Abstract

This paper presents a detailed numerical study on the performance of the standard phasing algorithms with random phase illumination (RPI). Phasing with high resolution RPI and the oversampling ratio σ=4 determines a unique phasing solution up to a global phase factor. Under this condition, the standard phasing algorithms converge rapidly to the true solution without stagnation. Excellent approximation is achieved after a small number of iterations, not just with high resolution but also low resolution RPI in the presence of additive as well multiplicative noises. It is shown that RPI with σ=2 is sufficient for phasing complex-valued images under a sector condition and σ=1 for phasing nonnegative images. The error-reduction algorithm with RPI is proved to converge to the true solution under proper conditions.

© 2012 Optical Society of America

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References

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  1. R. P. Millane, “Phase retrieval in crystallography and optics” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  2. J. C. Dainty and J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, 1987), pp. 231–275.
  3. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
    [CrossRef]
  4. M. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process 30, 140–154 (1982).
    [CrossRef]
  5. M. Hayes, “Reducible polynomials in more than one variables,” Proc. IEEE 70, 197–198 (1982).
    [CrossRef]
  6. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  7. 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]
  8. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
    [CrossRef]
  9. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
    [CrossRef]
  10. A. Fannjiang, “Absolute uniqueness of phase retrieval with random illumination,” Inverse Probl. 28, 075008 (2012).
  11. T. A. Pitts and J. F. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
    [CrossRef]
  12. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
    [CrossRef]
  13. P. F. Almoro, G. Pedrine, P. N. Gundu, W. Osten, and S. G. Hansom, “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser,” Opt. Lasers Eng. 49, 252–257 (2011).
    [CrossRef]
  14. A. Anand, G. Pedrini, W. Osten, and P. Almoro, “Wavefront sensing with random amplitude mask and phase retrieval,” Opt. Lett. 32, 1584–1586 (2007).
    [CrossRef]
  15. E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” preprint, August 2011.
  16. J. Miao and D. Sayre, “On possible extensions of X-ray crystallography through diffraction-pattern oversampling,” Acta Crystallogr. Sect. A 56, 596–605 (2000).
    [CrossRef]
  17. J. Miao, J. Kirz, and D. Sayre, “The oversampling phasing method,” Acta Crystallogr. Sect. D 56, 1312–1315 (2000).
    [CrossRef]
  18. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
    [CrossRef]
  19. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef]
  20. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237 (1972).
  21. A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. 1, 932–943 (1984).
    [CrossRef]

2012

A. Fannjiang, “Absolute uniqueness of phase retrieval with random illumination,” Inverse Probl. 28, 075008 (2012).

2011

P. F. Almoro, G. Pedrine, P. N. Gundu, W. Osten, and S. G. Hansom, “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser,” Opt. Lasers Eng. 49, 252–257 (2011).
[CrossRef]

2007

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[CrossRef]

A. Anand, G. Pedrini, W. Osten, and P. Almoro, “Wavefront sensing with random amplitude mask and phase retrieval,” Opt. Lett. 32, 1584–1586 (2007).
[CrossRef]

2003

T. A. Pitts and J. F. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef]

2002

2000

J. Miao and D. Sayre, “On possible extensions of X-ray crystallography through diffraction-pattern oversampling,” Acta Crystallogr. Sect. A 56, 596–605 (2000).
[CrossRef]

J. Miao, J. Kirz, and D. Sayre, “The oversampling phasing method,” Acta Crystallogr. Sect. D 56, 1312–1315 (2000).
[CrossRef]

1999

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

1998

1990

1987

1986

1984

A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. 1, 932–943 (1984).
[CrossRef]

1982

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

M. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process 30, 140–154 (1982).
[CrossRef]

M. Hayes, “Reducible polynomials in more than one variables,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

1972

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

Almoro, P.

Almoro, P. F.

P. F. Almoro, G. Pedrine, P. N. Gundu, W. Osten, and S. G. Hansom, “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser,” Opt. Lasers Eng. 49, 252–257 (2011).
[CrossRef]

Anand, A.

Bauschke, H. H.

Candès, E. J.

E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” preprint, August 2011.

Chapman, H. N.

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Combettes, P. L.

Dainty, J. C.

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

Eldar, Y.

E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” preprint, August 2011.

Fannjiang, A.

A. Fannjiang, “Absolute uniqueness of phase retrieval with random illumination,” Inverse Probl. 28, 075008 (2012).

Fienup, J. R.

Gerchberg, R. W.

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

Greenleaf, J. F.

T. A. Pitts and J. F. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef]

Gundu, P. N.

P. F. Almoro, G. Pedrine, P. N. Gundu, W. Osten, and S. G. Hansom, “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser,” Opt. Lasers Eng. 49, 252–257 (2011).
[CrossRef]

Hansom, S. G.

