Abstract

Simple analytical expressions are derived for the spatial resolution, contrast and signal-to-noise in X-ray projection images of a generic phase edge. The obtained expressions take into account the maximum phase shift generated by the sample and the sharpness of the edge, as well as such parameters of the imaging set-up as the wavelength spectrum and the size of the incoherent source, the source-to-object and object-to-detector distances and the detector resolution. Different asymptotic behavior of the expressions in the cases of large and small Fresnel numbers is demonstrated. The analytical expressions are compared with the results of numerical simulations using Kirchhoff diffraction theory, as well as with experimental X-ray measurements.

© 2008 Optical Society of America

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  1. R. Fitzgerald, "Phase-sensitive x-ray imaging," Physics Today 53, 23-26 (2000).
    [CrossRef]
  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, 5486-5492 (1995).
    [CrossRef]
  3. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays:," Nature 384, 335-338 (1996).
    [CrossRef]
  4. P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. J. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996).
    [CrossRef]
  5. A. Krol, J. C. Kieffer and E. Foerster, "Laser-driven x-ray source for diagnostic radiology. Applications of X-rays Generated from Lasers and Other Bright Sources," Proc. SPIE 3157, 156-163 (1997).
    [CrossRef]
  6. T. A. Shelkovenko, D. B. Sinars, S. A. Pikuz, K. M. Chandler and D. A. Hammer, "Point-projection x-ray radiography using an X pinch as the radiation source," Rev. Sci. Instrum. 72, 667-670 (2001).
    [CrossRef]
  7. C. J. Kotre, I. P. Birch and K. J. Robson, "Anomalous image quality phantom scores in magnification mammography: evidence of phase contrast enhancement," British J. Radiol. 75, 170-173 (2002).
  8. J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975), Sec.3.4.2.
  9. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin and Z. Barnea, "Quantitative phase imaging using hard X-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
    [CrossRef] [PubMed]
  10. Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093706 (2005).
    [CrossRef]
  11. M. R. Teague, "Deterministic Phase Retrieval: a Green's Function Solution," J. Opt. Soc. Am. 73, 1434-1441 (1983).
    [CrossRef]
  12. T. E. Gureyev and S. W. Wilkins, "On X-ray phase imaging with a point source," J. Opt. Soc. Am. A 15, 579-85 (1998).
    [CrossRef]
  13. 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]
  14. 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, 068103-1 - 068103-4 (2004).
    [CrossRef] [PubMed]
  15. T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
    [CrossRef]
  16. A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
    [CrossRef]
  17. J.-P. Guigay, "Fourier transform analysis of Fresnel diffraction patterns and in-line holograms," Optik 49, 121-125 (1977).
  18. A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus X-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
    [CrossRef]
  19. P. Cloetens, J.-P. Guigay, C. De Martino, and J. Baruchel, "Fractional Talbot imaging of phase gratings with hard x rays," Opt. Lett. 22, 1059-1061 (1997).
    [CrossRef] [PubMed]
  20. V. E. Cosslett and W. C. Nixon, "The X-Ray Shadow Microscope," J. Appl. Phys. 24, 616-623 (1953).
    [CrossRef]
  21. G. Margaritondo and G. Tromba, "Coherence-based edge diffraction sharpening of x-ray images: a simple model," J. Appl. Phys. 85, 3406-3408 (1999).
    [CrossRef]
  22. K. Levenberg, "A method for the solution of certain non-linear problems in least squares," Quart. Appl. Math. 2, 164-168 (1944).
  23. D. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," SIAM J. Appl. Math. 11, 431-441 (1963).
    [CrossRef]
  24. S. Geller, "Parameter interaction in least squares structure refinement," Acta Cryst. 14, 1026-1035 (1961).
    [CrossRef]
  25. J. S. Rollett, Computing Methods in Crystallography (Pergamon Press, Oxford, 1965).
  26. W. C. Hamilton, "Significance tests on the crystallographic R factor," Acta Cryst. 18, 502-510 (1965).
    [CrossRef]
  27. S. Brennan and P. L. Cowan, "A suite of programs for calculating X-ray absorption, reflection, and diffraction performance for a variety of materials at arbitrary wavelengths," Rev. Sci. Instrum. 63, 850-853 (1992).
    [CrossRef]
  28. Y. Amemiya, K. Wakabayashi, H. Tanaka, Y. Ueno, and J. Miyahara, "Laser-stimulated luminescence used to measure X-ray diffraction of a contracting striated muscle," Science 237, 164-168 (1987).
    [CrossRef] [PubMed]
  29. Y. Amemiya, "Imaging plates for use with synchrotron radiation," J. Synch. Rad. 2, 13-21 (1995).
    [CrossRef]

