Abstract

Deterministic phase retrieval is reinterpreted in terms of phase-space optics. A novel derivation of the transport-of-intensity equation is presented based on the Wigner distribution function and the ambiguity function. The phase retrieval problem is formulated as estimating the local first-order moment of the Wigner function from intensity information. A comparison with phase-space tomography suggests a generalization of deterministic phase retrieval that provides larger flexibility for signal recovery. In addition, one particular numerical implementation of generalized deterministic phase retrieval is presented. Simulated intensity data are used to validate the method.

© 2004 Optical Society of America

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References

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  1. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.
  2. A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
    [CrossRef]
  3. R. W. Harrison, “Phase problem in crystallography,” J. Opt. Soc. Am. A 10, 1046–1055 (1993).
    [CrossRef]
  4. J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. (Bellingham) 19, 297–305 (1980).
    [CrossRef]
  5. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  6. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  7. Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef] [PubMed]
  8. M. R. Teague, “Deterministic phase retrieval: a Green’s function approach,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  9. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  10. N. Jayshree, G. KeshavaDatta, R. M. Vasu, “Optical tomographic microscope for quantitative imaging of phase objects,” Appl. Opt. 39, 277–283 (2000).
    [CrossRef]
  11. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  12. T. Alieva, M. Bastiaans, L. Stanković, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
    [CrossRef]
  13. G. Gbur, E. Wolf, “Hybrid diffraction tomography without phase information,” J. Opt. Soc. Am. A 19, 2194–2202 (2002).
    [CrossRef]
  14. C. Dorrer, I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett. 28, 1481–1483 (2003).
    [CrossRef] [PubMed]
  15. M. J. Bastiaans, K. B. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. A 20, 1046–1049 (2003).
    [CrossRef]
  16. D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
    [CrossRef] [PubMed]
  17. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
    [CrossRef] [PubMed]
  18. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  19. D. Dragoman, M. Dragoman, K.-H. Brenner, “Amplitude and phase recovery of rotationally symmetric beams,” Appl. Opt. 41, 5512–5518 (2002).
    [CrossRef] [PubMed]
  20. X. Liu, K.-H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19–30 (2003).
    [CrossRef]
  21. M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbräuker, F. Hlawatsch, eds. (Elsevier, Amsterdam, 1997), pp. 375–426.
  22. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  23. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001).
  24. A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd. ed. (McGraw-Hill, New York, 1991), Chap. 5.5, pp. 115–117.
  25. J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
    [CrossRef]
  26. D. Dragoman, “Redundancy of phase-space distribution functions in complex field recovery problems,” Appl. Opt. 42, 1932–1937 (2003).
    [CrossRef] [PubMed]
  27. T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
    [CrossRef]
  28. M. E. Testorf, M. A. Fiddy, “Simulation of light propagation in planar-integrated free-space optics,” Opt. Commun. 176, 365–372 (2000).
    [CrossRef]
  29. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C–The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 105–128.

2004 (1)

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

2003 (5)

2002 (2)

2000 (2)

M. E. Testorf, M. A. Fiddy, “Simulation of light propagation in planar-integrated free-space optics,” Opt. Commun. 176, 365–372 (2000).
[CrossRef]

N. Jayshree, G. KeshavaDatta, R. M. Vasu, “Optical tomographic microscope for quantitative imaging of phase objects,” Appl. Opt. 39, 277–283 (2000).
[CrossRef]

1998 (1)

1997 (1)

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

1996 (1)

1995 (2)

1994 (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

1993 (3)

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

R. W. Harrison, “Phase problem in crystallography,” J. Opt. Soc. Am. A 10, 1046–1055 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

1983 (1)

1980 (1)

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

1978 (1)

1972 (1)

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

Alieva, T.

T. Alieva, M. Bastiaans, L. Stanković, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

Barty, A.

Bastiaans, M.

T. Alieva, M. Bastiaans, L. Stanković, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, K. B. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. A 20, 1046–1049 (2003).
[CrossRef]

M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbräuker, F. Hlawatsch, eds. (Elsevier, Amsterdam, 1997), pp. 375–426.

Beck, M.

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Brenner, K.-H.

X. Liu, K.-H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19–30 (2003).
[CrossRef]

D. Dragoman, M. Dragoman, K.-H. Brenner, “Amplitude and phase recovery of rotationally symmetric beams,” Appl. Opt. 41, 5512–5518 (2002).
[CrossRef] [PubMed]

Clarke, L.

Dainty, J. C.

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

Dorrer, C.

Dorsch, R. G.

Dragoman, D.

Dragoman, M.

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Fiddy, M. A.

M. E. Testorf, M. A. Fiddy, “Simulation of light propagation in planar-integrated free-space optics,” Opt. Commun. 176, 365–372 (2000).
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. (Bellingham) 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. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C–The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 105–128.

Gbur, G.

Gerchberg, R. W.

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

Gureyev, T. E.

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[CrossRef]

Harrison, R. W.

Jayshree, N.

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001).

Kang, I.

KeshavaDatta, G.

Liu, X.

X. Liu, K.-H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19–30 (2003).
[CrossRef]

Lohmann, A. W.

Mayer, A.

McAlister, D. F.

