Abstract

Phase-retrieval algorithms have been developed that handle a complicated optical system that requires multiple Fresnellike transforms to propagate from one end of the system to the other including the absorption by apertures in more than one plane and allowance for bad detector pixels. Gradient-search algorithms and generalizations of the iterative-transform phase-retrieval algorithms are derived. Analytic expressions for the gradient of an error metric, with respect to polynomial coefficients and with respect to point-by-point phase descriptions, are given. The entire gradient can be computed with the number of transforms required to propagate a wave front from one end of the optical system to the other and back again, independent of the number of coefficients or phase points. This greatly speeds the computation. The reconstruction of pupil amplitude is also given. A convergence proof of the generalized iterative transform algorithm is given. These improved algorithms permit a more accurate characterization of complicated optical systems from their point spread functions.

© 1993 Optical Society of America

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References

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  1. 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]
  2. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.
  3. G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Macmillan, London, 1968).
  4. C. Burrows, Hubble Space Telescope Optical Telescope Assembly Handbook (Space Telescope Science Institute, Baltimore, Md., 1990).
  5. R. Griffiths, Hubble Space Telescope Wide Field and Planetary Camera Instrument Handbook (Space Telescope Science Institute, Baltimore, Md., 1990).
  6. R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. A 66, 961–964 (1976).
    [CrossRef]
  7. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  8. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  9. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  10. 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]
  11. Y. Guozhen, W. Li, D. Bizhen, G. Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).
  12. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15 and 20.
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  14. D. P. Feder, “Automatic optical design,” Appl. Opt. 2, 1209–1226 (1963).
    [CrossRef]
  15. A. K. Rigler, R. J. Pegis, “Optimization methods in optics,” in B. R. Frieden, ed., The Computer in Optical Research (Springer-Verlag, New York, 1980), Chap. 4, pp. 211–268.
    [CrossRef]
  16. J. R. Fienup, C. C. Wackerman, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  17. J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 31, 1747–1767 (1993).
    [CrossRef]
  18. J. R. Fienup, “Phase retrieval for the Hubble Space Telescope using iterative propagation algorithms,” in Applications of Digital Image Processing XIV, Proc. Soc. Photo-Opt. Instrum. Eng. 1567, 33 (1991).
  19. J. R. Fienup, “HST aberrations and alignment determined by phase retrieval algorithms,” in Space Optics for Astrophysics and Earth and Planetary Remote Sensing, Vol. 19 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 19–21.

1993 (1)

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 31, 1747–1767 (1993).
[CrossRef]

1991 (1)

J. R. Fienup, “Phase retrieval for the Hubble Space Telescope using iterative propagation algorithms,” in Applications of Digital Image Processing XIV, Proc. Soc. Photo-Opt. Instrum. Eng. 1567, 33 (1991).

1989 (1)

1987 (1)

Y. Guozhen, W. Li, D. Bizhen, G. Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

1986 (1)

1982 (1)

1976 (1)

R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. A 66, 961–964 (1976).
[CrossRef]

1972 (1)

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

1963 (1)

Benyuan, G.

Y. Guozhen, W. Li, D. Bizhen, G. Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Bizhen, D.

Y. Guozhen, W. Li, D. Bizhen, G. Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Burrows, C.

C. Burrows, Hubble Space Telescope Optical Telescope Assembly Handbook (Space Telescope Science Institute, Baltimore, Md., 1990).

Cederquist, J. N.

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.

Feder, D. P.

Fienup, J. R.

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 31, 1747–1767 (1993).
[CrossRef]

J. R. Fienup, “Phase retrieval for the Hubble Space Telescope using iterative propagation algorithms,” in Applications of Digital Image Processing XIV, Proc. Soc. Photo-Opt. Instrum. Eng. 1567, 33 (1991).

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, 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. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.

J. R. Fienup, “HST aberrations and alignment determined by phase retrieval algorithms,” in Space Optics for Astrophysics and Earth and Planetary Remote Sensing, Vol. 19 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 19–21.

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]

Gerchberg, R. W.

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

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. A 66, 961–964 (1976).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Griffiths, R.

