (b) Phase closure is achieved in the power spectrum by taking pairs of vectors of opposite sign, −u and +u, to form a closed loop. (c) Approximate phase closure is achieved in the KT method by taking two vectors, u and u + Δu, and assuming that the pupil phase is constant over Δu. The bispectrum method adds a third vector, Δu, to form phase closure explicitly. (d) The third vector Δu is essential for phase closure when λΔu > r0. The KT method fails with this arrangement.

Fig. 7
Fig. 7

Graphs illustrating the retention of phase information in the KT and TC techniques. The results were obtained by using a computer simulation of imaging a point source through a 2-m telescope and approximately 0.7-arcsec seeing at a high light level. The point source is centroided and so has zero Fourier phase at all spatial frequencies. The phase-distribution statistics were obtained by using an ensemble of 5000 different atmospheric realizations. The fluctuation of phase between frames is plotted for only the first 150 realizations. The telescope pupil consists of three widely spaced subapertures of diameter r0. (a) The uniformly distributed Fourier phase associated with spatial frequencies greater than r0/λ. (b), (d) Fourier-phase-difference information is retained in both the KT and the TC techniques for Δu < r0/λ. (c) Fourier-phase information is lost if the KT is used with Δu > r0/λ. (e) Phase-difference information is again retained, even though Δu > r0/λ, because of the phase-closure property of the bispectrum.

Fig. 8
Fig. 8

Plots showing the moduli of corresponding subplanes of (a) the bispectrum (u1 = u1x, u2 = u2x) and (b) the KT (u1 = u1x, Δu = Δux) four-dimensional transfer functions for imaging through a 2-m telescope with approximately 0.7-arcsec seeing. The data were obtained by using a Monte Carlo computer simulation, generating 10,000 independent realizations of the atmosphere to obtain the ensemble average. Both (a) and (b) are plotted with the same natural logarithmic scale.

Fig. 9
Fig. 9

Graphs of cross sections through (a) the bispectrum (u1 = u1x, u2 = u2x) and (b) the KT (u1 = u1x, Δu = Δux) subplanes of their atmosphere–telescope combination transfer functions. The cross sections correspond to frequency difference vectors u2x and Δux of 0, 2, 4, 6, and 8 frequency sampling intervals. The telescope cutoff occurs at 50 and with r0/λ at approximately 4.5 frequency sampling intervals.

Fig. 10
Fig. 10

Plots of the ratio of the phase error ӨE to 1 / ( 2 SNR m ) for the subplanes of the KT and bispectrum results for the asteroid-type object shown in Fig. 3(c) and defined by u1 = u1x, u2 = u2x, Δux. These plots are obtained by computationally simulating the effect of imaging the asteroid shown in Fig. 3(c) with a 2-m telescope and 0.7-arcsec seeing. Departure from unity occurs only near the axes.

Fig. 11
Fig. 11

Plots of a single plane of (a) the KT and (b) the bispectrum (u1 = u1x, u2 = u2x, Δu = Δux) SNR’s at a high light level, using the computer simulation to obtain ensemble averages over 10,000 independent realizations of atmospheric turbulence.

Fig. 12
Fig. 12

Graphs of cross sections through (a) the KT subplane (u1 = u1x, Δu = Δux) and (b) the bispectrum subplane (u1 = u1x, u2 = u2x) subplanes of the SNR’s shown in Fig. 11. The cross sections correspond to frequency-difference vectors u2x and Δux of 0, 2, 4, 6, and 8 frequency sampling intervals. The telescope cutoff occurs at 50 and with r0/λ at approximately 4.5 frequency sampling intervals.

Fig. 13
Fig. 13

Graphs of the results of a Monte Carlo computer simulation whereby ensemble averages over the atmospheric statistics are taken for the photon-noise-dependent variances defined in Appendix A. (a) Dependency of the SNR of the bispectrum on n ¯s and N ¯. The relative SNR is the ratio of the SNR for the bispectrum point defined by (u1 = 0.4D/λ, u2 = 0) at r0 ≈ 0.185D to that at r0 ≈ 0.095D. This ratio of r0 corresponds to a ratio of 0.25 in the mean number of speckles per frame, n ¯s. The plot tends to 4 at very low light levels, corresponding to a 1/ n ¯s dependency [see relations (67) and (77)], and tends to unity at high light levels, corresponding to a SNR that is approximately independent of n ¯s [see relation (67)]. (b) A plot equivalent to (a) for the bispectrum point defined by (u1 = 0.6D/λ, u2 = 0.3D/λ). The plot tends to a value of 16 at very low light levels, corresponding to a 1/ n ¯s2 dependency [see relation (76)], and tends to 2 at high light levels, corresponding to a 1 / n ¯ s dependency [see relation 67)]. (c) The approximate linear dependence of the SNR of the same bispectrum point on N ¯ when the mean number of photons per speckle is approximately 1. The mean number of speckles per image, n ¯s, in this example is approximately 1000. (d) The approximate N ¯3/2 dependency of the same bispectrum point when the mean number of photons per speckle is much less than unity [see relation (76)]. The upper line is an N ¯3/2-dependent reference. (e) The linear dependence of the SNR’s ofthe KT and TC signals defined by (u1 = 0.4D/λ, u2 = 0, Δu = 0) on N ¯ at low light levels [see relations (77) and (78)]. The upper line is a linear reference. The KT signal has the slightly higher SNR; however, the lines appear to be superimposed.

Fig. 14
Fig. 14

Phase-closure relationships can be viewed in terms of spatial-frequency vectors forming closed loops in the telescope pupil. (a) Representation of the six vectors that must be considered when analyzing the modulus squared of the bispectrum at a point defined by (u1, u2). (b) How the vectors can be selected in conjugate pairs to cancel the phase, each pair being analogous to the power spectrum at that particular frequency. (c), (d) Other possible combinations of vectors that result in phase closure. (e), (f) Redundant configurations between those represented in (b)–(d).

