Abstract

The phase retrieval is an important task in x-ray phase contrast imaging. The robustness of phase retrieval is especially important for potential medical imaging applications such as phase contrast mammography. Recently the authors developed an iterative phase retrieval algorithm, the attenuation-partition based algorithm, for the phase retrieval in inline phase-contrast imaging [1]. Applied to experimental images, the algorithm was proven to be fast and robust. However, a quantitative analysis of the performance of this new algorithm is desirable. In this work, we systematically compared the performance of this algorithm with other two widely used phase retrieval algorithms, namely the Gerchberg-Saxton (GS) algorithm and the Transport of Intensity Equation (TIE) algorithm. The systematical comparison is conducted by analyzing phase retrieval performances with a digital breast specimen model. We show that the proposed algorithm converges faster than the GS algorithm in the Fresnel diffraction regime, and is more robust against image noise than the TIE algorithm. These results suggest the significance of the proposed algorithm for future medical applications with the x-ray phase contrast imaging technique.

© 2010 OSA

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References

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  1. A. Yan, X. Wu, and H. Liu, “An attenuation-partition based iterative phase retrieval algorithm for in-line phase-contrast imaging,” Opt. Express 16, 13,330 – 13,341 (2008).
    [Crossref]
  2. S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335 – 338 (1996).
    [Crossref]
  3. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Shelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486 – 5492 (1995).
    [Crossref]
  4. K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative Phase Imaging Using Hard X Rays,” Phy. Rev. Lett. 77, 2961 – 2965 (1996).
    [Crossref]
  5. A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774 – 2782 (1997).
    [Crossref]
  6. F. Arfelli and V. Bonvicini, and et al, “Mammography with synchrotron radiation: phase-detected Techniques,” Radiology 215, 286 – 293 (2000).
  7. D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33 – 40 (2002).
    [Crossref] [PubMed]
  8. S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289 – 2302 (2003).
    [Crossref] [PubMed]
  9. X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-ray Sci. and Tech. 11, 33 – 42 (2003).
  10. X. Wu and H. Liu, “Clinical implementation of phase-contrast x-ray imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169 – 2179 (2003).
    [Crossref] [PubMed]
  11. X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378 – 2384 (2004).
    [Crossref] [PubMed]
  12. E. Donnelly, R. Price, and D. Pickens, “Experimental validation of the Wigner distributions theory of phase-contrast imaging,” Med. Phys. 32, 928 – 931 (2005).
    [Crossref] [PubMed]
  13. D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55, 2249 – 2257 (2008).
    [Crossref] [PubMed]
  14. X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett. 30(4), 379 – 381 (2005).
    [Crossref] [PubMed]
  15. X. Wu and H. Liu, “X-Ray cone-beam phase tomography formulas based on phase-attenuation duality,” Opt. Express 13, 6000 – 6014 (2005).
    [Crossref] [PubMed]
  16. P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” PNAS 103, 14,626 – 14,630 (2006).
    [Crossref]
  17. X. Wu, H. Liu, and A. Yan, “Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval,” Eur. J. Radiology 68, S8 – S12 (2008).
    [Crossref]
  18. D. Paganin and K. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phy. Rev. Lett. 80, 2586 – 2589 (1998).
    [Crossref]
  19. X. Wu and H. Liu, “A dual detector approach for X-ray attenuation and phase imaging,” J. X-ray Sci. and Tech. 12, 35 – 42 (2004).
  20. X. Wu and H. Liu, “Phase-space evolution of x-ray coherence in phase-sensitive imaging,” Appl. Opt. 47, E44 – E52 (2008).
    [Crossref] [PubMed]
  21. M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434 – 1441 (1983).
    [Crossref]
  22. T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Comm. 259, 569 – 580 (2006).
    [Crossref]
  23. J. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617 – 1619 (2007).
    [Crossref] [PubMed]
  24. X. Wu and A. Yan, “Phase Retrieval From One Single Phase Contrast X-Ray Image,” Opt. Express p. Opt. Express 17, 11187 – 11196 (2009).
    [Crossref]
  25. L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Comm. 199, 65 – 75 (2001).
    [Crossref]
  26. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237 – 246 (1972).
  27. J. Fienup, “Reconstruction of an object from the modulus of its Fourier Transform,” Opt. Lett. 3, 27 – 29 (1978).
    [Crossref] [PubMed]
  28. J. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758 – 2769 (1982).
    [Crossref] [PubMed]
  29. N. Dyson, X-Rays in Atomic and Nuclear Physics (Longman Scientific and Technical, Essex, UK, 1973).
  30. X. Wu, A. Dean, and H. Liu, Biomedical Photonics Handbook, chap. 26, pp. 26-1–26-34 (CRC Press, Tampa, Fla., 2003).
  31. J. H. Hubbell, W. I. Veigele, and E. A. Briggs, et al., “Atomic form factors, incohoerent scattering functions, and photon scattering cross sections,” Journal of Physical Chemistry Reference Data 4, 471 – 538 (1975).
    [Crossref]
  32. L. Rudin, “Images, numerical analysis of singularities and shock filters,” Report #TR:5250:87, Caltech, C,S, Dept. (1987).
  33. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259 – 268 (1992).
    [Crossref]
  34. X. Wu, G.T. Barnes, and D.M. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143 – 148 (1991).
    [PubMed]
  35. X. Wu, E.L. Gingold, G.T. Barnes, and D.M. Tucker, “Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography,” Radiology 193, 83 – 89 (1994).
    [PubMed]
  36. J. Seldin and J. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7(3), 412 – 427 (1990).
    [Crossref]
  37. F. Roddier and C. Roddier, “Wavefront reconstruction using Iterative Fourier transforms,” Appl. Opt. 30, 1325 – 1327 (1991).
    [Crossref] [PubMed]
  38. C. Roddier and F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277 – 2287 (1993).
    [Crossref]
  39. T. Gureyev, A. Roberts, and K. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942 – 1946 (1995).
    [Crossref]
  40. T. Gureyev and K. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670 – 1682 (1996).
    [Crossref]
  41. A. Tychonoff and V. Arsenin, Solution of Ill-posed Problems (Winston & Sons, Washington, 1977).

