Abstract

An improved method for resolution of object reconstruction using phase retrieval by use of a scanning slit aperture is proposed. The reconstruction is based on measurements of the Fraunhofer diffraction intensities of wave fields transmitted through a scanning slit in the Fresnel-zone plane of an object. In the improved method, the measurement coordinates of the intensities depend not only on the slit's position used in a previous method but also on the slit's position scaled by the ratio between two distances among the object, Fresnel-zone, and detector planes. The spatial-frequency band for the object reconstruction, which is limited in a previous method by the extent of the Fourier transform of the slit function, can be extended to the bandwidth dependent on the scanning area with the slit. In addition, even in the measurement of the Fresnel diffraction intensities of wave fields transmitted through the slit, the improved resolution can be retained by compensation for a transverse shift of the intensities.

© 2006 Optical Society of America

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  1. K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54, 27-32 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).
  4. J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D x-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
    [CrossRef] [PubMed]
  5. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
    [CrossRef]
  6. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, "Atomic resolution imaging of a carbon nanotube from diffraction intensities," Science 300, 1419-1421 (2003).
    [CrossRef] [PubMed]
  7. N. Nakajima, "Phase-retrieval system using a shifted Gaussian filter," J. Opt. Soc. Am. A 15, 402-406 (1998).
    [CrossRef]
  8. N. Nakajima, "Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering," Appl. Opt. 37, 6219-6226 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670-1682 (1996).
    [CrossRef]
  15. N. Nakajima and M. Watanabe, "Phase retrieval from experimental far-field intensities by use of a Gaussian beam," Appl. Opt. 41, 4133-4139 (2002).
    [CrossRef] [PubMed]
  16. S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
    [CrossRef] [PubMed]
  17. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2005 (1)

2004 (1)

2003 (2)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, "Atomic resolution imaging of a carbon nanotube from diffraction intensities," Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

2002 (2)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D x-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

N. Nakajima and M. Watanabe, "Phase retrieval from experimental far-field intensities by use of a Gaussian beam," Appl. Opt. 41, 4133-4139 (2002).
[CrossRef] [PubMed]

2001 (1)

K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54, 27-32 (2001).
[CrossRef]

2000 (1)

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

1998 (2)

1996 (2)

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

T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670-1682 (1996).
[CrossRef]

1995 (1)

1983 (1)

1982 (2)

1976 (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London Ser. A 350, 191-212 (1976).
[CrossRef]

1972 (1)

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

Anderson, E. H.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D x-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Bajt, S.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

Barnea, Z.

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

Barty, A.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

Burge, R. E.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London Ser. A 350, 191-212 (1976).
[CrossRef]

Chapman, H. N.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Cookson, D. F.

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

Fiddy, M. A.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London Ser. A 350, 191-212 (1976).
[CrossRef]

Fienup, J. R.

Gao, M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, "Atomic resolution imaging of a carbon nanotube from diffraction intensities," Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Gerchberg, R. W.

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

Greenaway, A. H.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London Ser. A 350, 191-212 (1976).
[CrossRef]

Gureyev, T. E.

Hau-Riege, S. P.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

He, H.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Hodgson, K. O.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D x-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Howells, M. R.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Ishikawa, T.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D x-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Johnson, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D x-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Lai, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D x-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Marchesini, S.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

McCartney, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

Miao, J.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D x-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Nagahara, L. A.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, "Atomic resolution imaging of a carbon nanotube from diffraction intensities," Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Nakajima, N.

Noy, A.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Nugent, K. A.

K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54, 27-32 (2001).
[CrossRef]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670-1682 (1996).
[CrossRef]

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

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials," J. Opt. Soc. Am. A 12, 1932-1941 (1995).
[CrossRef]

Paganin, D.

K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54, 27-32 (2001).
[CrossRef]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

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

Roberts, A.

Ross, G.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London Ser. A 350, 191-212 (1976).
[CrossRef]

Saxton, W. O.

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

Spence, J. C. H.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Teague, M. R.

Vartanyants, I.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, "Atomic resolution imaging of a carbon nanotube from diffraction intensities," Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Wall, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

Watanabe, M.

Weierstrall, U.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Zhang, R.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, "Atomic resolution imaging of a carbon nanotube from diffraction intensities," Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Zuo, J. M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, "Atomic resolution imaging of a carbon nanotube from diffraction intensities," Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

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

Optik (1)

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

Phys. Rev. B (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Phys. Rev. Lett. (2)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D x-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

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

Phys. Today (1)

K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54, 27-32 (2001).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The phase problem," Proc. R. Soc. London Ser. A 350, 191-212 (1976).
[CrossRef]

Science (1)

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, "Atomic resolution imaging of a carbon nanotube from diffraction intensities," Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Ultramicroscopy (1)

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

Other (1)

N. Nakajima, "Phase retrieval using the properties of entire functions," in Advances in Imaging and Electron Physics, P.W.Hawkes, ed. (Academic, 1995), Vol. 93, pp. 109-171.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematic of the geometry of the phase-retrieval system. While the slit aperture is scanning in the Fresnel-zone plane, intensity data for the phase retrieval are recorded at two coordinates in the detector plane as a function of the slit position s.

