Abstract

The transport of intensity equation (TIE) has long been recognized as a quantitative method for phase retrieval and phase contrast imaging. However, it is shown that the most widely accepted fast Fourier transform (FFT) based solutions do not provide an exact solution to the TIE in general. The root of the problem lies in the so-called “Teague’s assumption” that the transverse flux is considered to be a conservative field, which cannot be satisfied for a general object. In this work, we present the theoretical analysis of the phase discrepancy owing to the Teague’s assumption, and derive the necessary and sufficient conditions for the FFT-based solution to coincide with the exact phase. An iterative algorithm is then proposed aiming to compensate such phase discrepancy in a simple yet effective manner.

© 2014 Optical Society of America

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    [CrossRef]
  34. J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).

2014

2013

2011

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[CrossRef]

2010

2005

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54(3), 191–197 (2005).
[CrossRef] [PubMed]

2004

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

2001

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef] [PubMed]

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[CrossRef]

1999

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

1998

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[CrossRef]

1997

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

1996

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

1995

1992

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals,” P IEEE 80(4), 520–538 (1992).
[CrossRef]

1988

1986

1983

1979

1977

A. J. Jerri, “The Shannon sampling theorem—Its various extensions and applications: A tutorial review,” P IEEE 65(11), 1565–1596 (1977).
[CrossRef]

Allen, L. J.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[CrossRef]

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef] [PubMed]

Allman, B.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54(3), 191–197 (2005).
[CrossRef] [PubMed]

Anastasio, M. A.

Asundi, A.

Ayubi, G. A.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: Validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[CrossRef]

Barbastathis, G.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
[CrossRef] [PubMed]

Bastiaans, M. J.

Boashash, B.

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals,” P IEEE 80(4), 520–538 (1992).
[CrossRef]

Börrnert, F.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of Intensity Phase Retrieval of Arbitrary Wave Fields Including Vortices,” Phys. Rev. Lett. 111(17), 173902 (2013).
[CrossRef] [PubMed]

Carney, P. S.

Chen, Q.

Cookson, D. J.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Crozier, K. B.

Falaggis, K.

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Optimum measurement criteria for the axial derivative intensity used in transport of intensity-equation-based solvers,” Opt. Lett. 39(2), 182–185 (2014).
[CrossRef] [PubMed]

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).

Faulkner, H. M.

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef] [PubMed]

Ferrari, J. A.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: Validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[CrossRef]

Flores, J. L.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: Validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[CrossRef]

Gureyev, T.

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

Gureyev, T. E.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[CrossRef]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
[CrossRef]

Guzzinati, G.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of Intensity Phase Retrieval of Arbitrary Wave Fields Including Vortices,” Phys. Rev. Lett. 111(17), 173902 (2013).
[CrossRef] [PubMed]

Ishizuka, K.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54(3), 191–197 (2005).
[CrossRef] [PubMed]

Jerri, A. J.

A. J. Jerri, “The Shannon sampling theorem—Its various extensions and applications: A tutorial review,” P IEEE 65(11), 1565–1596 (1977).
[CrossRef]

Kou, S. S.

Kozacki, T.

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Optimum measurement criteria for the axial derivative intensity used in transport of intensity-equation-based solvers,” Opt. Lett. 39(2), 182–185 (2014).
[CrossRef] [PubMed]

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).

Kujawinska, M.

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).

Lubk, A.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of Intensity Phase Retrieval of Arbitrary Wave Fields Including Vortices,” Phys. Rev. Lett. 111(17), 173902 (2013).
[CrossRef] [PubMed]

Martinez-Carranza, J.

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Optimum measurement criteria for the axial derivative intensity used in transport of intensity-equation-based solvers,” Opt. Lett. 39(2), 182–185 (2014).
[CrossRef] [PubMed]

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[CrossRef]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
[CrossRef]

Orth, A.

Oxley, M. P.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[CrossRef]

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef] [PubMed]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Paganin, D. M.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[CrossRef]

Pavlov, K. M.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[CrossRef]

Perciante, C. D.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: Validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[CrossRef]

Qu, W.

Raven, C.

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

Reed Teague, M.

Roberts, A.

Roddier, F.

Schmalz, J. A.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[CrossRef]

Schoonover, R. W.

Sheppard, C. J. R.

Snigirev, A.

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

Snigireva, I.

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

Tian, L.

Verbeeck, J.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of Intensity Phase Retrieval of Arbitrary Wave Fields Including Vortices,” Phys. Rev. Lett. 111(17), 173902 (2013).
[CrossRef] [PubMed]

Waller, L.

Welford, W. T.

Wilkins, S.

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

Winston, R.

Yu, Y.

Zuo, C.

Zysk, A. M.

Appl. Opt.

J. Electron Microsc. (Tokyo)

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54(3), 191–197 (2005).
[CrossRef] [PubMed]

J. Microsc.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D Appl. Phys.

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

Opt. Commun.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[CrossRef]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: Validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[CrossRef]

Opt. Express

Opt. Lett.

P IEEE

A. J. Jerri, “The Shannon sampling theorem—Its various extensions and applications: A tutorial review,” P IEEE 65(11), 1565–1596 (1977).
[CrossRef]

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals,” P IEEE 80(4), 520–538 (1992).
[CrossRef]

Phys. Rev. A

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

L. J. Allen, H. M. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001).
[CrossRef] [PubMed]

Phys. Rev. Lett.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of Intensity Phase Retrieval of Arbitrary Wave Fields Including Vortices,” Phys. Rev. Lett. 111(17), 173902 (2013).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Proc. SPIE

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).

