Abstract

Phase-contrast tomography (PCT) allows three-dimensional imaging of objects that display insufficient contrast for conventional absorption-based tomography. We prove that PCT is stable with respect to high-frequency noise in experimental phase-contrast data, unlike conventional tomography, which is known to be mildly unstable. We use known properties of the three-dimensional x-ray transform and transport-of-intensity equation to construct a matrix representation of the forward PCT operator. We then invert this formula to show that, under natural boundary conditions, the PCT reconstruction operator exists and leads to a unique solution. We show that the singular values sn of the reconstruction operator have asymptotic behavior sn=O(n32), guaranteeing the mathematical stability of the reconstruction process.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Wells, J. R. Davis, and M. J. Morgan, "Computed tomography," Mater. Forum 18, 111-133 (1994).
  2. F. Natterer, The Mathematics of Computerized Tomography (Society for Industrial and Applied Mathematics, 2001).
  3. R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53, 23-26 (2000).
  4. D. M. Paganin, Coherent X-Ray Optics (Oxford U. Press, 2006).
  5. M. R. Teague, "Deterministic phase retrieval: a Green's function solution," J. Opt. Soc. Am. 73, 1434-1441 (1983).
  6. T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942-1946 (1995).
  7. S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. M. Paganin, A. Pogany, A. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
  8. A. Groso, R. Abela, and M. Stampanoni, "Implementation of a fast method for high resolution phase contrast tomography," Opt. Express 14, 8103-8110 (2006).
    [CrossRef]
  9. A. J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Probl. 2, 161-183 (1986).
    [CrossRef]
  10. 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]
  11. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
    [CrossRef]
  12. E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
    [CrossRef]
  13. G. Gbur and E. Wolf, "Diffraction tomography without phase information," Opt. Lett. 27, 1890-1892 (2002).
  14. M. A. Anastasio, D. Shi, D. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2003).
    [CrossRef]
  15. M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," Proc. SPIE 5535, 361-368 (2004).
  16. M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Phase-contrast tomography using incident spherical waves," Proc. SPIE 5535, 724-732 (2004).
  17. A. V. Bronnikov, "Reconstruction formulas in phase-contrast tomography," Opt. Commun. 171, 239-242 (1999).
    [CrossRef]
  18. A. V. Bronnikov, "Theory of quantitative phase-contrast computed tomography," J. Opt. Soc. Am. A 19, 472-480 (2002).
  19. X. Wu and H. Liu, "X-ray cone-beam phase tomography formulas based on phase-attenuation duality," Opt. Express 13, 6000-6014 (2005).
    [CrossRef]
  20. P. Maass, "The X-ray transform: singular value decomposition and resolution," Inverse Probl. 3, 729-741 (1987).
    [CrossRef]
  21. T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
    [CrossRef]
  22. 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, 1932-1940 (1995).
  23. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed. (Oxford U. Press, 1979), pp. 7-8.
  24. I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators (American Mathematical Society, 1969).
  25. T. E. Gureyev and S. W. Wilkins, "On x-ray phase imaging with a point source," J. Opt. Soc. Am. A 15, 579-585 (1998).
  26. D. M. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy. III. The effects of noise," J. Microsc. 214, 51-61 (2004).
    [CrossRef]
  27. T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
    [CrossRef]
  28. D. M. 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]
  29. A. Erdelyi, W. Magnus, R. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953).
  30. W. W. Bell, Special Functions for Scientists and Engineers (Dover, 1968) p. 199.

2006

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

A. Groso, R. Abela, and M. Stampanoni, "Implementation of a fast method for high resolution phase contrast tomography," Opt. Express 14, 8103-8110 (2006).
[CrossRef]

2005

2004

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

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

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

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Phase-contrast tomography using incident spherical waves," Proc. SPIE 5535, 724-732 (2004).

2003

M. A. Anastasio, D. Shi, D. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2003).
[CrossRef]

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

2002

A. V. Bronnikov, "Theory of quantitative phase-contrast computed tomography," J. Opt. Soc. Am. A 19, 472-480 (2002).

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

D. M. 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]

2000

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53, 23-26 (2000).

