Abstract

Phase-contrast imaging methods exploit variations in an object’s refractive index distribution to permit the visualization of subtle features that may have very similar optical absorption properties. Although phase-contrast is often viewed as being desirable in many biomedical applications, its relative influence on signal detectability when both absorption- and phase-contrast are present remains relatively unexplored. In this work, we investigate the ideal Bayesian observer signal-to-noise ratio in phase-contrast imaging for a signal-known-exactly/background-known exactly detection task involving a weak signal. We demonstrate that this signal detectability measure can be decomposed into three contributions that have distinct interpretations associated with the imaging physics.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
    [CrossRef] [PubMed]
  2. K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
    [CrossRef] [PubMed]
  3. V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
    [CrossRef]
  4. D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
    [CrossRef] [PubMed]
  5. M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
    [CrossRef]
  6. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica (Amsterdam) 9, 686–698 (1942).
    [CrossRef]
  7. T. Wilhein, B. Kaulich, E. D. Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast X-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082 (2001).
    [CrossRef]
  8. T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
    [CrossRef] [PubMed]
  9. X. Wu and H. Liu, “Clinical implementation of x-ray phase-contrast imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169–2179 (2003).
    [CrossRef] [PubMed]
  10. E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (2003).
    [CrossRef] [PubMed]
  11. 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]
  12. 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]
  13. P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D 29, 133–146 (1996).
    [CrossRef]
  14. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
    [CrossRef]
  15. C. J. Kotre, I. P. Birch, and K. J. Robson, “Anomalous image quality phantom scores in magnification mammography: Evidence of phase contrast enhancement,” Br. J. Radiol. 75, 170–173 (2002).
    [PubMed]
  16. D. 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] [PubMed]
  17. B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
    [CrossRef]
  18. X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Methods Phys. Res. B 234, 563–572 (2005).
    [CrossRef]
  19. Y. 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]
  20. T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
    [CrossRef] [PubMed]
  21. H. H. Barrett and K. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics (Wiley, 2004).
  22. H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 7, 1266–1278 (1990).
    [CrossRef] [PubMed]
  23. H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. U.S.A. 90, 9758–9765 (1993).
    [CrossRef] [PubMed]
  24. R. F. Wagner and D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
    [CrossRef]
  25. M. F. Insana and T. J. Hall, “Visual detection efficiency in ultrasonic imaging: A framework for objective assessment of image quality,” J. Acoust. Soc. Am. 95, 2081 (1994).
    [CrossRef]
  26. J. G. Brankov, A. Saiz-Herranz, and M. N. Wernick, “Noise analysis for diffraction enhanced imaging,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2004), Vol. 2, pp. 1428–1431.
  27. J. G. Brankov, A. A. Saiz-Herranz, and M. N. Wernick, “Task-based evaluation of diffraction-enhanced imaging,” in 2005 IEEE Nuclear Science Symposium Conference Record (2006), Vol. 3, pp. 1539–1541.
    [CrossRef]
  28. K. Majidi, J. G. Brankov, and M. N. Wernick, “Sampling strategies in multiple-image radiography,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2008), pp. 688–691.
  29. C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line x-ray phase-contrast imaging,” Opt. Express , 17, 14466–14480 (2009).
    [CrossRef] [PubMed]
  30. C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. 37, 270–281 (2010).
    [CrossRef] [PubMed]
  31. ICRU, Medical Imaging—The Assessment of Image Quality, Rep. 54 (International Commission on Radiation Units and Measurements, 1996).
  32. D. M. Paganin, Coherent X-Ray Optics (Oxford U. Press, 2006).
    [CrossRef]
  33. D. M. Paganin and T. E. Gureyev, “Phase-contrast, phase retrieval and aberration balancing in shift-invariant linear imaging systems,” Opt. Commun. 281, 965–981 (2008).
    [CrossRef]
  34. D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
    [CrossRef]
  35. H. M. L. Faulkner, L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration determination and correction,” Opt. Commun. 216, 89–98 (2003).
    [CrossRef]
  36. K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D: Appl. Phys. 37, 2746–2750 (2004).
    [CrossRef]
  37. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2004).
  38. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  39. J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik (Stuttgart) 49, 121–125 (1977).
  40. D. Shi and M. A. Anastasio, “Relationships between smooth- and small-phase conditions in X-ray phase-contrast imaging,” IEEE Trans. Med. Imaging 28, 1969–1973 (2009).
    [CrossRef] [PubMed]
  41. P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” Proc. SPIE 4503, 82–91 (2002).
    [CrossRef]
  42. T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Stability and locality of amplitude and phase contrast tomographies,” Proc. SPIE 6318, 63180V (2006).
  43. D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, 1966).
  44. I. A. Cunningham and R. Shaw, “Signal-to-noise optimization of medical imaging systems,” J. Opt. Soc. Am. A 16, 621–632 (1999).
    [CrossRef]
  45. K. M. Hanson, “Variations in task and the ideal observer,” Proc. SPIE 419, 60–67 (1983).
  46. 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]
  47. Y. Nesterets, T. Gureyev, and S. Wilkins, “Polychromaticity in the combined PB/AB phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
    [CrossRef]
  48. P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. 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]
  49. M. Phelps, E. Hoffman, and M. M. Ter-Pogossian, “Attenuation coefficients of various body tissues, fluids, and lesions at photon energies of 18 to 136 keV,” Radiology 117, 573–583 (1975).
    [PubMed]
  50. J. A. Kalef-Ezra, A. H. Karantanas, T. Koligliatis, A. Boziari, and P. Tsekeris, “Electron density of tissues and breast cancer radiotherapy: a quantitative CT study,” Int. J. Radiat. Oncol., Biol., Phys. 41, 1209–1214 (1998).
    [CrossRef]
  51. G. Ullman, M. Sandborg, R. Hunt, D. R. Dance, and G. A. Carlsson, Implementation of Pathologies in the Monte Carlo Model Chest and Breast, Tech. Rep. 94 (Department of Radiation Physics, Linkoping University, 2003).
  52. R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004).
    [CrossRef] [PubMed]
  53. Y. Nesterets, T. Gureyev, D. Paganin, K. Pavlov, and S. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 37, 1262–1274 (2004).
    [CrossRef]
  54. J. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to x-ray analyser-based imaging,” Opt. Commun. 270, 180–188 (2007).
    [CrossRef]
  55. T. E. Gureyev and S. W. Wilkins, “Regimes of X-ray phase-contrast imaging with perfect crystals,” Nuovo Cimento D 19, 545–552 (1997).
    [CrossRef]
  56. J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics (Wiley, 2001).
  57. A. R. Pineda and H. H. Barrett, “What does DQE say about lesion detectability in digital radiography?” Proc. SPIE 4320, 561–569 (2001).
    [CrossRef]

