Abstract

It is difficult to reconstruct an image of a complex-valued object from the modulus of its Fourier transform (i.e., retrieve the Fourier phase) except in some special cases. By using additionally a low-resolution intensity image from a telescope with a small aperture, a fine-resolution image of a general object can be reconstructed in a two-step approach. First the Fourier phase over the small aperture is retrieved, using the Gerchberg–Saxton algorithm. Then that phase is used, in conjunction with the Fourier modulus data over a large aperture together with a support constraint on the object, to reconstruct a fine-resolution image (retrieve the phase over the large aperture) by the iterative Fourier-transform algorithm. The second step requires a modified algorithm that employs an expanding weighting function on the Fourier modulus.

© 1990 Optical Society of America

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References

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  1. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  2. J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front phase estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
    [CrossRef]
  3. T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3–11 (1990).
    [CrossRef]
  4. R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
    [CrossRef]
  5. R. G. Paxman, J. R. Fienup, J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 184–189 (1987).
    [CrossRef]
  6. R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
    [CrossRef]
  7. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  8. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  9. J. R. Fienup, C. C. Wackerman, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  10. E. N. Leith, J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962).
    [CrossRef]
  11. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  12. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  13. J. R. Fienup, “Improved synthesis and computational methods for computer-generated holograms,” Ph.D. dissertation, University Microfilms No. 75-25523 (Stanford University, Stanford, Calif., May1975), Chap. 5.
  14. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  15. J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds. Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  17. R. Rolleston, N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. 25, 178–183 (1986).
    [CrossRef] [PubMed]
  18. J. R. Fienup, “Phase retrieval from Fourier intensity data,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 13–17 (1987).
    [CrossRef]
  19. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 2.
  20. J. R. Fienup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,” J. Opt. Soc. Am. 72, 610–624 (1982).
    [CrossRef]
  21. B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
    [CrossRef]
  22. D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
    [CrossRef]
  23. R. G. Lane, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
    [CrossRef]
  24. J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object using a low-resolution image,” in Digest of Topcial Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 142–145.
  25. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, New York, 1980), p. 480.Note that the earlier edition was in error.

1990

1989

J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front phase estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
[CrossRef]

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

1987

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[CrossRef]

1986

1982

1980

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

1978

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1962

Bates, R. H. T.

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

Cederquist, J. N.

Clinthorne, J. T.

R. G. Paxman, J. R. Fienup, J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 184–189 (1987).
[CrossRef]

Crimmins, T. R.

Fienup, J. R.

T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3–11 (1990).
[CrossRef]

J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front phase estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
[CrossRef]

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[CrossRef]

J. R. Fienup, C. C. Wackerman, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,” J. Opt. Soc. Am. 72, 610–624 (1982).
[CrossRef]

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

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

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object using a low-resolution image,” in Digest of Topcial Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 142–145.

J. R. Fienup, “Phase retrieval from Fourier intensity data,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 13–17 (1987).
[CrossRef]

R. G. Paxman, J. R. Fienup, J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 184–189 (1987).
[CrossRef]

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds. Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
[CrossRef]

J. R. Fienup, “Improved synthesis and computational methods for computer-generated holograms,” Ph.D. dissertation, University Microfilms No. 75-25523 (Stanford University, Stanford, Calif., May1975), Chap. 5.

George, N.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 2.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, New York, 1980), p. 480.Note that the earlier edition was in error.

Holsztynski, W.

Izraelevitz, D.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

Kowalczyk, A. M.

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object using a low-resolution image,” in Digest of Topcial Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 142–145.

Kryskowski, D.

Lane, R. G.

R. G. Lane, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

Leith, E. N.

Lim, J. S.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

McCallum, B. C.

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

Paxman, R. G.

R. G. Paxman, J. R. Fienup, J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 184–189 (1987).
[CrossRef]

Robinson, S. R.

Rolleston, R.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, New York, 1980), p. 480.Note that the earlier edition was in error.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Tan, D. G. H.

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

Thelen, B. J.

Upatnieks, J.

Wackerman, C. C.

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

J. Mod. Opt.

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Opt. Lett.

Optik

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Other

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

J. R. Fienup, “Improved synthesis and computational methods for computer-generated holograms,” Ph.D. dissertation, University Microfilms No. 75-25523 (Stanford University, Stanford, Calif., May1975), Chap. 5.

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds. Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

R. G. Paxman, J. R. Fienup, J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 184–189 (1987).
[CrossRef]

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object using a low-resolution image,” in Digest of Topcial Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 142–145.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, New York, 1980), p. 480.Note that the earlier edition was in error.

J. R. Fienup, “Phase retrieval from Fourier intensity data,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 13–17 (1987).
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 2.

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Figures (13)

Fig. 1
Fig. 1

Optical sensor configuration. Data collected for a coherently illuminated object (not shown, located far to the right) include aperture-plane (Fourier) intensity and a low-resolution diffraction-limited intensity image from a small-aperture telescope.

Fig. 2
Fig. 2

Data-processing steps to reconstruct a fine-resolution image (retrieve the phase in the aperture plane) from the intensity measurements.

Fig. 3
Fig. 3

Thresholding the low-resolution intensity image to estimate a support constraint, with no weighting on the small aperture.(A) Diffraction-limited low-resolution image modulus (overexposed in order to show the sidelobes that extend beyond the support of the object); (B)–(D) thresholded images, with threshold values equal to (B) 0.078, (C) 0.157, and (D) 0.392 of the maximum value of the image.

Fig. 4
Fig. 4

Thresholding the low-resolution intensity image to estimate a support constraint,,with a weighting on the small aperture. (A) Diffraction-limited low-resolution image (not overexposed); (B)–(D) thresholded images, with threshold values equal to (B) 0.078, (C) 0.157, and (D) 0.392 of the maximum value of the image.