1999

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

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

1998

1996

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]

1995

1994

P. Wells, J. R. Davis, and M. J. Morgan, "Computed tomography," Mater. Forum 18, 111-133 (1994).

1987

P. Maass, "The X-ray transform: singular value decomposition and resolution," Inverse Probl. 3, 729-741 (1987).
[CrossRef]

1986

A. J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Probl. 2, 161-183 (1986).
[CrossRef]

1983

1969

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

Abela, R.

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).

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Phase-contrast tomography using incident spherical waves," Proc. SPIE 5535, 724-732 (2004).

M. A. Anastasio, D. Shi, D. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2003).
[CrossRef]

Barty, A.

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

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Bell, W. W.

W. W. Bell, Special Functions for Scientists and Engineers (Dover, 1968) p. 199.

Bronnikov, A. V.

A. V. Bronnikov, "Theory of quantitative phase-contrast computed tomography," J. Opt. Soc. Am. A 19, 472-480 (2002).

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

Cloetens, P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Davis, J. R.

P. Wells, J. R. Davis, and M. J. Morgan, "Computed tomography," Mater. Forum 18, 111-133 (1994).

Davis, T. J.

De Carlo, D.

M. A. Anastasio, D. Shi, D. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2003).
[CrossRef]

Devaney, A. J.

A. J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Probl. 2, 161-183 (1986).
[CrossRef]

Erdelyi, A.

A. Erdelyi, W. Magnus, R. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953).

Fitzgerald, R.

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53, 23-26 (2000).

Gbur, G.

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Phase-contrast tomography using incident spherical waves," Proc. SPIE 5535, 724-732 (2004).

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

Gohberg, I. C.

I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators (American Mathematical Society, 1969).

Groso, A.

Guigay, J. P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Gureyev, T. E.

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

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

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

D. M. 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]

T. E. Gureyev and S. W. Wilkins, "On x-ray phase imaging with a point source," J. Opt. Soc. Am. A 15, 579-585 (1998).

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

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, 1932-1940 (1995).

Hardy, G. H.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed. (Oxford U. Press, 1979), pp. 7-8.

Huang, Y.

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Phase-contrast tomography using incident spherical waves," Proc. SPIE 5535, 724-732 (2004).

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, 1826-1828 (1996).
[CrossRef]

Krein, M. G.

I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators (American Mathematical Society, 1969).

Liu, H.

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Maass, P.

P. Maass, "The X-ray transform: singular value decomposition and resolution," Inverse Probl. 3, 729-741 (1987).
[CrossRef]

Magnus, W.

A. Erdelyi, W. Magnus, R. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953).

Mayo, S. C.

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

D. M. 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]

McMahon, P. J.

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

Miller, P. R.

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

D. M. 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]

Morgan, M. J.

P. Wells, J. R. Davis, and M. J. Morgan, "Computed tomography," Mater. Forum 18, 111-133 (1994).

Myers, G. R.

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

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Society for Industrial and Applied Mathematics, 2001).

Nesterets, Y. I.

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

Nugent, K. A.

Oberhettinger, R.

A. Erdelyi, W. Magnus, R. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953).

Paganin, D. M.

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

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

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

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

D. M. 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]

D. M. Paganin, Coherent X-Ray Optics (Oxford U. Press, 2006).

Pan, X.

M. A. Anastasio, D. Shi, D. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2003).
[CrossRef]

Pogany, A.

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

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

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]

Roberts, A.

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Shi, D.

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Phase-contrast tomography using incident spherical waves," Proc. SPIE 5535, 724-732 (2004).

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

M. A. Anastasio, D. Shi, D. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2003).
[CrossRef]

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, 1826-1828 (1996).
[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, 1826-1828 (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]

Stampanoni, M.

Stevenson, A.

Teague, M. R.

Tricomi, F.

A. Erdelyi, W. Magnus, R. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953).

Van Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Van Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Wells, P.

P. Wells, J. R. Davis, and M. J. Morgan, "Computed tomography," Mater. Forum 18, 111-133 (1994).

Wilkins, S. W.

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

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

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

D. M. 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]

T. E. Gureyev and S. W. Wilkins, "On x-ray phase imaging with a point source," J. Opt. Soc. Am. A 15, 579-585 (1998).

Wolf, E.

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

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

Wright, E. M.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed. (Oxford U. Press, 1979), pp. 7-8.

Wu, X.

Appl. Phys. Lett.

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]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

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

Inverse Probl.