R. Griffiths, Hubble Space Telescope Wide Field and Planetary Camera Instrument Handbook (Space Telescope Science Institute, Baltimore, Md., 1990).

Guozhen, Y.

Y. Guozhen, W. Li, D. Bizhen, G. Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Jensen, L. H.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Macmillan, London, 1968).

Kryskowski, D.

Li, W.

Y. Guozhen, W. Li, D. Bizhen, G. Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Marron, J. C.

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 31, 1747–1767 (1993).
[CrossRef]

Pegis, R. J.

A. K. Rigler, R. J. Pegis, “Optimization methods in optics,” in B. R. Frieden, ed., The Computer in Optical Research (Springer-Verlag, New York, 1980), Chap. 4, pp. 211–268.
[CrossRef]

Rigler, A. K.

A. K. Rigler, R. J. Pegis, “Optimization methods in optics,” in B. R. Frieden, ed., The Computer in Optical Research (Springer-Verlag, New York, 1980), Chap. 4, pp. 211–268.
[CrossRef]

Robinson, S. R.

Saxton, W. O.

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

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

Schulz, T. J.

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 31, 1747–1767 (1993).
[CrossRef]

Seldin, J. H.

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 31, 1747–1767 (1993).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15 and 20.

Stout, G. H.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Macmillan, London, 1968).

Wackerman, C. C.

Appl. Opt. (3)

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 31, 1747–1767 (1993).
[CrossRef]

D. P. Feder, “Automatic optical design,” Appl. Opt. 2, 1209–1226 (1963).
[CrossRef]

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

Applications of Digital Image Processing XIV (1)

J. R. Fienup, “Phase retrieval for the Hubble Space Telescope using iterative propagation algorithms,” in Applications of Digital Image Processing XIV, Proc. Soc. Photo-Opt. Instrum. Eng. 1567, 33 (1991).

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

Optik (Stuttgart) (2)

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

Y. Guozhen, W. Li, D. Bizhen, G. Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Other (10)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15 and 20.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

A. K. Rigler, R. J. Pegis, “Optimization methods in optics,” in B. R. Frieden, ed., The Computer in Optical Research (Springer-Verlag, New York, 1980), Chap. 4, pp. 211–268.
[CrossRef]

J. R. Fienup, “HST aberrations and alignment determined by phase retrieval algorithms,” in Space Optics for Astrophysics and Earth and Planetary Remote Sensing, Vol. 19 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 19–21.

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]

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

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Macmillan, London, 1968).

C. Burrows, Hubble Space Telescope Optical Telescope Assembly Handbook (Space Telescope Science Institute, Baltimore, Md., 1990).

R. Griffiths, Hubble Space Telescope Wide Field and Planetary Camera Instrument Handbook (Space Telescope Science Institute, Baltimore, Md., 1990).

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Equations (67)

