Abstract

We investigate a variant of the reconstruction technique for the in-line X-ray phase-contrast tomography data. This technique uses a newly introduced quantity, which represents a particular combination of the real and imaginary parts of the complex refractive index n. This quantity coincides with the real part of (1-n) in the case of objects having negligible absorption. The advantage of the proposed approach is in the significantly simplified form of the reconstruction algorithm for the introduced quantity. As demonstrated by our numerical experiments, the newly introduced quantity can be predictably associated with a particular refractive index.

© 2012 OSA

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2012 (2)

M. Langer, P. Cloetens, A. Pacureanu, and F. Peyrin, “X-ray in-line phase tomography of multimaterial objects,” Opt. Lett.37(11), 2151–2153 (2012).
[CrossRef] [PubMed]

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic X-ray in-line phase-contrast tomography for soft tissue,” Europhys. Lett.98(1), 14001 (2012).
[CrossRef]

2011 (3)

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol.56(23), 7353–7369 (2011).
[CrossRef] [PubMed]

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. A84(2), 023808 (2011).
[CrossRef]

T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE8141, 81410B, 81410B-14 (2011).
[CrossRef]

2010 (2)

M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express18(7), 6423–6436 (2010).
[CrossRef] [PubMed]

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process.19(9), 2428–2436 (2010).
[CrossRef] [PubMed]

2007 (3)

2006 (3)

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett.89(3), 034102 (2006).
[CrossRef]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE6318, 63180S, 63180S-10 (2006).
[CrossRef]

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

2005 (2)

2004 (3)

D. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Y. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc.214(3), 315–327 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93(6), 068103 (2004).
[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(1), 121–144 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (2)

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 Appl. Phys.34(10A), A168–A172 (2001).
[CrossRef]

2000 (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(4), 933–946 (2000).
[CrossRef] [PubMed]

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys.33(13), 1596–1605 (2000).
[CrossRef]

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun.175(4–6), 329–336 (2000).
[CrossRef]

1999 (1)

A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun.171(4-6), 239–244 (1999).
[CrossRef]

1998 (2)

T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun.147(4-6), 229–232 (1998).
[CrossRef]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80(12), 2586–2589 (1998).
[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(9), 5878–5886 (1997).
[CrossRef]

1996 (2)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (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(13), 1826–1828 (1996).
[CrossRef]

1995 (2)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum.66(12), 5486–5492 (1995).
[CrossRef]

A. Momose, T. Takeda, and Y. Itai, “X-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum.66(2), 1434–1436 (1995).
[CrossRef]

1985 (1)

G. Harding, J. M. Kosanetzky, and U. Neitzel, “Elastic scatter computed tomography,” Phys. Med. Biol.30(2), 183–186 (1985).
[CrossRef] [PubMed]

1983 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun.1(4), 153–156 (1969).
[CrossRef]

Anastasio, M. A.

Barty, A.

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun.175(4–6), 329–336 (2000).
[CrossRef]

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(9), 5878–5886 (1997).
[CrossRef]

Beltran, M. A.

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol.56(23), 7353–7369 (2011).
[CrossRef] [PubMed]

M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express18(7), 6423–6436 (2010).
[CrossRef] [PubMed]

Bronnikov, A. V.

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A19(3), 472–480 (2002).
[CrossRef] [PubMed]

A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun.171(4-6), 239–244 (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(9), 5878–5886 (1997).
[CrossRef]

Bunk, O.

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE6318, 63180S, 63180S-10 (2006).
[CrossRef]

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(4), 933–946 (2000).
[CrossRef] [PubMed]

Cloetens, P.

M. Langer, P. Cloetens, A. Pacureanu, and F. Peyrin, “X-ray in-line phase tomography of multimaterial objects,” Opt. Lett.37(11), 2151–2153 (2012).
[CrossRef] [PubMed]

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process.19(9), 2428–2436 (2010).
[CrossRef] [PubMed]

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(9), 5878–5886 (1997).
[CrossRef]

David, C.