2009 (1)

X. Wu and A. Yan, “Phase Retrieval From One Single Phase Contrast X-Ray Image,” Opt. Express p. Opt. Express 17, 11187 – 11196 (2009).
[Crossref]

2008 (4)

X. Wu and H. Liu, “Phase-space evolution of x-ray coherence in phase-sensitive imaging,” Appl. Opt. 47, E44 – E52 (2008).
[Crossref] [PubMed]

A. Yan, X. Wu, and H. Liu, “An attenuation-partition based iterative phase retrieval algorithm for in-line phase-contrast imaging,” Opt. Express 16, 13,330 – 13,341 (2008).
[Crossref]

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55, 2249 – 2257 (2008).
[Crossref] [PubMed]

X. Wu, H. Liu, and A. Yan, “Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval,” Eur. J. Radiology 68, S8 – S12 (2008).
[Crossref]

2007 (1)

2006 (2)

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Comm. 259, 569 – 580 (2006).
[Crossref]

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” PNAS 103, 14,626 – 14,630 (2006).
[Crossref]

2005 (3)

2004 (2)

X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378 – 2384 (2004).
[Crossref] [PubMed]

X. Wu and H. Liu, “A dual detector approach for X-ray attenuation and phase imaging,” J. X-ray Sci. and Tech. 12, 35 – 42 (2004).

2003 (3)

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289 – 2302 (2003).
[Crossref] [PubMed]

X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-ray Sci. and Tech. 11, 33 – 42 (2003).

X. Wu and H. Liu, “Clinical implementation of phase-contrast x-ray imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169 – 2179 (2003).
[Crossref] [PubMed]

2002 (1)

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. 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)

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Comm. 199, 65 – 75 (2001).
[Crossref]

2000 (1)

F. Arfelli and V. Bonvicini, and et al, “Mammography with synchrotron radiation: phase-detected Techniques,” Radiology 215, 286 – 293 (2000).