Fig. 5
Fig. 5

Removal of nulls due to speckles in the image. (A) Thresholded image from Fig. 4(C), (B) convolution of (A) with a circle of diameter 7 pixels (about half the diameter of a speckle), (C) thresholding of (B) at 0.58 of its peak, (D) enlarged version of (C) that may be used to ensure that the object fits within it.

Fig. 6
Fig. 6

Block diagram of the Gerchberg–Saxton algorithms. The object-domain constraint is the square root of the measured intensity of the low-resolution image, and the Fourier constraint is the square root of the measured intensity over the small aperture. FFT denotes fast Fourier transform.

Fig. 7
Fig. 7

Convergence of the Gerchberg–Saxton (GS) and accelerated Gerchberg–Saxton (GS2) algorithms. Twenty iterations of either GS2 or GS were followed by twenty iterations of GS. GS2 with feedback parameter β = 1.5 to β = 2 converged fastest.

Fig. 8
Fig. 8

Rms error (ABSERR) of the complex-valued reconstructed low-resolution image as a function of iteration number of a variety of light levels.

Fig. 9
Fig. 9

Rms error of the complex-valued reconstructed low-resolution image as a function of light level.

Fig. 10
Fig. 10

Black diagram of the iterative Fourier-transform algorithm. The object domain constraint is a support constraint derived from the measured data, and the Fourier-domain constraint are the square root of the measured intensity over the entire large aperture and the phase retrieved by the Gerchberg–Saxton algorithm over the small aperture.

Fig. 11
Fig. 11

Image-reconstruction example. (A) Fourier modulus data over a large circular aperture—the black circle shows the area of the small aperture, (B) low-resolution image from the small aperture, (C) object support constraint derived from (B), (D) image reconstructed by the Gerchberg–Saxton algorithm followed by the iterative Fourier-transform algorithm using (A)–(C), (E) ideal image for comparison.

Fig. 12
Fig. 12

Intermediate reconstruction results with different weightings on the Fourier modulus. Top row: low-resolution image from the small-aperture telescope; bottom row: intermediate reconstructed images; middle row: ideal images with the same Fourier weighting. Diameter of weighting function in pixels: (B), (C) 21; (D), (E) 31; (F), (G) 43; (H), (I) 63.

Fig. 13
Fig. 13

Convergence of the iterative Fourier-transform algorithm: ODEM (□) and ABSERR (Δ) (solid curves, scale at left) and the diameter of the Fourier-modulus weighting function (dotted curve, scale at right) as a function of iteration number.

Equations (18)

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g k + 1 ( x ) = g k + β [ 2 | g ( x ) | g k ( x ) | g k ( x ) | g k ( x ) | g ( x ) | g k ( x ) | g k ( x ) | ] ,
ODEM 2 = x [ | g k ( x ) | | g ( x ) | ] 2 x | g ( x ) | 2 ,
ABSERR 2 = x | α g ( x x 0 ) g ( x ) | 2 x | g ( x ) | 2 ,
ABSERR 2 1 exp ( σ ϕ 2 ) .
ODEM 2 = x S | g k ( x ) | 2 x | g k ( x ) | 2 ,
G ( u ) = F ( u ) exp [ i ϕ e ( u ) ]
E 2 = A 1 | G ( u ) F ( u ) | 2 d 2 u = A 1 | F ( u ) | 2 | 1 exp [ i ϕ e ( u ) ] | 2 d 2 u = A 1 | F ( u ) | 2 4 sin 2 [ ϕ e ( u ) / 2 ] d 2 u ,
E 2 4 | F ( u ) | 2 4 sin 2 [ ϕ e ( u ) / 2 ] 4 | F ( u ) | 2 4 sin 2 [ ϕ e ( u ) / 2 ] .
0 exp ( p x 2 ) sin 2 ( a x ) d x = ( 1 / 4 ) π / ρ [ 1 exp ( a 2 / p ) ] , p > 0 ,
sin 2 [ ϕ e ( u ) / 2 ] = sin 2 ( ϕ e / 2 ) 1 2 π σ ϕ exp ( ϕ e 2 / 2 σ ϕ 2 ) d ϕ e = ( 1 / 2 ) [ 1 exp ( σ ϕ 2 / 2 ) ] .
e 2 E 2 | F ( u ) | 2 2 [ 1 exp ( σ ϕ 2 / 2 ) ] .
e 2 σ ϕ 2 for σ ϕ 2 1 .
E 2 = A 1 | α G ( u ) F ( u ) | 2 d 2 u ,
α = G * ( u ) F ( u ) d 2 u | G ( u ) | 2 d 2 u .
α = G * F | G | 2 = | F | 2 exp ( i ϕ e ) | F | 2 .
= exp ( i ϕ e ) = exp ( σ ϕ 2 / 2 ) .
E 2 A 1 | F ( u ) | 2 | exp ( σ ϕ 2 / 2 ) exp [ i ϕ e ( u ) ] 1 | 2 d 2 u = A 1 | F ( u ) | 2 { exp ( σ ϕ 2 ) + 1 2 exp ( σ ϕ 2 / 2 ) cos [ ϕ e ( u ) ] } d 2 u | F ( u ) | 2 { exp ( σ ϕ 2 ) + 1 2 exp ( σ ϕ 2 / 2 ) cos [ ϕ e ( u ) ] } = | F ( u ) | 2 [ exp ( σ ϕ 2 ) + 1 2 exp ( σ ϕ 2 / 2 ) exp ( σ ϕ 2 / 2 ) ] = | F ( u ) | 2 [ 1 exp ( σ ϕ 2 ) ]
e 2 = E 2 | F ( u ) | 2 1 exp ( σ ϕ 2 ) .

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