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE6318, 63180S, 63180S-10 (2006).
[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 Appl. Phys.34(10A), A168–A172 (2001).
[CrossRef]

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys.33(13), 1596–1605 (2000).
[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(1), 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(4), 933–946 (2000).
[CrossRef] [PubMed]

Fouras, A.

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol.56(23), 7353–7369 (2011).
[CrossRef] [PubMed]

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

Gbur, G.

Gu, M.

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic X-ray in-line phase-contrast tomography for soft tissue,” Europhys. Lett.98(1), 14001 (2012).
[CrossRef]

Guo, X.

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic X-ray in-line phase-contrast tomography for soft tissue,” Europhys. Lett.98(1), 14001 (2012).
[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. A84(2), 023808 (2011).
[CrossRef]

T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE8141, 81410B, 81410B-14 (2011).
[CrossRef]

G. R. Myers, T. E. Gureyev, and D. M. Paganin, “Stability of phase-contrast tomography,” J. Opt. Soc. Am. A24(9), 2516–2526 (2007).
[CrossRef] [PubMed]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A76(4), 045804 (2007).
[CrossRef]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett.89(3), 034102 (2006).
[CrossRef]

D. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Y. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc.214(3), 315–327 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93(6), 068103 (2004).
[CrossRef] [PubMed]

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. Express11(19), 2289–2302 (2003).
[CrossRef] [PubMed]

T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun.147(4-6), 229–232 (1998).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

Harding, G.

G. Harding, J. M. Kosanetzky, and U. Neitzel, “Elastic scatter computed tomography,” Phys. Med. Biol.30(2), 183–186 (1985).
[CrossRef] [PubMed]

Hattori, T.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

Hooper, S. B.

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol.56(23), 7353–7369 (2011).
[CrossRef] [PubMed]

Huang, S.

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic X-ray in-line phase-contrast tomography for soft tissue,” Europhys. Lett.98(1), 14001 (2012).
[CrossRef]

Huang, Y.

Itai, Y.

A. Momose, T. Takeda, and Y. Itai, “X-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum.66(2), 1434–1436 (1995).
[CrossRef]

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 Appl. Phys.34(10A), A168–A172 (2001).
[CrossRef]

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys.33(13), 1596–1605 (2000).
[CrossRef]

Kitchen, M. J.

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol.56(23), 7353–7369 (2011).
[CrossRef] [PubMed]

M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express18(7), 6423–6436 (2010).
[CrossRef] [PubMed]

Kohn, V.

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(13), 1826–1828 (1996).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum.66(12), 5486–5492 (1995).
[CrossRef]

Kosanetzky, J. M.

G. Harding, J. M. Kosanetzky, and U. Neitzel, “Elastic scatter computed tomography,” Phys. Med. Biol.30(2), 183–186 (1985).
[CrossRef] [PubMed]

Kottler, C.

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE6318, 63180S, 63180S-10 (2006).
[CrossRef]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum.66(12), 5486–5492 (1995).
[CrossRef]

Langer, M.

M. Langer, P. Cloetens, A. Pacureanu, and F. Peyrin, “X-ray in-line phase tomography of multimaterial objects,” Opt. Lett.37(11), 2151–2153 (2012).
[CrossRef] [PubMed]

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process.19(9), 2428–2436 (2010).
[CrossRef] [PubMed]

Liu, B.

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic X-ray in-line phase-contrast tomography for soft tissue,” Europhys. Lett.98(1), 14001 (2012).
[CrossRef]

Liu, H.

Liu, X.

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic X-ray in-line phase-contrast tomography for soft tissue,” Europhys. Lett.98(1), 14001 (2012).
[CrossRef]

Mayo, S. C.

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A76(4), 045804 (2007).
[CrossRef]

D. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Y. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc.214(3), 315–327 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93(6), 068103 (2004).
[CrossRef] [PubMed]

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. Express11(19), 2289–2302 (2003).
[CrossRef] [PubMed]

Miller, P. R.