1998 (1)

D. Paganin and K. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phy. Rev. Lett. 80, 2586 – 2589 (1998).
[Crossref]

1997 (1)

A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774 – 2782 (1997).
[Crossref]

1996 (3)

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative Phase Imaging Using Hard X Rays,” Phy. Rev. Lett. 77, 2961 – 2965 (1996).
[Crossref]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335 – 338 (1996).
[Crossref]

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

1995 (2)

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

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

1994 (1)

X. Wu, E.L. Gingold, G.T. Barnes, and D.M. Tucker, “Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography,” Radiology 193, 83 – 89 (1994).
[PubMed]

1993 (1)

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259 – 268 (1992).
[Crossref]

1991 (2)

X. Wu, G.T. Barnes, and D.M. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143 – 148 (1991).
[PubMed]

F. Roddier and C. Roddier, “Wavefront reconstruction using Iterative Fourier transforms,” Appl. Opt. 30, 1325 – 1327 (1991).
[Crossref] [PubMed]

1990 (1)

1983 (1)

1982 (1)

1978 (1)

1975 (1)

J. H. Hubbell, W. I. Veigele, and E. A. Briggs, et al., “Atomic form factors, incohoerent scattering functions, and photon scattering cross sections,” Journal of Physical Chemistry Reference Data 4, 471 – 538 (1975).
[Crossref]

1972 (1)

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

Allen, L.

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Comm. 199, 65 – 75 (2001).
[Crossref]

Archer, A.

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55, 2249 – 2257 (2008).
[Crossref] [PubMed]

Arfelli, F.

F. Arfelli and V. Bonvicini, and et al, “Mammography with synchrotron radiation: phase-detected Techniques,” Radiology 215, 286 – 293 (2000).

Arsenin, V.

A. Tychonoff and V. Arsenin, Solution of Ill-posed Problems (Winston & Sons, Washington, 1977).

Barnea, Z.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative Phase Imaging Using Hard X Rays,” Phy. Rev. Lett. 77, 2961 – 2965 (1996).
[Crossref]

Barnes, G.T.

X. Wu, E.L. Gingold, G.T. Barnes, and D.M. Tucker, “Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography,” Radiology 193, 83 – 89 (1994).
[PubMed]

X. Wu, G.T. Barnes, and D.M. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143 – 148 (1991).
[PubMed]

Boistel, R.

Bonvicini, V.

F. Arfelli and V. Bonvicini, and et al, “Mammography with synchrotron radiation: phase-detected Techniques,” Radiology 215, 286 – 293 (2000).

Briggs, E. A.

J. H. Hubbell, W. I. Veigele, and E. A. Briggs, et al., “Atomic form factors, incohoerent scattering functions, and photon scattering cross sections,” Journal of Physical Chemistry Reference Data 4, 471 – 538 (1975).
[Crossref]

Cloetens, P.

J. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617 – 1619 (2007).
[Crossref] [PubMed]

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” PNAS 103, 14,626 – 14,630 (2006).
[Crossref]

Cookson, D.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative Phase Imaging Using Hard X Rays,” Phy. Rev. Lett. 77, 2961 – 2965 (1996).
[Crossref]

Davis, T.

Dean, A.

X. Wu, A. Dean, and H. Liu, Biomedical Photonics Handbook, chap. 26, pp. 26-1–26-34 (CRC Press, Tampa, Fla., 2003).

Donnelly, E.

E. Donnelly, R. Price, and D. Pickens, “Experimental validation of the Wigner distributions theory of phase-contrast imaging,” Med. Phys. 32, 928 – 931 (2005).
[Crossref] [PubMed]

Donvan, M.

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55, 2249 – 2257 (2008).
[Crossref] [PubMed]

Dyson, N.

N. Dyson, X-Rays in Atomic and Nuclear Physics (Longman Scientific and Technical, Essex, UK, 1973).

Fajardo, L.

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55, 2249 – 2257 (2008).
[Crossref] [PubMed]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259 – 268 (1992).
[Crossref]

Fienup, J.

Gao, D.

A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774 – 2782 (1997).
[Crossref]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335 – 338 (1996).
[Crossref]

Gerchberg, R. W.

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

Gingold, E.L.

X. Wu, E.L. Gingold, G.T. Barnes, and D.M. Tucker, “Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography,” Radiology 193, 83 – 89 (1994).
[PubMed]

Guigay, J.

Gureyev, T.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Comm. 259, 569 – 580 (2006).
[Crossref]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289 – 2302 (2003).
[Crossref] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33 – 40 (2002).
[Crossref] [PubMed]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative Phase Imaging Using Hard X Rays,” Phy. Rev. Lett. 77, 2961 – 2965 (1996).
[Crossref]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335 – 338 (1996).
[Crossref]

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

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

Hubbell, J. H.

