Abstract

A detailed comparison of the original Gerchberg–Saxton and the Yang–Gu algorithms for the reconstruction of model images from two intensity measurements in a nonunitary transform system is presented. The Yang–Gu algorithm is a generalization of the Gerchberg–Saxton algorithm and is effective in solving the general amplitude–phase-retrieval problem in any linear unitary or nonunitary transform system. For a unitary transform system the Yang–Gu algorithm is identical to the Gerchberg–Saxton algorithm. The reconstruction of images from data corrupted with random noise is also investigated. The simulation results show that the Yang–Gu algorithm is relatively insensitive to the presence of noise in data. In all cases studied the Yang–Gu algorithm always resulted in a highly accurate recovered phase.

© 1994 Optical Society of America

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References

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  1. H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1978), pp. 13–19.
    [CrossRef]
  2. E. L. O'Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
    [CrossRef]
  3. B. J. Hoenders, “On the solution of the phase retrieval problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
  4. R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).
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  6. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
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    [CrossRef]
  8. R. W. Gerchberg, W. O. Saxton, “Wave phase from image and diffraction plane pictures,” in Image Processing and Computer-Aided Design in Electron Optics, P. W. Hawkes, ed. (Academic, New York, 1973), pp. 66–81.
  9. D. L. Misell, “The phase problem in electron microscopy,” in Advances in Optical and Electron Microscopy, V. E. Cosslett, R. Barber, eds. (Academic, London, 1978), Vol. 7, pp. 185–276.
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    [CrossRef]
  14. D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
    [CrossRef]
  15. C. H. Slump, H. A. Ferwerda, “Statistical analysis of low-dose reconstruction of weak phase-amplitude objects from two defocused images. I.,” Optik 62, 93–104 (1982).
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    [CrossRef]
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  28. R. Barakat, G. Newsam, “Algorithm for reconstruction of partially known, band-limited Fourier-transform pairs from noisy data,” J. Opt. Soc. Am. A 2, 2027–3039 (1985).
    [CrossRef]
  29. B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).
  30. G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).
  31. G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involved nonunitary transformation,” Optik 75, 68–74 (1987).
  32. B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
    [CrossRef] [PubMed]
  33. E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1986).
  34. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  35. G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 457–479 (1992).
  36. J. Ximen, J. Yan, “Computational trials for phase retrieval in electron microscopy from image and diffraction patterns,” Acta Phys. Sin. 32, 762–769 (1983).

1990

1987

1986

1985

1984

1983

1982

1981

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).

1977

A. M. J. Huiser, P. Van Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).

P. Van Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. IV. Checking of algorithm by means of simulated objects,” Optik 47, 123–134 (1977).

H. A. Ferwerda, B. J. Hoenders, A. M. J. Huiser, P. Van Toorn, “On the phase reconstruction problem in light and electron microscopy,” Photogr. Sc. Eng. 21, 282–289 (1977).

1976

B. J. Hoenders, “On the solution of the phase retrieval problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).

1974

R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1973

R. W. Gerchberg, W. O. Saxton, “Comment on ‘A method for the solution of the phase problem in electron microscopy,’” J. Phys. D 6, L31–L32 (1973);“A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
[CrossRef]

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

1971

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

1963

E. L. O'Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

Barakat, R.

Bates, R. H. T.

Boucher, R. H.

R. H. Boucher, “Convergence of algorithm for phase retrieval from two intensity distributions,” in 1980 International Optical Computing Conference I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–141 (1980).

Crimmins, T. R.

Dong, B.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involved nonunitary transformation,” Optik 75, 68–74 (1987).

B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
[CrossRef] [PubMed]

G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 457–479 (1992).

Fenup, J. R.

Ferwerda, H. A.

C. H. Slump, H. A. Ferwerda, “Statistical analysis of low-dose reconstruction of weak phase-amplitude objects from two defocused images. I.,” Optik 62, 93–104 (1982).