P. F. Almoro, G. Pedrine, P. N. Gundu, W. Osten, and S. G. Hansom, “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser,” Opt. Lasers Eng. 49, 252–257 (2011).
[CrossRef]

Hayes, M.

M. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process 30, 140–154 (1982).
[CrossRef]

M. Hayes, “Reducible polynomials in more than one variables,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Kirz, J.

J. Miao, J. Kirz, and D. Sayre, “The oversampling phasing method,” Acta Crystallogr. Sect. D 56, 1312–1315 (2000).
[CrossRef]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Levi, A.

A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. 1, 932–943 (1984).
[CrossRef]

Luke, D. R.

Marchesini, S.

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[CrossRef]

Miao, J.

J. Miao and D. Sayre, “On possible extensions of X-ray crystallography through diffraction-pattern oversampling,” Acta Crystallogr. Sect. A 56, 596–605 (2000).
[CrossRef]

J. Miao, J. Kirz, and D. Sayre, “The oversampling phasing method,” Acta Crystallogr. Sect. D 56, 1312–1315 (2000).
[CrossRef]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

Millane, R. P.

Osten, W.

P. F. Almoro, G. Pedrine, P. N. Gundu, W. Osten, and S. G. Hansom, “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser,” Opt. Lasers Eng. 49, 252–257 (2011).
[CrossRef]

A. Anand, G. Pedrini, W. Osten, and P. Almoro, “Wavefront sensing with random amplitude mask and phase retrieval,” Opt. Lett. 32, 1584–1586 (2007).
[CrossRef]

Pedrine, G.

P. F. Almoro, G. Pedrine, P. N. Gundu, W. Osten, and S. G. Hansom, “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser,” Opt. Lasers Eng. 49, 252–257 (2011).
[CrossRef]

Pedrini, G.

Pitts, T. A.

T. A. Pitts and J. F. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef]

Saxton, W. O.

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

Sayre, D.

J. Miao and D. Sayre, “On possible extensions of X-ray crystallography through diffraction-pattern oversampling,” Acta Crystallogr. Sect. A 56, 596–605 (2000).
[CrossRef]

J. Miao, J. Kirz, and D. Sayre, “The oversampling phasing method,” Acta Crystallogr. Sect. D 56, 1312–1315 (2000).
[CrossRef]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

Stark, H.

A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. 1, 932–943 (1984).
[CrossRef]

Strohmer, T.

E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” preprint, August 2011.

Voroninski, V.

E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” preprint, August 2011.

Wackerman, C. C.

Acta Crystallogr. Sect. A

J. Miao and D. Sayre, “On possible extensions of X-ray crystallography through diffraction-pattern oversampling,” Acta Crystallogr. Sect. A 56, 596–605 (2000).
[CrossRef]

Acta Crystallogr. Sect. D

J. Miao, J. Kirz, and D. Sayre, “The oversampling phasing method,” Acta Crystallogr. Sect. D 56, 1312–1315 (2000).
[CrossRef]

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process

M. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process 30, 140–154 (1982).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control

T. A. Pitts and J. F. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef]

Inverse Probl.

A. Fannjiang, “Absolute uniqueness of phase retrieval with random illumination,” Inverse Probl. 28, 075008 (2012).

J. Opt. Soc. Am.

A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. 1, 932–943 (1984).
[CrossRef]

J. Opt. Soc. Am. A

Nature

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Opt. Lasers Eng.

P. F. Almoro, G. Pedrine, P. N. Gundu, W. Osten, and S. G. Hansom, “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser,” Opt. Lasers Eng. 49, 252–257 (2011).
[CrossRef]

Opt. Lett.

Optik

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

Proc. IEEE

M. Hayes, “Reducible polynomials in more than one variables,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Rev. Sci. Instrum.

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[CrossRef]

Other

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

E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” preprint, August 2011.

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Figures (8)

Fig. 1.
Fig. 1.

Error reduction algorithm with random illumination.

Fig. 2.
Fig. 2.

Test images of loose support: (a) 269×269 cameraman, (b) 200×200 phantom, where the dark borders represent loose support.

Fig. 3.
Fig. 3.

(a) Recovery by 1651 ER iterations with UI and σ=4; (b) r(fk) versus k with r(f^)5.49%; (c) recovery by 1000HIO+103ER with UI and σ=4; (d) r(fk) versus k with r(f^)0.49%; (e) recovery by 1587 ER steps with one low resolution RPI with σ=2; (f) r(fk) versus k with r(f^)0.52% and e(f^)2.51%; (g) recovery by 33HIO+24ER steps with low resolution RPI with σ=2; (h) r(fk) versus k with r(f^)0.05% and e(f^)0.32%; (i) recovery by 5512 ER steps with high resolution RPI with σ=1; (j) r(fk) versus k with r(f^)0.19% and e(f^)3.27%; (k) recovery by 77HIO+67ER steps with high resolution RPI with σ=1; (l) r(fk) versus k with r(f^)0.10% and e(f^)1.39%.

