Abstract

Optical systems capable of three-dimensional transmission imaging are considered; these systems employ a conventional tomographic setup with an added linear shift-invariant optical system between the sample and the detector. A theoretical analysis is presented of image formation and sample reconstruction in such systems, examples of which include diffraction tomography and phase-contrast tomography with the use of analyzer crystals. An example is introduced in which the image is obtained by scanning the beam along the line orthogonal to the optic axis and to the axis of rotation with a one-dimensional slit or grating parallel to the rotation axis. We show that under certain conditions the proposed system may allow quantitative local (region-of-interest) tomography.

© 2007 Optical Society of America

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  2. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).
  3. F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).
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    [CrossRef]
  5. G. Gbur and W. Wolf, "Diffraction tomography without phase information," Opt. Lett. 27, 1890-1892 (2002).
    [CrossRef]
  6. M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," Proc. SPIE 5535, 361-368 (2004).
    [CrossRef]
  7. A. Momose, T. Takeda, and Y. Itai, "X-ray computed tomography for observing biological specimens and organic materials," Rev. Sci. Instrum. 66, 1434-1436 (1995).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  16. K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D 34, A168-A172 (2001).
    [CrossRef]
  17. I. Koyama, A. Momose, J. Wu, T. T. Lwin, and T. Takeda, "Biological imaging by x-ray phase tomography using diffraction enhanced imaging," Jpn. J. Appl. Phys. 44, 8219-8221 (2005).
    [CrossRef]
  18. A. G. Ramm and A. I. Katsevich, The Radon Transform and Local Tomography (CRC Press, 1996).
  19. M. A. Anastasio, D. Shi, F. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2004).
    [CrossRef] [PubMed]
  20. F. Noo, R. Clackdoyle, and J. D. Pack, "A two-step Hilbert transform method for 2D image reconstruction," Phys. Med. Biol. 49, 3903-3923 (2004).
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    [CrossRef]
  23. Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging," J. Phys. D 38, 4259-4271 (2005).
    [CrossRef]
  24. T. E. Gureyev, G. R. Myers, Ya. I. Nesterets, D. M. Paganin, K. M. Pavlov, S. C. Mayo, and S. W. Wilkins, "Stability and locality of amplitude and phase contrast tomographies," Proc. SPIE 6318, 63180V (2006).
  25. Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "General reconstruction formulas for analyzer-based computed tomography," Appl. Phys. Lett. 89, 264103 (2006).
    [CrossRef]
  26. Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Local tomography with a scanning slit aperture, in preparation." (Tim.Gureyev@csiro.au)
  27. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
    [CrossRef] [PubMed]
  28. X. Wu and H. Liu, "A reconstruction formula for soft tissue x-ray phase tomography," J. X-Ray Sci. Technol. 12, 273-279 (2004).
  29. T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
    [CrossRef]
  30. T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two-component samples using in-line hard x-ray images," J. Synchrotron Radiat. 9, 148-153 (2002).
    [CrossRef] [PubMed]

2006 (3)

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
[CrossRef]

T. E. Gureyev, G. R. Myers, Ya. I. Nesterets, D. M. Paganin, K. M. Pavlov, S. C. Mayo, and S. W. Wilkins, "Stability and locality of amplitude and phase contrast tomographies," Proc. SPIE 6318, 63180V (2006).

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "General reconstruction formulas for analyzer-based computed tomography," Appl. Phys. Lett. 89, 264103 (2006).
[CrossRef]

2005 (2)

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging," J. Phys. D 38, 4259-4271 (2005).
[CrossRef]

I. Koyama, A. Momose, J. Wu, T. T. Lwin, and T. Takeda, "Biological imaging by x-ray phase tomography using diffraction enhanced imaging," Jpn. J. Appl. Phys. 44, 8219-8221 (2005).
[CrossRef]

2004 (5)

M. A. Anastasio, D. Shi, F. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2004).
[CrossRef] [PubMed]

F. Noo, R. Clackdoyle, and J. D. Pack, "A two-step Hilbert transform method for 2D image reconstruction," Phys. Med. Biol. 49, 3903-3923 (2004).
[CrossRef] [PubMed]

M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," Proc. SPIE 5535, 361-368 (2004).
[CrossRef]

X. Wu and H. Liu, "A reconstruction formula for soft tissue x-ray phase tomography," J. X-Ray Sci. Technol. 12, 273-279 (2004).