2010 (1)

C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. 37, 270–281 (2010).
[CrossRef] [PubMed]

2009 (2)

D. Shi and M. A. Anastasio, “Relationships between smooth- and small-phase conditions in X-ray phase-contrast imaging,” IEEE Trans. Med. Imaging 28, 1969–1973 (2009).
[CrossRef] [PubMed]

C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line x-ray phase-contrast imaging,” Opt. Express , 17, 14466–14480 (2009).
[CrossRef] [PubMed]

2008 (3)

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
[CrossRef] [PubMed]

K. Majidi, J. G. Brankov, and M. N. Wernick, “Sampling strategies in multiple-image radiography,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2008), pp. 688–691.

D. M. Paganin and T. E. Gureyev, “Phase-contrast, phase retrieval and aberration balancing in shift-invariant linear imaging systems,” Opt. Commun. 281, 965–981 (2008).
[CrossRef]

2007 (1)

J. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to x-ray analyser-based imaging,” Opt. Commun. 270, 180–188 (2007).
[CrossRef]

2006 (4)

J. G. Brankov, A. A. Saiz-Herranz, and M. N. Wernick, “Task-based evaluation of diffraction-enhanced imaging,” in 2005 IEEE Nuclear Science Symposium Conference Record (2006), Vol. 3, pp. 1539–1541.
[CrossRef]

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

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

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

2005 (4)

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Methods Phys. Res. B 234, 563–572 (2005).
[CrossRef]

Y. 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]

Y. Nesterets, T. Gureyev, and S. Wilkins, “Polychromaticity in the combined PB/AB phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
[CrossRef]

2004 (9)

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004).
[CrossRef] [PubMed]

Y. Nesterets, T. Gureyev, D. Paganin, K. Pavlov, and S. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 37, 1262–1274 (2004).
[CrossRef]

J. G. Brankov, A. Saiz-Herranz, and M. N. Wernick, “Noise analysis for diffraction enhanced imaging,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2004), Vol. 2, pp. 1428–1431.

H. H. Barrett and K. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics (Wiley, 2004).

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D: Appl. Phys. 37, 2746–2750 (2004).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2004).

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

D. 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] [PubMed]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
[CrossRef]

2003 (5)

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

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

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (2003).
[CrossRef] [PubMed]

H. M. L. Faulkner, L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration determination and correction,” Opt. Commun. 216, 89–98 (2003).
[CrossRef]

G. Ullman, M. Sandborg, R. Hunt, D. R. Dance, and G. A. Carlsson, Implementation of Pathologies in the Monte Carlo Model Chest and Breast, Tech. Rep. 94 (Department of Radiation Physics, Linkoping University, 2003).

2002 (2)

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” Proc. SPIE 4503, 82–91 (2002).
[CrossRef]

C. J. Kotre, I. P. Birch, and K. J. Robson, “Anomalous image quality phantom scores in magnification mammography: Evidence of phase contrast enhancement,” Br. J. Radiol. 75, 170–173 (2002).
[PubMed]

2001 (3)

T. Wilhein, B. Kaulich, E. D. Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast X-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics (Wiley, 2001).

A. R. Pineda and H. H. Barrett, “What does DQE say about lesion detectability in digital radiography?” Proc. SPIE 4320, 561–569 (2001).
[CrossRef]

1999 (2)

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. 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]

I. A. Cunningham and R. Shaw, “Signal-to-noise optimization of medical imaging systems,” J. Opt. Soc. Am. A 16, 621–632 (1999).
[CrossRef]

1998 (1)

J. A. Kalef-Ezra, A. H. Karantanas, T. Koligliatis, A. Boziari, and P. Tsekeris, “Electron density of tissues and breast cancer radiotherapy: a quantitative CT study,” Int. J. Radiat. Oncol., Biol., Phys. 41, 1209–1214 (1998).
[CrossRef]

1997 (3)

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]

T. E. Gureyev and S. W. Wilkins, “Regimes of X-ray phase-contrast imaging with perfect crystals,” Nuovo Cimento D 19, 545–552 (1997).
[CrossRef]

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

1996 (4)

K. A. Nugent, T. E. Gureyev, D. 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. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D 29, 133–146 (1996).
[CrossRef]

ICRU, Medical Imaging—The Assessment of Image Quality, Rep. 54 (International Commission on Radiation Units and Measurements, 1996).