Momose, A.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

A. Momose, T. Takeda, and Y. Itai, “X-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum.66(2), 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 Appl. Phys.34(10A), A168–A172 (2001).
[CrossRef]

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys.33(13), 1596–1605 (2000).
[CrossRef]

Myers, G. R.

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A76(4), 045804 (2007).
[CrossRef]

G. R. Myers, T. E. Gureyev, and D. M. Paganin, “Stability of phase-contrast tomography,” J. Opt. Soc. Am. A24(9), 2516–2526 (2007).
[CrossRef] [PubMed]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett.89(3), 034102 (2006).
[CrossRef]

Neitzel, U.

G. Harding, J. M. Kosanetzky, and U. Neitzel, “Elastic scatter computed tomography,” Phys. Med. Biol.30(2), 183–186 (1985).
[CrossRef] [PubMed]

Nesterets, Y.

T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE8141, 81410B, 81410B-14 (2011).
[CrossRef]

Nesterets, Y. I.

D. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Y. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc.214(3), 315–327 (2004).
[CrossRef] [PubMed]

Nesterets, Ya. I.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett.89(3), 034102 (2006).
[CrossRef]

Ni, C.

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic X-ray in-line phase-contrast tomography for soft tissue,” Europhys. Lett.98(1), 14001 (2012).
[CrossRef]

Nugent, K.

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun.175(4–6), 329–336 (2000).
[CrossRef]

Nugent, K. A.

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

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(4), 933–946 (2000).
[CrossRef] [PubMed]

Pacureanu, A.

Paganin, D.

D. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Y. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc.214(3), 315–327 (2004).
[CrossRef] [PubMed]

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. Express11(19), 2289–2302 (2003).
[CrossRef] [PubMed]

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun.175(4–6), 329–336 (2000).
[CrossRef]

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

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. A84(2), 023808 (2011).
[CrossRef]

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol.56(23), 7353–7369 (2011).
[CrossRef] [PubMed]

M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express18(7), 6423–6436 (2010).
[CrossRef] [PubMed]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A76(4), 045804 (2007).
[CrossRef]

G. R. Myers, T. E. Gureyev, and D. M. Paganin, “Stability of phase-contrast tomography,” J. Opt. Soc. Am. A24(9), 2516–2526 (2007).
[CrossRef] [PubMed]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett.89(3), 034102 (2006).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93(6), 068103 (2004).
[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(1), 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(9), 5878–5886 (1997).
[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. A84(2), 023808 (2011).
[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 Appl. Phys.34(10A), A168–A172 (2001).
[CrossRef]

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys.33(13), 1596–1605 (2000).
[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(9), 5878–5886 (1997).
[CrossRef]

Peyrin, F.

M. Langer, P. Cloetens, A. Pacureanu, and F. Peyrin, “X-ray in-line phase tomography of multimaterial objects,” Opt. Lett.37(11), 2151–2153 (2012).
[CrossRef] [PubMed]

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process.19(9), 2428–2436 (2010).
[CrossRef] [PubMed]

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(9), 5878–5886 (1997).
[CrossRef]

Pfeiffer, F.

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE6318, 63180S, 63180S-10 (2006).
[CrossRef]

Pogany, A.

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. Express11(19), 2289–2302 (2003).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

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(13), 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(4), 933–946 (2000).
[CrossRef] [PubMed]

Reser, D. H.

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol.56(23), 7353–7369 (2011).
[CrossRef] [PubMed]

Roberts, A.

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun.175(4–6), 329–336 (2000).
[CrossRef]

Sakellariou, A.

T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE8141, 81410B, 81410B-14 (2011).
[CrossRef]

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum.66(12), 5486–5492 (1995).
[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(9), 5878–5886 (1997).
[CrossRef]

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. A84(2), 023808 (2011).
[CrossRef]

Shi, D.

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, “Image reconstruction in spherical-wave intensity diffraction tomography,” J. Opt. Soc. Am. A22(12), 2651–2661 (2005).
[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(1), 121–144 (2004).
[CrossRef] [PubMed]

Siu, K. K. W.