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

2003 (1)

2002 (3)

G. Gbur and W. Wolf, "Diffraction tomography without phase information," Opt. Lett. 27, 1890-1892 (2002).
[CrossRef]

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two-component samples using in-line hard x-ray images," J. Synchrotron Radiat. 9, 148-153 (2002).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

2001 (1)

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D 34, A168-A172 (2001).
[CrossRef]

2000 (1)

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

1999 (1)

A. V. Bronnikov, "Reconstruction formulas in phase-contrast tomography," Opt. Commun. 171, 239-242 (1999).
[CrossRef]

1997 (1)

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, "Observation of microstructure and damage in materials by phase sensitive radiography and tomography," J. Appl. Phys. 81, 5878-5886 (1997).
[CrossRef]

1996 (3)

U. Bonse and F. Busch, "X-ray computed microtomography (μCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
[CrossRef] [PubMed]

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x-rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy x-rays," Nature 384, 49-51 (1996).
[CrossRef]

1995 (1)

A. Momose, T. Takeda, and Y. Itai, "X-ray computed tomography for observing biological specimens and organic materials," Rev. Sci. Instrum. 66, 1434-1436 (1995).
[CrossRef]

1977 (1)

J.-P. Guigay, "Fourier transform analysis of Fresnel diffraction patterns and in-line holograms," Optik (Stuttgart) 49, 121-125 (1977).

1969 (1)

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Anastasio, M. A.

M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," Proc. SPIE 5535, 361-368 (2004).
[CrossRef]

M. A. Anastasio, D. Shi, F. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2004).
[CrossRef] [PubMed]

Baruchel, J.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, "Observation of microstructure and damage in materials by phase sensitive radiography and tomography," J. Appl. Phys. 81, 5878-5886 (1997).
[CrossRef]

Bonse, U.

U. Bonse and F. Busch, "X-ray computed microtomography (μCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Bronnikov, A. V.

A. V. Bronnikov, "Reconstruction formulas in phase-contrast tomography," Opt. Commun. 171, 239-242 (1999).
[CrossRef]

Buffiere, J. Y.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, "Observation of microstructure and damage in materials by phase sensitive radiography and tomography," J. Appl. Phys. 81, 5878-5886 (1997).
[CrossRef]

Busch, F.

U. Bonse and F. Busch, "X-ray computed microtomography (μCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
[CrossRef] [PubMed]

Chapman, L. D.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Clackdoyle, R.

F. Noo, R. Clackdoyle, and J. D. Pack, "A two-step Hilbert transform method for 2D image reconstruction," Phys. Med. Biol. 49, 3903-3923 (2004).
[CrossRef] [PubMed]

Cloetens, P.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, "Observation of microstructure and damage in materials by phase sensitive radiography and tomography," J. Appl. Phys. 81, 5878-5886 (1997).
[CrossRef]

Davis, J. R.

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D 34, A168-A172 (2001).
[CrossRef]

Davis, T. J.

De Carlo, F.

M. A. Anastasio, D. Shi, F. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2004).
[CrossRef] [PubMed]

Dilmanian, F. A.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Gbur, G.

Guigay, J.-P.

J.-P. Guigay, "Fourier transform analysis of Fresnel diffraction patterns and in-line holograms," Optik (Stuttgart) 49, 121-125 (1977).

Gureyev, T. E.

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
[CrossRef]

T. E. Gureyev, G. R. Myers, Ya. I. Nesterets, D. M. Paganin, K. M. Pavlov, S. C. Mayo, and S. W. Wilkins, "Stability and locality of amplitude and phase contrast tomographies," Proc. SPIE 6318, 63180V (2006).

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "General reconstruction formulas for analyzer-based computed tomography," Appl. Phys. Lett. 89, 264103 (2006).
[CrossRef]

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging," J. Phys. D 38, 4259-4271 (2005).
[CrossRef]

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two-component samples using in-line hard x-ray images," J. Synchrotron Radiat. 9, 148-153 (2002).
[CrossRef] [PubMed]

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Local tomography with a scanning slit aperture, in preparation." (Tim.Gureyev@csiro.au)

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections. The Fundamentals of Computerized Tomography (Academic, 1980).

Itai, Y.

A. Momose, T. Takeda, and Y. Itai, "X-ray computed tomography for observing biological specimens and organic materials," Rev. Sci. Instrum. 66, 1434-1436 (1995).
[CrossRef]

Katsevich, A. I.

A. G. Ramm and A. I. Katsevich, The Radon Transform and Local Tomography (CRC Press, 1996).

Kewish, C. M.