2007

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

2006

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

2005

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093706 (2005).
[CrossRef]

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]

2002

C. J. Kotre, I. P. Birch and K. J. Robson, "Anomalous image quality phantom scores in magnification mammography: evidence of phase contrast enhancement," British J. Radiol. 75, 170-173 (2002).

2001

T. A. Shelkovenko, D. B. Sinars, S. A. Pikuz, K. M. Chandler and D. A. Hammer, "Point-projection x-ray radiography using an X pinch as the radiation source," Rev. Sci. Instrum. 72, 667-670 (2001).
[CrossRef]

2000

R. Fitzgerald, "Phase-sensitive x-ray imaging," Physics Today 53, 23-26 (2000).
[CrossRef]

1999

G. Margaritondo and G. Tromba, "Coherence-based edge diffraction sharpening of x-ray images: a simple model," J. Appl. Phys. 85, 3406-3408 (1999).
[CrossRef]

1998

T. E. Gureyev and S. W. Wilkins, "On X-ray phase imaging with a point source," J. Opt. Soc. Am. A 15, 579-85 (1998).
[CrossRef]

1997

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus X-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

P. Cloetens, J.-P. Guigay, C. De Martino, and J. Baruchel, "Fractional Talbot imaging of phase gratings with hard x rays," Opt. Lett. 22, 1059-1061 (1997).
[CrossRef] [PubMed]

A. Krol, J. C. Kieffer and E. Foerster, "Laser-driven x-ray source for diagnostic radiology. Applications of X-rays Generated from Lasers and Other Bright Sources," Proc. SPIE 3157, 156-163 (1997).
[CrossRef]

1996

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays:," Nature 384, 335-338 (1996).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. J. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996).
[CrossRef]

1995

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, 5486-5492 (1995).
[CrossRef]

Y. Amemiya, "Imaging plates for use with synchrotron radiation," J. Synch. Rad. 2, 13-21 (1995).
[CrossRef]

1992

S. Brennan and P. L. Cowan, "A suite of programs for calculating X-ray absorption, reflection, and diffraction performance for a variety of materials at arbitrary wavelengths," Rev. Sci. Instrum. 63, 850-853 (1992).
[CrossRef]

1987

Y. Amemiya, K. Wakabayashi, H. Tanaka, Y. Ueno, and J. Miyahara, "Laser-stimulated luminescence used to measure X-ray diffraction of a contracting striated muscle," Science 237, 164-168 (1987).
[CrossRef] [PubMed]

1983

M. R. Teague, "Deterministic Phase Retrieval: a Green's Function Solution," J. Opt. Soc. Am. 73, 1434-1441 (1983).
[CrossRef]

1977

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

1965

W. C. Hamilton, "Significance tests on the crystallographic R factor," Acta Cryst. 18, 502-510 (1965).
[CrossRef]

1963

D. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," SIAM J. Appl. Math. 11, 431-441 (1963).
[CrossRef]

1961

S. Geller, "Parameter interaction in least squares structure refinement," Acta Cryst. 14, 1026-1035 (1961).
[CrossRef]

1953

V. E. Cosslett and W. C. Nixon, "The X-Ray Shadow Microscope," J. Appl. Phys. 24, 616-623 (1953).
[CrossRef]

1944

K. Levenberg, "A method for the solution of certain non-linear problems in least squares," Quart. Appl. Math. 2, 164-168 (1944).

Amemiya, Y.

Y. Amemiya, "Imaging plates for use with synchrotron radiation," J. Synch. Rad. 2, 13-21 (1995).
[CrossRef]

Y. Amemiya, K. Wakabayashi, H. Tanaka, Y. Ueno, and J. Miyahara, "Laser-stimulated luminescence used to measure X-ray diffraction of a contracting striated muscle," Science 237, 164-168 (1987).
[CrossRef] [PubMed]

Barnea, Z.