J. H. Hubbell, W. I. Veigele, and E. A. Briggs, et al., “Atomic form factors, incohoerent scattering functions, and photon scattering cross sections,” Journal of Physical Chemistry Reference Data 4, 471 – 538 (1975).
[Crossref]

Kohn, V.

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

Kuznetsov, S.

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

Langer, M.

Lerbs-Mache, S.

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” PNAS 103, 14,626 – 14,630 (2006).
[Crossref]

Liu, H.

X. Wu, H. Liu, and A. Yan, “Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval,” Eur. J. Radiology 68, S8 – S12 (2008).
[Crossref]

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55, 2249 – 2257 (2008).
[Crossref] [PubMed]

A. Yan, X. Wu, and H. Liu, “An attenuation-partition based iterative phase retrieval algorithm for in-line phase-contrast imaging,” Opt. Express 16, 13,330 – 13,341 (2008).
[Crossref]

X. Wu and H. Liu, “Phase-space evolution of x-ray coherence in phase-sensitive imaging,” Appl. Opt. 47, E44 – E52 (2008).
[Crossref] [PubMed]

X. Wu and H. Liu, “X-Ray cone-beam phase tomography formulas based on phase-attenuation duality,” Opt. Express 13, 6000 – 6014 (2005).
[Crossref] [PubMed]

X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett. 30(4), 379 – 381 (2005).
[Crossref] [PubMed]

X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378 – 2384 (2004).
[Crossref] [PubMed]

X. Wu and H. Liu, “A dual detector approach for X-ray attenuation and phase imaging,” J. X-ray Sci. and Tech. 12, 35 – 42 (2004).

X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-ray Sci. and Tech. 11, 33 – 42 (2003).

X. Wu and H. Liu, “Clinical implementation of phase-contrast x-ray imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169 – 2179 (2003).
[Crossref] [PubMed]

X. Wu, A. Dean, and H. Liu, Biomedical Photonics Handbook, chap. 26, pp. 26-1–26-34 (CRC Press, Tampa, Fla., 2003).

Mache, R.

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” PNAS 103, 14,626 – 14,630 (2006).
[Crossref]

Mayo, S.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289 – 2302 (2003).
[Crossref] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. 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.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289 – 2302 (2003).
[Crossref] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33 – 40 (2002).
[Crossref] [PubMed]

Nesterets, Y.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Comm. 259, 569 – 580 (2006).
[Crossref]

Nugent, K.

D. Paganin and K. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phy. Rev. Lett. 80, 2586 – 2589 (1998).
[Crossref]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative Phase Imaging Using Hard X Rays,” Phy. Rev. Lett. 77, 2961 – 2965 (1996).
[Crossref]

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

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

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259 – 268 (1992).
[Crossref]

Oxley, M.

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Comm. 199, 65 – 75 (2001).
[Crossref]

Paganin, D.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Comm. 259, 569 – 580 (2006).
[Crossref]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33 – 40 (2002).
[Crossref] [PubMed]

D. Paganin and K. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phy. Rev. Lett. 80, 2586 – 2589 (1998).
[Crossref]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative Phase Imaging Using Hard X Rays,” Phy. Rev. Lett. 77, 2961 – 2965 (1996).
[Crossref]

Pickens, D.

E. Donnelly, R. Price, and D. Pickens, “Experimental validation of the Wigner distributions theory of phase-contrast imaging,” Med. Phys. 32, 928 – 931 (2005).
[Crossref] [PubMed]

Poganin, D.

Pogany, A.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Comm. 259, 569 – 580 (2006).
[Crossref]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289 – 2302 (2003).
[Crossref] [PubMed]

A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774 – 2782 (1997).
[Crossref]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335 – 338 (1996).
[Crossref]

Price, R.

E. Donnelly, R. Price, and D. Pickens, “Experimental validation of the Wigner distributions theory of phase-contrast imaging,” Med. Phys. 32, 928 – 931 (2005).
[Crossref] [PubMed]

Roberts, A.

Roddier, C.

Roddier, F.

Rudin, L.

L. Rudin, “Images, numerical analysis of singularities and shock filters,” Report #TR:5250:87, Caltech, C,S, Dept. (1987).

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259 – 268 (1992).
[Crossref]

Saxton, W. O.

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

Schlenker, M.

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” PNAS 103, 14,626 – 14,630 (2006).
[Crossref]

Seldin, J.

Shelokov, I.

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

Snigirev, A.

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

Snigireva, I.