A. J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Probl. 2, 161-183 (1986).
[CrossRef]

P. Maass, "The X-ray transform: singular value decomposition and resolution," Inverse Probl. 3, 729-741 (1987).
[CrossRef]

J. Microsc.

D. M. 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]

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Mater. Forum

P. Wells, J. R. Davis, and M. J. Morgan, "Computed tomography," Mater. Forum 18, 111-133 (1994).

Opt. Commun.

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

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

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

Opt. Express

Opt. Lett.

Phys. Med. Biol.

M. A. Anastasio, D. Shi, D. De Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2003).
[CrossRef]

Phys. Today

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53, 23-26 (2000).

Proc. SPIE

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

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Phase-contrast tomography using incident spherical waves," Proc. SPIE 5535, 724-732 (2004).

Other

D. M. Paganin, Coherent X-Ray Optics (Oxford U. Press, 2006).

F. Natterer, The Mathematics of Computerized Tomography (Society for Industrial and Applied Mathematics, 2001).

A. Erdelyi, W. Magnus, R. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953).

W. W. Bell, Special Functions for Scientists and Engineers (Dover, 1968) p. 199.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed. (Oxford U. Press, 1979), pp. 7-8.

I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators (American Mathematical Society, 1969).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Arrangement of the object and image planes.

Fig. 2
Fig. 2

Relation between the coordinate systems r and r . The Dirchlet boundary conditions we impose on the object (see Section 4) mean that it will lie entirely within the indicated unit sphere.

Equations (94)