Fig. 2
Fig. 2

Examples of inverse Fourier transforms of slit functions with quadratic phases shown in Eq. (18). Solid curves in (a) and (b) [or (c) and (d)] are the modulus and phase of the transformed function, respectively, in the case of a Fresnel number of N F = 0.112 (or N F = 0.759 ). Dotted curves in (a) and (c) represent the Gaussian function that is used for the approximation of the sinc function in Eq. (5).

Fig. 3
Fig. 3

Object reconstruction from Fraunhofer diffraction intensities of the transmitted wave through the slit: (a) modulus of an original object and (b) modulus of a reconstructed object from two series of noiseless intensities at the coordinates depending only on the slit's position s. (c) and (d) are the moduli of the reconstructed objects from noiseless and noisy (SNR = 156), respectively, intensities at the coordinates depending on the sum of the slit's position and the slit's position scaled by the ratio between the distances z and l. The Fresnel number defined by Eq. (21) is N F = 0.112 .

Fig. 4
Fig. 4

Object reconstruction from Fresnel diffraction intensities of the transmitted wave through the slit: (a) modulus and (b) phase of an original object and (c) modulus and (d) phase of a reconstructed object from noiseless intensities at the coordinates depending on the sum of the slit's position and the slit's position scaled by the ratio between the distances z and l but without compensation for a transverse shift of the intensities. The Fresnel number defined by Eq. (21) is N F = 0.759 .

Fig. 5
Fig. 5

Object reconstruction under the same conditions as in Fig. 4 except for the use of the compensation method for a transverse shift of measured intensities: (a) and (b) [or (c) and (d)] are the modulus and phase, respectively, of a reconstructed object from noiseless [or noisy ( SNR = 151 ) ] intensities.

Equations (27)

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K ( x ) = σ f ( u ) exp [ i π λ z ( x u ) 2 ] d u ,
H ( ξ ) = K ( x ) R ( x s ) exp [ i π λl ( ξ x ) 2 ] d x ,
| H ( s ) | 2 = | K ( x ) R ( x s ) d x | 2 ,
| H ( s + τ ) | 2 = | K ( x ) R ( x s ) exp ( i 2 π λ l x τ ) d x | 2 .
R ( x ) exp ( i 2 π α x ) d x = 2 a   sinc ( 2 π a α ) 2 a exp [ ( 2 π a α ) 2 / 6 ] ,
| H ( s + τ ) | 2 = exp ( 3 c 2 a 2 ) | H ( s i c ) | 2 ,
ln [ | H ( s + τ ) | | M ( s i c ) | ] + 3 c 2 2 a 2 = ϕ I ( s ) ,
H ( s ) exp ( i 2 π α s ) d s = F ( α ) i λ z exp ( i π λ z α 2 ) × 2 a   sinc ( 2 π a α ) ,
F ( α ) = f ( u ) exp ( i 2 π α u ) d u .
K ( x ) = exp ( i π λ z x 2 ) σ f ( u ) exp ( i π λ z u 2 ) × exp ( i 2 π λ z x u ) d u .
H [ s ( 1 + l z ) ] = q F ( x ) R ( x s ) × exp [ i π λ l ( 1 + l z ) ( x s ) 2 ] d x ,
F ( x ) = σ f ( u ) exp ( i π λ z u 2 ) exp ( i 2 π λ z x u ) d u .
| H [ s ( 1 + l z ) ] | 2 = | F ( x ) R ( x s ) d x | 2 .
| H [ s ( 1 + l z ) + τ ] | 2 = | F ( x ) R ( x s ) × exp ( i 2 π λ l x τ ) d x | 2 .
| H [ s ( 1 + l z ) + τ ] | 2 = exp ( 3 c 2 a 2 ) | H [ ( s i c ) ( 1 + l z ) ] | 2 .
H [ s ( 1 + l z ) ] exp ( i 2 π α s ) d s = [ F ( x ) R ( x s ) d x ] exp ( i 2 π α s ) d s = f ( λ z α ) exp ( i π λ z α 2 ) 2 a   sinc ( 2 π a α ) .
H [ s ( 1 + l z ) ] = q F ( x ) R ( x s ) d x ,
R ( x ) = R ( x ) exp [ i π λ l ( 1 + l z ) x 2 ] .
| H [ s ( 1 + l z ) ] | 2 = | F ( x ) R ( x s ) d x | 2 .
1 { H [ s ( 1 + l z ) ] } = 1 [ F ( x ) R ( x s ) d x ] = f ( λ z α ) exp ( i π λ z α 2 ) 1 [ R ( x ) ] ,
N F = a 2 λ l ,
1 [ R ( x ) ] 2 a exp [ - ( 2 π a α ) 2 6 ] exp ( - ib α 2 ) ,
| H [ s ( 1 + l z ) + τ ] | 2
= exp ( 3 c 2 a 2 ) | H [ ( s + d i c ) ( 1 + l z ) ] | 2 ,
d = b τ λ l π .
1 { H [ s ( 1 + l z ) ] } = f ( λ z α ) exp ( i π λ z α 2 ) 2 a × exp [ - ( 2 π ) 2 6 ] exp ( i b α 2 ) ,
ER = [ u , v σ | f ( u , v ) f r ( u , v ) | 2 u , v σ | f ( u , v ) | 2 ] 1 / 2 ,

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