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

Stevenson, A.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289 – 2302 (2003).
[Crossref] [PubMed]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335 – 338 (1996).
[Crossref]

Teague, M.

Tucker, D.M.

X. Wu, E.L. Gingold, G.T. Barnes, and D.M. Tucker, “Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography,” Radiology 193, 83 – 89 (1994).
[PubMed]

X. Wu, G.T. Barnes, and D.M. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143 – 148 (1991).
[PubMed]

Tychonoff, A.

A. Tychonoff and V. Arsenin, Solution of Ill-posed Problems (Winston & Sons, Washington, 1977).

Veigele, W. I.

J. H. Hubbell, W. I. Veigele, and E. A. Briggs, et al., “Atomic form factors, incohoerent scattering functions, and photon scattering cross sections,” Journal of Physical Chemistry Reference Data 4, 471 – 538 (1975).
[Crossref]

Wilkins, S.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Comm. 259, 569 – 580 (2006).
[Crossref]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289 – 2302 (2003).
[Crossref] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33 – 40 (2002).
[Crossref] [PubMed]

A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774 – 2782 (1997).
[Crossref]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335 – 338 (1996).
[Crossref]

Wu, X.

X. Wu and A. Yan, “Phase Retrieval From One Single Phase Contrast X-Ray Image,” Opt. Express p. Opt. Express 17, 11187 – 11196 (2009).
[Crossref]

X. Wu and H. Liu, “Phase-space evolution of x-ray coherence in phase-sensitive imaging,” Appl. Opt. 47, E44 – E52 (2008).
[Crossref] [PubMed]

A. Yan, X. Wu, and H. Liu, “An attenuation-partition based iterative phase retrieval algorithm for in-line phase-contrast imaging,” Opt. Express 16, 13,330 – 13,341 (2008).
[Crossref]

X. Wu, H. Liu, and A. Yan, “Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval,” Eur. J. Radiology 68, S8 – S12 (2008).
[Crossref]

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55, 2249 – 2257 (2008).
[Crossref] [PubMed]

X. Wu and H. Liu, “X-Ray cone-beam phase tomography formulas based on phase-attenuation duality,” Opt. Express 13, 6000 – 6014 (2005).
[Crossref] [PubMed]

X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett. 30(4), 379 – 381 (2005).
[Crossref] [PubMed]

X. Wu and H. Liu, “A dual detector approach for X-ray attenuation and phase imaging,” J. X-ray Sci. and Tech. 12, 35 – 42 (2004).

X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378 – 2384 (2004).
[Crossref] [PubMed]

X. Wu and H. Liu, “Clinical implementation of phase-contrast x-ray imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169 – 2179 (2003).
[Crossref] [PubMed]

X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-ray Sci. and Tech. 11, 33 – 42 (2003).

X. Wu, E.L. Gingold, G.T. Barnes, and D.M. Tucker, “Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography,” Radiology 193, 83 – 89 (1994).
[PubMed]

X. Wu, G.T. Barnes, and D.M. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143 – 148 (1991).
[PubMed]

X. Wu, A. Dean, and H. Liu, Biomedical Photonics Handbook, chap. 26, pp. 26-1–26-34 (CRC Press, Tampa, Fla., 2003).

Yan, A.

X. Wu and A. Yan, “Phase Retrieval From One Single Phase Contrast X-Ray Image,” Opt. Express p. Opt. Express 17, 11187 – 11196 (2009).
[Crossref]

X. Wu, H. Liu, and A. Yan, “Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval,” Eur. J. Radiology 68, S8 – S12 (2008).
[Crossref]

A. Yan, X. Wu, and H. Liu, “An attenuation-partition based iterative phase retrieval algorithm for in-line phase-contrast imaging,” Opt. Express 16, 13,330 – 13,341 (2008).
[Crossref]

X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett. 30(4), 379 – 381 (2005).
[Crossref] [PubMed]

Zhang, D.

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55, 2249 – 2257 (2008).
[Crossref] [PubMed]

Appl. Opt. (3)

Eur. J. Radiology (1)

X. Wu, H. Liu, and A. Yan, “Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval,” Eur. J. Radiology 68, S8 – S12 (2008).
[Crossref]

IEEE Trans. Biomed. Eng. (1)

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55, 2249 – 2257 (2008).
[Crossref] [PubMed]

J. Microsc. (1)

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33 – 40 (2002).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. X-ray Sci. and Tech. (2)

X. Wu and H. Liu, “A dual detector approach for X-ray attenuation and phase imaging,” J. X-ray Sci. and Tech. 12, 35 – 42 (2004).