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

ψ ( r ) = I ( r ) exp { i [ k z + ϕ ( r ) ] } .
I z 2 = I z 1 z 2 z 1 k ( I z 1 ϕ z 1 ) .
P ( x λ z 2 z 1 ξ , y λ z 2 z 1 η ) = P ( x , y ) λ z 2 z 1 [ ξ ( x P ) ( x , y ) + η ( y P ) ( x , y ) ] ,
I in I z 2 ( x , y ) I in = z 2 k 2 ϕ 0 ( x , y ) ,
n ( r ) = 1 Δ ( r ) + i β ( r ) .
A ξ x ̂ = x ̂ = cos ( ξ ) x ̂ sin ( ξ ) z ̂ ,
A ξ y ̂ = y ̂ = y ̂ ,
A ξ z ̂ = z ̂ = cos ( ξ ) z ̂ + sin ( ξ ) x ̂ .
ln [ I 0 ( r , θ , ξ ) I in ] = 2 k ( P β ) ( r , θ , ξ ) ,
ϕ 0 ( r , θ , ξ ) = k ( P Δ ) ( r , θ , ξ ) ,
( P f ) ( x , y , ξ ) = 1 1 f [ A ξ 1 ( x , y , z + s ) ] d s .
W ( r ) = ( 1 ρ 2 ) 1 2 , ρ r , r Ω 3
( P V ̂ m , l 0 , l 1 ) ( r , θ , ξ ) = σ m , l 0 , l 1 U ̂ m , l 0 , l 1 ( r , θ , ξ ) ,
m l 0 l 1 0 , l 0 + l 1 = even.
g ( r , θ , ξ ) = z 2 2 ( P Δ ) ( r , θ , ξ ) .
g ( r , θ , ξ ) = I z 2 ( r , θ , ξ ) I in I in .
g ( r , θ , ξ ) = m = 0 m = N ( l 0 = 0 , m + l 0 = even ) l 0 = m l 1 = l 0 l 1 = l 0 g m , l 0 , l 1 U ̂ m , l 0 , l 1 ( r , θ , ξ ) ,
g m , l 0 , l 1 = g ( r , θ , ξ ) U ̂ m , l 0 , l 1 ( r , θ , ξ ) L 2 ( T , 1 ) ,
m l 0 l 1 0 , m + l 0 = even ,
Δ ( r ) = p = 0 p = N ( q 0 = 0 , p + q 0 = even ) q 0 = p q 1 = q 0 q 1 = q 0 c p , q 0 , q 1 V ̂ p , q 0 , q 1 ( r ) ,
c p , q 0 , q 1 = Δ ( r ) V ̂ p , q 0 , q 1 ( r ) L 2 ( Ω 3 , W ( r ) ) ,
p q 0 q 1 0 , p + q 0 = even.
g m , l 0 , l 1 = z 2 p = m p = N 2 M p + 2 , m l 0 , l 1 c p + 2 , l 0 , l 1 ,
c p + 2 , l 0 , l 1 = z 2 1 m = p m = N 2 [ M 1 ] m , p + 2 l 0 , l 1 g m , l 0 , l 1 ,
N 2 m l 0 l 1 0 , m + l 0 = even ,
N 2 p l 0 l 1 0 , p + l 0 = even.
δ q 0 , l 0 δ q 1 , l 1 M p , m l 0 , l 1 σ p , l 0 , l 1 = 2 U ̂ p , q 0 , q 1 ( r , θ , ξ ) U ̂ m , l 0 , l 1 ( r , θ , ξ ) L 2 ( T , 1 ) ,
h m , l 0 , l 1 ( P Δ ) ( r , θ , ξ ) U ̂ m , l 0 , l 1 ( r , θ , ξ ) L 2 ( T , 1 ) .
( P Δ ) ( r , θ , ξ ) = p = 0 p = N ( q 0 = 0 , p + q 0 = even ) q 0 = p q 1 = q 0 q 1 = q 0 h p , q 0 , q 1 U ̂ p , q 0 , q 1 ( r , θ , ξ )
g ( x , y , ξ ) = z 2 2 ( P Δ ) ( r , θ , ξ ) .
g m , l 0 , l 1 = z 2 p = 0 p = N ( q 0 = 0 , p + q 0 = even ) q 0 = p q 1 = q 0 q 1 = q 0 2 U ̂ p , q 0 , q 1 ( r , θ , ξ ) U ̂ m , l 0 , l 1 ( r , θ , ξ ) L 2 ( T , 1 ) h p , q 0 , q 1 ,
m l 0 l 1 0 , m + l 0 = even ,
p l 0 l 1 0 , p + l 0 = even.
g m , l 0 , l 1 = z 2 p = l 0 p = N M p , m l 0 , l 1 σ p , l 0 , l 1 h p , l 0 , l 1 ,
N m l 0 l 1 0 , m + l 0 = even ,
N p l 0 l 1 0 , p + l 0 = even.