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

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. J. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996).
[CrossRef]

Baruchel, J.

P. Cloetens, J.-P. Guigay, C. De Martino, and J. Baruchel, "Fractional Talbot imaging of phase gratings with hard x rays," Opt. Lett. 22, 1059-1061 (1997).
[CrossRef] [PubMed]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. J. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996).
[CrossRef]

Birch, I. P.

C. J. Kotre, I. P. Birch and K. J. Robson, "Anomalous image quality phantom scores in magnification mammography: evidence of phase contrast enhancement," British J. Radiol. 75, 170-173 (2002).

Brennan, S.

S. Brennan and P. L. Cowan, "A suite of programs for calculating X-ray absorption, reflection, and diffraction performance for a variety of materials at arbitrary wavelengths," Rev. Sci. Instrum. 63, 850-853 (1992).
[CrossRef]

Chandler, K. M.

T. A. Shelkovenko, D. B. Sinars, S. A. Pikuz, K. M. Chandler and D. A. Hammer, "Point-projection x-ray radiography using an X pinch as the radiation source," Rev. Sci. Instrum. 72, 667-670 (2001).
[CrossRef]

Cloetens, P.

P. Cloetens, J.-P. Guigay, C. De Martino, and J. Baruchel, "Fractional Talbot imaging of phase gratings with hard x rays," Opt. Lett. 22, 1059-1061 (1997).
[CrossRef] [PubMed]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. J. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996).
[CrossRef]

Coman, I.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Cookson, D. F.

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

Cosslett, V. E.

V. E. Cosslett and W. C. Nixon, "The X-Ray Shadow Microscope," J. Appl. Phys. 24, 616-623 (1953).
[CrossRef]

Cowan, P. L.

S. Brennan and P. L. Cowan, "A suite of programs for calculating X-ray absorption, reflection, and diffraction performance for a variety of materials at arbitrary wavelengths," Rev. Sci. Instrum. 63, 850-853 (1992).
[CrossRef]

De Martino, C.

Fitzgerald, R.

R. Fitzgerald, "Phase-sensitive x-ray imaging," Physics Today 53, 23-26 (2000).
[CrossRef]

Foerster, E.

A. Krol, J. C. Kieffer and E. Foerster, "Laser-driven x-ray source for diagnostic radiology. Applications of X-rays Generated from Lasers and Other Bright Sources," Proc. SPIE 3157, 156-163 (1997).
[CrossRef]

Gao, D.

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus X-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays:," Nature 384, 335-338 (1996).
[CrossRef]

Geller, S.

S. Geller, "Parameter interaction in least squares structure refinement," Acta Cryst. 14, 1026-1035 (1961).
[CrossRef]

Guigay, J.-P.

P. Cloetens, J.-P. Guigay, C. De Martino, and J. Baruchel, "Fractional Talbot imaging of phase gratings with hard x rays," Opt. Lett. 22, 1059-1061 (1997).
[CrossRef] [PubMed]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. J. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996).
[CrossRef]

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

Gureyev, T.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Gureyev, T. E.

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

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093706 (2005).
[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]

T. E. Gureyev and S. W. Wilkins, "On X-ray phase imaging with a point source," J. Opt. Soc. Am. A 15, 579-85 (1998).
[CrossRef]

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays:," Nature 384, 335-338 (1996).
[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, 068103-1 - 068103-4 (2004).
[CrossRef] [PubMed]

Hamilton, W. C.

W. C. Hamilton, "Significance tests on the crystallographic R factor," Acta Cryst. 18, 502-510 (1965).
[CrossRef]

Hammer, D. A.

T. A. Shelkovenko, D. B. Sinars, S. A. Pikuz, K. M. Chandler and D. A. Hammer, "Point-projection x-ray radiography using an X pinch as the radiation source," Rev. Sci. Instrum. 72, 667-670 (2001).
[CrossRef]

Kieffer, J. C.