A. M. J. Huiser, P. Van Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).

H. A. Ferwerda, B. J. Hoenders, A. M. J. Huiser, P. Van Toorn, “On the phase reconstruction problem in light and electron microscopy,” Photogr. Sc. Eng. 21, 282–289 (1977).

P. Van Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. IV. Checking of algorithm by means of simulated objects,” Optik 47, 123–134 (1977).

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1978), pp. 13–19.
[CrossRef]

Fienup, J. R.

Fright, W. R.

Gerchberg, R. W.

R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

R. W. Gerchberg, W. O. Saxton, “Comment on ‘A method for the solution of the phase problem in electron microscopy,’” J. Phys. D 6, L31–L32 (1973);“A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
[CrossRef]

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

R. W. Gerchberg, W. O. Saxton, “Wave phase from image and diffraction plane pictures,” in Image Processing and Computer-Aided Design in Electron Optics, P. W. Hawkes, ed. (Academic, New York, 1973), pp. 66–81.

Goodman, J. W.

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

Gu, B.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involved nonunitary transformation,” Optik 75, 68–74 (1987).

B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
[CrossRef] [PubMed]

G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 457–479 (1992).

Hoenders, B. J.

H. A. Ferwerda, B. J. Hoenders, A. M. J. Huiser, P. Van Toorn, “On the phase reconstruction problem in light and electron microscopy,” Photogr. Sc. Eng. 21, 282–289 (1977).

B. J. Hoenders, “On the solution of the phase retrieval problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).

Holsztynski, W.

Huiser, A. M. J.

A. M. J. Huiser, P. Van Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).

H. A. Ferwerda, B. J. Hoenders, A. M. J. Huiser, P. Van Toorn, “On the phase reconstruction problem in light and electron microscopy,” Photogr. Sc. Eng. 21, 282–289 (1977).

Kreyszig, E.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1986).

LaHaie, I.

Levi, A.

Millane, R. P.

Misell, D. L.

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

D. L. Misell, “The phase problem in electron microscopy,” in Advances in Optical and Electron Microscopy, V. E. Cosslett, R. Barber, eds. (Academic, London, 1978), Vol. 7, pp. 185–276.

Newsam, G.

O'Neill, E. L.

E. L. O'Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “Comment on ‘A method for the solution of the phase problem in electron microscopy,’” J. Phys. D 6, L31–L32 (1973);“A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
[CrossRef]

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

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

W. O. Saxton, “Recovery of specimen information for strongly scattering objects,” in Computer Processing of Electron Microscope Image, P. W. Hawkes, ed. (Springer-Verlag, Berlin, 1980), pp. 35–87.
[CrossRef]

R. W. Gerchberg, W. O. Saxton, “Wave phase from image and diffraction plane pictures,” in Image Processing and Computer-Aided Design in Electron Optics, P. W. Hawkes, ed. (Academic, New York, 1973), pp. 66–81.

Slump, C. H.

C. H. Slump, H. A. Ferwerda, “Statistical analysis of low-dose reconstruction of weak phase-amplitude objects from two defocused images. I.,” Optik 62, 93–104 (1982).

Stark, H.

Van Toorn, P.

A. M. J. Huiser, P. Van Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).

P. Van Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. IV. Checking of algorithm by means of simulated objects,” Optik 47, 123–134 (1977).

H. A. Ferwerda, B. J. Hoenders, A. M. J. Huiser, P. Van Toorn, “On the phase reconstruction problem in light and electron microscopy,” Photogr. Sc. Eng. 21, 282–289 (1977).

Walther, A.

E. L. O'Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

Wang, L.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involved nonunitary transformation,” Optik 75, 68–74 (1987).

Ximen, J.

J. Ximen, J. Yan, “Computational trials for phase retrieval in electron microscopy from image and diffraction patterns,” Acta Phys. Sin. 32, 762–769 (1983).

Yan, J.

J. Ximen, J. Yan, “Computational trials for phase retrieval in electron microscopy from image and diffraction patterns,” Acta Phys. Sin. 32, 762–769 (1983).