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D 34, A168-A172 (2001).
[CrossRef]

Kohn, V.

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy x-rays," Nature 384, 49-51 (1996).
[CrossRef]

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x-rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

Koyama, I.

I. Koyama, A. Momose, J. Wu, T. T. Lwin, and T. Takeda, "Biological imaging by x-ray phase tomography using diffraction enhanced imaging," Jpn. J. Appl. Phys. 44, 8219-8221 (2005).
[CrossRef]

Lengeler, B.

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy x-rays," Nature 384, 49-51 (1996).
[CrossRef]

Liu, H.

X. Wu and H. Liu, "A reconstruction formula for soft tissue x-ray phase tomography," J. X-Ray Sci. Technol. 12, 273-279 (2004).

Lwin, T. T.

I. Koyama, A. Momose, J. Wu, T. T. Lwin, and T. Takeda, "Biological imaging by x-ray phase tomography using diffraction enhanced imaging," Jpn. J. Appl. Phys. 44, 8219-8221 (2005).
[CrossRef]

Mayo, S. C.

T. E. Gureyev, G. R. Myers, Ya. I. Nesterets, D. M. Paganin, K. M. Pavlov, S. C. Mayo, and S. W. Wilkins, "Stability and locality of amplitude and phase contrast tomographies," Proc. SPIE 6318, 63180V (2006).

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

Miller, P. R.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

Momose, A.

I. Koyama, A. Momose, J. Wu, T. T. Lwin, and T. Takeda, "Biological imaging by x-ray phase tomography using diffraction enhanced imaging," Jpn. J. Appl. Phys. 44, 8219-8221 (2005).
[CrossRef]

A. Momose, T. Takeda, and Y. Itai, "X-ray computed tomography for observing biological specimens and organic materials," Rev. Sci. Instrum. 66, 1434-1436 (1995).
[CrossRef]

Morgan, M. J.

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D 34, A168-A172 (2001).
[CrossRef]

Myers, G. R.

T. E. Gureyev, G. R. Myers, Ya. I. Nesterets, D. M. Paganin, K. M. Pavlov, S. C. Mayo, and S. W. Wilkins, "Stability and locality of amplitude and phase contrast tomographies," Proc. SPIE 6318, 63180V (2006).

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

Nesterets, Ya. I.

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
[CrossRef]

T. E. Gureyev, G. R. Myers, Ya. I. Nesterets, D. M. Paganin, K. M. Pavlov, S. C. Mayo, and S. W. Wilkins, "Stability and locality of amplitude and phase contrast tomographies," Proc. SPIE 6318, 63180V (2006).

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "General reconstruction formulas for analyzer-based computed tomography," Appl. Phys. Lett. 89, 264103 (2006).
[CrossRef]

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging," J. Phys. D 38, 4259-4271 (2005).
[CrossRef]

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Local tomography with a scanning slit aperture, in preparation." (Tim.Gureyev@csiro.au)

Noo, F.

F. Noo, R. Clackdoyle, and J. D. Pack, "A two-step Hilbert transform method for 2D image reconstruction," Phys. Med. Biol. 49, 3903-3923 (2004).
[CrossRef] [PubMed]

Orion, I.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Pack, J. D.

F. Noo, R. Clackdoyle, and J. D. Pack, "A two-step Hilbert transform method for 2D image reconstruction," Phys. Med. Biol. 49, 3903-3923 (2004).
[CrossRef] [PubMed]

Paganin, D.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

Paganin, D. M.

T. E. Gureyev, G. R. Myers, Ya. I. Nesterets, D. M. Paganin, K. M. Pavlov, S. C. Mayo, and S. W. Wilkins, "Stability and locality of amplitude and phase contrast tomographies," Proc. SPIE 6318, 63180V (2006).

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
[CrossRef]

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two-component samples using in-line hard x-ray images," J. Synchrotron Radiat. 9, 148-153 (2002).
[CrossRef] [PubMed]

Pan, X.

M. A. Anastasio, D. Shi, F. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2004).
[CrossRef] [PubMed]

Pateyron-Salome, M.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, "Observation of microstructure and damage in materials by phase sensitive radiography and tomography," J. Appl. Phys. 81, 5878-5886 (1997).
[CrossRef]

Pavlov, K. M.