1995 (2)

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

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
[CrossRef]

1994 (1)

M. F. Insana and T. J. Hall, “Visual detection efficiency in ultrasonic imaging: A framework for objective assessment of image quality,” J. Acoust. Soc. Am. 95, 2081 (1994).
[CrossRef]

1993 (1)

H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. U.S.A. 90, 9758–9765 (1993).
[CrossRef] [PubMed]

1990 (1)

1985 (1)

R. F. Wagner and D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

1983 (1)

K. M. Hanson, “Variations in task and the ideal observer,” Proc. SPIE 419, 60–67 (1983).

1980 (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

1977 (1)

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

1975 (1)

M. Phelps, E. Hoffman, and M. M. Ter-Pogossian, “Attenuation coefficients of various body tissues, fluids, and lesions at photon energies of 18 to 136 keV,” Radiology 117, 573–583 (1975).
[PubMed]

1966 (1)

D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, 1966).

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica (Amsterdam) 9, 686–698 (1942).
[CrossRef]

Allen, L. J.

H. M. L. Faulkner, L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration determination and correction,” Opt. Commun. 216, 89–98 (2003).
[CrossRef]

Als-Nielsen, J.

J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics (Wiley, 2001).

Anastasio, M. A.

C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. 37, 270–281 (2010).
[CrossRef] [PubMed]

C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line x-ray phase-contrast imaging,” Opt. Express , 17, 14466–14480 (2009).
[CrossRef] [PubMed]

D. Shi and M. A. Anastasio, “Relationships between smooth- and small-phase conditions in X-ray phase-contrast imaging,” IEEE Trans. Med. Imaging 28, 1969–1973 (2009).
[CrossRef] [PubMed]

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Ardeli, F.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Arhatari, B. D.

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
[CrossRef]

Barnea, Z.

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

Barrett, H. H.

H. H. Barrett and K. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics (Wiley, 2004).

A. R. Pineda and H. H. Barrett, “What does DQE say about lesion detectability in digital radiography?” Proc. SPIE 4320, 561–569 (2001).
[CrossRef]

H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. U.S.A. 90, 9758–9765 (1993).
[CrossRef] [PubMed]

H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 7, 1266–1278 (1990).
[CrossRef] [PubMed]

Barrett, R.

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

Barty, A.

D. 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] [PubMed]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. 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]

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

Beliaevskaya, E. A.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
[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,” Br. J. Radiol. 75, 170–173 (2002).
[PubMed]

Boller, E.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” Proc. SPIE 4503, 82–91 (2002).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Boziari, A.

J. A. Kalef-Ezra, A. H. Karantanas, T. Koligliatis, A. Boziari, and P. Tsekeris, “Electron density of tissues and breast cancer radiotherapy: a quantitative CT study,” Int. J. Radiat. Oncol., Biol., Phys. 41, 1209–1214 (1998).
[CrossRef]

Brankov, J. G.

K. Majidi, J. G. Brankov, and M. N. Wernick, “Sampling strategies in multiple-image radiography,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2008), pp. 688–691.

J. G. Brankov, A. A. Saiz-Herranz, and M. N. Wernick, “Task-based evaluation of diffraction-enhanced imaging,” in 2005 IEEE Nuclear Science Symposium Conference Record (2006), Vol. 3, pp. 1539–1541.
[CrossRef]

J. G. Brankov, A. Saiz-Herranz, and M. N. Wernick, “Noise analysis for diffraction enhanced imaging,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2004), Vol. 2, pp. 1428–1431.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Brown, D. G.

R. F. Wagner and D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Cabrini, S.

T. Wilhein, B. Kaulich, E. D. Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast X-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Carlsson, G. A.

G. Ullman, M. Sandborg, R. Hunt, D. R. Dance, and G. A. Carlsson, Implementation of Pathologies in the Monte Carlo Model Chest and Breast, Tech. Rep. 94 (Department of Radiation Physics, Linkoping University, 2003).

Chapman, D.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Chou, C. -Y.

C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. 37, 270–281 (2010).
[CrossRef] [PubMed]

C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line x-ray phase-contrast imaging,” Opt. Express , 17, 14466–14480 (2009).
[CrossRef] [PubMed]

Cloetens, P.

J. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to x-ray analyser-based imaging,” Opt. Commun. 270, 180–188 (2007).
[CrossRef]

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” Proc. SPIE 4503, 82–91 (2002).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. 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]

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

Cookson, D.

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

Cunningham, I. A.

Dance, D. R.

G. Ullman, M. Sandborg, R. Hunt, D. R. Dance, and G. A. Carlsson, Implementation of Pathologies in the Monte Carlo Model Chest and Breast, Tech. Rep. 94 (Department of Radiation Physics, Linkoping University, 2003).

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Donnelly, E. F.

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (2003).
[CrossRef] [PubMed]

Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. 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]

Fabrizio, E. D.

T. Wilhein, B. Kaulich, E. D. Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast X-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Faulkner, H. M. L.

H. M. L. Faulkner, L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration determination and correction,” Opt. Commun. 216, 89–98 (2003).
[CrossRef]

Furukawa, A.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

Galatsanos, N. P.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[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]

Gmur, N.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2004).

Green, D. M.

D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, 1966).

Guigay, J.

J. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to x-ray analyser-based imaging,” Opt. Commun. 270, 180–188 (2007).
[CrossRef]

Guigay, J. P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. 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]

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

Guigay, J. -P.

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

Gureyev, T.