Yang, G.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involved nonunitary transformation,” Optik 75, 68–74 (1987).

B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
[CrossRef] [PubMed]

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).

G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 457–479 (1992).

Acta Opt. Sin.

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

Acta Phys. Sin.

G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).

J. Ximen, J. Yan, “Computational trials for phase retrieval in electron microscopy from image and diffraction patterns,” Acta Phys. Sin. 32, 762–769 (1983).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

R. W. Gerchberg, W. O. Saxton, “Comment on ‘A method for the solution of the phase problem in electron microscopy,’” J. Phys. D 6, L31–L32 (1973);“A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
[CrossRef]

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Opt. Acta

E. L. O'Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Optik

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involved nonunitary transformation,” Optik 75, 68–74 (1987).

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

C. H. Slump, H. A. Ferwerda, “Statistical analysis of low-dose reconstruction of weak phase-amplitude objects from two defocused images. I.,” Optik 62, 93–104 (1982).

A. M. J. Huiser, P. Van Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).

P. Van Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. IV. Checking of algorithm by means of simulated objects,” Optik 47, 123–134 (1977).

Photogr. Sc. Eng.

H. A. Ferwerda, B. J. Hoenders, A. M. J. Huiser, P. Van Toorn, “On the phase reconstruction problem in light and electron microscopy,” Photogr. Sc. Eng. 21, 282–289 (1977).

Proc. R. Soc. London Ser. A

B. J. Hoenders, “On the solution of the phase retrieval problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).

Other

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1978), pp. 13–19.
[CrossRef]

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

W. O. Saxton, “Recovery of specimen information for strongly scattering objects,” in Computer Processing of Electron Microscope Image, P. W. Hawkes, ed. (Springer-Verlag, Berlin, 1980), pp. 35–87.
[CrossRef]

R. W. Gerchberg, W. O. Saxton, “Wave phase from image and diffraction plane pictures,” in Image Processing and Computer-Aided Design in Electron Optics, P. W. Hawkes, ed. (Academic, New York, 1973), pp. 66–81.

D. L. Misell, “The phase problem in electron microscopy,” in Advances in Optical and Electron Microscopy, V. E. Cosslett, R. Barber, eds. (Academic, London, 1978), Vol. 7, pp. 185–276.

R. H. Boucher, “Convergence of algorithm for phase retrieval from two intensity distributions,” in 1980 International Optical Computing Conference I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–141 (1980).

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1986).

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

G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 457–479 (1992).

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

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Figures (8)

Fig. 1
Fig. 1

Schematic of a general imaging system: G, optical modulators.

Fig. 2
Fig. 2

Flow charts of the iterative algorithms for the phase-retrieval problem from two intensities: (a) the YG algorithm, (b) the GS algorithm.

Fig. 3
Fig. 3

Sketch of an optical imaging system with diffraction loss, corresponding to a nonunitary transform system.

Fig. 4
Fig. 4

Comparison of the recovered phases obtained by use of the YG and the GS algorithms in an optical system with a = 2: (a) image f 1(x 1), (b) image f 2(x 1), (c) image f 3(x 1). In these figures, all the curves marked by character a correspond to the results obtained by the YG algorithm, curves b correspond to results of the GS algorithm, and curves c to correspond the correct phase.

Fig. 5
Fig. 5

Recovered phases for the model image f 1(x 1) with different sizes of the aperture: (a) a = 1, (b) a = 4, (c) a = 16. Curves a correspond to the results of the YG algorith, curves b correspond to results of the GS algorithm, and curves c correspond to the correct phase.

Fig. 6
Fig. 6

Reconstruction of the model image f 2 with the power spectrum corrupted by random noise: (a) the YG algorithm, (b) the GS algorithm. Curves a correspond to the recovered phases in the presence of noise, curves b correspond to the recovered phases in the absence of noise, and curves c correspond to the correct phases.