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E = u W ( u ) [ | G ( u ) | | F ( u ) | ] 2 ,
g ( x 1 ) = U 1 ( x 1 ) = m 1 ( x 1 ) exp [ i θ ( x 1 ) ] ,
U n + 1 ( x n + 1 ) = exp ( i α n x n + 1 2 ) 1 N x n = 0 N 1 U n ( x n ) × exp ( i β n x n 2 ) exp ( i 2 π x n x n + 1 / N ) = exp ( i α n x n + 1 2 ) [ U n ( x n ) exp ( i β n x n 2 ) ] ,
[ U n ( x n ) ] = 1 N x n = 0 N 1 U n ( x n ) exp ( i 2 π x n x n + 1 / N ) .
G ( u ) = P [ g ( x 1 ) ] = exp ( i α d 1 u 2 ) 1 N x d 1 m d 1 ( x d 1 ) exp ( i β d 1 x d 1 2 ) × exp ( i 2 π x d 1 u / N ) exp ( i α d 2 x d 1 2 ) × 1 N x d 2 m d 2 ( x d 2 ) exp ( i β d 2 x d 2 2 ) × exp ( i 2 π x d 2 x d 1 / N ) exp ( i α d 3 x d 2 2 ) × 1 N x d 3 exp ( i α 1 x 2 2 ) × 1 N x 1 g ( x 1 ) exp ( i 2 π x 1 x 2 / N ) = exp ( i α d 1 u 2 ) 1 N x d 1 exp ( i 2 π x d 1 u / N ) × M d 1 ( x d 1 ) 1 N x d 2 exp ( i 2 π x d 2 x d 1 u / N ) × M d 2 ( x d 2 ) M 2 ( x 2 ) 1 N × x 1 exp ( i 2 π x 1 x 2 / N ) g ( x 1 ) = exp ( i α d 1 u 2 ) { M d 1 ( x d 1 ) × { M d 2 ( x d 2 ) M 2 ( x 2 ) [ g ( x 1 ) ] } } ,
M n ( x n ) = exp [ i ( β n + α n 1 ) x n 2 ] m n ( x n )
E p = 2 u W ( u ) [ | G ( u ) | | F ( u ) | ] | G ( u ) | p = u W ( u ) [ G * ( u ) | F ( u ) | | G ( u ) | G * ( u ) ] G ( u ) p + c . c . = u G w * ( u ) G ( u ) p + c . c . ,
G w ( u ) = W ( u ) [ | F ( u ) | G ( u ) | G ( u ) | G ( u ) ] .
G ( u ) p = p [ P [ g ( x 1 ) ] ] = P [ g 1 ( x 1 ) p ] .
E p = u G w * ( u ) P [ g ( x 1 ) p ] + c . c .
E p = u G w * ( u ) exp ( i α d 1 u 2 ) 1 N × x d 1 exp ( i 2 π u x d 1 / N ) M d 1 ( x d 1 ) × 1 N x d 2 exp ( i 2 π x d 2 x d 1 / N ) × M d 2 ( x d 2 ) × 1 N x 1 exp ( i 2 π x 1 x 2 / N ) g ( x 1 ) p + c . c . = x 1 g ( x 1 ) p × 1 N x 2 exp ( i 2 π x 1 x 2 / N ) M 2 ( x 2 ) M d 2 ( x d 2 ) × 1 N x d 1 exp ( i 2 π x d 2 x d 1 / N ) M d 1 ( x d 1 ) × 1 N × u exp ( i 2 π u x d 1 / N ) exp ( i α d 1 u 2 ) G w * ( u ) + c . c .
E p = x 1 g ( x 1 ) p P 1 [ G w * ( u ) ] + c . c . = 2 Re { x 1 g ( x 1 ) p P 1 [ G w * ( u ) ] } ,
P 1 [ G w * ( u ) ] = { M 2 ( x 2 ) { M d 2 ( x d 2 ) × { M d 1 ( x d 1 ( [ G w * ( u ) ] } } } .
E p = x 1 [ g ( x 1 ) p ] * P [ G w ( u ) ] + c . c . = 2 Re { x 1 [ g ( x 1 ) p ] * P [ G w ( u ) ] } , = 2 Re ( x 1 g ( x 1 ) p { P [ G w ( u ) ] } * ) ,
P [ G w ( u ) ] = 1 { M 2 * ( x 2 ) 1 { M d 2 * ( x d 2 ) × 1 { M d 1 * ( x d 1 ) 1 [ G w ( u ) ] } } } .