Other

S. R. Crouch and J. D. Ingle, Spectrochemical Analysis (Prentice Hall, 1988).

V. Berinde, “Approximating fixed points of weak contractions using the Picard iteration,” in Nonlinear Analysis Forum (2004), 43–54.

H. Jeffreys and B. Jeffreys, Methods of Mathematical Physics (Cambridge University, 1999).

M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics: Fundamentals and Applications: Fundamentals and Applications: Fundamentals and Applications (McGraw Hill Professional, 2009).

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Figures (10)

Fig. 1
Fig. 1

Phase retrieval of a strongly absorbing object (MEMS diaphragm) using the Paganin and Nugent’s solution. (a) Phase distribution. (b) Intensity distribution. (c) Phase retrieved by the Paganin and Nugent’s solution (d) Phase discrepancy. Scale bar 500 μm

Fig. 2
Fig. 2

Block diagram of the iterative compensation method.

Fig. 3
Fig. 3

Simulated intensity and phase distributions. (a) Intensity distribution defined by Eq. (25). (b) Phase distribution defined by Eq. (26).

Fig. 4
Fig. 4

Helmholtz decomposition of the transverse flux field. The x and y components of the vector fields are shown in the first row and the second row, respectively.

Fig. 5
Fig. 5

Simulation results of the iterative compensation algorithm for a smooth wavefront. (a) Phase error of the Paganin and Nugent’s solution before compensation (RMSE 9.53%). (b) Phase error after the first iteration of the compensation (RMSE 0.99%). (c) Phase error after the third iteration of the compensation (RMSE 0.016%). (d) One-line profile of the RMSE verse the iteration times of the proposed compensation approach.

Fig. 6
Fig. 6

Simulation results of the compensation algorithm with the existence of noise. (a) Phase error of the Paganin and Nugent’s solution before compensation (RMSE 10.64%). (b) Phase error after three iterations of the compensation (RMSE 1.90%).

Fig. 7
Fig. 7

RMSE versus iterative number under different levels of noise. The red/green boxes outline the raw phase errors, the residual errors after compensation, and the corresponding correction terms for the cases of high/medium noise levels, respectively.

Fig. 8
Fig. 8

Simulation result of complex intensity and phase distribution. (a) Intensity distribution. (b) Phase distribution. (c) Phase error of the Paganin and Nugent’s solution before compensation (RMSE 15.91%). (d) Phase error after five iterations of the compensation (RMSE 3.66%).

Fig. 9
Fig. 9

RMSE versus iterative number with different defocus distance. The red/green boxes outlines the raw phase errors, the residual errors after compensation, and the correction terms for the cases of medium /large defocus distance, respectively.

Fig. 10
Fig. 10

RMSE versus iterative number for a realistic MEMS example. When the value range of the intensity is [0.05, 1], our approach can converge to the exact solution (blue curve). When the minimum intensity is close to zero (0.001), the iterative process will diverge (red curve). This problem can be addressed by setting an intensity threshold (green curve).

Equations (26)

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W( x,u )= Γ( x+ x 2 ,x x 2 )exp( i2πu x )d x ,
j x ( x )=λ uW( x,u )du ,
j z ( x )= 1 k k 2 4 π 2 | u | 2 W( x,u )du ,
j z ( x ) z = j x ( x ),
j z ( x ) W( x,u )du=I( x ) .
I( x ) z = j x ( x ).
I( x ) z = 1 k [ I( x )ϕ( x ) ],
uW( x,u )du W( x,u )du = I 1 uW( x,u )du = 1 2π ϕ( x ).
j x ( x )= k 1 I( x )ϕ( x ),
j x ( x )= k 1 I( x )[ ϕ s ( x )+× ϕ v ( x ) ].
j x ( x )= k 1 I( x )ϕ( x )= k 1 ψ( x ),
I( x ) z = 1 k 2 ψ( x ),
[ I ( x ) 1 ψ( x ) ]= 2 ϕ( x ),
ϕ( x )=k 2 [ 1 I( x ) 2 I( x ) z ],
j x ( x )= k 1 I( x )ϕ( x )= k 1 [ ψ( x )+×η( x ) ],
2 ψ( x )=[ I( x )ϕ( x ) ],
2 η( x )=I( x )×ϕ( x ),
2 ϕ( x )=[ I ( x ) 1 ψ( x ) ]+I ( x ) 1 ×η( x ).
I ( x ) 1 ×η( x )=0.
I ( x ) 1 × 2 { [ I( x )×ϕ( x ) ] }=0.
f 2 = | f( x ) | 2 dx / dx .
I( x ) z = 1 k [ I( x )ϕ( x ) ] = 1 k [ I( x )ϕ( x )+I( x ) 2 ϕ( x ) ] 1 k I( x ) 2 ϕ( x ).
I ( x ) 1 ψ ( x ) = ϕ 0 ( x ) + × η 0 ( x ) ,
j x 0 ( x ) = k 1 I ( x ) ϕ 0 ( x ) = k 1 [ ψ ( x ) + I ( x ) × η 0 ( x ) ] .
I ( x , y ) = exp ( a 0 x 2 b 0 y 2 ) ,
ϕ ( x , y ) = a 0 x 2 b 0 y 2 a 1 x 8 + b 1 y 8 ,

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