Y. Nesterets, T. Gureyev, and S. Wilkins, “Polychromaticity in the combined PB/AB phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
[CrossRef]

Y. Nesterets, T. Gureyev, D. Paganin, K. Pavlov, and S. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 37, 1262–1274 (2004).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
[CrossRef] [PubMed]

D. M. Paganin and T. E. Gureyev, “Phase-contrast, phase retrieval and aberration balancing in shift-invariant linear imaging systems,” Opt. Commun. 281, 965–981 (2008).
[CrossRef]

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

Y. 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]

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D: Appl. Phys. 37, 2746–2750 (2004).
[CrossRef]

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

T. E. Gureyev and S. W. Wilkins, “Regimes of X-ray phase-contrast imaging with perfect crystals,” Nuovo Cimento D 19, 545–552 (1997).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. 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]

Hall, T. J.

M. F. Insana and T. J. Hall, “Visual detection efficiency in ultrasonic imaging: A framework for objective assessment of image quality,” J. Acoust. Soc. Am. 95, 2081 (1994).
[CrossRef]

Hanson, K. M.

K. M. Hanson, “Variations in task and the ideal observer,” Proc. SPIE 419, 60–67 (1983).

Helfen, L.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” Proc. SPIE 4503, 82–91 (2002).
[CrossRef]

Hoffman, E.

M. Phelps, E. Hoffman, and M. M. Ter-Pogossian, “Attenuation coefficients of various body tissues, fluids, and lesions at photon energies of 18 to 136 keV,” Radiology 117, 573–583 (1975).
[PubMed]

Honda, C.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Hunt, R.

G. Ullman, M. Sandborg, R. Hunt, D. R. Dance, and G. A. Carlsson, Implementation of Pathologies in the Monte Carlo Model Chest and Breast, Tech. Rep. 94 (Department of Radiation Physics, Linkoping University, 2003).

Ingal, V. N.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
[CrossRef]

Insana, M. F.

M. F. Insana and T. J. Hall, “Visual detection efficiency in ultrasonic imaging: A framework for objective assessment of image quality,” J. Acoust. Soc. Am. 95, 2081 (1994).
[CrossRef]

Johnston, R.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Kalef-Ezra, J. A.

J. A. Kalef-Ezra, A. H. Karantanas, T. Koligliatis, A. Boziari, and P. Tsekeris, “Electron density of tissues and breast cancer radiotherapy: a quantitative CT study,” Int. J. Radiat. Oncol., Biol., Phys. 41, 1209–1214 (1998).
[CrossRef]

Karantanas, A. H.

J. A. Kalef-Ezra, A. H. Karantanas, T. Koligliatis, A. Boziari, and P. Tsekeris, “Electron density of tissues and breast cancer radiotherapy: a quantitative CT study,” Int. J. Radiat. Oncol., Biol., Phys. 41, 1209–1214 (1998).
[CrossRef]

Kaulich, B.

T. Wilhein, B. Kaulich, E. D. Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast X-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Kitchen, M.

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[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]

Koligliatis, T.

J. A. Kalef-Ezra, A. H. Karantanas, T. Koligliatis, A. Boziari, and P. Tsekeris, “Electron density of tissues and breast cancer radiotherapy: a quantitative CT study,” Int. J. Radiat. Oncol., Biol., Phys. 41, 1209–1214 (1998).
[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,” Br. J. Radiol. 75, 170–173 (2002).
[PubMed]

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]

Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. 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]

Lewis, R.

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004).
[CrossRef] [PubMed]

Lewis, R. A.

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D: Appl. Phys. 37, 2746–2750 (2004).
[CrossRef]

Liu, H.

X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Methods Phys. Res. B 234, 563–572 (2005).
[CrossRef]

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

Ludwig, W.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” Proc. SPIE 4503, 82–91 (2002).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. 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]

Mache, R.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” Proc. SPIE 4503, 82–91 (2002).
[CrossRef]

Majidi, K.

K. Majidi, J. G. Brankov, and M. N. Wernick, “Sampling strategies in multiple-image radiography,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2008), pp. 688–691.

Mancuso, A. P.

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
[CrossRef]

Matsuo, S.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Mcmahon, P. J.

D. 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] [PubMed]

McMorrow, D.

J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics (Wiley, 2001).

Menk, R.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Miller, P. R.

Morgan, M. J.

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D: Appl. Phys. 37, 2746–2750 (2004).
[CrossRef]

Muehleman, C.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Murata, K.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Myers, G. R.

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

Myers, K.

H. H. Barrett and K. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics (Wiley, 2004).

Myers, K. J.

H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. U.S.A. 90, 9758–9765 (1993).
[CrossRef] [PubMed]

Nesterets, Y.

Y. Nesterets, T. Gureyev, and S. Wilkins, “Polychromaticity in the combined PB/AB phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
[CrossRef]

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D: Appl. Phys. 37, 2746–2750 (2004).
[CrossRef]

Y. Nesterets, T. Gureyev, D. Paganin, K. Pavlov, and S. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 37, 1262–1274 (2004).
[CrossRef]

Nesterets, Y. I.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
[CrossRef] [PubMed]

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

Y. 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]

Nitta, N.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Noma, K.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Nugent, K. A.

D. 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] [PubMed]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
[CrossRef]

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

Ohara, H.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Oltulu, O.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Ota, S.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Oxley, M. P.

H. M. L. Faulkner, L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration determination and correction,” Opt. Commun. 216, 89–98 (2003).
[CrossRef]

Paganin, D.