T. E. Gureyev, G. R. Myers, Ya. I. Nesterets, D. M. Paganin, K. M. Pavlov, S. C. Mayo, and S. W. Wilkins, "Stability and locality of amplitude and phase contrast tomographies," Proc. SPIE 6318, 63180V (2006).

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D 34, A168-A172 (2001).
[CrossRef]

Peix, G.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, "Observation of microstructure and damage in materials by phase sensitive radiography and tomography," J. Appl. Phys. 81, 5878-5886 (1997).
[CrossRef]

Peyrin, F.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, "Observation of microstructure and damage in materials by phase sensitive radiography and tomography," J. Appl. Phys. 81, 5878-5886 (1997).
[CrossRef]

Pogany, A.

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
[CrossRef]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

Ramm, A. G.

A. G. Ramm and A. I. Katsevich, The Radon Transform and Local Tomography (CRC Press, 1996).

Raven, C.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x-rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

Ren, B.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Rudolf, D.

G. Schmahl and D. Rudolf, "Proposal for a phase contrast x-ray microscope" in X-ray Microscopy, P.Cheng and G.J.Jan, eds., (Springer-Verlag, 1987), pp. 231-238.
[CrossRef]

Schlenker, M.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, "Observation of microstructure and damage in materials by phase sensitive radiography and tomography," J. Appl. Phys. 81, 5878-5886 (1997).
[CrossRef]

Schmahl, G.

G. Schmahl and D. Rudolf, "Proposal for a phase contrast x-ray microscope" in X-ray Microscopy, P.Cheng and G.J.Jan, eds., (Springer-Verlag, 1987), pp. 231-238.
[CrossRef]

Shi, D.

M. A. Anastasio, D. Shi, F. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2004).
[CrossRef] [PubMed]

M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," Proc. SPIE 5535, 361-368 (2004).
[CrossRef]

Snigirev, A.

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two-component samples using in-line hard x-ray images," J. Synchrotron Radiat. 9, 148-153 (2002).
[CrossRef] [PubMed]

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x-rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy x-rays," Nature 384, 49-51 (1996).
[CrossRef]

Snigireva, I.

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two-component samples using in-line hard x-ray images," J. Synchrotron Radiat. 9, 148-153 (2002).
[CrossRef] [PubMed]

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x-rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy x-rays," Nature 384, 49-51 (1996).
[CrossRef]

Souvorov, A.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x-rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

Spanne, P.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x-rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

Stevenson, A. W.

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two-component samples using in-line hard x-ray images," J. Synchrotron Radiat. 9, 148-153 (2002).
[CrossRef] [PubMed]

Takeda, T.

I. Koyama, A. Momose, J. Wu, T. T. Lwin, and T. Takeda, "Biological imaging by x-ray phase tomography using diffraction enhanced imaging," Jpn. J. Appl. Phys. 44, 8219-8221 (2005).
[CrossRef]

A. Momose, T. Takeda, and Y. Itai, "X-ray computed tomography for observing biological specimens and organic materials," Rev. Sci. Instrum. 66, 1434-1436 (1995).
[CrossRef]

Thomlinson, W. C.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Weitkamp, T.

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two-component samples using in-line hard x-ray images," J. Synchrotron Radiat. 9, 148-153 (2002).
[CrossRef] [PubMed]

Wilkins, S. W.

T. E. Gureyev, G. R. Myers, Ya. I. Nesterets, D. M. Paganin, K. M. Pavlov, S. C. Mayo, and S. W. Wilkins, "Stability and locality of amplitude and phase contrast tomographies," Proc. SPIE 6318, 63180V (2006).

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "General reconstruction formulas for analyzer-based computed tomography," Appl. Phys. Lett. 89, 264103 (2006).
[CrossRef]

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
[CrossRef]

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging," J. Phys. D 38, 4259-4271 (2005).
[CrossRef]

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two-component samples using in-line hard x-ray images," J. Synchrotron Radiat. 9, 148-153 (2002).
[CrossRef] [PubMed]

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Local tomography with a scanning slit aperture, in preparation." (Tim.Gureyev@csiro.au)

Wolf, E.

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Wolf, W.

Wu, J.

I. Koyama, A. Momose, J. Wu, T. T. Lwin, and T. Takeda, "Biological imaging by x-ray phase tomography using diffraction enhanced imaging," Jpn. J. Appl. Phys. 44, 8219-8221 (2005).
[CrossRef]

Wu, X.

X. Wu and H. Liu, "A reconstruction formula for soft tissue x-ray phase tomography," J. X-Ray Sci. Technol. 12, 273-279 (2004).