Equations (126)

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SNR = | S ( u ) | σ S ,
I ( u ) = + i ( x ) exp ( 2 π j u x ) d x .
i TC ( x 1 , x 2 ) = + i * ( x ) i ( x + x 1 ) i ( x + x 2 ) d x ,
I TC ( u 1 , u 2 ) = I ( u 1 ) I * ( u 1 + u 2 ) I ( u 2 ) .
i KT ( x 1 , Δ u ) = + i * ( x ) i ( x + x 1 ) exp ( 2 π j Δ u x ) d x ,
I KT ( u 1 , Δ u ) = I ( u 1 ) I * ( u 1 + Δ u ) ,
I TC ( u 1 , Δ u ) = I ( u 1 ) I * ( u 1 + Δ u ) I ( Δ u ) .
i TC ( x 1 , Δ u ) = I ( Δ u ) + i * ( x ) i ( x + x 1 ) exp ( 2 π j Δ u x ) d x = I ( Δ u ) i KT ( x 1 , Δ u ) .
i A ( x 1 ) = i KT ( x 1 , 0 ) = + i * ( x ) i ( x + x 1 ) d x ,
I A ( u 1 ) = I KT ( u 1 , 0 ) = I ( u 1 ) I * ( u 1 ) .
arg { I A ( u 1 ) } = Φ A ( u 1 ) = ϕ ( u 1 ) ϕ ( u 1 ) = 0 ,
arg { I KT ( u 1 , Δ u ) } = Φ KT ( u 1 , Δ u ) = ϕ ( u 1 ) ϕ ( u 1 + Δ u ) ,
arg { I TC ( u 1 , u 2 ) } = Φ TC ( u 1 , u 2 ) = ϕ ( u 1 ) ϕ ( u 1 + u 2 ) + ϕ ( u 2 ) ,
Φ TC ( u 1 , Δ u ) = Φ KT ( u 1 , Δ u ) + ϕ ( Δ u ) .
I PG ( u 1 , a ) = [ Im { I ( u 1 ) I * ( u 1 ) } ] a π = [ | I ( u 1 ) | 2 ϕ ( u 1 ) ] a π ,
i PG ( x 1 , + a ) = + i * ( x ) i ( x + x 1 ) ( + a · x + b ) d x , i PG ( x 1 , a ) = + i * ( x ) i ( x + x 1 ) ( a · x + b ) d x
I PG ( u 1 , a ) = [ FT { i PG ( x 1 , a ) } ] * FT { i PG ( x 1 , + a ) } = { [ I ( u 1 ) I * ( u 1 ) ] * I ( u 1 ) I * ( u 1 ) } a 2 π = [ Im { I ( u 1 ) I * ( u 1 ) } ] a π = [ | I ( u 1 ) | 2 ϕ ( u 1 ) ] a π .
Im { I ( u 1 ) I * ( u 1 ) } = 1 Δ u Im { I KT ( u 1 , Δ u ) } Δ u 0 = 1 Δ u Im { I ( u 1 ) I * ( u 1 + Δ u ) } Δ u 0 1 Δ u Im { I ( u 1 ) [ I ( u 1 ) + I ( u 1 ) Δ u ] * } | I ( u 1 ) | 2 ϕ ( u 1 ) .
d p ( x ) = k = 1 N p δ ( x x p k ) ,
D p ( u ) = k = 1 N p δ ( x x p k ) exp ( 2 π j u x ) d x ,
i KT ( x 1 , Δ u ) = k 1 = 1 N p k 2 = 1 N p δ ( x p k 2 x p k 1 + x 1 ) exp ( 2 π j Δ u x p k 2 ) .
i TC ( x 1 , Δ u ) = k = 1 N p δ ( x x p k ) exp ( 2 π j Δ u x p k ) d x × k 1 = 1 N p k 2 = 1 N p δ ( x p k 2 x p k 1 + x 1 ) exp ( 2 π j Δ u x p k 2 ) .
i PG ( x 1 , a ) = k 1 = 1 N p k 2 = 1 N p δ ( x p k 2 x p k 1 + x 1 ) ( a x p k 2 + b ) .
i TC ( x 1 , Δ u ) = k = 1 , k k 1 , k k 2 N p δ ( x x p k ) exp ( 2 π j Δ u x p k ) d x × k 1 = 1 , k 1 k 2 N p k 2 = 1 , k 2 k 1 N p δ ( x p k 2 x p k 1 + x 1 ) × exp ( 2 π j Δ u x p k 2 ) .
D p ( u 1 ) D p * ( u 1 ) D p ( 0 ) .
D p ( u 1 ) D p * ( u 1 + Δ u ) D p * ( Δ u ) .
D p ( u 1 ) D p * ( u 1 + Δ u ) D p ( Δ u ) | D p ( u 1 ) | 2 | D p ( u 1 + Δ u ) | 2 | D p ( Δ u ) | 2 + 2 N p .
j 2 Δ u [ D p ( u 1 ) D p * ( u 1 + Δ u ) D p * ( Δ u ) D p * ( u 1 ) D p ( u 1 + Δ u ) + D p ( Δ u ) ] .
SNR m = | mean signal | standard deviation of signal M = | S | σ m M ,
tan Ө E = [ σ I 2 cos 2 ϕ + σ R 2 sin 2 ϕ Cov ( I , R ) sin 2 ϕ ] 1 / 2 | S | 1 M ,
Ө E = [ σ I 2 cos 2 ϕ + σ R 2 sin 2 ϕ Cov ( I , R ) sin 2 ϕ ] 1 / 2 | S | 1 M .
SNR ( KT ) = N ¯ 2 ( 4 N ¯ 3 + 2 N ¯ 2 ) 1 / 2
SNR ( TC ) = N ¯ 3 ( 9 N ¯ 5 + 18 N ¯ 4 + 6 N ¯ 3 ) 1 / 2
lim N ¯ SNR ( KT ) = N ¯ 1 / 2 2 ,
lim N ¯ SNR ( TC ) = N ¯ 1 / 2 3 ,
SNR ( KT ) = N ¯ 2 , N ¯ 1 ,