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

Y. Nesterets, T. Gureyev, D. Paganin, K. Pavlov, and S. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 37, 1262–1274 (2004).
[CrossRef]

D. 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] [PubMed]

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D: Appl. Phys. 37, 2746–2750 (2004).
[CrossRef]

H. M. L. Faulkner, L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration determination and correction,” Opt. Commun. 216, 89–98 (2003).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. 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.

D. M. Paganin and T. E. Gureyev, “Phase-contrast, phase retrieval and aberration balancing in shift-invariant linear imaging systems,” Opt. Commun. 281, 965–981 (2008).
[CrossRef]

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

Pagot, E.

J. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to x-ray analyser-based imaging,” Opt. Commun. 270, 180–188 (2007).
[CrossRef]

Pavlov, K.

Y. Nesterets, T. Gureyev, D. Paganin, K. Pavlov, and S. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 37, 1262–1274 (2004).
[CrossRef]

Pavlov, K. M.

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

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D: Appl. Phys. 37, 2746–2750 (2004).
[CrossRef]

Peele, A. G.

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
[CrossRef]

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Phelps, M.

M. Phelps, E. Hoffman, and M. M. Ter-Pogossian, “Attenuation coefficients of various body tissues, fluids, and lesions at photon energies of 18 to 136 keV,” Radiology 117, 573–583 (1975).
[PubMed]

Pickens, D. R.

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (2003).
[CrossRef] [PubMed]

Pineda, A. R.

A. R. Pineda and H. H. Barrett, “What does DQE say about lesion detectability in digital radiography?” Proc. SPIE 4320, 561–569 (2001).
[CrossRef]

Pisano, E.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Pogany, A.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
[CrossRef] [PubMed]

Y. 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]

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]

Price, R. R.

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (2003).
[CrossRef] [PubMed]

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,” Br. J. Radiol. 75, 170–173 (2002).
[PubMed]

Rolland, J. P.

H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. U.S.A. 90, 9758–9765 (1993).
[CrossRef] [PubMed]

Romanato, F.

T. Wilhein, B. Kaulich, E. D. Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast X-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Saiz-Herranz, A.

J. G. Brankov, A. Saiz-Herranz, and M. N. Wernick, “Noise analysis for diffraction enhanced imaging,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2004), Vol. 2, pp. 1428–1431.

Saiz-Herranz, A. A.

J. G. Brankov, A. A. Saiz-Herranz, and M. N. Wernick, “Task-based evaluation of diffraction-enhanced imaging,” in 2005 IEEE Nuclear Science Symposium Conference Record (2006), Vol. 3, pp. 1539–1541.
[CrossRef]

Sakashita, Y.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Salvo, L.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” Proc. SPIE 4503, 82–91 (2002).
[CrossRef]

Sandborg, M.

G. Ullman, M. Sandborg, R. Hunt, D. R. Dance, and G. A. Carlsson, Implementation of Pathologies in the Monte Carlo Model Chest and Breast, Tech. Rep. 94 (Department of Radiation Physics, Linkoping University, 2003).

Sayers, D.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

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.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” Proc. SPIE 4503, 82–91 (2002).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. 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]

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

Shaw, R.

Shi, D.

D. Shi and M. A. Anastasio, “Relationships between smooth- and small-phase conditions in X-ray phase-contrast imaging,” IEEE Trans. Med. Imaging 28, 1969–1973 (2009).
[CrossRef] [PubMed]

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. W.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
[CrossRef] [PubMed]

Y. 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]

Susini, J.

T. Wilhein, B. Kaulich, E. D. Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast X-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Swets, J. A.

D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, 1966).

Takahashi, M.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Tanaka, T.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Ter-Pogossian, M. M.

M. Phelps, E. Hoffman, and M. M. Ter-Pogossian, “Attenuation coefficients of various body tissues, fluids, and lesions at photon energies of 18 to 136 keV,” Radiology 117, 573–583 (1975).
[PubMed]

Thomlinson, W.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Tsekeris, P.

J. A. Kalef-Ezra, A. H. Karantanas, T. Koligliatis, A. Boziari, and P. Tsekeris, “Electron density of tissues and breast cancer radiotherapy: a quantitative CT study,” Int. J. Radiat. Oncol., Biol., Phys. 41, 1209–1214 (1998).
[CrossRef]

Tsuchiya, K.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Ullman, G.

G. Ullman, M. Sandborg, R. Hunt, D. R. Dance, and G. A. Carlsson, Implementation of Pathologies in the Monte Carlo Model Chest and Breast, Tech. Rep. 94 (Department of Radiation Physics, Linkoping University, 2003).

Wagner, R. F.

R. F. Wagner and D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

Washburn, D.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Wernick, M. N.

K. Majidi, J. G. Brankov, and M. N. Wernick, “Sampling strategies in multiple-image radiography,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2008), pp. 688–691.

J. G. Brankov, A. A. Saiz-Herranz, and M. N. Wernick, “Task-based evaluation of diffraction-enhanced imaging,” in 2005 IEEE Nuclear Science Symposium Conference Record (2006), Vol. 3, pp. 1539–1541.
[CrossRef]

J. G. Brankov, A. Saiz-Herranz, and M. N. Wernick, “Noise analysis for diffraction enhanced imaging,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2004), Vol. 2, pp. 1428–1431.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Wilhein, T.

T. Wilhein, B. Kaulich, E. D. Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast X-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Wilkins, S.