Wu, X. Y.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Zhong, Z.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x-rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "General reconstruction formulas for analyzer-based computed tomography," Appl. Phys. Lett. 89, 264103 (2006).
[CrossRef]

J. Appl. Phys. (1)

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, "Observation of microstructure and damage in materials by phase sensitive radiography and tomography," J. Appl. Phys. 81, 5878-5886 (1997).
[CrossRef]

J. Microsc. (1)

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

J. Phys. D (2)

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging," J. Phys. D 38, 4259-4271 (2005).
[CrossRef]

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D 34, A168-A172 (2001).
[CrossRef]

J. Synchrotron Radiat. (1)

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two-component samples using in-line hard x-ray images," J. Synchrotron Radiat. 9, 148-153 (2002).
[CrossRef] [PubMed]

J. X-Ray Sci. Technol. (1)

X. Wu and H. Liu, "A reconstruction formula for soft tissue x-ray phase tomography," J. X-Ray Sci. Technol. 12, 273-279 (2004).

Jpn. J. Appl. Phys. (1)

I. Koyama, A. Momose, J. Wu, T. T. Lwin, and T. Takeda, "Biological imaging by x-ray phase tomography using diffraction enhanced imaging," Jpn. J. Appl. Phys. 44, 8219-8221 (2005).
[CrossRef]

Nature (1)

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy x-rays," Nature 384, 49-51 (1996).
[CrossRef]

Opt. Commun. (4)

A. V. Bronnikov, "Reconstruction formulas in phase-contrast tomography," Opt. Commun. 171, 239-242 (1999).
[CrossRef]

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optik (Stuttgart) (1)

J.-P. Guigay, "Fourier transform analysis of Fresnel diffraction patterns and in-line holograms," Optik (Stuttgart) 49, 121-125 (1977).

Phys. Med. Biol. (3)

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

M. A. Anastasio, D. Shi, F. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2004).
[CrossRef] [PubMed]

F. Noo, R. Clackdoyle, and J. D. Pack, "A two-step Hilbert transform method for 2D image reconstruction," Phys. Med. Biol. 49, 3903-3923 (2004).
[CrossRef] [PubMed]

Proc. SPIE (2)

M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," Proc. SPIE 5535, 361-368 (2004).
[CrossRef]

T. E. Gureyev, G. R. Myers, Ya. I. Nesterets, D. M. Paganin, K. M. Pavlov, S. C. Mayo, and S. W. Wilkins, "Stability and locality of amplitude and phase contrast tomographies," Proc. SPIE 6318, 63180V (2006).

Prog. Biophys. Mol. Biol. (1)

U. Bonse and F. Busch, "X-ray computed microtomography (μCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

A. Momose, T. Takeda, and Y. Itai, "X-ray computed tomography for observing biological specimens and organic materials," Rev. Sci. Instrum. 66, 1434-1436 (1995).
[CrossRef]

Other (6)

G. Schmahl and D. Rudolf, "Proposal for a phase contrast x-ray microscope" in X-ray Microscopy, P.Cheng and G.J.Jan, eds., (Springer-Verlag, 1987), pp. 231-238.
[CrossRef]

A. G. Ramm and A. I. Katsevich, The Radon Transform and Local Tomography (CRC Press, 1996).

G. T. Herman, Image Reconstruction from Projections. The Fundamentals of Computerized Tomography (Academic, 1980).

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Local tomography with a scanning slit aperture, in preparation." (Tim.Gureyev@csiro.au)

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Figures (7)

Fig. 1
Fig. 1

Geometry of the CT setup with an added optical system.

Fig. 2
Fig. 2

Geometry of the CT setup with a scanning 1D slit.

Fig. 3
Fig. 3

Distribution of the real part of the refractive index decrement in the phantom.

Fig. 4
Fig. 4

(a) Example of calculated intensity distributions I θ SG ( x ; R ) (thick curve) and I θ PBI ( x ; R ) (thin curve); (b) Corresponding contrast function K θ SG ( x ; R ) at the same view angle as in (a).

Fig. 5
Fig. 5

Contrast sinogram of the vicinity of the ROI in the phantom obtained with a scanning 1D grating.

Fig. 6
Fig. 6

(a) Backprojection of the contrast sinogram of Fig. 5; (b) the result of application of the operator ( R d ) ( 4 π γ λ R 2 ) 1 to the central part of (a).