Mendlovic, D.

Nugent, K. A.

Paganin, D.

Paganin, D. M.

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd. ed. (McGraw-Hill, New York, 1991), Chap. 5.5, pp. 115–117.

Pogany, A.

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C–The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 105–128.

Raymer, M. G.

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Roberts, A.

Saxton, W. O.

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

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001).

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Stankovic, L.

T. Alieva, M. Bastiaans, L. Stanković, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Tamura, S.

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Teague, M. R.

Testorf, M. E.

M. E. Testorf, M. A. Fiddy, “Simulation of light propagation in planar-integrated free-space optics,” Opt. Commun. 176, 365–372 (2000).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C–The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 105–128.

Tu, J.

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Vasu, R. M.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C–The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 105–128.

Wilkins, S. W.

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

Wolf, E.

Wolf, K. B.

Zalevsky, Z.

Appl. Opt. (3)

IEEE Trans. Signal Process. (1)

T. Alieva, M. Bastiaans, L. Stanković, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (4)

X. Liu, K.-H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19–30 (2003).
[CrossRef]

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

M. E. Testorf, M. A. Fiddy, “Simulation of light propagation in planar-integrated free-space optics,” Opt. Commun. 176, 365–372 (2000).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Opt. Eng. (Bellingham) (1)

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

Opt. Lett. (5)

Optik (Stuttgart) (1)

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

Phys. Rev. E (1)

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Other (5)

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

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C–The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 105–128.

M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbräuker, F. Hlawatsch, eds. (Elsevier, Amsterdam, 1997), pp. 375–426.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001).

A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd. ed. (McGraw-Hill, New York, 1991), Chap. 5.5, pp. 115–117.

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Figures (7)

Fig. 1
Fig. 1

Model system. The wave front in plane z=0 is recovered from intensities recorded in planes z (dashed lines).

Fig. 2
Fig. 2

Phase-space diagram (AF) of DPR. The AFs along the ν¯ axis and along line Pδz correspond to the Fourier transformation of the intensities at z=0 and z=δz. The grid with cell size δx¯×δν¯ marks the sampling distance required to represent the AF by a set of discrete values.

Fig. 3
Fig. 3

Signal recovery from samples of the AF. The dotted lines Pn correspond to intensity measurements required for PST. PST uses samples along line t to compute the signal at one single point x¯. GDPR processes samples along line g simultaneously.

Fig. 4
Fig. 4

Program structure diagram of the GDPR algorithm.

Fig. 5
Fig. 5

Phase functions recovered with the GDPR. The intensity data are simulated assuming a rectangular beam aperture at z=0 (aperture width 256λ). Simulated noise-free intensity data were used as input for the GDPR algorithm.

Fig. 6
Fig. 6

Error of the estimated phase for M intensity samples. The simulated diffraction intensities of signal (a) in Fig. 5 are corrupted with additive noise and used as input for the GDPR algorithm. Ab is the estimate of a range for which the signal reconstruction improves significantly as compared with DPR.

Fig. 7
Fig. 7

Error of the estimated phase for a Gaussian intensity envelope for various amplitudes of additive noise. The doted curve corresponds to the signal intensity as a function of the transverse coordinate.  

Equations (24)

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

W(x, ν)=-ux+x2u*x-x2exp(-i2πxν)dx.
u(x)u*(0)=-W(x/2, ν)exp(i2πxν)dν.
I(x)=|u(x)|2=-W(x, ν)dν.
mν(x)=-νW(x, ν)dν=12π I(x) ddx ϕ(x).
W(x, ν; z)=W(x-λνz, ν; 0),
I(x, δz)=-W(x-λνδz, ν)dν-W(x, ν)dν-δzλ x-νW(x, ν)dν.
I(x, δz)-I(x, 0)δz-λ x-νW(x, ν)dν.
2πλz I(x, z)=-xI(x, z) x ϕ(x, z),
A(ν¯, x¯)=-ux+x¯2u*x-x¯2exp(-i2πxν¯)dx,
A(ν¯, x¯)=--W(x, ν)exp[-i2π(ν¯x-νx¯)]dxdν.
I˜(ν¯, z)=-I(x, z)exp(-i2πν¯x)dx=A(ν¯, -λν¯z).
tan θ=-λ δz.
mν(x)=-νW(x, ν)dν=-i2π-A(ν¯, 0)exp(i2πν¯x)dν¯,
u(x¯)u*(0)=-A(ν¯, x¯)exp(i2πν¯x¯/2)dν¯,
N2=Δx¯δx¯Δν¯δν¯.
δz<2λΔv2.
δzA¯A(ν¯, 0)λν¯.
x¯m(νn)=-λν¯nzm.
Aˆ(ν¯n, x¯)=p=0M-11p!px¯p A(ν¯n, x¯)|x¯=0x¯p,
A1(ν¯n, x¯m)=A(ν¯n, x¯m)-A(ν¯n, 0)x¯m.
A^1(ν¯n, x¯)=p=1M-11p!px¯p A(ν¯n, x¯)|x¯=0x¯p-1.
ϕ(x)=ϕ(0)+0xϕ(x)dx,
nGDPR=(M+2)N log N.
nPST=2N2log N.

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