A. Krol, J. C. Kieffer and E. Foerster, "Laser-driven x-ray source for diagnostic radiology. Applications of X-rays Generated from Lasers and Other Bright Sources," Proc. SPIE 3157, 156-163 (1997).
[CrossRef]

Kieffer, J.-C.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Kincaid, R.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Kohn, V.

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, 5486-5492 (1995).
[CrossRef]

Kotre, C. J.

C. J. Kotre, I. P. Birch and K. J. Robson, "Anomalous image quality phantom scores in magnification mammography: evidence of phase contrast enhancement," British J. Radiol. 75, 170-173 (2002).

Krol, A.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

A. Krol, J. C. Kieffer and E. Foerster, "Laser-driven x-ray source for diagnostic radiology. Applications of X-rays Generated from Lasers and Other Bright Sources," Proc. SPIE 3157, 156-163 (1997).
[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, 5486-5492 (1995).
[CrossRef]

Levenberg, K.

K. Levenberg, "A method for the solution of certain non-linear problems in least squares," Quart. Appl. Math. 2, 164-168 (1944).

Lipson, E.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Margaritondo, G.

G. Margaritondo and G. Tromba, "Coherence-based edge diffraction sharpening of x-ray images: a simple model," J. Appl. Phys. 85, 3406-3408 (1999).
[CrossRef]

Marquardt, D.

D. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," SIAM J. Appl. Math. 11, 431-441 (1963).
[CrossRef]

Mayo, S. C.

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, 068103-1 - 068103-4 (2004).
[CrossRef] [PubMed]

Miyahara, J.

Y. Amemiya, K. Wakabayashi, H. Tanaka, Y. Ueno, and J. Miyahara, "Laser-stimulated luminescence used to measure X-ray diffraction of a contracting striated muscle," Science 237, 164-168 (1987).
[CrossRef] [PubMed]

Nesterets, Y.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Nesterets, Ya. I.

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

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093706 (2005).
[CrossRef]

Nixon, W. C.

V. E. Cosslett and W. C. Nixon, "The X-Ray Shadow Microscope," J. Appl. Phys. 24, 616-623 (1953).
[CrossRef]

Nugent, K. A.

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

Paganin, D.

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

Paganin, D. M.

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

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[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, 068103-1 - 068103-4 (2004).
[CrossRef] [PubMed]

Pikuz, S. A.

T. A. Shelkovenko, D. B. Sinars, S. A. Pikuz, K. M. Chandler and D. A. Hammer, "Point-projection x-ray radiography using an X pinch as the radiation source," Rev. Sci. Instrum. 72, 667-670 (2001).
[CrossRef]

Pogany, A.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

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

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093706 (2005).
[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. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus X-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays:," Nature 384, 335-338 (1996).
[CrossRef]

Robson, K. J.

C. J. Kotre, I. P. Birch and K. J. Robson, "Anomalous image quality phantom scores in magnification mammography: evidence of phase contrast enhancement," British J. Radiol. 75, 170-173 (2002).

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, 5486-5492 (1995).
[CrossRef]

Schlenker, M. J.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. J. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996).
[CrossRef]

Servol, M.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Shelkovenko, T. A.

T. A. Shelkovenko, D. B. Sinars, S. A. Pikuz, K. M. Chandler and D. A. Hammer, "Point-projection x-ray radiography using an X pinch as the radiation source," Rev. Sci. Instrum. 72, 667-670 (2001).
[CrossRef]

Sinars, D. B.

T. A. Shelkovenko, D. B. Sinars, S. A. Pikuz, K. M. Chandler and D. A. Hammer, "Point-projection x-ray radiography using an X pinch as the radiation source," Rev. Sci. Instrum. 72, 667-670 (2001).
[CrossRef]

Snigirev, A.

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, 5486-5492 (1995).
[CrossRef]

Snigireva, 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, 5486-5492 (1995).
[CrossRef]

Stevenson, A.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Stevenson, A. W.

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093706 (2005).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays:," Nature 384, 335-338 (1996).
[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, 068103-1 - 068103-4 (2004).
[CrossRef] [PubMed]

Tanaka, H.

Y. Amemiya, K. Wakabayashi, H. Tanaka, Y. Ueno, and J. Miyahara, "Laser-stimulated luminescence used to measure X-ray diffraction of a contracting striated muscle," Science 237, 164-168 (1987).
[CrossRef] [PubMed]

Teague, M. R.