Y. Nesterets, T. Gureyev, and S. Wilkins, “Polychromaticity in the combined PB/AB phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
[CrossRef]

Y. Nesterets, T. Gureyev, D. Paganin, K. Pavlov, and S. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 37, 1262–1274 (2004).
[CrossRef]

Wilkins, S. W.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
[CrossRef] [PubMed]

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

Y. 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 and S. W. Wilkins, “Regimes of X-ray phase-contrast imaging with perfect crystals,” Nuovo Cimento D 19, 545–552 (1997).
[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]

Wirjadi, O.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Wu, X.

X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Methods Phys. Res. B 234, 563–572 (2005).
[CrossRef]

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

Yamada, A.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Yamasaki, M.

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

Yan, A.

X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Methods Phys. Res. B 234, 563–572 (2005).
[CrossRef]

Yang, Y.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Yao, J.

H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. U.S.A. 90, 9758–9765 (1993).
[CrossRef] [PubMed]

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica (Amsterdam) 9, 686–698 (1942).
[CrossRef]

Zhong, Z.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

T. Wilhein, B. Kaulich, E. D. Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast X-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. 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]

Br. J. Radiol. (1)

C. J. Kotre, I. P. Birch, and K. J. Robson, “Anomalous image quality phantom scores in magnification mammography: Evidence of phase contrast enhancement,” Br. J. Radiol. 75, 170–173 (2002).
[PubMed]

IEEE Trans. Med. Imaging (1)

D. Shi and M. A. Anastasio, “Relationships between smooth- and small-phase conditions in X-ray phase-contrast imaging,” IEEE Trans. Med. Imaging 28, 1969–1973 (2009).
[CrossRef] [PubMed]

Int. J. Radiat. Oncol., Biol., Phys. (1)

J. A. Kalef-Ezra, A. H. Karantanas, T. Koligliatis, A. Boziari, and P. Tsekeris, “Electron density of tissues and breast cancer radiotherapy: a quantitative CT study,” Int. J. Radiat. Oncol., Biol., Phys. 41, 1209–1214 (1998).
[CrossRef]

Invest. Radiol. (1)

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[CrossRef] [PubMed]

J. Acoust. Soc. Am. (1)

M. F. Insana and T. J. Hall, “Visual detection efficiency in ultrasonic imaging: A framework for objective assessment of image quality,” J. Acoust. Soc. Am. 95, 2081 (1994).
[CrossRef]

J. Microsc. (1)

D. 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] [PubMed]

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

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

J. Phys. D: Appl. Phys. (4)

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
[CrossRef]

Y. Nesterets, T. Gureyev, and S. Wilkins, “Polychromaticity in the combined PB/AB phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
[CrossRef]

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D: Appl. Phys. 37, 2746–2750 (2004).
[CrossRef]

Y. Nesterets, T. Gureyev, D. Paganin, K. Pavlov, and S. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 37, 1262–1274 (2004).
[CrossRef]

Med. Phys. (3)

C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. 37, 270–281 (2010).
[CrossRef] [PubMed]

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

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (2003).
[CrossRef] [PubMed]

Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Nature (2)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[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]

Nucl. Instrum. Methods Phys. Res. B (1)

X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Methods Phys. Res. B 234, 563–572 (2005).
[CrossRef]

Nuovo Cimento D (1)

T. E. Gureyev and S. W. Wilkins, “Regimes of X-ray phase-contrast imaging with perfect crystals,” Nuovo Cimento D 19, 545–552 (1997).
[CrossRef]

Opt. Commun. (4)

J. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to x-ray analyser-based imaging,” Opt. Commun. 270, 180–188 (2007).
[CrossRef]

D. M. Paganin and T. E. Gureyev, “Phase-contrast, phase retrieval and aberration balancing in shift-invariant linear imaging systems,” Opt. Commun. 281, 965–981 (2008).
[CrossRef]

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

H. M. L. Faulkner, L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration determination and correction,” Opt. Commun. 216, 89–98 (2003).
[CrossRef]

Opt. Express (2)

Optik (Stuttgart) (1)

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

Phys. Med. Biol. (4)

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004).
[CrossRef] [PubMed]

R. F. Wagner and D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

Phys. Rev. Lett. (1)

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

Physica (Amsterdam) (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica (Amsterdam) 9, 686–698 (1942).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. U.S.A. 90, 9758–9765 (1993).
[CrossRef] [PubMed]

Proc. SPIE (4)

A. R. Pineda and H. H. Barrett, “What does DQE say about lesion detectability in digital radiography?” Proc. SPIE 4320, 561–569 (2001).
[CrossRef]

K. M. Hanson, “Variations in task and the ideal observer,” Proc. SPIE 419, 60–67 (1983).

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” Proc. SPIE 4503, 82–91 (2002).
[CrossRef]

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

Radiology (1)

M. Phelps, E. Hoffman, and M. M. Ter-Pogossian, “Attenuation coefficients of various body tissues, fluids, and lesions at photon energies of 18 to 136 keV,” Radiology 117, 573–583 (1975).
[PubMed]

Rev. Sci. Instrum. (4)

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]

Y. 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]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
[CrossRef]

Other (11)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2004).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, 1966).

ICRU, Medical Imaging—The Assessment of Image Quality, Rep. 54 (International Commission on Radiation Units and Measurements, 1996).

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

J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics (Wiley, 2001).

G. Ullman, M. Sandborg, R. Hunt, D. R. Dance, and G. A. Carlsson, Implementation of Pathologies in the Monte Carlo Model Chest and Breast, Tech. Rep. 94 (Department of Radiation Physics, Linkoping University, 2003).

J. G. Brankov, A. Saiz-Herranz, and M. N. Wernick, “Noise analysis for diffraction enhanced imaging,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2004), Vol. 2, pp. 1428–1431.