h m , l 0 , l 1 = σ m , l 0 , l 1 c m , l 0 , l 1 .
g m , l 0 , l 1 = z 2 p = l 0 p = N M p , m l 0 , l 1 c p , l 0 , l 1 ,
N m l 0 l 1 0 , m + l 0 = even ,
N p l 0 l 1 0 , p + l 0 = even.
dim ( V N ) = p = 0 p = N ( l 0 = 0 , p + l 0 = even ) l 0 = p ( 2 l 0 + 1 ) ,
dim ( V N ) = dim ( V N 2 ) + l 0 = 0 l 0 = N ( 2 l 0 + 1 ) = dim ( V N 2 ) + ( N + 1 ) 2 .
p = m + 2 p = N M p , m l 0 , l 1 c p , l 0 , l 1 = 0 m , l 0 , l 1 z 2 0 .
dim ( Null ) = l 0 = 0 l 0 = N ( 2 l 0 + 1 ) = ( N + 1 ) 2 .
dim ( Range ) = dim ( Domain ) dim ( Null )
dim ( Range ) = dim ( V N 2 ) = dim ( U N 2 ) .
Δ ( r ) = l 0 = 0 l 0 = N l 1 = l 0 l 1 = l 0 c l 0 , l 0 , l 1 V ̂ l 0 , l 0 , l 1 ( r ) + m = 0 m = N 2 ( l 0 = 0 , m + l 0 = even ) l 0 = m l 1 = l 0 l 1 = l 0 c m + 2 , l 0 , l 1 V ̂ m + 2 , l 0 , l 1 ( r ) .
c l 0 , l 0 , l 1 = m = l 0 , m + l 0 = even m = N 2 { ( m + 3 ) ( l 0 + 1 ) B [ l 0 + 1 , 1 2 ] D [ m l 0 + 3 2 , l 0 + 1 2 , 1 2 ] } 1 2 c m + 2 , l 0 , l 1 , c N 1 , N 1 , l 1 = 0 , c N , N , l 1 = 0 .
U m , l 0 , l 1 ( r , θ , ξ ) = j 1 = l 0 , j 1 + l 0 = even j 1 = l 0 d m , l 0 , l 1 , j 1 d l 0 , j 1 l 0 , l 1 ( A ξ ) V ¯ m , j 1 ( r , θ ) ,
N 1 a 1 .
m = 0 m = N 2 ( m + 1 ) = N ( N 1 ) 2
N ( N 1 ) 2 ( a 2 ) 2 .
a 1 ( a 1 + 1 ) = 2 ( a 2 ) 2 .
h = W N 1 W N
N W λ z 2 = W r 1 F ,
O U ̂ m , l 0 , l 1 ( r , θ , ξ ) = V ̂ m + 2 , l 0 , l 1 ( r ) .
O * V ̂ m + 2 , l 0 , l 1 ( r ) = U ̂ m , l 0 , l 1 ( r , θ , ξ ) ,
O * Q N U ̂ m , l 0 , l 1 ( r , θ , ξ ) U ̂ p , q 0 , q 1 ( r , θ , ξ ) L 2 ( T , 1 ) = z 2 1 [ M 1 ] m , p + 2 l 0 , l 1 δ l 0 , q 0 δ l 1 , q 1 .
( z 2 M n + 2 , n p , l 1 ) 2 = [ λ n , p , l 1 ( O * Q N ) ] 2 = λ n , p , l 1 ( ( O * Q N ) * O * Q N ) = λ n , p , l 1 ( Q N * O O * Q N ) = [ s n , p , l 1 ( Q N ) ] 2 .
s n , p , l 1 ( Q N ) = ( z 2 M n + 2 , n p , l 1 ) 1 , N 2 n p | l 1 | 0 .
lim x Γ [ x + y ] Γ [ x ] x y , for x > 0 , y ,
[ M n + 2 , n p , l 1 ] 1 = O ( n 3 2 ) .
[ M n + 2 , n p , l 1 ] 1 = O ( n 7 4 ) .
[ M n + 2 , n p , l 1 ] 1 = O ( n 3 2 ) .
[ M n + 2 , n p , l 1 ] 1 = O ( n 3 2 ) .
s n , p , l 1 ( Q N ) = O ( n 3 2 ) .
B N 2 = max ( ( m + 3 ) ( l 0 + 1 ) B [ l 0 + 1 , 1 2 ] D [ m l 0 + 3 2 , l 0 + 1 2 , 1 2 ] ) + 1 .
V m , l 0 , l 1 ( r ) = [ W ( r ) ] 1 ρ l 0 P ( m l 0 ) 2 1 2 , l 0 + 1 2 ( 2 ρ 2 1 ) Y l 0 , l 1 ( θ , φ ) .
Y l 1 ( θ ) = e i l 1 θ ,
Y l 0 , l 1 ( θ , φ ) = e i l 1 θ ( sin φ ) l 1 C l 0 l 1 + 1 2 ( cos φ ) ,