M. R. Teague, "Deterministic Phase Retrieval: a Green's Function Solution," J. Opt. Soc. Am. 73, 1434-1441 (1983).
[CrossRef]

Toth, R.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Tromba, G.

G. Margaritondo and G. Tromba, "Coherence-based edge diffraction sharpening of x-ray images: a simple model," J. Appl. Phys. 85, 3406-3408 (1999).
[CrossRef]

Ueno, Y.

Y. Amemiya, K. Wakabayashi, H. Tanaka, Y. Ueno, and J. Miyahara, "Laser-stimulated luminescence used to measure X-ray diffraction of a contracting striated muscle," Science 237, 164-168 (1987).
[CrossRef] [PubMed]

Wakabayashi, K.

Y. Amemiya, K. Wakabayashi, H. Tanaka, Y. Ueno, and J. Miyahara, "Laser-stimulated luminescence used to measure X-ray diffraction of a contracting striated muscle," Science 237, 164-168 (1987).
[CrossRef] [PubMed]

Wilkins, S.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Wilkins, S. W.

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

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093706 (2005).
[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]

T. E. Gureyev and S. W. Wilkins, "On X-ray phase imaging with a point source," J. Opt. Soc. Am. A 15, 579-85 (1998).
[CrossRef]

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus X-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays:," Nature 384, 335-338 (1996).
[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, 068103-1 - 068103-4 (2004).
[CrossRef] [PubMed]

Ye, H.

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Acta Cryst.

S. Geller, "Parameter interaction in least squares structure refinement," Acta Cryst. 14, 1026-1035 (1961).
[CrossRef]

W. C. Hamilton, "Significance tests on the crystallographic R factor," Acta Cryst. 18, 502-510 (1965).
[CrossRef]

British J. Radiol.

C. J. Kotre, I. P. Birch and K. J. Robson, "Anomalous image quality phantom scores in magnification mammography: evidence of phase contrast enhancement," British J. Radiol. 75, 170-173 (2002).

J. Appl. Phys.

G. Margaritondo and G. Tromba, "Coherence-based edge diffraction sharpening of x-ray images: a simple model," J. Appl. Phys. 85, 3406-3408 (1999).
[CrossRef]

J. Appl. Phys.

V. E. Cosslett and W. C. Nixon, "The X-Ray Shadow Microscope," J. Appl. Phys. 24, 616-623 (1953).
[CrossRef]

J. Opt. Soc. Am.

M. R. Teague, "Deterministic Phase Retrieval: a Green's Function Solution," J. Opt. Soc. Am. 73, 1434-1441 (1983).
[CrossRef]

J. Opt. Soc. Am. A

T. E. Gureyev and S. W. Wilkins, "On X-ray phase imaging with a point source," J. Opt. Soc. Am. A 15, 579-85 (1998).
[CrossRef]

J. Phys. D: Appl. Phys.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. J. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996).
[CrossRef]

J. Synch. Rad.

Y. Amemiya, "Imaging plates for use with synchrotron radiation," J. Synch. Rad. 2, 13-21 (1995).
[CrossRef]

Nature

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays:," Nature 384, 335-338 (1996).
[CrossRef]

Opt. Commun.

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]

Opt. Lett.

Opt. Commun.

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

Optik

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

Phys. Rev. Lett.

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

Physics Today

R. Fitzgerald, "Phase-sensitive x-ray imaging," Physics Today 53, 23-26 (2000).
[CrossRef]

Proc. SPIE

A. Krol, J. C. Kieffer and E. Foerster, "Laser-driven x-ray source for diagnostic radiology. Applications of X-rays Generated from Lasers and Other Bright Sources," Proc. SPIE 3157, 156-163 (1997).
[CrossRef]

A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007).
[CrossRef]

Quart. Appl. Math.

K. Levenberg, "A method for the solution of certain non-linear problems in least squares," Quart. Appl. Math. 2, 164-168 (1944).

Rev. Sci. Instrum.

A. Pogany, D. Gao, and S. W. 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. Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Rev. Sci. Instrum.