Fig. 4.
Fig. 4.

(a) Recovery by 4140 ER iterations with UI with σ=4; (b) r(fk) versus k with r(f^)14.71%; (c) recovery by 1000HIO+421ER steps with one UI with σ=4; (d) r(fk) versus k with r(f^)3.94%; (e) recovery by 460 ER with one low resolution RPI with σ=2; (f) r(fk) versus k with r(f^)0.03% and e(f^)0.09%; (g) recovery by 103HIO+11ER steps with one low resolution RPI with σ=2; (h) r(fk) versus k with r(f^)0.03% and e(f^)0.12%; (i) recovery by 966 ER steps with one high resolution RPI with σ=1; (j) r(fk) versus k with r(f^)0.06% and e(f^)0.40%; (k) recovery by 94HIO+16ER steps with one high resolution RPI with σ=1; (l) r(fk) versus k with r(f^)0.04% and e(f^)0.19%.

Fig. 5.
Fig. 5.

(a) Relative error with one RPI for nonnegative-valued phantom; (b) relative error with one RPI for complex-valued phantom with phases randomly distributed in [0,π/2]; (c) relative error by 200HIO+300ER with one RPI and UI for complex-valued phantom with phases randomly distributed in [0,2π].

Fig. 6.
Fig. 6.

Phasing with σ=2 and one high resolution RPI: (a) recovery by 18HIO+10ER with 5% Gaussian noise; (b) r(fk) versus k with r(f^)2.62% and e(f^)4.20%; (c) recovery by 19HIO+10ER with 5% Gaussian noise; (d) r(fk) versus k with r(f^)2.85% and e(f^)3.51%; (e) recovery by 16HIO+10ER with 5% Poisson noise; (f) r(fk) versus k with r(f^)3.71% and e(f^)5.89%; (g) recovery by 17HIO+10ER with 5% Poisson noise; (h) r(fk) versus k with r(f^)4.05% and e(f^)4.84%; (i) recovery by 14HIO+10ER with 5% illuminator noise; (j) r(fk) versus k with r(f^)5.28% and e(f^)7.75%; (k) recovery by 16HIO+10ER with 5% illuminator noise; (l) r(fk) versus k with r(f^)5.48% and e(f^)6.35%.

Fig. 7.
Fig. 7.

Phasing with σ=2 and one low (40×40) resolution RPI: (a) recovery by 27HIO+10ER with 5% Gaussian noise; (b) r(fk) versus k with r(f^)2.50% and e(f^)7.37%; (c) recovery by 35HIO+10ER with 5% Gaussian noise; (d) r(fk) versus k with r(f^)2.85% and e(f^)4.18%; (e) recovery by 22HIO+10ER with 5% Poisson noise; (f) r(fk) versus k with r(f^)3.77% and e(f^)6.27%; (g) recovery by 100HIO+10ER with 5% Poisson noise; (h) r(fk) versus k with r(f^)4.24% and e(f^)5.09%; (i) recovery by 20HIO+10ER with 5% illuminator noise; (j) r(fk) versus k with r(f^)4.00% and e(f^)13.14%; (k) recovery by 34HIO+10ER with 5% illuminator noise; (l) r(fk) versus k with r(f^)5.48% and e(f^)9.46%.

Fig. 8.
Fig. 8.

(a) Relative error for nonnegative-valued phantom and σ=2; (b) relative error for complex-valued phantom with phases randomly distributed in [0,π/2] and σ=4; (c) relative error for complex-valued phantom with phases randomly distributed in [0,2π] and σ=3.

Equations (88)