Fig. 7
Fig. 7

Vertical cross-section along the dashed square in Fig. 6b, showing the reconstructed distribution of the real part of the refractive index in the ROI.

Tables (1)

Tables Icon

Table 1 Chemical Composition, Density and X-Ray Refractive Indexes (at E = 25 keV ) for Materials Used in the Simulated Phantom

Equations (57)

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

μ θ ( x , y ) = k ( P θ β ) ( x , y ) ,
φ θ ( x , y ) = k ( P θ Δ ) ( x , y ) ,
β ( x , y , z ) = k 1 0 π exp { i 2 π [ ξ ( x sin θ + z cos θ ) + η y ] } ( F μ θ ) ( ξ , η ) ξ d ξ d η d θ ,
Δ ( x , y , z ) = k 1 0 π exp { i 2 π [ ξ ( x sin θ + z cos θ ) + η y ] } ( F φ θ ) ( ξ , η ) ξ d ξ d η d θ .
u θ det ( x , y ) = u θ obj ( x ̃ , y ̃ ) G ( x x ̃ , y y ̃ ) d x ̃ d y ̃ ,
μ θ ( x , y ) 1 ,
φ θ ( x , y ) φ θ ( x ̃ , y ̃ ) 1 ,
exp [ μ θ ( x 1 , y 1 ) μ θ ( x 2 , y 2 ) + i φ θ ( x 1 , y 1 ) i φ θ ( x 2 , y 2 ) ] 1 μ θ ( x 1 , y 1 ) μ θ ( x 2 , y 2 ) + i φ θ ( x 1 , y 1 ) i φ θ ( x 2 , y 2 ) .
( F I θ det ) ( ξ , η ) = I in { F G 2 ( 0 , 0 ) δ ( ξ , η ) ( F μ θ ) ( ξ , η ) [ ( F G * ) ( 0 , 0 ) ( F G ) ( ξ , η ) + ( F G ) ( 0 , 0 ) ( F G * ) ( ξ , η ) ] + ( F φ θ ) ( ξ , η ) [ i ( F G * ) ( 0 , 0 ) ( F G ) ( ξ , η ) i ( F G ) ( 0 , 0 ) ( F G * ) ( ξ , η ) ] } .
K θ ( x , y ) 1 I θ det ( x , y ) [ I in T ( 0 , 0 ) 2 ] .
( F K θ ) ( ξ , η ) = 2 T ̃ a ( ξ , η ) ( F μ θ ) ( ξ , η ) + 2 T ̃ p ( ξ , η ) ( F φ θ ) ( ξ , η ) ,
T ̃ a ( ξ , η ) = 1 2 { T ̃ ( ξ , η ) + T ̃ * ( ξ , η ) } ,
T ̃ p ( ξ , η ) = i 2 { T ̃ ( ξ , η ) + T ̃ * ( ξ , η ) } .
( F μ θ ) ( ξ , η ) = T ̃ 2 p ( ξ , η ) ( F K θ , 1 ) ( ξ , η ) T ̃ 1 p ( ξ , η ) ( F K θ , 2 ) ( ξ , η ) 2 [ T ̃ 1 a ( ξ , η ) T ̃ 2 p ( ξ , η ) T ̃ 2 a ( ξ , η ) T ̃ 1 p ( ξ , η ) ] ,
( F φ θ ) ( ξ , η ) = T ̃ 2 a ( ξ , η ) ( F K θ , 1 ) ( ξ , η ) + T ̃ 1 a ( ξ , η ) ( F K θ , 2 ) ( ξ , η ) 2 [ T ̃ 1 a ( ξ , η ) T ̃ 2 p ( ξ , η ) T ̃ 2 a ( ξ , η ) T ̃ 1 p ( ξ , η ) ] ,
β ( x , y , z ) = 0 π exp { i 2 π [ ξ ( x sin θ + z cos θ ) + η y ] } × T ̃ 2 p ( F K θ , 1 ) T ̃ 1 p ( F K θ , 2 ) 2 k ( T ̃ 1 a T ̃ 2 p T ̃ 1 p T ̃ 2 a ) ( ξ , η ) ξ d ξ d η d θ ,