J. G. Brankov, A. A. Saiz-Herranz, and M. N. Wernick, “Task-based evaluation of diffraction-enhanced imaging,” in 2005 IEEE Nuclear Science Symposium Conference Record (2006), Vol. 3, pp. 1539–1541.
[CrossRef]

K. Majidi, J. G. Brankov, and M. N. Wernick, “Sampling strategies in multiple-image radiography,” in Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging (2008), pp. 688–691.

H. H. Barrett and K. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics (Wiley, 2004).

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 (8)

Fig. 1
Fig. 1

Schematic of a generic x-ray phase-contrast imaging system.

Fig. 2
Fig. 2

Schematic of the object model employed in our signal detection studies.

Fig. 3
Fig. 3

Plot of the values of the SNR 2 components as a function of beam energy in propagation-based imaging.

Fig. 4
Fig. 4

Plot of the relative values of the SNR 2 components as a function of beam energy in propagation-based imaging.

Fig. 5
Fig. 5

Plot of the relative values of the SNR 2 components as a function of the system blur FWHM in propagation-based imaging.

Fig. 6
Fig. 6

Plot of the values of the SNR 2 components as a function of beam energy in analyzer-based imaging.

Fig. 7
Fig. 7

Plot of the relative values of the SNR 2 components as a function of beam energy in analyzer-based imaging.

Fig. 8
Fig. 8

Plot of the relative values of the SNR 2 components as a function of the system blur FWHM in analyzer-based imaging.

Equations (72)