g w ( x 1 ) = P [ G w ( u ) ] ,
P 1 [ G w * ( u ) ] = g w * ( x 1 ) = { P [ G w ( u ) ] } * .
θ ( x 1 ) = j = 1 J a j Z j ( x 1 ) .
g ( x 1 ) a j = g ( x 1 ) i Z j ( x 1 ) .
E a j = 2 Re [ x 1 i g ( x 1 ) Z j ( x 1 ) g w * ( x 1 ) ] = 2 Im [ x 1 g ( x 1 ) Z j ( x 1 ) g w * ( x 1 ) ] .
g ( x 1 ) θ ( x ) = i g ( x ) δ ( x , x 1 ) ,
δ ( x , x 1 ) = { 1 , x 1 = x , 0 , otherwise ,
E θ ( x ) = 2 Re [ x 1 i g ( x ) δ ( x , x 1 ) g w * ( x 1 ) ] = 2 Im [ g ( x ) g w * ( x ) ] ,
g ( x 1 ) m 1 ( x ) = exp [ i θ ( x ) ] δ ( x , x 1 ) .
E m 1 ( x ) = 2 Re { x 1 exp [ i θ ( x ) ] δ ( x , x 1 ) g w * ( x 1 ) } = 2 Re { exp [ i θ ( x ) ] g w * ( x 1 ) } .
g ( x 1 ) = g R ( x 1 ) + i g I ( x 1 ) ,
g ( x 1 ) g R ( x ) = δ ( x , x 1 ) ,
g ( x 1 ) g I ( x 1 ) = i δ ( x , x 1 ) .
E g R ( x ) = 2 Re [ x 1 δ ( x , x 1 ) g w * ( x 1 ) ] = 2 Re [ g w * ( x ) ] = 2 Re [ g w ( x ) ] ,
E g I ( x ) = 2 Re [ x 1 i δ ( x , x 1 ) g w * ( x 1 ) ] = 2 Im [ g w * ( x ) ] = 2 Im [ g w ( x ) ] .
E g ( x ) E g R ( x ) + i E g I ( x ) = 2 g w ( x ) ,
g ( x 1 ; s ) = g ( x 1 ) s E g ( x 1 ) = g ( x 1 ) + 2 s g w ( x 1 ) .
E q = u G w * ( u ) G ( u ) q + c . c . ,
G ( u ) q = 1 N x d 1 x n exp ( i 2 π x n x n 1 / N ) M n ( x n ) q 1 N x 1 exp ( i 2 π x 1 x 2 / N ) g ( x 1 ) .
E q = x n m n ( x n ) q P 1 n [ g ( x 1 ) ] P d n 1 [ G w * ( u ) ] + c . c . = x n m n ( x n ) q P 1 n [ g ( x 1 ) ] { P d n [ G w ( u ) ] } * + c . c . ,
P 1 n [ g ( x 1 ) ] = exp ( i α n 1 x n 2 ) 1 N × x n 1 exp ( i 2 π x n 1 x n / N ) M n 1 ( x n 1 ) 1 N x 1 exp ( i 2 π x 1 x 2 / N ) g ( x 1 ) ,
P d n 1 [ G w * ( u ) ] = exp ( i β n x n 2 ) 1 N × x n + 1 exp ( i 2 π x n x n + 1 / N ) M n + 1 ( x n + 1 ) 1 N u exp ( i 2 π x d 1 u / N ) G w * ( u ) .
m n ( x n ) = | m n ( x n ) | exp [ i θ n ( x n ) ]
θ n ( x n ) = j = 1 b j Z j ( x n ) ,
m n ( x n ) b j = m n ( x n ) i Z j ( x n ) ,
E b j = 2 Im ( x n m n ( x n ) Z j ( x n ) P 1 n [ g ( x 1 ) ] × { P d n [ G w ( u ) ] } * ) .
m n ( x n ) θ n ( x ) = i m n ( x ) δ ( x , x n ) ,
E θ n ( x ) = 2 Im ( m n ( x ) P 1 n [ g ( x 1 ) ] { P d n [ G w ( u ) ] } * ) ,
m n ( x n ) | m n ( x ) | = exp [ i θ n ( x n ) ] δ ( x , x n ) ,
E | m n ( x ) | = 2 Re ( exp [ i θ n ( x ) ] P 1 n [ g ( x 1 ) ] × { P d n [ G w ( u ) ] } * ) ,
E 2 = u W ( u ) [ | G ( u ) | 2 | F ( u ) | 2 ] 2 .