SNR ( TC ) = N ¯ 3 / 2 6 , N ¯ 1.
Ө E 1 2 SNR m ;
Ө E 1 SNR m ( 1 + 2 2 ) 1 / 2 .
Ө E = σ I | S | 1 M 1 / 2 .
p ( a ) = 1 ( 2 π ) 1 / 2 σ a exp ( σ a 2 2 σ a 2 ) ,
T s ( u ) = exp ( 2 π j u a ) exp [ 2 π j ( u + Δ u ) a ] p ( a ) d a = exp ( 2 π 2 Δ u 2 σ a 2 ) .
I ˜ ( u ) = I ( u ) [ I * ( Δ u ) | I ( Δ u ) | ] u / Δ u ,
I ˜ ( u ) I ˜ * ( u + Δ u ) = I ( u ) [ I * ( Δ u ) | I ( Δ u ) | ] u / Δ u × I * ( u + Δ u ) [ I ( Δ u ) | I ( Δ u ) | ] ( u + Δ u ) / Δ u = I ( u ) I * ( u + Δ u ) I ( Δ u ) | I ( Δ u ) | .
N p ( N p 1 ) I ˆ ( u + Δ u N p ) I ˆ * ( u + Δ u Δ u N p ) I ˆ * ( Δ u N p ) N p 2 ,
N p ( N p 1 ) I ˆ ( u ) I ˆ * ( u + Δ u ) ,
arg { I ( u ) } = ϕ ( u ) + Φ A Φ B ,
I ( u ) = 0 , u > r 0 / λ ,
arg { I ( u ) I ( u ) } = ϕ ( u ) + Φ A Φ B + ϕ ( u ) Φ A + Φ B = 0
I ( u ) I ( u ) = | I ( u ) | 2 0.
arg { I ( u ) ( I ( u Δ u ) } = ϕ ( u ) + Φ A Φ B + ϕ ( u Δ u ) Φ A + Φ B = ϕ ( u ) ϕ ( u + Δ u )
I ( u ) I ( u Δ u ) 0 , Δ u < r 0 / λ .
arg { I ( u ) I ( u Δ u ) } = ϕ ( u ) + Φ A Φ B ϕ ( u Δ u ) Φ A + Φ C = ϕ ( u ) ϕ ( u + Δ u ) Φ B + Φ C ,
I ( u ) I ( u Δ u ) = 0 , Δ u < r 0 / λ .
arg { I ( u ) I ( Δ u ) I ( u Δ u ) } = ϕ ( u ) + Φ A Φ B + ϕ ( Δ u ) + Φ B Φ C + ϕ ( u Δ u ) Φ A + Φ C = ϕ ( u ) ϕ ( u + Δ u ) + ϕ ( Δ u ) .
I p ( u ) = O ( u ) S p ( u ) ,
T ( KT ) ( u 1 , Δ u ) = S p ( u 1 ) S p * ( u 1 + Δ u )
T ( TC ) ( u 1 , u 2 ) = S p ( u 1 ) S p * ( u 1 + u 1 ) S p ( u 2 ) ,
SNR ( KT ) ( u 1 , Δ u ) = | I p ( u 1 ) I p * ( u 1 + Δ u ) | σ ( KT ) M ,
σ ( KT ) 2 = | I p ( u 1 ) I p * ( u 1 + Δ u ) | 2 | I p ( u 1 ) I p * ( u 1 + Δ u ) | 2 ,
SNR ( TC ) ( u 1 , u 2 ) = | I p ( u 1 ) I p * ( u 1 + u 2 ) I p ( u 2 ) | σ ( TC ) M ,
σ ( TC ) 2 = | I p ( u 1 ) I p * ( u 1 + u 2 ) I p ( u 2 ) | 2 | I p ( u 1 ) I p * ( u 1 + u 2 ) I p ( u 2 ) | 2
S p ( u 1 ) S p * ( u 1 + u 2 ) S p ( u 2 ) 2 n ¯ s 2 T ( 3 ) ( u 1 , u 2 ) ,
| S p ( u 1 ) S p * ( u 1 + u 2 ) S p ( u 2 ) | 2 1 n ¯ s 3 T ( 2 ) ( u 1 ) T ( 2 ) ( u 2 ) × T ( 2 ) ( u 1 + u 2 ) ,
S p ( u 1 ) S p * ( u 1 + Δ u ) 0 ;
S p ( u 1 ) S p * ( u 1 + u 2 ) S p ( u 2 ) 1 n ¯ s T ( 2 ) ( u 1 ) S p ( u 2 ) ,
S p ( u 1 ) S p * ( u 1 + Δ u ) 1 n ¯ s T ( 2 ) ( u 1 ) ,
| S p ( u 1 ) S p * ( u 1 + u 2 ) S p ( u 2 ) | 2 2 n ¯ s 2 | T ( 2 ) ( u 1 ) | 2 | S p ( u 2 ) | 2 ,
| S p ( u 1 ) S p * ( u 1 + Δ u ) | 2 2 n ¯ s 2 | T ( 2 ) ( u 1 ) | 2 ,
SNR ( TC ) 2 T ( 3 ) ( u 1 , u 2 ) n ¯ s 1 / 2 [ T ( 2 ) ( u 1 ) T ( 2 ) ( u 1 + u 2 ) T ( 2 ) ( u 2 ) ] 1 / 2 M , u 1 , u 2 > r 0 / λ M , u 2 < r 0 / λ
SNR ( KT ) 0 , M , u 1 , Δ u > r 0 / λ Δ u < r 0 / λ .
σ ( TC ) 2 N ¯ 3 + T ( 2 ) ( u 2 ) n ¯ s N ¯ 4 + T ( 2 ) ( u 1 ) n ¯ s N ¯ 4 + T ( 2 ) ( u 1 + u 2 ) n ¯ s N ¯ 4 + T ( 2 ) ( u 1 ) n ¯ s 2 T ( 2 ) ( u 2 ) N ¯ 5 + T ( 2 ) ( u 1 ) n ¯ s 2 T ( 2 ) ( u 1 + u 2 ) N ¯ 5 + T ( 2 ) ( u 1 + u 2 ) n ¯ s 2 T ( 2 ) ( u 2 ) N ¯ 5 + T ( 2 ) ( u 1 ) n ¯ s 3 × T ( 2 ) ( u 1 + u 2 ) T ( 2 ) ( u 2 ) N ¯ 6 , N ¯ > 1 ,
σ ( TC ) 2 N ¯ 3 , N ¯ > 1 , n ¯ 1 ,
σ ( KT ) 2 0.