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol.56(23), 7353–7369 (2011).
[CrossRef] [PubMed]

Snigirev, 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(13), 1826–1828 (1996).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum.66(12), 5486–5492 (1995).
[CrossRef]

Snigireva, I.

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(13), 1826–1828 (1996).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum.66(12), 5486–5492 (1995).
[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(13), 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(13), 1826–1828 (1996).
[CrossRef]

Stevenson, A. W.

T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE8141, 81410B, 81410B-14 (2011).
[CrossRef]

D. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Y. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc.214(3), 315–327 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93(6), 068103 (2004).
[CrossRef] [PubMed]

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. Express11(19), 2289–2302 (2003).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

Suzuki, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

Takeda, T.

A. Momose, T. Takeda, and Y. Itai, “X-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum.66(2), 1434–1436 (1995).
[CrossRef]

Takeda, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

Taylor, J. A.

T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE8141, 81410B, 81410B-14 (2011).
[CrossRef]

Teague, M. R.

Ternovski, D.

T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE8141, 81410B, 81410B-14 (2011).
[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(4), 933–946 (2000).
[CrossRef] [PubMed]

Thompson, D.

T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE8141, 81410B, 81410B-14 (2011).
[CrossRef]

Uesugi, K.

Weitkamp, T.

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE6318, 63180S, 63180S-10 (2006).
[CrossRef]

Wilkins, S. W.

T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE8141, 81410B, 81410B-14 (2011).
[CrossRef]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A76(4), 045804 (2007).
[CrossRef]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett.89(3), 034102 (2006).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93(6), 068103 (2004).
[CrossRef] [PubMed]

D. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Y. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc.214(3), 315–327 (2004).
[CrossRef] [PubMed]

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. Express11(19), 2289–2302 (2003).
[CrossRef] [PubMed]

T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun.147(4-6), 229–232 (1998).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

Wolf, E.

G. Gbur and E. Wolf, “Diffraction tomography without phase information,” Opt. Lett.27(21), 1890–1892 (2002).
[CrossRef] [PubMed]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun.1(4), 153–156 (1969).
[CrossRef]

Wu, X.

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(4), 933–946 (2000).
[CrossRef] [PubMed]

Yan, A.

Yashiro, W.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

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(4), 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(13), 1826–1828 (1996).
[CrossRef]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett.89(3), 034102 (2006).
[CrossRef]

Europhys. Lett. (1)

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic X-ray in-line phase-contrast tomography for soft tissue,” Europhys. Lett.98(1), 14001 (2012).
[CrossRef]

IEEE Trans. Image Process. (1)

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process.19(9), 2428–2436 (2010).
[CrossRef] [PubMed]

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(9), 5878–5886 (1997).
[CrossRef]

J. Microsc. (1)

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Figures (6)

Fig. 1
Fig. 1

Generic layout of the in-line CT experimental setup.

Fig. 2
Fig. 2

The structure of the object—and its bounding cube—that was simulated. It consisted of twelve spheres arranged in three layers of four, each parallel to the base. The bottom and top layers had the spheres situated in the corners, and the middle layer had them in the centre of each face. The lightness of the spheres decreases with increasing β for the constant Δ objects, Δ for the constant β objects, and β and Δ for the objects with both varying.

Fig. 3
Fig. 3

The reconstructed values using Δ ˜ for the three objects with constant β (closed circles, squares and diamond symbols). The dashed line is Δ ˜ =Δ . The uncertainty is taken to be 1 standard deviation of the values in the centre of the spheres in the reconstructed slices. Also shown are the curves of best fit Δ ˜ =Δexp{2k <Pβ> β β} for closed circles, squares and diamond symbols, which parameters are shown in Table 2. The triangular symbols show the result of a “homogenous” TIE reconstruction.

Fig. 4
Fig. 4

The reconstructed values using Δ ˜ for the three objects with constant β and 5% Poisson noise. The dashed line is Δ ˜ =Δ . The uncertainty is taken to be 1 standard deviation of the values in the centre of the spheres in the reconstructed slices. Also shown are the same curves of best fit as in Fig. 3.