X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-ray Sci. and Tech. 11, 33 – 42 (2003).

Journal of Physical Chemistry Reference Data (1)

J. H. Hubbell, W. I. Veigele, and E. A. Briggs, et al., “Atomic form factors, incohoerent scattering functions, and photon scattering cross sections,” Journal of Physical Chemistry Reference Data 4, 471 – 538 (1975).
[Crossref]

Med. Phys. (3)

X. Wu and H. Liu, “Clinical implementation of phase-contrast x-ray imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169 – 2179 (2003).
[Crossref] [PubMed]

X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378 – 2384 (2004).
[Crossref] [PubMed]

E. Donnelly, R. Price, and D. Pickens, “Experimental validation of the Wigner distributions theory of phase-contrast imaging,” Med. Phys. 32, 928 – 931 (2005).
[Crossref] [PubMed]

Nature (1)

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335 – 338 (1996).
[Crossref]

Opt. Comm. (2)

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Comm. 199, 65 – 75 (2001).
[Crossref]

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Comm. 259, 569 – 580 (2006).
[Crossref]

Opt. Express (3)

Opt. Express p. Opt. Express (1)

X. Wu and A. Yan, “Phase Retrieval From One Single Phase Contrast X-Ray Image,” Opt. Express p. Opt. Express 17, 11187 – 11196 (2009).
[Crossref]

Opt. Lett. (3)

Optik (1)

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

Phy. Rev. Lett. (2)

D. Paganin and K. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phy. Rev. Lett. 80, 2586 – 2589 (1998).
[Crossref]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative Phase Imaging Using Hard X Rays,” Phy. Rev. Lett. 77, 2961 – 2965 (1996).
[Crossref]

Physica D (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259 – 268 (1992).
[Crossref]

PNAS (1)

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” PNAS 103, 14,626 – 14,630 (2006).
[Crossref]

Radiology (3)

F. Arfelli and V. Bonvicini, and et al, “Mammography with synchrotron radiation: phase-detected Techniques,” Radiology 215, 286 – 293 (2000).

X. Wu, G.T. Barnes, and D.M. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143 – 148 (1991).
[PubMed]

X. Wu, E.L. Gingold, G.T. Barnes, and D.M. Tucker, “Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography,” Radiology 193, 83 – 89 (1994).
[PubMed]

Rev. Sci. Instrum. (2)

A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774 – 2782 (1997).
[Crossref]

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

Other (4)

L. Rudin, “Images, numerical analysis of singularities and shock filters,” Report #TR:5250:87, Caltech, C,S, Dept. (1987).

N. Dyson, X-Rays in Atomic and Nuclear Physics (Longman Scientific and Technical, Essex, UK, 1973).

X. Wu, A. Dean, and H. Liu, Biomedical Photonics Handbook, chap. 26, pp. 26-1–26-34 (CRC Press, Tampa, Fla., 2003).

A. Tychonoff and V. Arsenin, Solution of Ill-posed Problems (Winston & Sons, Washington, 1977).

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Figures (9)

Fig. 1.
Fig. 1.

Flow chart of APBA

Fig. 2.
Fig. 2.

Example “Lena” images used in measuring the closeness between images. (a) ϕ 1, (b) ϕ 2, (c) ϕ 3

Fig. 3.
Fig. 3.

Image manifest of (a) A 2 pe,coh, (b) A 2 KN and (c) A 2 0 when x-ray energy equals 35.5 keV.

Fig. 4.
Fig. 4.

Profiles of A 2 pe,coh, the solid lines, and A 2 KN, the dotted lines, when x-ray energy equals (a) 18.5 keV, (b) 35.5 keV and (c) 59.5 keV.

Fig. 5.
Fig. 5.

Image representation of the inputs generated from the simulation model and Fresnel propagation. (a) the phase map ϕ; (b) the attenuation map A 2 0; and (c) the normalized Fresnel propagated phase contrast image I with object to detector distance R 2 = 26 in (0.66 m).

Fig. 6.
Fig. 6.