σ ( TC ) 2 N ¯ 3 + N ¯ 4 | S p ( u 2 ) | 2 + N ¯ 4 n ¯ s T ( 2 ) ( u 1 ) + N ¯ 4 n ¯ s T ( 2 ) ( u 1 + u 2 ) + N ¯ 5 n ¯ s T ( 2 ) ( u 1 ) | S p ( u 2 ) | 2 + N ¯ 5 n ¯ s T ( 2 ) ( u 1 + u 2 ) | S p ( u 2 ) | 2 + N ¯ 6 n ¯ s 2 T ( 2 ) ( u 1 ) T ( 2 ) ( u 1 + u 2 ) | S p ( u 2 ) | 2 , N ¯ > 1 ,
σ ( TC ) 2 N ¯ 3 + N ¯ 4 | S p ( u 2 ) | 2 , N ¯ > 1 , n ¯ 1 ,
σ ( KT ) 2 N ¯ 2 + N ¯ 3 n ¯ s T ( 2 ) ( u 1 ) + N ¯ 3 n ¯ s T ( 2 ) ( u 1 + Δ u ) + N ¯ 4 n ¯ s 2 T ( 2 ) ( u 1 ) T ( 2 ) ( u 1 + Δ u ) , N ¯ > 1 ,
σ ( KT ) 2 N ¯ 2 , N ¯ > 1 , n ¯ 1.
N ¯ 3 T ( TC ) ( u 1 + u 2 ) , N ¯ 2 T ( KT ) ( u 1 , Δ u )
SNR ( TC ) N ¯ 3 / 2 T ( TC ) ( u 1 , u 2 ) M 2 n ¯ s 2 N ¯ 3 / 2 T ( 3 ) ( u 1 , u 2 ) M , n ¯ 1.
SNR ( TC ) N ¯ 3 T ( 2 ) ( u 1 ) S p ( u 2 ) n ¯ s [ N ¯ 3 + N ¯ 4 | S p ( u 2 ) | 2 ] 1 / 2 M
SNR ( KT ) N ¯ n ¯ s T ( 2 ) ( u 2 ) M .
Ө E 1 SNR m ( 1 + 2 2 ) 1 / 2
D ( KT ) ( u 1 , Δ u ) = D p ( u 1 ) D p * ( u 1 + Δ u ) D p * ( Δ u ) ,
1 4 [ D p ( KT ) ( u 1 , Δ u ) + D p ( KT ) * ( u 1 , Δ u ) ] 2 ,
1 4 [ I ( u 1 ) I ( u 1 ) I ( u 1 + Δ u ) I ( u 1 + Δ u ) N ¯ 4 + c . c . + 2 I ( u 1 Δ u ) I ( u 1 ) I ( u 1 ) I ( u 1 + Δ u ) N ¯ 4 + I ( u 1 ) I ( u 1 ) I ( 2 u 1 + 2 Δ u ) N ¯ 3 + c . c . + 2 I ( u 1 ) I ( u 1 ) N ¯ 3 + 2 I ( u 1 ) I ( Δ u ) I ( u 1 + Δ u ) N ¯ 3 + c . c . + 2 I ( u 1 Δ u ) I ( u 1 ) I ( 2 u 1 + Δ u ) N ¯ 3 + c . c . + I ( u 1 Δ u ) I ( u 1 Δ u ) I ( 2 u 1 ) N ¯ 3 + c . c . + 2 I ( u 1 Δ u ) I ( u 1 + Δ u ) N ¯ 3 + I ( Δ u ) I ( Δ u ) N ¯ 2 + c . c . + I ( 2 u 1 ) I ( 2 u 1 + 2 Δ u ) N ¯ 2 + c . c . + 2 I ( 2 u 1 Δ u ) I ( 2 u 1 + Δ u ) N ¯ 2 + 2 N ¯ 2 ] ,
1 4 [ D p ( KT ) ( u 1 , Δ u ) D p ( KT ) * ( u 1 , Δ u ) ] 2 ,
1 4 [ I ( u 1 ) I ( u 1 ) I ( u 1 + Δ u ) I ( u 1 + Δ u ) N ¯ 4 + c . c . 2 I ( u 1 Δ u ) I ( u 1 ) I ( u 1 ) I ( u 1 + Δ u ) N ¯ 4 + I ( u 1 ) I ( u 1 ) I ( 2 u 1 + 2 Δ u ) N ¯ 3 + c . c . 2 I ( u 1 ) I ( u 1 ) N ¯ 3 + 2 I ( u 1 ) I ( Δ u ) I ( u 1 + Δ u ) N ¯ 3 + c . c . 2 I ( u 1 Δ u ) I ( u 1 ) I ( 2 u 1 + Δ u ) N ¯ 3 + c . c . + I ( u 1 Δ u ) I ( u 1 Δ u ) I ( 2 u 1 ) N ¯ 3 + c . c . 2 I ( u 1 Δ u ) I ( u 1 + Δ u ) N ¯ 3 + I ( Δ u ) I ( Δ u ) N ¯ 2 + c . c . + I ( 2 u 1 ) I ( 2 u 1 + 2 Δ u ) N ¯ 2 + c . c . 2 I ( 2 u 1 Δ u ) I ( 2 u 1 + Δ u ) N ¯ 2 2 N ¯ 2 ] .
j / 4 [ D p ( KT ) ( u 1 , Δ u ) D p ( KT ) * ( u 1 , Δ u ) ] × [ D p ( KT ) * ( u 1 , Δ u ) + D p ( KT ) * ( u 1 , Δ u ) ] ,
j / 4 [ I ( u 1 Δ u ) I ( u 1 Δ u ) I ( u 1 ) I ( u 1 ) N ¯ 4 c . c . + I ( u 1 Δ u ) I ( u 1 Δ u ) I ( 2 u 1 ) N ¯ 3 c . c . + 2 I ( u 1 Δ u ) × I ( Δ u ) I ( u 1 ) N ¯ 3 c . c . + I ( 2 u 1 2 Δ u ) I ( u 1 ) I ( u 1 ) N ¯ 3 c . c . + I ( Δ u ) I ( Δ u ) N ¯ 2 c . c . + I ( 2 u 1 2 Δ u ) × I ( 2 u 1 ) N ¯ 2 c . c . ] .
D p ( TC ) ( u 1 , u 2 ) = D p ( u 1 ) D p * ( u 1 + u 2 ) D p ( u 2 ) | D p ( u 1 ) | 2 | D p ( u 1 + u 2 ) | 2 | D p ( u 2 ) | 2 + 2 N p ,
1 4 [ D p ( TC ) ( u 1 , u 2 ) + D p ( TC ) * ( u 1 , u 2 ) ] 2 ,