Fig. 5
Fig. 5

Δ ˜ (triangular, open circles and inverted triangular symbols) and the reconstructed values of Δ (see Eq. (22)) (closed circles, diamond and square symbols) using Δ ˜ for the three objects with constant Δ. The uncertainty is taken to be 1 standard deviation of the values in the centre of the spheres in the reconstructed slices. Also shown are the curves of best fit Δ ˜ =Δexp{2k <Pβ> β β} , which parameters are given in Table 3.

Fig. 6
Fig. 6

(a) Typical projection with an object-to-detector projection distance of 10 cm; (b) reconstructed slice of Δ through the bottom layer of spheres and (c) a cross-section indicated by the solid line through the bottom row in (a). This particular object had spheres with constant Δ=5× 10 7 and β varying from 1 × 10−9 to 2 × 10−8.

Tables (3)

Tables Icon

Table 1 The Components of Refractive Indices of Certain Real Materials at a Wavelength of 1 Å [35].

Tables Icon

Table 2 The Best-Fit Parameters Obtained from Figs. 3 and 4.

Tables Icon

Table 3 The Best-Fit Parameters Obtained from Fig. 5.

Equations (22)

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φ θ (x',y)=k( P θ Δ)(x',y),
I θ (x',y)= I in exp{2k( P θ β)(x',y)},
( P θ f)(x',y)= f(x,y,z)δ(x'xsinθzcosθ)dxdz
I θ R ( x ,y)= I θ ( x ,y)(R/k) [ I θ ( x ,y) φ θ ( x ,y)],
ψ θ ( x ,y)= I θ ( x ,y) φ θ ( x ,y).
2 ψ θ ( x ,y)=(k/R)[ I θ R ( x ,y) I θ ( x ,y)].
2 φ θ ( x ,y)= [ I θ 1 ( x ,y) ψ θ ( x ,y)],
f(x,y,z)=[(F P θ f)( ξ ,η)](x,y,z),
(Fh)( ξ ,η)= exp[i2π( x ξ +yη)]h( x ,y)d x dy ,
[g](x,y,z)= 0 π exp{i2π[ ξ (xsinθ+zcosθ)+ηy]}g( ξ ,η,θ)| ξ |d ξ dηdθ
Δ(x,y,z)=(1/k)[(F φ θ )( ξ ,η)](x,y,z),
β(x,y,z)=(1/2k){F[ln( I θ / I in )]( ξ ,η)}(x,y,z).
Δ(x,y,z)=[1/(4 π 2 R)]{F[ln( I θ R / I in )]( ξ ,η)/( ξ 2 + η 2 )}(x,y,z).
2 Δ(x,y,z)= R 1 {F[ln( I θ R / I in )]( ξ ,η)}(x,y,z),
Δ(x,y,z)= R 1 2 {F[ln( I θ R / I in )]( ξ ,η)}(x,y,z).
ψ θ (x',y)=k I in ( P θ Δ ˜ )(x',y),
Δ ˜ (x,y,z)= 1 4 π 2 R { F[( I θ R I θ )]( ξ ,η) I in ( ξ 2 + η 2 ) }(x,y,z).
Δ ˜ (x,y,z)= R 1 2 {[F( I θ R / I in )( ξ ,η)][F( I θ / I in )( ξ ,η)]}(x,y,z).
Δ ˜ (x,y,z)= 1 4 π 2 { F( [exp{2k( P θ β)(x',y)} ( P θ Δ)( x ,y)])( ξ ,η) ξ 2 + η 2 }(x,y,z).
2 ( P θ Δ ˜ )(x',y)= [exp{2k( P θ β)(x',y)} ( P θ Δ)( x ,y)] 2 [exp{2k( P θ β)(x',y)}( P θ Δ)( x ,y)].
( P θ Δ ˜ )(x',y)exp{2k<Pβ>(y)}( P θ Δ)( x ,y)=( P θ [exp{2k<Pβ>(y)}Δ])( x ,y),
Δ(x,y,z) Δ ˜ (x,y,z)exp{2k<Pβ>(y)}.

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