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

N={0nN},N=(N1,N2,,Nd).
F(z)=nf(n)zn.
F(ei2πω)=nf(n)e2πiω·n
|F(ei2πw)|2=n=NNm+nNf(m+n)f(m)¯ei2πn·w,
Cf(n)=mNf(m+n)f(m)¯
L={ω=(ω1,,ωd)|ωj=0,12Nj+1,22Nj+1,,2Nj2Nj+1},
f(·)f(·+t)for sometZd,
f(·)f(N·)¯,
f(·)eiθf(·).
σ=Fourier magnitude data numberunknown-valued image pixel number
g(n)=λ(n)f(n),
λ(n)=eiϕ(n),
infgDgf=hf.
G(ω)=T{G}(ω)={Y(ω)eiG(ω)if|G(ω)|>0Y(ω)if|G(ω)|=0.
Pf=Λ1Φ1TΦΛ.
Pfθ=Λ1Φ1TθΦΛ,
Tθ{G}(ω)={Y(ω)eiG(ω)if|G(ω)|>0Y(ω)eiθ(ω)if|G(ω)|=0.
Po{h}(n)=Pα{h(n)}max{J(h(n))sinα+R(h(n))cosα,0}eiα.
ifβαπ,Po{h}(n)={h(n)ifαh(n)βPβ{h(n)}ifβh(n)[β+π/2]R(Pα{h(n)})if[απ/2]h(n)α0else,
ifβα>π,Po{h}(n)={h(n)ifαh(n)βPβ{h(n)}ifβh(n)[(α+β)/2+π]Pα{h(n)}if[(α+β)/2+π]h(n)α,
{aθbifabaθ<2πor0θbifa>b.
Po{h}(n)=R(h(n)).
Po{h}(n)=max{R(h(n)),0}.
R(Po{h}(n))=max(R(h(n)),0),J(Po{h}(n))=max(J(h(n)),0).
Po{h}(n)={h(n)ifnS0else.
εo(h)=Po{h}h,εf(h)=Pf{h}h
εf(h)=Pf{h}h=TΦΛhΦΛh=Y|ΦΛh|.
σ=image pixel number+zero-padding pixel numberimage pixel number.
fk+1=PoPffk,
εf(fk+1)εf(fk).
εf(fk)=fkfkfk+1fk=Gk+1GkGk+1Gk+1=fk+1fk+1=εf(fk+1).
R(fk+1(n))=R(fk(n)),
J(fk+1(n))=J(fk(n))β·J(fk(n)).
R(fk+1(n))={R(fk(n))ifR(fk(n))0R(fk(n))β·R(fk(n))ifR(fk(n))<0,
J(fk+1(n))=J(fk(n))β·J(fk(n)).
R(fk+1(n))={R(fk(n))ifR(fk(n))0R(fk(n))β·R(fk(n))ifR(fk(n))<0,
J(fk+1(n))={J(fk(n))ifJ(fk(n))0J(fk(n))β·J(fk(n))ifJ(fk(n))<0.
P1=Λ11Φ1T1ΦΛ1
P2=Λ21Φ1T2ΦΛ2.
fk+1=PoP2P1fk.
e(f^)={ff^/fif absolute uniqueness holdsminν[0,2π)feiνf^/fif uniqueness holds only up to a global phase,
r(f^)=Y|ΦΛPo{f^}|Y,
limkΦΛfk(ω)=ΦΛf(ω),ω.
limkΦΛfk=ΦΛf,
limkGk=limkTΦΛfk=TΦΛf.
limkfk+1=PoPff,
f=PoPff.
limjPoΛ1Φ1TΦΛfkj=PoΛ1Φ1TθΦΛf;
limjfkj+1=limjPoPffkj=PoPfθf,
f=PoPfθf.
fm+(·)=f(m+·),fm(·)=f(m·).
F(z)=nf(n)zn
F(z)=αzn0k=1pFk(z),
X(z)=eiθzkX(z¯1)¯.
X(z)=A(z)·zNA(z¯1)¯
h(n)=f(n)g(n)¯
H(z)=F(z)G(z¯1)¯.
H(e2πiω)=F(e2πiω)G(e2πiω),
H(z)=H(z¯1)¯.
F(z)G(z¯1)¯=F(z¯1)¯G(z).
F(z)G(z¯1)¯zN=F(z¯1)¯G(z)zN.
F(z)=akFk(z),
zNF(z¯1)¯=azmkF~k(z),
G(z)=bznG(z),
zNG(z¯1)¯=bznG˜(z),
abznkFk(z)G˜(z)=abzmnkF˜k(z)G(z).
G(z)=Q(z)kFk(z)=1aQ(z)F(z).
a¯aQ(z)=Q(z¯1)¯.
f˜t+(n)=λ(t+n)f(t+n)
F˜t+(z)=nλ(t+n)f(t+n)zn.
F˜t+(z)=eiθzkF˜t+(z¯1)¯
nλ(t+n)f(t+n)zn=eiθzknλ(t+n)f(t+n)¯zn,
λ(t+n)f(t+n)=eiθλ(t+kn)f(t+kn)¯,n.
Φft+(ω)=e2πit·ωΦf(ω),
2πt·ω+Φf(ω)(mod2π)=Φh(ω)
Φf(ω)=Φh(ω)2πt·ω(mod2π),
Φf=Φh(t)+.
Poh=h,
|ΦΛh|=|ΦΛf|,
ΦΛh=ΦΛh.
h=eiνΛ1Λm+fm+
h=eiνΛ1Λmfm¯.
eiνΦΛm+fm+=ΦΛh.
eiνΦΛf=ΦΛ(m)+h(m)+.
γeiνΛf=Λ(m)+h(m)+
h(n)=γeiνλ(n+m)λ(n)f(n+m)
h=±γf
h=γeiνf

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