Fig. 7
Fig. 7

Results for the 2-D image f 1(x 1, y 1) with a = 4: (a) the original phase of the image, (b) the recovered phase obtained by the YG algorithm, (c) the recovered phase obtained by the GS algorithm.

Fig. 8
Fig. 8

Results for the 2-D image f 2(x 1, y 1) with a = 4: (a) the original phase of the image, (b) the YG algorithm, (c) the GS algorithm.

Tables (3)

Tables Icon

Table 1 Degree of Deviation of G ̂ from a Unitary Transform for Different Aperture Widths a 2a

Tables Icon

Table 2 Comparison of the Yang–Gu and the Gerchberg–Saxton Algorithms for Phase Retrieval of Three One-Dimensional Images in an Optical Diffraction-Limited System with Aperture Width 2a = 4 a

Tables Icon

Table 3 Comparison of the Yang–Gu and the Gerchberg–Saxton Algorithms for Phase Retrieval of Two Two-Dimensional Images in an Optical Diffraction-Limited System with Aperture Width 2a = 8 a

Equations (61)

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

U 1 ( X 1 ) = ρ 1 ( X 1 ) exp [ i ϕ 1 ( X 1 ) ] ,
U 2 ( X 2 ) = ρ 2 ( X 2 ) exp [ i ϕ 2 ( X 2 ) ] .
U 2 ( X 2 ) = G ( X 2 , X 1 ) U 1 ( X 1 ) d X 1 .
U 2 ( X 2 ) = G ̂ U 1 ( X 1 ) ,
G ̂ + G ̂ = A ̂ I ̂ ,
U 1 l = ρ 1 l exp ( i ϕ 1 l ) , l = 1 , 2 , 3 , , N 1
U 2 m = ρ 2 m exp ( i ϕ 2 m ) ,
U 2 m = l = 1 N 1 G m l U 1 l , m = 1 , 2 , 3 , , N 2 .
D ( ρ 1 , ϕ 1 ; ρ 2 , ϕ 2 ) = U 2 G ̂ U 1 = [ d x 2 | U 2 ( x 2 ) G ̂ U 1 | 2 ] 1 / 2 = [ m = 1 N 2 | U 2 m ( G ̂ U 1 ) m | 2 ] 1 / 2 .
δ ρ 1 D 2 = 0 , δ ϕ 1 D 2 = 0 , δ ρ 2 D 2 = 0 , δ ϕ 2 D 2 = 0 ,
ρ 1 k = 1 A k k Re { j G j k * ρ 2 j exp [ i ( ϕ 2 j ϕ 1 k ) ] j k A k j ρ 1 j exp [ i ( ϕ 1 j ϕ 1 k ) ] } ,
ρ 2 k = Re { j G k j ρ 1 j exp [ i ( ϕ 1 j ϕ 2 k ) ] } ,
ϕ 1 k = 1 A k k arg [ j G j k * ρ 2 j exp ( i ϕ 2 j ) j k ' A k j ρ 1 j exp ( i ϕ 1 j ) ] ,
ϕ 2 k = arg [ j G k j ρ 1 j exp ( i ϕ 1 j ) ] ,
ρ 1 = Re { A ̂ D 1 [ exp ( i arg ( U 1 + ) ) G ̂ + U 2 exp ( i arg ( U 1 + ) ) A ̂ N D U 1 ] } ,