T. A. Shelkovenko, D. B. Sinars, S. A. Pikuz, K. M. Chandler and D. A. Hammer, "Point-projection x-ray radiography using an X pinch as the radiation source," Rev. Sci. Instrum. 72, 667-670 (2001).
[CrossRef]

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093706 (2005).
[CrossRef]

S. Brennan and P. L. Cowan, "A suite of programs for calculating X-ray absorption, reflection, and diffraction performance for a variety of materials at arbitrary wavelengths," Rev. Sci. Instrum. 63, 850-853 (1992).
[CrossRef]

Science

Y. Amemiya, K. Wakabayashi, H. Tanaka, Y. Ueno, and J. Miyahara, "Laser-stimulated luminescence used to measure X-ray diffraction of a contracting striated muscle," Science 237, 164-168 (1987).
[CrossRef] [PubMed]

SIAM J. Appl. Math.

D. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," SIAM J. Appl. Math. 11, 431-441 (1963).
[CrossRef]

Other

J. S. Rollett, Computing Methods in Crystallography (Pergamon Press, Oxford, 1965).

J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975), Sec.3.4.2.

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, 068103-1 - 068103-4 (2004).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1.
Fig. 1.

In-line imaging geometry.

Fig. 2.
Fig. 2.

Phase shift distribution in the object plane after transmission of the incident plane X-ray wave through a blurred phase edge.

Fig. 3.
Fig. 3.

Plot of the spectral density distribution in the vicinity of the geometric image of the edge feature for different values of the Fresnel number. The hatched area indicates the location of the object.

Fig. 4.
Fig. 4.

Plot of the spatial resolution in in-line imaging as predicted by Eq. (16) (solid line), Eq. (17) (dotted line) and Eq. (6) (dash-dotted line).

Fig. 5.
Fig. 5.

Plot of the maximum image contrast in in-line imaging as predicted by Eq. (18) (solid line), Eq. (19) (dotted line) and Eq. (5) (dash-dotted line).

Fig. 6.
Fig. 6.

Plot of the ratio of the SNR to the product of the maximum image contrast and the square root of the integral X-ray intensity incident on the region of interest.

Fig. 7.
Fig. 7.

Incident X-ray spectrum used to calculate the values in Tables 2 and 3.

Fig. 8.
Fig. 8.

An image of a phantom used in the experimental tests acquired at R1=10 cm.

Tables (6)

Tables Icon

Table 1. Validity conditions for various approximations.

Tables Icon

Table 2. Comparison of the image characteristics obtained with Kirchhoff diffraction theory (“Exact”) and those obtained using analytical formulae derived in the present paper, Eqs. (6), (8)-(9) (“TIE”), using I in=10 ph/µm2, Ly =1 mm and R=2 m. The source size was 4 µm (FWHM) and the detector resolution was 100 µm (FWHM).

Tables Icon

Table 3. Comparison of the image characteristics obtained with Kirchhoff diffraction theory (“Exact”) and those obtained using analytical formulae derived in the present paper, Eqs. (6), (8)-(9) (“TIE”), using I in=10 ph/µm2 and Ly =1 mm and R=2 m. The source size was 100 µm (FWHM) and the detector resolution was 4 µm (FWHM).

Tables Icon

Table 4. Comparison of the image contrast and spatial resolution obtained with Kirchhoff diffraction theory (“Exact”) with those obtained using analytical formulae derived in the present paper. Analytical results are given for the general formulae, Eq. (16) and (18) (“WO”), as well as for the limiting cases of very large Fresnel numbers, Eq. (5) and Eq. (6) (“TIE”) and very small Fresnel numbers, Eq. (17) and (19) (“Far”). The source size was 4 µm (FWHM), the detector resolution was 100 µm (FWHM), λ=0.4 Å, |φ|max=1 rad and M=25.3.

Tables Icon

Table 5. Observed and calculated (corresponding to refined parameters in least-squares analysis) values of contrast and resolution, for both vertical and horizontal directions.

Tables Icon

Table 6. Refined parameter values from least-squares analysis of experimental data. Values in italics (and without an esd) were fixed. The values in square brackets are expressed as FWHM rather than σ (FWHM≈2.35σ) for convenience. RH is Hamilton’s R-factor and provides a measure of the agreement between theory and experiment [26].