1 4 [ I ( u 1 ) I ( u 1 ) I ( u 2 ) I ( u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 5 + c . c . + 2 I ( u 1 ) × I ( u 1 ) I ( u 2 ) I ( u 1 ) I ( u 1 + u 2 ) N ¯ 5 + c . c . + I ( u 1 ) I ( u 1 ) × I ( 2 u 2 ) I ( u 1 + u 2 ) I ( u 1 + u 2 ) N ¯ 5 + c . c . + 2 I ( u 1 ) I ( u 2 ) × I ( u 2 ) I ( u 2 ) I ( u 1 + u 2 ) N ¯ 5 + c . c . + 2 I ( u 1 ) I ( u 2 ) I ( u 2 ) × I ( u 1 ) N ¯ 5 + 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) I ( u 1 ) × I ( u 1 + 2 u 2 ) N ¯ 5 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) × I ( u 1 + u 2 ) I ( u 1 + u 2 ) N ¯ 5 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 ) × I ( u 2 ) I ( u 2 ) I ( 2 u 1 + u 2 ) N ¯ 5 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 ) × I ( u 1 ) I ( u 1 + u 2 ) N ¯ 5 + 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) I ( u 1 u 2 ) × I ( u 1 + u 2 ) N ¯ 5 + c . c . + I ( u 1 u 2 ) I ( u 1 u 2 ) I ( u 2 ) I ( u 2 ) × I ( 2 u 1 ) N ¯ 5 + c . c . + 2 I ( u 1 u 2 ) I ( u 2 ) I ( u 2 ) I ( u 1 + u 2 ) N ¯ 5 + I ( u 1 ) I ( u 1 ) I ( 2 u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 4 + c . c . + 2 I ( u 1 ) × I ( u 1 ) I ( u 1 ) I ( u 1 ) N ¯ 4 + 4 I ( u 1 ) I ( u 2 ) I ( u 2 ) I ( u 1 ) N ¯ 4 + 2 I ( u 1 ) I ( 2 u 2 ) I ( u 2 ) I ( u 1 + u 2 ) N ¯ 4 + c . c . + 2 I ( u 1 ) I ( u 1 ) N ¯ 4 + 2 I ( u 1 ) I ( u 2 ) I ( u 1 u 2 ) N ¯ 4 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 ) × I ( u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 4 + c . c . + 4 I ( u 1 u 2 ) I ( u 1 ) I ( u 1 ) × I ( u 1 + u 2 ) N ¯ 4 + 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 1 u 2 ) × I ( u 1 + 2 u 2 ) N ¯ 4 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) × I ( 2 u 1 ) N ¯ 4 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 ) × I ( 2 u 1 + u 2 ) N ¯ 4 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 + u 2 ) I ( u 2 ) × I ( 2 u 1 + u 2 ) N ¯ 4 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 + u 2 ) × I ( u 1 u 2 ) I ( u 1 + u 2 ) N ¯ 4 + 2 I ( u 1 u 2 ) I ( u 1 u 2 ) × I ( u 1 + u 2 ) I ( u 1 + u 2 ) N ¯ 4 + I ( u 1 u 2 ) I ( u 1 u 2 ) I ( 2 u 2 ) × I ( 2 u 1 ) N ¯ 4 + c . c . + 2 I ( u 1 u 2 ) I ( u 2 ) × I ( u 1 + 2 u 2 ) N ¯ 4 + c . c . + 4 I ( u 1 u 2 ) I ( u 2 ) I ( u 2 ) × I ( u 1 + u 2 ) N ¯ 4 + 2 I ( u 1 u 2 ) I ( u 1 + u 2 ) N ¯ 4 + 2 I ( u 1 2 u 2 ) I ( u 1 ) I ( u 1 ) I ( u 1 + 2 u 2 ) N ¯ 4 + 2 I ( u 1 2 u 2 ) × I ( u 1 ) I ( u 2 ) I ( 2 u 1 + u 2 ) N ¯ 4 + c . c . + 2 I ( u 2 ) I ( u 2 ) I ( u 2 ) × I ( u 2 ) N ¯ 4 + 2 I ( u 2 ) I ( u 2 ) N ¯ 4 + I ( 2 u 1 ) I ( u 2 ) I ( u 2 ) × I ( 2 u 1 + 2 u 2 ) N ¯ 4 + c . c . + 2 I ( 2 u 1 u 2 ) I ( u 2 ) I ( u 2 ) × I ( 2 u 1 + u 2 ) N ¯ 4 + I ( u 1 ) I ( u 1 ) I ( 2 u 1 ) N ¯ 3 + c . c . + 2 I ( u 1 ) × I ( u 2 ) I ( u 1 + u 2 ) N ¯ 3 + c . c . + 2 I ( u 1 + u 2 ) I ( u 1 u 2 ) N ¯ 3 + I ( u 1 u 2 ) I ( u 1 u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 3 + c . c . + 2 I ( u 1 + 2 u 2 ) I ( u 1 + u 2 ) I ( 2 u 1 + u 2 ) N ¯ 3 + c . c . + 2 I ( u 1 2 u 2 ) I ( u 1 + 2 u 2 ) N ¯ 3 + I ( u 2 ) I ( u 2 ) × I ( 2 u 2 ) N ¯ 3 + c . c . + I ( 2 u 1 ) I ( 2 u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 3 + c . c . + 2 I ( 2 u 1 u 2 ) I ( 2 u 1 + u 2 ) N ¯ 3 + 2 N ¯ 3 + I ( u 1 ) I ( u 1 ) I ( u 2 ) I ( u 2 ) I ( u 1 + u 2 ) × I ( u 1 + u 2 ) N ¯ 6 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) I ( u 2 ) × I ( u 1 ) I ( u 1 + u 2 ) N ¯ 6 ] .