ρ 2 = Re [ exp ( i arg ( U 2 + ) ) G ̂ U 1 ] ,
[ p ( x 1 ) + A ̂ D ] U 1 ( x 1 ) = G ̂ + U 2 A ̂ N D U 1 ,
q ( x 2 ) U 2 ( x 2 ) = G ̂ U 1 ,
ρ 2 = G ̂ U 1 ,
ρ 1 = G ̂ + U 2 ,
ϕ 2 = arg ( G ̂ U 1 ) ,
ϕ 1 = arg ( G ̂ + U 2 ) .
ϕ 2 ( x 2 ) = arg [ G ̂ ρ 1 exp ( i ϕ 1 ) ] ,
ϕ 1 ( x 1 ) = arg { A ̂ D 1 [ G ̂ + ρ 2 exp ( i ϕ 2 ) A ̂ N D ρ 1 exp ( i ϕ 1 ) ] } .
d x 1 | ϕ 1 ( 0 , m 1 ) ϕ 1 ( 0 , m 1 + 1 ) | 1 ,
SSE = [ ρ 2 ( x 2 ) G ̂ ρ 1 exp ( i ϕ 1 ( n , 0 ) ) ] 2 d x 2 / ρ 2 2 ( x 2 ) d x 2 ,
G ̂ ( x 2 , x 1 ) = G ̂ 0 ( l 2 ) T ̂ P ̂ a G ̂ 0 ( l 1 ) ,
G 0 ( l ) = G 0 ( x 2 , x 1 ; l ) = 1 + cos θ i 2 λ r exp ( ikr ) ,
T ( x ) = exp ( i π x 2 / λ l f ) .
P a ( x ) = { 1 for | x | a 0 otherwise .
G 0 ( x 2 , x 1 ; l ) = ( 1 i λ l ) 1 / 2 exp ( i 2 π l / λ ) exp [ i π ( x 2 x 1 ) 2 / λ l ] .
G ( x 2 , x 1 ) = ( 1 i λ l f ) 1 / 2 exp ( i 4 π l f / λ ) exp ( i 2 π x 1 x 2 / λ l f ) × ( 1 2 i ) 1 / 2 ξ 1 ξ 2 exp ( i π ξ 2 / 2 ) d ξ ,
ξ 2 = ( 2 λ l f ) 1 / 2 ( a x 1 x 2 ) , ξ 1 = ( 2 λ l f ) 1 / 2 ( a + x 1 + x 2 ) ,
lim a ( 1 2 i ) 1 / 2 ξ 1 ξ 2 exp ( i π ξ 2 / 2 ) d ξ = ( 1 2 i ) 1 / 2 + exp ( i π ξ 2 / 2 ) d ξ = 1 .
G ( x 2 , x 1 ) = ( 1 i λ l f ) 1 / 2 exp ( i 4 π l f / λ ) exp ( i 2 π x 1 x 2 / λ l f ) .
G ( x 2 , x 1 ) = ( 1 2 ) 1 / 2 exp ( i 2 π x 1 x 2 ) ξ 1 ¯ ξ 2 ¯ exp ( i π ξ 2 / 2 ) d ξ , ξ 1 ¯ = ( λ l f ) 1 / 2 ξ 1 , ξ 2 ¯ = ( λ l f ) 1 / 2 ξ 2 .
B ( a ) = 1 N i = 1 N | A ii | ,
C ( a ) = 1 N ( N 1 ) i = 1 N j i N | A i j | .
f 1 ( x 1 ) = ξ ( x 1 ) exp ( i π x 1 2 / 2 ) , f 2 ( x 1 ) = η ( x 1 ) exp [ i sin ( 2 π x 1 ) ] , f 3 ( x 1 ) = η ( x 1 ) exp [ i exp ( x 1 ) ] ,
ξ ( x ) = { 1 for | x | 0.5 0 otherwise ,
η ( x ) = { 1 + x for | x | 0.5 0 otherwise ,
Δ ϕ ¯ 1 = 1 N k = 1 N | ϕ 1 k ϕ 1 k | .
f 1 ( x 1 , y 1 ) = ξ ( x 1 , y 1 ) exp [ i ( x 1 2 + y 1 2 ) 2 ] , f 2 ( x 1 , y 1 ) = ξ ( x 1 , y 1 ) exp [ i ( x 1 3 3 x 1 y 1 2 ) ] ,