Equations (27)

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φ ( x , y , ν ) = ( H * P obj ) ( x ) φ max ( ν ) ,
S ( M x , M y , R 2 , ν ) = M 2 S in ( ν ) [ 1 ( R k ) 2 φ ( x , y , ν ) * P sys ( x , y ) ] ,
S ( M x , M y , R 2 , ν ) = M 2 S in ( ν ) { 1 R ( k σ M 3 2 π ) φ max x exp [ x 2 ( 2 σ M 2 ) ] } .
( ν , A ) M L y Ma Ma S ( x , y , R 2 , ν ) S 0 ( ν ) d x = 2 S 0 ( ν ) M L y R k φ max ( ν ) 0 Ma x M 2 π σ M 3
× exp ( x 2 2 M 2 σ M 2 ) d x = 2 S in ( ν ) L y R k σ M 2 π φ max ( ν ) [ 1 exp ( a 2 2 σ M 2 ) ] .
SNR TIE ( ν ) = C 1 [ D TIE ( ν ) ] 1 2 N F 1 φ max ( ν ) ,
C max TIE ( ν ) = C 2 N F 1 φ max ( ν ) = C 3 SNR TIE ( ν ) [ D TIE ( ν ) ] 1 2 ,
( Δ x ) min TIE = 2 σ sys ( M ) .
I ( M x , M y , R 2 ) = M 2 I in [ 1 R I in 1 2 S in ( ν ) k 1 φ ( x , y , ν ) d ν * P sys ( x , y ) ] .
SNR TIE = C 1 ( D TIE ) 1 2 σ M 2 R ψ ̅ max ,
C max TIE = C 2 σ M 2 R ψ ̅ max = C 3 ( D TIE ) 1 2 SNR TIE ,
SNR TIE = γ L y 1 2 R σ M 3 2 S in ( ν ) φ max ( ν ) k 1 d ν ,
SNR TIE = γ R σ M 3 2 = γ R ( M 1 ) M 1 2 [ M 2 σ obj 2 + ( M 1 ) 2 σ src 2 + σ det 2 ] 3 4 ,
φ ( r + , ν ) φ ( r , ν ) 1 ,
S ̂ ( ξ M , η M , R 2 , ν ) = S in ( ν ) { δ ( ξ , η ) + sin [ π λ R ( ξ 2 + η 2 ) ] φ ̂ ( ξ , η , ν ) P ̂ sys ( ξ , η , M ) } ,
S ( M x , M y , R 2 , ν ) = M 2 S in ( ν ) { 1 φ max ( ν ) 2 n F π F [ x ( σ M n F ) , N F ] } ,
F ( x , N F ) = 0 x exp ( t 2 2 ) sin [ ( 1 2 ) arctan N F 1 t 2 ( 2 N F ) ] d t ,
( Δ x ) min = 2 σ sys ( M ) ( N F + N F 1 ) arctan N F 1 .
( Δ x ) min far = 2 σ sys ( M ) ( π 2 ) N F 1 = R λ .
C max ( ν ) = 2 π ( 1 + N F 2 ) 1 4 F ( N F arctan N F 1 , N F ) φ max ( ν ) .
C max far ( ν ) = 0.4876 φ max ( ν ) .
M L y 2 M x 0 2 M x 0 + S ( x , y , ν ) S 0 ( ν ) d x = S in ( ν ) L y φ max ( ν ) 8 n F π 0 2 x 0 + F [ x ( σ M n F ) , N F ] d x
= S in ( ν ) L y φ max ( ν ) σ M 8 n F 3 π 0 2 N F arctan N F 1 F ( t , N F ) d t .
SNR ( ν ) = α N F [ D ( ν ) ] 1 2 φ max ( ν ) ,
SNR far ( ν ) 0.1325 [ D far ( ν ) ] 1 2 φ max ( ν ) 0.2717 C max far ( ν ) [ D far ( ν ) ] 1 2 ,
γ ( N F ) SNR ( ν ) [ D ( ν ) ] 1 2 C max 1 ( ν ) ,
M 2 I ( M x , M y , R 2 ) = I in S in ( ν ) φ max ( ν ) 2 n F π F [ x ( σ M n F ) , N F ] d ν .

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