1 4 D P ( TC ) ( u 1 , u 2 ) D P ( TC ) * ( u 1 , u 2 ) ] 2 ,
1 4 [ I ( u 1 ) I ( u 1 ) I ( u 2 ) I [ 2 u 1 + 2 u 2 ) N ¯ 5 + c . c . + 2 I ( u 1 ) I ( u 1 ) I ( u 2 ) I ( u 1 ) I ( u 1 + u 2 ) N ¯ 5 + c . c . + I ( u 1 ) × I ( u 1 ) I ( 2 u 2 ) I ( u 1 + u 2 ) I ( u 1 + u 2 ) N ¯ 5 + c . c . + 2 I ( u 1 ) × I ( u 2 ) I ( u 2 ) I ( u 2 ) I ( u 1 + u 2 ) N ¯ 5 + c . c . 2 I ( u 1 ) I ( u 2 ) × I ( u 2 ) I ( u 1 ) N ¯ 5 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) I ( u 1 ) × I ( u 1 + 2 u 2 ) N ¯ 5 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) × I ( u 1 + u 2 ) I ( u 1 + u 2 ) N ¯ 5 + c . c . 2 I ( u 1 u 2 ) I ( u 1 ) × I ( u 2 ) I ( u 2 ) I ( 2 u 1 + u 2 ) N ¯ 5 + c . c . 2 I ( u 1 u 2 ) I ( u 1 ) × I ( u 1 ) I ( u 1 + u 2 ) N ¯ 5 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) I ( u 1 u 2 ) × I ( u 1 + u 2 ) N ¯ 5 + c . c . + I ( u 1 u 2 ) I ( u 1 u 2 ) I ( u 2 ) I ( u 2 ) × I ( 2 u 1 ) N ¯ 5 + c . c . 2 I ( u 1 u 2 ) I ( u 2 ) I ( u 2 ) I ( u 1 + u 2 ) N ¯ 5 + I ( u 1 ) I ( u 1 ) I ( 2 u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 4 + c . c . + 2 I ( u 1 ) × I ( u 1 ) I ( u 1 ) I ( u 1 ) N ¯ 4 + 4 I ( u 1 ) I ( u 2 ) I ( u 2 ) I ( u 1 ) N ¯ 4 + 2 I ( u 1 ) I ( 2 u 2 ) I ( u 2 ) I ( u 1 + u 2 ) N ¯ 4 + c . c . 2 I ( u 1 ) × I ( u 1 ) N ¯ 4 2 I ( u 1 ) I ( u 2 ) I ( u 1 u 2 ) N ¯ 4 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 4 + c . c . + 4 I ( u 1 u 2 ) I ( u 1 ) I ( u 1 ) I ( u 1 + u 2 ) N ¯ 4 + c . c . 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 1 u 2 ) I ( u 1 + 2 u 2 ) N ¯ 4 + c . c . + 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) I ( 2 u 1 ) N ¯ 4 + c . c . 2 I ( u 1 u 2 ) × I ( u 1 ) I ( 2 u 1 + u 2 ) N ¯ 4 + c . c . 2 I ( u 1 u 2 ) I ( u 1 + u 2 ) × I ( u 2 ) I ( 2 u 1 + u 2 ) N ¯ 4 + c . c . 2 I ( u 1 u 2 ) I ( u 1 + u 2 ) × I ( u 1 u 2 ) I ( u 1 + u 2 ) N ¯ 4 + 2 I ( u 1 u 2 ) I ( u 1 u 2 ) × I ( u 1 + u 2 ) I ( u 1 + u 2 ) N ¯ 4 + I ( u 1 u 2 ) I ( u 1 u 2 ) I ( 2 u 2 ) × I ( 2 u 1 ) N ¯ 4 + c . c . 2 I ( u 1 u 2 ) I ( u 2 ) × I ( u 1 + 2 u 2 ) N ¯ 4 + c . c . + 4 I ( u 1 u 2 ) I ( u 2 ) I ( u 2 ) × I ( u 1 + u 2 ) N ¯ 4 2 I ( u 1 u 2 ) I ( u 1 + u 2 ) N ¯ 4 + c . c . 2 I ( u 1 2 u 2 ) I ( u 1 ) I ( u 1 ) I ( u 1 + 2 u 2 ) N ¯ 4 2 I ( u 1 2 u 2 ) I ( u 1 ) I ( u 2 ) I ( 2 u 1 + u 2 ) N ¯ 4 + c . c . + 2 I ( u 2 ) I ( u 2 ) I ( u 2 ) I ( u 2 ) N ¯ 4 2 I ( u 2 ) I ( u 2 ) N ¯ 4 + I ( 2 u 1 ) I ( u 2 ) I ( u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 4 + c . c . 2 I ( 2 u 1 u 2 ) I ( u 2 ) I ( u 2 ) I ( 2 u 1 + u 2 ) N ¯ 4 + I ( u 1 ) × I ( u 1 ) I ( 2 u 1 ) N ¯ 3 + c . c . + 2 I ( u 1 ) I ( u 2 ) × I ( u 1 + u 2 ) N ¯ 3 + c . c . 2 I ( u 1 + u 2 ) I ( u 1 u 2 ) N ¯ 3 + I ( u 1 u 2 ) I ( u 1 u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 3 + c . c . 2 I ( u 1 2 u 2 ) I ( u 1 + u 2 ) I ( 2 u 1 + u 2 ) N ¯ 3 + c . c . 2 I ( u 1 2 u 2 ) I ( u 1 + 2 u 2 ) N ¯ 3 + I ( u 2 ) I ( u 2 ) × I ( 2 u 2 ) N ¯ 3 + c . c . + I ( 2 u 1 ) I ( 2 u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 3 + c . c . 2 I ( 2 u 1 u 2 ) I ( 2 u 1 + u 2 ) N ¯ 3 2 N ¯ 3 + I ( u 1 ) I ( u 1 ) × I ( u 2 ) I ( u 2 ) I ( u 1 + u 2 ) I ( u 1 + u 2 ) N ¯ 6 + c . c . 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) I ( u 2 ) I ( u 1 ) I ( u 1 + u 2 ) N ¯ 6 ] .