U m , l 0 , l 1 ( r , θ , ξ ) = j 1 = l 0 , j 1 + l 0 = even j 1 = l 0 d m , l 0 , l 1 , j 1 d l 0 , j 1 l 0 , l 1 ( A ξ ) V ¯ m , j 1 ( r , θ ) ,
l 0 l 1 0 .
V ¯ m , j 1 ( r , θ ) = r j 1 P ( m j 1 ) 2 0 , j 1 ( 2 r 2 1 ) Y j 1 ( θ ) ,
d m , l 0 , l 1 , j 0 , j 1 = C l 0 j 1 j 1 + 1 2 ( 0 ) B [ m l 0 2 + 1 2 , 1 2 ] ,
Y l 0 , l 1 ( A ξ ω ) = j 1 = l 0 j 1 = l 0 d l 0 , j 1 l 0 , l 1 ( A ξ ) Y l 0 , j 1 ( ω ) ,
( σ m , l 0 , l 1 ) 2 = U m , l 0 , l 1 L 2 ( T , 1 ) 2 V m , l 0 , l 1 L 2 ( Ω 3 , W ( r ) ) 2 = 2 π 2 Y l 0 , l 1 L 2 ( S ( 2 ) , 1 ) 2 D [ m l 0 2 + 1 2 , l 0 + 1 2 , 1 2 ] × j 1 = l 0 , j 1 + l 0 = even j 1 = l 0 [ C l 0 j 1 j 1 + 1 2 ( 0 ) ] 2 0 2 π d l 0 , j 1 l 0 , l 1 ( A ξ ) 2 d ξ , D [ a , b , c ] = Γ [ a ] Γ [ a + b ] Γ [ a + c ] Γ [ a + b + c ] .
P [ ( γ β ) 2 ] α α , β + α ( x ) = f = 0 [ ( γ β ) 2 ] α q = 0 [ ( γ β ) 2 ] α f ϵ α , γ , β , f , q P [ ( γ β ) 2 ] α f q α 1 , β + α 1 ( x ) ,
P [ ( γ β ) 2 ] α α , β + α ( x ) = C ( f , q , s , p ) [ ( γ β ) 2 ] α , η α , γ , β , f , q , s , p P [ ( γ β ) 2 ] α f q s p α 2 , β + α 2 ( x ) ,
ϵ α , γ , β , f , q = ( 1 ) f ( γ 2 ( f + q ) 1 ) ( γ 2 f ) Γ [ γ β 2 + 1 ] Γ [ γ β 2 f + 1 ]
× Γ [ γ + β 2 f ] Γ [ α + γ + β 2 ( f + q ) 1 ] Γ [ α + γ + β 2 + 1 ] Γ [ γ + β 2 ( f + q ) ] ,
η α , γ , β , f , q , s , p = ϵ α , γ , β , f , q ϵ α 1 , γ 2 q 2 f , β , s , p .
C ( f , q , s , p ) α f = 0 α q = 0 α f s = 0 α f q p = 0 α f q s , .
V m , l 0 , l 1 L 2 ( Ω 3 , W ( r ) ) 2 = Y l 0 , l 1 L 2 ( S ( 2 ) , 1 ) 2 1 2 ( m + 1 ) Γ [ m l 0 2 + 1 2 ] Γ [ m + l 0 2 + 3 2 ] Γ [ m l 0 2 + 1 ] Γ [ m + l 0 2 + 1 ] ,
V ¯ m , j 1 L 2 ( R 2 , 1 ) 2 = π 1 ( m + 1 ) ,
Y l 0 , l 1 L 2 ( S ( 2 ) , 1 ) 2 = π 2 2 1 2 l 1 Γ [ l 0 + l 1 + 1 ] ( l 0 l 1 ) ! ( l 0 + 1 2 ) ( Γ [ l 1 + 1 2 ] ) 2 ,
σ m , l 0 , l 1 2 = U m , l 0 , l 1 L 2 ( T , 1 ) 2 V m , l 0 , l 1 L 2 ( Ω 3 , W ( r ) ) 2 .
2 V ¯ a , b ( r , θ ) = Y b ( θ ) [ 2 ( b + 1 ) ( a + b + 2 ) r b × P [ ( a b ) 2 ] 1 1 , b + 1 ( 2 r 2 1 ) + ( a + b + 2 ) ( a + b + 4 ) r b + 2 × P [ ( a b ) 2 ] 2 2 , b + 2 ( 2 r 2 1 ) ] .
2 V ¯ a , b ( r , θ ) = 2 ( b + 1 ) ( a + b + 2 ) C ( f , q ) [ ( a b ) 2 ] 1 ϵ 1 , a , b , f , q V ¯ a 2 ( f + q + 1 ) , b ( r , θ ) + ( a + b + 2 ) ( a + b + 4 ) C ( f , q , s , p ) [ ( a b ) 2 ] 2 [ ι a , b , f , q , s , p V ¯ a 2 ( f + q + s + p + 1 ) , b ( r , θ ) + ϖ a , b , f , q , s , p V ¯ a 2 ( f + q + s + p + 2 ) , b ( r , θ ) + τ a , b , f , q , s , p V ¯ a 2 ( f + q + s + p + 3 ) , b ( r , θ ) ] , ι a , b , f , q , s , p = [ a b 2 ( f + q + s + p + 1 ) ] [ a + b 2 ( f + q + s + p + 1 ) ] [ a 2 ( f + p + q + s ) 3 ] [ a 2 ( f + p + q + s + 1 ) ] η 2 , a , b , f , q , s , p , ϖ a , b , f , q , s , p = { b 2 2 [ a 2 ( f + p + s + q + 2 ) ] [ a 2 ( f + p + s + q + 1 ) ] 1 } η 2 , a , b , f , q , s , p , τ a , b , f , q , s , p = [ a b 2 ( f + q + s + p + 2 ) ] [ a + b 2 ( f + q + s + p + 2 ) ] [ a 2 ( f + p + q + s ) 3 ] [ a 2 ( f + p + q + s + 2 ) ] η 2 , a , b , f , q , s , p .
2 V ¯ a , b ( r , θ ) V ¯ m , n ( r , θ ) L 2 ( R 2 , 1 ) = δ b , n V ¯ m , n L 2 ( R 2 , 1 ) 2 { 2 ( b + 1 ) ( a + b + 2 ) C ( f , q ) [ ( a b ) 2 ] 1 ϵ 1 , a , b , f , q δ a 2 ( f + q + 1 ) , m + ( a + b + 2 ) ( a + b + 4 ) C ( f , q , s , p ) [ ( a b ) 2 ] 2 [ ι a , b , f , q , s , p δ a 2 ( f + q + s + p + 1 ) , m + ϖ a , b , f , q , s , p δ a 2 ( f + q + s + p + 2 ) , m + τ a , b , f , q , s , p δ a 2 ( f + q + s + p + 3 ) , m ] }
δ b , n M a , m b .
δ q 0 , l 0 δ q 1 , l 1 M p , m l 0 , l 1 σ p , l 0 , l 1 = 2 U ̂ p , q 0 , q 1 ( r , θ , ξ ) U ̂ m , l 0 , l 1 ( r , θ , ξ ) L 2 ( T , 1 ) .
2 U ̂ p , q 0 , q 1 ( r , θ , ξ ) U ̂ m , l 0 , l 1 ( r , θ , ξ ) L 2 ( T , 1 ) = ( j 1 = l 0 , j 1 + l 0 = even ) j 1 = l 0 ( t 1 = q 0 , t 1 + q 0 = even ) t 1 = q 0 d m , l 0 , l 1 , j 1 d p , q 0 , q 1 , t 1 U p , q 0 , q 1 L 2 ( T , 1 ) U m , l 0 , l 1 L 2 ( T , 1 ) 0 2 π d l 0 , j 1 l 0 , l 1 ( A ξ ) d q 0 , t 1 q 0 , q 1 ( A ξ ) d ξ R 2 V ¯ m , j 1 ( r , θ ) 2 V ¯ p , t 1 ( r , θ ) d r d θ , l 0 j 1 0 , q 0 t 1 0 ,
2 U ̂ p , q 0 , q 1 ( r , θ , ξ ) U ̂ m , l 0 , l 1 ( r , θ , ξ ) L 2 ( T , 1 ) = δ q 0 , l 0 δ q 1 , l 1 σ p , l 0 , l 1 { 2 π ( p + 1 ) ( m + 1 ) Y l 0 , l 1 L 2 ( S ( 2 ) , 1 ) 2 D [ p l 0 + 1 2 , l 0 + 1 2 , 1 2 ] } 1 2 { t 1 = l 0 , t 1 + l 0 = even t 1 = l 0 [ C l 0 t 1 t 1 + 1 2 ( 0 ) ] 2 0 2 π d l 0 , t 1 l 0 , l 1 ( A ξ ) 2 d ξ } 1 2 j 1 = l 0 , j 1 + l 0 = even j 1 = l 0 M p , m j 1 [ C l 0 j 1 j 1 + 1 2 ( 0 ) ] 2 0 2 π d l 0 , j 1 l 0 , l 1 ( A ξ ) 2 d ξ .
M p , m l 0 , l 1 = { 2 π ( p + 1 ) ( m + 1 ) Y l 0 , l 1 L 2 ( S ( 2 ) , 1 ) 2 D [ p l 0 + 1 2 , l 0 + 1 2 , 1 2 ] } 1 2 × { t 1 = l 0 , t 1 + l 0 = even t 1 = l 0 [ C l 0 t 1 t 1 + 1 2 ( 0 ) ] 2 0 2 π d l 0 , t 1 l 0 , l 1 ( A ξ ) 2 d ξ } 1 2 j 1 = l 0 , j 1 + l 0 = even j 1 = l 0 M p , m j 1 [ C l 0 j 1 j 1 + 1 2 ( 0 ) ] 2 0 2 π d l 0 , j 1 l 0 , l 1 ( A ξ ) 2 d ξ .

Metrics