Δ ( x , y , z ) = 0 π exp { i 2 π [ ξ ( x sin θ + z cos θ ) + η y ] } × T ̃ 2 a ( F K θ , 1 ) T ̃ 1 a ( F K θ , 2 ) 2 k ( T ̃ 1 a T ̃ 2 p T ̃ 1 p T ̃ 2 a ) ( ξ , η ) ξ d ξ d η d θ .
T ̃ TIE a ( ξ , η ) = cos [ π λ R ( ξ 2 + η 2 ) ] ,
T ̃ TIE p ( ξ , η ) = sin [ π λ R ( ξ 2 + η 2 ) ] .
Δ ( x , y , z ) = 0 π exp { i 2 π [ ξ ( x sin θ + z cos θ ) + η y ] } ξ 2 k T ̃ p ( ξ , η ) ( F K θ ) ( ξ , η ) d ξ d η d θ .
T ̃ p ( ξ , η ) = p ξ ,
Δ ( x , y , z ) = ( 2 p k ) 1 0 π K θ ( x sin θ + z cos θ , y ) d θ .
( F I θ det ) ( ξ , η ; R ; x ̃ ) = I in + + exp ( i 2 π [ x ξ + y η ] ) q θ ( x + λ R ξ 2 , y + λ R η 2 ) q θ * ( x λ R ξ 2 , y λ R η 2 ) χ d ( x x ̃ + λ R ξ 2 ) χ d * ( x x ̃ λ R ξ 2 ) d x d y ,
( F I θ SS ) ( ξ , η ; R ) d 1 + ( F I θ det ) ( ξ , η ; R ; x ̃ ) d x ̃ = ( F I θ PBI ) ( ξ , η ; R ) Λ Ξ ( ξ ) ,
Λ Ξ ( ξ ) d 1 + χ d ( x ) χ d ( x + λ R ξ ) d x = { 1 ξ Ξ ξ < Ξ 0 ξ Ξ
K θ SS ( x , y ; R ) [ I θ PBI ( x , y ; R ) I θ SS ( x , y ; R ) ] I in ,
( F K θ SS ) ( ξ , η ; R ) = [ ( F I θ PBI ) ( ξ , η ; R ) I in ] ξ Ξ , when ξ < Ξ .
( F I θ PBI ) ( ξ , η ; R ) = I in { δ ( ξ , η ) 2 cos [ π λ R ( ξ 2 + η 2 ) ] ( F μ θ ) ( ξ , η ) + 2 sin [ π λ R ( ξ 2 + η 2 ) ] ( F φ θ ) ( ξ , η ) } .
( F I θ PBI ) ( ξ , η ; R ) = I in δ ( ξ , η ) 2 I in ( F μ θ ) ( ξ , η ) [ 1 + γ π λ R ( ξ 2 + η 2 ) ] .
( F K θ SS ) ( ξ , η ; R ) = 2 ( F μ θ ) ( ξ , η ) [ 1 + γ π λ R ( ξ 2 + η 2 ) ] ξ Ξ , ξ < Ξ .
β ( x , y , z ) = d 4 π R 0 π exp { i 2 π [ ξ ( x sin θ + z cos θ ) + η y ] } F K θ SS ( ξ , η ; R ) 1 + γ π λ R ( ξ 2 + η 2 ) d ξ d η d θ ,
( R d ) ( 4 π γ λ R 2 ) β ( x , y , z ) = 0 π exp { i 2 π [ ξ ( x sin θ + z cos θ ) + η y ] } ( F K θ SS ) ( ξ , η ; R ) d ξ d η d θ .
( R d ) ( 4 π γ λ R 2 ) β ( x , y , z ) = 0 π K θ SS ( x sin θ + z cos θ , y ; R ) d θ .
( R d ) χ 2 A ( x , z ) ( 4 π γ λ R 2 ) β ( x , y , z ) = χ 2 A ( x , z ) 0 π χ 1 A ( x sin θ + z cos θ ) K θ SS ( x sin θ + z cos θ , y ; R ) d θ ,
β ( x , y , z ) = ( d R ) ( 4 π γ λ R 2 ) 1 0 π K θ SS ( x sin θ + z cos θ , y ; R ) d θ .
( 4 π R d ) β ( x , y , z ) 0 π K θ SS ( x sin θ + z cos θ , y ; R ) d θ .
( F I θ SG ) ( ξ , η ; R ) d 1 d + d ( F I ̃ θ det ) ( ξ , η ; R ; x ̃ ) d x ̃ = ( F I θ PBI ) ( ξ , η ; R ) m = M M Λ Ξ ( ξ + 2 m Ξ ) ,