Comparison of the performance of the GS algorithm and APBA. (a) plot of the accuracy measures with respect to iteration steps. The plot with solid line represents the APBA. The one with dashed line represents the GS algorithm; (b) recovered phase map with the GS algorithm after 100 iterations; (c) recovered phase map with APBA after 100 iterations.

Fig. 7.
Fig. 7.

Comparison of APBA and the TIE algorithm with pure data. (a) plot of the accuracy measure with respect to iteration steps. The plot with solid line represents the APBA. The one with dashed line represents the TIE algorithm. It needs 1110 steps for the TV measure, 0.00215226, of APBA to achieve to the TV measure, 0.00215269, of TIE; (b) recovered phase using the TIE algorithm; (c) recovered phase using APBA after 1500 iteration steps.

Fig. 8.
Fig. 8.

Comparison of the TIE algorithm and APBA with noise added. (a) True phase map ϕ; (b) attenuation map A 2 0; (c) the normalized phase contrast image I; (d) recovered phase map with the TIE algorithm, no Tikhonov regularization is used; (e) recovered phase map with the TIE algorithm with Tikhonov regularization; (f) recovered phase map with APBA after 10 iteration steps. In the simulation, the acquired data is assumed to have a level of δb = 0.03% detector noise and one pixel misalignment between A 2 0 and I horizontally.

Fig. 9.
Fig. 9.

Profiles, along a line passing through the microcalcifications, of the recovered phase using APBA, the solid line, and using the TIE algorithm with Tikhonov regularization, the dashed line, in Case 3. The dash-dotted line is the true phase.

Tables (1)

Tables Icon

Table 1. TV comparison of the TIE algorithm and APBA. In the table, κ is the Tikhonov regularization parameter, Δ represents the sampling step-size in FT-space.

Equations (19)

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

ϕ ( r ) = ( h c E ) r e ρ e ( r , z ) dz = ( hc E ) r e ρ e , p ( r ) ,
̂ ( I ) ( u M ; R 1 + R 2 ) = I in { cos ( π λ R 2 M u 2 ) · ̂ ( A 0 2 ) +
+ [ 2 sin ( π λ R 2 M u 2 ) ( 2 π λ R 2 M u 2 ) · cos ( π λ R 2 M u 2 ) ] · ̂ ( A 0 2 ϕ )
cos ( π λ R 2 M u 2 ) · λ R 2 2 π M · ̂ ( · ( A 0 2 ϕ ) )
λ R 2 4 π M sin ( π λ R 2 M u 2 ) · ̂ ( 2 A 0 2 ) } ,
I ( r ; R 1 + R 2 ) = I in M 2 { A 0 2 ( r M ) λ R 2 2 π M · ( A 0 2 ϕ ) ( r M ) } .
A KN ( r ) = exp [ σ KN 2 ρ e , p ( r ) ] , ϕ ( r ) = λ r e ρ e , p ,
σ KN ( E photon ) = 2 π r e 2 { 1 + η η 2 [ 2 ( 1 + η ) 1 + 2 η 1 η log ( 1 + 2 η ) ] +
+ 1 2 η log ( 1 + 2 η ) 1 + 3 η ( 1 + 2 η ) 2 } ,
A KN 2 ( r ) = 𝔇 ( I ) = ̂ 1 ( ̂ ( I ) 1 + 4 π 2 k ˜ u 2 ) , ϕ ( r ) = ( λ r e σ KN ) ln ( A KN 2 ( r ) ) ,
k ˜ = λ R 2 2 π M · λ r e σ KN ,
A 0 ( r ) = A KN ( r ) · A pe , coh ( r ) ,
A 0 = A KN δ A , δ A = A KN ( 1 A pe , coh ) ,
δ A = A KN · ( 1 P ) , P = A 0 A KN .
𝔉 𝔯 ( T ) ( r ) = 1 λ R 2 2 exp [ i π M λ R 2 ( r M ξ ) 2 ] T ( ξ ) d ξ .
std ( g , f ) : = [ Ω ( g ( r ) f ( r ) μ ) 2 d r ] 1 2 V ( Ω )
TV ( g , f ) : = Ω ( g f ) d r V ( Ω ) = Ω ( ( g f ) x ) 2 + ( ( g f ) y ) 2 d r V ( Ω ) ,
ϕ ( r ) = 2 π M λ R 2 2 { · [ [ 2 ( I A 0 2 ) ] A 0 2 ] } ,
x κ = arg min x X A x y Y 2 + κ x X 2 .

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