E 2 p = 2 u W ( u ) [ | G ( u ) | 2 | F ( u ) | 2 ] G * ( u ) G ( u ) p + c . c . = 4 Re { u [ | G ( u ) | 2 | F ( u ) | 2 ] G * ( u ) P [ g ( x 1 ) p ] } = 4 Re ( x 1 g ( x 1 ) p { P [ | G ( u ) | 2 G * ( u ) | F ( u ) | 2 G * ( u ) ] } * ) ,
G ( u ) = | F ( u ) | G ( u ) / | G ( u ) | .
G p ( u ) = W ( u ) | F ( u ) | G ( u ) | G ( u ) | + [ 1 W ( u ) ] G ( u )
= G ( u ) { W ( u ) [ | F ( u ) | | G ( u ) | 1 ] + 1 } = W ( u ) G ( u ) [ | F ( u ) | | G ( u ) | 1 ] + G ( u ) = G w ( u ) + G ( u ) ,
P d n + + [ G p ( u ) ] = | m n ( x n ) | P d n [ G p ( u ) ] + [ 1 | m n ( x n ) | ] P 1 n [ g ( x 1 ) ] ,
P d n + + [ G p ( u ) ] = | m n ( x n ) | P d n [ G w ( u ) ] + | m n ( x n ) | P d n [ G ( u ) ] + [ 1 | m n ( x n ) | ] P 1 n [ g ( x 1 ) ] = | m n ( x n ) | P d n [ G w ( u ) ] P 1 n [ g ( x 1 ) ] ,
| m n ( x n ) | P d n [ G ( u ) ] = | m n ( x n ) | P 1 n [ g ( x 1 ) ]
g p ( x 1 ) = P [ G w ( u ) ] + g ( x 1 ) = g w ( x 1 ) + g ( x 1 ) .
E Fk = u W ( u ) [ | G k ( u ) | F ( u ) ] 2 = u W ( u ) | G k ( u ) | G k ( u ) | | G k ( u ) | G k ( u ) | G k ( u ) | | F ( u ) | | 2 = u | G k w ( u ) | 2 = u | G k w ( u ) | 2 .
E Fk = u | G k w ( u ) | 2 = x 1 | g w ( x 1 ) | 2 + C 1 ,
C 1 = n = 2 d 1 x n [ 1 | m n ( x n ) | ] | P d n [ G k w ( u ) ] | 2
E Fk = x 1 | g k w ( x 1 ) | 2 + C 1 = x 1 | g k p ( x 1 ) g k ( x 1 ) | 2 + C 1 x 1 | g k p ( x 1 ) g k + 1 ( x 1 ) | 2 + C 1 E ok + C 1 ,
E Fk x 1 | g k w ( x 1 ) + g k ( x 1 ) g k + 1 ( x 1 ) | 2 + C 1 = u | G k w ( u ) + G k ( u ) G k + 1 ( u ) | 2 + C 2 ,
C 2 = n = 2 d 1 x n [ 1 | m n ( x n ) | ] | P 1 n [ g k ( x 1 ) g k + 1 ( x 1 ) ] | 2
E Fk u | G k p ( u ) G k + 1 ( u ) | 2 + C 2 u | G k + 1 p ( u ) G k + 1 ( u ) | 2 + C 2 = u | W ( u ) G k + 1 ( u ) [ | F ( u ) | | G k + 1 ( u ) | 1 ] | 2 + C 2 = u W ( u ) [ | G k + 1 ( u ) | | F ( u ) | ] 2 + C 2 = E F ( k + 1 ) + C 2 E F ( k + 1 ) ,
g k + 1 ( x 1 ) = m 1 ( x 1 ) g k p ( x 1 ) ,
g k + 1 ( x 1 ) = m 1 ( x 1 ) g k p ( x 1 ) + [ 1 m 1 ( x 1 ) ] [ g k ( x 1 ) β g k p ( x 1 ) ] ,
g k + 1 ( x 1 ) = m 1 ( x 1 ) g k p ( x 1 ) | g k p ( x 1 ) | .
g k + 1 ( x 1 ) = m 1 ( x 1 ) [ g k w ( x 1 ) + g k ( x 1 ) ] .
g k + 1 ( x 1 ) = g k ( x 1 ) + 2 ( 1 / 2 ) g k w ( x 1 ) = g k w ( x ) + g k ( x 1 )
g k + 1 ( x 1 ) = m 1 ( x 1 ) [ g k w ( x 1 ) + g k ( x 1 ) ] .

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