ξ ( x 1 , y 1 ) = { 1 for | x 1 | , | y 1 | 0.5 0 otherwise
D 2 = U 2 G ̂ U 1 2 = Tr ( U 2 + U 2 U 2 + G ̂ U 1 U 1 + G ̂ + U 2 + U 1 + G ̂ + G ̂ U 1 ) .
δ ϕ 1 ( D 2 ) = Tr [ i ( δ ϕ 1 U 1 + ) ( G ̂ + U 2 G ̂ + G ̂ U 1 ) i ( U 2 + G ̂ U 1 + G ̂ + G ̂ ) ( U 1 δ ϕ 1 ) ] = Tr [ i ( δ ϕ 1 U 1 + ) ( G ̂ + U 2 G ̂ + G ̂ U 1 ) + c . c . ] ,
δ ϕ 2 ( D 2 ) = Tr [ i ( δ ϕ 2 U 2 + ) G ̂ U 1 i U 1 + G ̂ + ( U 2 δ ϕ 2 ) ] = Tr [ i ( δ ϕ 2 U 2 + ) G ̂ U 1 + c . c . ] .
Im [ U 1 + ( G ̂ + U 2 G ̂ + G ̂ U 1 ) ] = 0 ,
Im ( U 2 + G ̂ U 1 ) = 0 .
p ( X 1 ) U 1 ( X 1 ) = G ̂ + U 2 G ̂ + G ̂ U 1 = G ̂ + U 2 A ̂ U 1 ,
q ( X 2 ) U 2 ( X 2 ) = G ̂ U 1 ,
ϕ 1 ( X 1 ) = arg [ A ̂ D 1 ( G ̂ + U 2 A ̂ N D U 1 ) ] ,
ϕ 2 ( X 2 ) = arg ( G ̂ U 1 ) .
δ ρ 1 ( D 2 ) = Tr { δ ρ 1 [ exp ( i arg ( U 1 + ) ) G ̂ + G ̂ U 1 exp ( i arg ( U 1 + ) ) G ̂ + U 2 ] + c . c . } = 0 ,
δ ρ 2 ( D 2 ) = Tr { δ ρ 2 [ exp ( i arg ( U 2 + ) ) U 2 exp ( i arg ( U 2 + ) ) G ̂ U 1 ] + c . c . } = 0 .
Re [ exp ( i arg ( U 1 + ) ) ( A ̂ U 1 G ̂ + U 2 ) ] = 0 ,
Re [ exp ( i arg ( U 2 + ) ) ( U 2 G ̂ U 1 ) ] = 0 .
ρ 1 = Re { A ̂ D 1 [ exp ( i arg ( U 1 + ) ) G ̂ + U 2 exp ( i arg ( U 1 + ) ) A ̂ N D U 1 ] } ,
ρ 2 = Re [ exp ( i arg ( U 2 + ) ) U 2 ] = Re [ exp ( i arg ( U 2 + ) ) G ̂ U 1 ] .
δ ϕ 1 2 ( D 2 ) = Tr { 2 ( δ ϕ 1 ) 2 U 1 + G ̂ + G ̂ U 1 + [ ( δ ϕ 1 ) 2 U 1 + ( G ̂ + U 2 A ̂ U 1 ) + c . c . ] } = Tr { 2 ( δ ϕ 1 ) 2 G ̂ U 1 2 + ( δ ϕ 1 ) 2 × ( U 1 + p U 1 + U 1 + p U 1 ) } = Tr { 2 ( δ ϕ 1 ) 2 [ G ̂ U 1 2 + p ( X 1 ) U 1 2 ] } 0 ,
δ ϕ 2 2 ( D 2 ) = Tr [ ( δ ϕ 2 ) 2 U 2 + G ̂ U 1 + c . c . ] = ( δ ϕ 2 ) 2 2 q ( X 2 ) U 2 2 0 , δ ρ 1 2 ( D 2 ) = 2 Tr [ ( δ ρ 1 ) 2 G ̂ exp ( i arg ( U 1 ) ) 2 ] 0 , δ ρ 2 2 ( D 2 ) = Tr [ 2 ( δ ρ 2 ) 2 exp ( i arg ( U 2 + ) ) exp ( i arg ( U 2 ) ) ] = 2 ( δ ρ 2 ) 2 0 ,

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