u θ PBI ( x ; R ; A ) = u in + exp ( i 2 π x ξ ) exp ( i π λ R ξ 2 ) ( F Q θ ) ( ξ ) d ξ ,
( F I θ PBI ) ( ξ ; R ; A ) = I in + exp [ i 2 π λ R ξ ζ ] ( F Q θ ) ( ζ + ξ 2 ) ( F Q θ ) * ( ζ ξ 2 ) d ζ
= I in + exp ( i 2 π x ξ ) Q θ ( x + λ R ξ 2 ) Q θ * ( x λ R ξ 2 ) d x ,
( F I θ PBI ) ( ξ ; R ; A ) I in = ( F I FF PBI ) ( ξ ; R ; A ) + exp ( i 2 π x ξ ) [ M θ ( x + λ R ξ 2 ) χ A * ( x λ R ξ 2 ) + M θ * ( x λ R ξ 2 ) χ A ( x + λ R ξ 2 ) ] d x + i + exp ( i 2 π x ξ ) [ Φ θ ( x + λ R ξ 2 ) χ A * ( x λ R ξ 2 ) Φ θ * ( x λ R ξ 2 ) χ A ( x + λ R ξ 2 ) ] d x ,
( F I FF PBI ) ( ξ ; R ; A ) + exp ( i 2 π x ξ ) χ A ( x + λ R ξ 2 ) χ A * ( x λ R ξ 2 ) d x
F ( ξ ) = + exp ( i 2 π x ξ ) M θ ( x + λ R ξ 2 ) χ A * ( x λ R ξ 2 ) d x = exp ( i π λ R ξ 2 ) + exp [ i 2 π λ R ξ ζ ] ( F M θ ) ( ζ + ξ ) ( F χ A ) * ( ζ ) d ζ .
( F M θ ) ( ζ + ξ ) = n = 0 ζ n n ! ( F M θ ) ( n ) ( ξ ) ,
F ( ξ ) = exp ( i π λ R ξ 2 ) n = 0 1 n ! ( 2 π i ) n ( F M θ ) ( n ) ( ξ ) ( χ A * ) ( n ) ( λ R ξ ) .
F ( ξ ) = exp ( i π λ R ξ 2 ) ( F M θ ) ( ξ ) .
( F I θ PBI ) ( ξ ; R ; A ) I in = ( F I FF PBI ) ( ξ ; R ; A ) [ exp ( i π λ R ξ 2 ) ( F M θ ) ( ξ ) + exp ( i π λ R ξ 2 ) ( F M θ ) * ( ξ ) ] + i [ exp ( i π λ R ξ 2 ) ( F Φ θ ) ( ξ ) exp ( i π λ R ξ 2 ) ( F Φ θ ) * ( ξ ) ] .
( F I θ PBI ) ( ξ ; R ; A ) = I in { ( F I FF PBI ) ( ξ ; R ; A ) 2 cos ( π λ R ξ 2 ) ( F M θ ) ( ξ ) + 2 sin ( π λ R ξ 2 ) ( F Φ θ ) ( ξ ) } .
I θ PBI ( x ; R ; A ) = I in { I FF PBI ( x ; R ; A ) 2 [ 1 γ λ R ( 4 π ) 2 ] ( μ θ χ A ) ( x ) } .
I θ SS ( x ; R ; A ) = I in Λ ̂ Ξ { I FF PBI ( x ; R ; A ) 2 [ 1 γ λ R ( 4 π ) 2 ] ( μ θ χ A ) ( x ) } ,
( Λ ̂ Ξ f ) ( x ) = Ξ A A f ( y ) sinc 2 [ π Ξ ( x y ) ] d y + Ξ A f ( y ) sinc 2 [ π Ξ ( x y ) ] d y + Ξ A f ( y ) sinc 2 [ π Ξ ( x y ) ] d y .
A f ( y ) sinc 2 [ π Ξ ( x y ) ] d y A f ( y ) [ π Ξ ( x y ) ] 2 d y
f max ( π 2 Ξ 2 A A ) ;
I err 2 f max ( π 2 Ξ A A ) .
I θ SS ( x ; R ; A ) I θ SS ( x ; R ) 4 I in C ( χ A , μ θ ) ( π 2 Ξ A A ) ,
C ( χ A , μ θ ) { 1 + ( R k ) max [ χ A , ( χ A ) 2 ] } { 1 + γ max μ θ + γ ( R k ) max [ μ θ 2 , 2 μ θ ] } .
I θ SS ( x ; R ; A ) I θ SS ( x ; R ) ( 16 + 8 max φ θ ) I in ( π 2 Ξ A A ) .

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