j / 4 [ D p ( TC ) ( u 1 , u 2 ) D p ( TC ) * ( u 1 , u 2 ) ] × [ D p ( TC ) ( u 1 , u 2 ) + D p ( TC ) * ( u 1 , u 2 ] ,
j / 4 [ I ( u 1 ) I ( u 1 ) I ( u 2 ) I ( u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 5 c . c . 2 I ( u 1 ) I ( u 1 ) I ( u 2 ) I ( u 1 ) I ( u 1 + u 2 ) N ¯ 5 c . c . I ( u 1 ) × I ( u 1 ) I ( 2 u 2 ) I ( u 1 + u 2 ) I ( u 1 + u 2 ) N ¯ 5 c . c . 2 I ( u 1 ) × I ( u 2 ) I ( u 2 ) I ( u 2 ) I ( u 1 + u 2 ) N ¯ 5 c . c . 2 I ( u 1 u 2 ) × I ( u 1 ) I ( u 2 ) I ( u 1 + u 2 ) I ( u 1 + u 2 ) N ¯ 5 c . c . + I ( u 1 u 2 ) I ( u 1 u 2 ) I ( u 2 ) I ( u 2 ) I ( 2 u 1 ) N ¯ 5 c . c . I ( u 1 ) I ( u 1 ) I ( 2 u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 4 c . c . 2 I ( u 1 ) × I ( 2 u 2 ) I ( u 2 ) I ( u 1 + u 2 ) N ¯ 4 c . c . 2 I ( u 1 u 2 ) I ( u 1 ) × I ( u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 4 c . c . + 2 I ( u 1 u 2 ) I ( u 1 ) I ( u 2 ) × I ( 2 u 1 ) N ¯ 4 c . c . + I ( u 1 u 2 ) I ( u 1 u 2 ) I ( 2 u 2 ) × I ( 2 u 1 ) N ¯ 4 c . c . I ( 2 u 1 ) I ( u 2 ) I ( u 2 ) × I ( 2 u 1 + 2 u 2 ) N ¯ 4 c . c . + I ( u 1 ) I ( u 1 ) I ( 2 u 1 ) N ¯ 3 c . c . + 2 I ( u 1 ) I ( u 2 ) I ( u 1 + u 2 ) N ¯ 3 c . c . I ( u 1 u 2 ) × I ( u 1 u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 3 c . c . + I ( u 2 ) I ( u 2 ) × I ( 2 u 2 ) N ¯ 3 c . c . I ( 2 u 1 ) I ( 2 u 2 ) I ( 2 u 1 + 2 u 2 ) N ¯ 3 c . c . I ( u 1 ) I ( u 1 ) I ( u 2 ) I ( u 2 ) I ( u 1 + u 2 ) × I ( u 1 + u 2 ) N ¯ 6 c . c . ]
N p ( X p x g ) = k = 1 N p ( x k x g ) ,
X p = k = 1 N p x k N p .
D p ( u 1 ) D p * ( u 1 + Δ u ) D p * ( Δ u ) ,
D p ( u ) exp ( 2 π j X p u ) ,
D p ( u 1 ) exp ( 2 π j X p u 1 ) D p * ( u 1 + Δ u ) exp [ 2 π j X p ( u 1 + Δ u ) ] D p * ( Δ u ) exp ( 2 π j X p Δ u ) .
N p ( N p 1 ) I ˆ ( u + Δ u N p ) I ˆ * ( u + Δ u Δ u N p ) I ˆ * ( Δ u N p ) N p 2 N p I ˆ * ( Δ u Δ u N p ) I ˆ ( Δ u N p ) N p 1 + N p I ˆ * ( Δ u Δ u N p ) I ˆ ( Δ u N p ) N p 1 ,
N p ( N p 1 ) I ˆ ( u 1 ) I ˆ * ( u 1 + Δ u ) ,
E [ D p ( u ) exp ( 2 π j X p u ) ] = N p I ˆ ( u u N p ) I ˆ * ( u N p ) N p 1 ,
S p ( u ) = pupil H 0 ( x ) H 0 * ( x + λ u ) A ( x ) A * ( x + λ u ) d x pupil | H 0 ( x ) | 2 d x .
S p ( u ) = 1 N ( 0 ) s = 1 N ( u ) exp [ j ( Φ s Φ s ) ] ,
| S p ( u 1 ) S p * ( u 1 + u 2 ) S p ( u 2 ) | 2 = 1 N ( 0 ) 6 q = 1 N ( u 1 ) r = 1 N ( u 1 + u 2 ) s = 1 N ( u 2 ) exp [ j ( Φ q Φ q ) + j ( Φ r Φ r ) + j ( Φ s Φ s ) ] q = 1 N ( u 1 ) r = 1 N ( u 1 + u 2 ) s = 1 N ( u 2 ) exp [ j ( Φ q Φ q ) + j ( Φ r Φ r ) + j ( Φ s Φ s ) ] .
( Φ q Φ q ) + ( Φ r Φ r ) + ( Φ s Φ s ) + ( Φ q Φ q ) + ( Φ r Φ r ) + ( Φ s Φ s ) = 0 ,
C A ( x ) = A ( x ) A * ( x + x ) = δ ( x ) .
| S p ( u 1 ) S p * ( u 1 + u 2 ) S p ( u 2 ) | 2 = 1 Y H 0 ( x 1 ) H 0 * ( x 1 + λ u 1 ) H 0 ( x 2 ) × H 0 * [ x 2 λ ( u 1 + u 2 ) ] H 0 ( x 3 ) H 0 * ( x 3 + λ u 2 ) H 0 * ( x 4 ) × H 0 ( x 4 + λ u 1 ) H 0 * ( x 5 ) H 0 [ x 5 λ ( u 1 + u 2 ) ] H 0 * ( x 6 ) × H 0 ( x 6 + λ u 2 ) A ( x 1 ) A * ( x 1 + λ u 1 ) A ( x 2 ) × A * [ x 2 λ ( u 1 + u 2 ) ] A ( x 3 ) A * ( x 3 + λ u 2 ) A * ( x 4 ) × A ( x 4 + λ u 1 ) A * ( x 5 ) A [ x 5 λ ( u 1 + u 2 ) ] A * ( x 6 ) × A ( x 6 + λ u 2 ) d x 1 d x 2 d x 3 d x 4 d x 5 d x 6 ,
Y = [ | H 0 ( x 1 ) | 2 | H 0 ( x 2 ) | 2 | H 0 ( x 3 ) | 2 d x 1 d x 2 d x 3 ] 2 .
( Φ q Φ q ) + ( Φ q Φ q ) = 0 , ( Φ r Φ r ) + ( Φ r Φ r ) = 0 , ( Φ s Φ s ) + ( Φ s Φ s ) = 0.
A ( x 1 ) A * ( x 4 ) A ( x 4 + λ u 1 ) A * ( x 1 + λ u 1 ) A ( x 2 ) A * ( x 5 ) × A [ x 5 λ ( u 1 + u 2 ) ] A * [ x 2 λ ( u 1 + u 2 ) ] × A ( x 3 ) A * ( x 6 ) A ( x 6 + λ u 2 ) A * ( x 3 + λ u 2 ) = δ ( x 4 x 1 ) δ ( x 5 x 2 ) δ ( x 6 x 3 ) .
1 Y pupil | H 0 ( x 1 ) | 2 | H 0 ( x 1 + λ u 1 ) | 2 | H 0 ( x 2 ) | 2 × | H 0 ( x 1 λ ) ( u 1 + u 2 ) ] | 2 | H 0 ( x 3 ) | 2 | H 0 ( x 3 + λ u 2 ) | 2 × d x 1 d x 2 d x 3 = 1 n ¯ s 3 T ( 2 ) ( u 1 ) T ( 2 ) ( u 2 ) T ( 2 ) ( u 1 + u 2 ) ,
n ¯ s pupil | H 0 ( x ) | 2 d x | C A ( x ) | 2 d x .
( Φ q Φ q ) + ( Φ q Φ q ) = 0 , ( Φ r Φ r ) + ( Φ r Φ r ) + ( Φ s Φ s ) + ( Φ s Φ s ) = 0 ,
A ( x 1 ) A * ( x 4 ) A ( x 4 + λ u 1 ) A * ( x 1 + λ u 1 ) A ( x 2 ) A * ( x 6 ) × A ( x 3 ) A * [ x 2 λ ( u 1 + u 2 ) ] A [ x 5 λ ( u 1 + u 2 ) ] × A * ( x 3 + λ u 2 ) A ( x 6 + λ u 2 ) A * ( x 5 ) = δ ( x 4 x 1 ) δ ( x 6 x 2 ) δ ( x 5 x 6 λ u 2 ) × δ [ x 2 x 3 λ ( u 1 + u 2 ) ] δ [ x 3 x 5 + λ ( u 1 + u 2 ) λ u 2 ]
1 Y pupil | H 0 ( x 1 ) | 2 | H 0 ( x 1 + λ u 1 ) | 2 d x 1 pupil | H 0 ( x 2 ) | 2 × | H 0 ( x 2 λ u 1 ) | 2 | H 0 [ x 1 λ ( u 1 + u 2 ) ] | 2 × | H 0 ( x 2 + λ u 2 ) | 2 d x 2 1 n ¯ s 4 T ( 2 ) ( u 1 ) T ( 4 ) ( u 2 , u 1 + u 2 ) .
2 n ¯ s 4 T ( 2 ) ( u 2 ) T ( 4 ) ( u 1 , u 1 + u 2 ) , 2 n ¯ s 4 T ( 2 ) ( u 1 + u 2 ) T ( 4 ) ( u 1 , u 2 ) .
( Φ q Φ q ) + ( Φ r Φ r ) + ( Φ s Φ s ) = 0 , ( Φ q Φ q ) + ( Φ r Φ r ) + ( Φ s Φ s ) = 0.
A ( x 3 ) A * ( x 1 + λ u 1 ) A ( x 1 ) A * [ x 2 λ ( u 1 + u 2 ) ] × A ( x 2 ) A * ( x 3 + λ u 2 ) A ( x 4 + λ u 1 ) A * ( x 6 ) × A [ x 5 λ ( u 1 + u 2 ) ] A * ( x 4 ) A ( x 6 + λ u 2 ) A * ( x 5 ) = δ ( x 1 x 3 + λ u 1 ) δ [ x 2 x 1 λ ( u 1 + u 2 ) ] δ ( x 3 x 2 + λ u 2 ) × δ ( x 6 x 4 λ u 1 ) δ [ x 4 x 5 + λ ( u 1 + u 2 ) ] × δ ( x 5 x 6 λ u 2 ) .
{ 2 Y pupil | H 0 ( x 1 ) | 2 | H 0 ( x 1 + λ u 1 ) | 2 | H 0 [ x 1 + λ ( u 1 + u 2 ) ] | 2 d x 1 } 2 4 n ¯ s 4 T ( 3 ) ( u 1 , u 2 ) T ( 3 ) ( u 1 , u 2 ) .
2 n ¯ s 5 T ( 4 ) ( u 2 , u 1 + u 2 ) .
4 n ¯ s 5 [ T ( 4 ) ( u 2 , u 1 + u 2 ) + T ( 4 ) ( u 1 , u 1 + u 2 ) + T ( 4 ) ( u 1 , u 2 ) ] + 2 n ¯ s 5 T ( 3 ) ( u 1 , u 2 ) .
1 Y pupil | H 0 ( x 1 ) | 4 | H 0 ( x 1 + λ u 1 ) | 4 | H 0 [ x 1 + λ ( u 1 + u 2 ) ] | 4 d x 1 T ( 3 ) ( u 1 , u 2 ) n ¯ s 5 .

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