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

n ( r ) 1 Δ ( r ) + j β ( r ) ,
μ ( r ) = 2 k β ( r ) ,
U o ( x , y ) = T ( x , y ) U i ,
T ( x , y ) = M ( x , y ) exp [ j ϕ ( x , y ) ] .
A ( x , y ) = k d z β ( r )
ϕ ( x , y ) = k d z Δ ( r ) ,
U m ( x , y ) = G m ( x , y ) U o ( x , y ) ,
I m ( x , y ) = | U m ( x , y ) | 2 ,
| A ( x , y ) | 1 ,
| ϕ ( x , y ) ϕ ( x ̂ , y ̂ ) | 1 ,
I ̃ m ( u , v ) = d x d y I m ( x , y ) exp [ j 2 π ( u x + v y ) ] F 2 { I m ( x , y ) } ,
I ̃ m ( u , v ) | G ̃ m ( 0 , 0 ) | 2 I i = δ ( u , v ) 2 G ̃ m a ( u , v ) A ̃ ( u , v ) 2 G ̃ m p ( u , v ) ϕ ̃ ( u , v ) ,
G ̃ m a ( u , v ) = 1 2 [ G ̃ m ( u , v ) G ̃ m ( 0 , 0 ) + G ̃ m ( u , v ) G ̃ m ( 0 , 0 ) ] ,
G ̃ m p ( u , v ) = 1 2 j [ G ̃ m ( u , v ) G ̃ m ( 0 , 0 ) G ̃ m ( u , v ) G ̃ m ( 0 , 0 ) ] .
g ( x , y ) = f ( x , y ) h ( x , y ) + n ( x , y ) ,
H 0 : g ( x , y ) = f ( x , y ) h ( x , y ) + n ( x , y ) ,
H 1 : g ( x , y ) = ( f ( x , y ) + Δ f ( x , y ) ) h ( x , y ) + n ( x , y ) ,
t ( g ) = p r ( g H 1 ) p r ( g H 0 ) ,
SNR t 1 t 0 1 2 σ 1 2 + 1 2 σ 0 2 ,
NEQ ( u , v ) MTF 2 ( u , v ) N ( u , v )
SNR h 2 = d u d v | Δ f ˜ ( u , v ) | 2 W ( u , v ) NEQ ( u , v ) ,
n s ( r ) 1 Δ s ( r ) + j β s ( r )
n b ( z ) 1 Δ b ( z ) + j β b ( z ) ,
A d ( x , y ) k t s ( x , y ) d z ( β s ( r ) β b ( z ) ) ,
ϕ d ( x , y ) k t s ( x , y ) d z ( Δ s ( r ) Δ b ( z ) ) ,
A b = k L d z β b ( z ) .
I ̃ m , 1 ( u , v ) = I i | G ̃ m ( 0 , 0 ) | 2 exp ( 2 A b ) { δ ( u , v ) 2 G ̃ m a ( u , v ) A ̃ d ( u , v ) 2 G ̃ m p ( u , v ) ϕ ̃ d ( u , v ) } ,
I ̃ m , 0 ( u , v ) = I i | G ̃ m ( 0 , 0 ) | 2 exp ( 2 A b ) δ ( u , v ) ,
g m ( x , y ) = I m ( x , y ) h ( x , y ) + n ( x , y ) ,
H 0 : g m ( x , y ) = I m , 0 ( x , y ) h ( x , y ) + n ( x , y ) ,
H 1 : g m ( x , y ) = I m , 1 ( x , y ) h ( x , y ) + n ( x , y ) .
Δ I m ˜ ( u , v ) = 2 I i | G ̃ m ( 0 , 0 ) | 2 exp ( 2 A b ) { G ̃ m a ( u , v ) A ̃ d ( u , v ) + G ̃ m p ( u , v ) ϕ ̃ d ( u , v ) } .
SNR 2 = d u d v | Δ I m ˜ ( u , v ) | 2 NEQ ( u , v ) .
SNR 2 = SNR A 2 + SNR ϕ 2 + SNR c c 2 ,
SNR A 2 = 4 I i 2 | G ̃ m ( 0 , 0 ) | 4 exp ( 4 A b ) d u d v | G ̃ m a ( u , v ) A ̃ d ( u , v ) | 2 NEQ ( u , v ) ,
SNR ϕ 2 = 4 I i 2 | G ̃ m ( 0 , 0 ) | 4 exp ( 4 A b ) d u d v | G ̃ m p ( u , v ) ϕ ̃ d ( u , v ) | 2 NEQ ( u , v ) ,
SNR c c 2 = 8 I i 2 | G ̃ m ( 0 , 0 ) | 4 exp ( 4 A b ) d u d v   Re { ( G ̃ m a ( u , v ) A ̃ d ( u , v ) ) ( G ̃ m p ( u , v ) ϕ ̃ d ( u , v ) ) } NEQ ( u , v ) ,
I m , 1 ( x , y ) = I i | G ̃ m ( 0 , 0 ) | 2 exp ( 2 A b ) I m A ( x , y ) I m P ( x , y ) ,
I m A ( x , y ) 2 I i | G ̃ m ( 0 , 0 ) | 2 exp ( 2 A b ) F 2 1 { G ̃ m a ( u , v ) A ̃ d ( u , v ) } ,
I m P ( x , y ) 2 I i | G ̃ m ( 0 , 0 ) | 2 exp ( 2 A b ) F 2 1 { G ̃ m p ( u , v ) ϕ ̃ d ( u , v ) } .
G ̃ m a ( u , v ) = cos ( π λ z m ( u 2 + v 2 ) ) ,
G ̃ m p ( u , v ) = sin ( π λ z m ( u 2 + v 2 ) ) .
SNR A 2 = 4 I i 2   exp ( 4 A b ) d u d v | A ̃ d ( u , v ) | 2 cos 2 ( π λ z m ( u 2 + v 2 ) ) NEQ ( u , v ) ,
SNR ϕ 2 = 4 I i 2   exp ( 4 A b ) d u d v | ϕ ̃ d ( u , v ) | 2 sin 2 ( π λ z m ( u 2 + v 2 ) ) NEQ ( u , v ) ,
SNR c c 2 = 8 I i 2   exp ( 4 A b ) d u d v   Re { A ̃ d ( u , v ) ϕ ̃ d ( u , v ) } cos ( π λ z m ( u 2 + v 2 ) ) sin ( π λ z m ( u 2 + v 2 ) ) NEQ ( u , v ) .
G ̃ m a ( u , v ) 1 ,
G ̃ m p ( u , v ) π λ z m ( u 2 + v 2 ) ,
Δ I m ˜ ( u , v ) 2 I i   exp ( 2 A b ) { A ̃ d ( u , v ) π λ z m ( u 2 + v 2 ) ϕ ̃ d ( u , v ) } .
SNR A 2 = 4 I i 2   exp ( 4 A b ) d u d v | A ̃ d ( u , v ) | 2 NEQ ( u , v ) ,
Δ I m ( x , y ) 2 I i   exp ( 2 A b ) { A d ( x , y ) + λ z m 4 π 2 ϕ d ( x , y ) } ,
n e q 1 / 2 ( x , y ) F 2 1 { MTF ( u , v ) N ( u , v ) } ,
SNR A 2 = 4 I i 2   exp ( 4 A b ) d x d y | A d ( x , y ) n e q 1 / 2 ( x , y ) | 2 ,
SNR ϕ 2 = I i 2   exp ( 4 A b ) λ 2 z m 2 4 π 2 d x d y | 2 ϕ d ( x , y ) n e q 1 / 2 ( x , y ) | 2 ,
SNR c c 2 = 2 π I i 2   exp ( 4 A b ) λ z m d x d y   Re { ( A d ( x , y ) n e q 1 / 2 ( x , y ) ) ( 2 ϕ d ( x , y ) n e q 1 / 2 ( x , y ) ) } .
NEQ ( u , v ) = MTF det 2 ( u , v ) MTF coh 2 ( u , v ) N ( u , v ) ,
A ̃ d ( u , v ) = π k R 2 β ( E ) [ J 2 ( 2 π R u ) + J 0 ( 2 π R u ) ] δ ( v ) ,
ϕ ̃ d ( u , v ) = π k R 2 Δ ( E ) [ J 2 ( 2 π R u ) + J 0 ( 2 π R u ) ] δ ( v ) ,
G ̃ m a ( u ) = 1 2 [ G ̃ 0 ( u + ω m ) G ̃ 0 ( ω m ) + G ̃ 0 ( ω m u ) G ̃ 0 ( ω m ) ] ,
G ̃ m p ( u ) = j 2 [ G ̃ 0 ( u + ω m ) G ̃ 0 ( ω m ) G ̃ 0 ( ω m u ) G ̃ 0 ( ω m ) ] ,
G ̃ 0 ( u + ω m ) G ̃ 0 ( ω m ) + | u d G ̃ 0 ( u ) d u | u = ω m ,
G ̃ 0 ( ω m u ) G ̃ 0 ( ω m ) | u d G ̃ 0 ( u ) d u | u = ω m .
G ̃ m a ( u ) 1 + j b m u ,
G ̃ m p ( u ) | j u 2 R ( ω m ) d R ( u ) d u | u = ω m .
b m 1 R ( ω m ) Im { | G ̃ 0 ( ω m ) d G ̃ 0 ( u ) d u | u = ω m } ,
Δ I m ˜ ( u , v ) 2 I i R ( ω m ) exp ( 2 A b ) { ( 1 + j b m u ) A ̃ d ( u , v ) + | j u 2 R ( ω m ) d R ( u ) d u | u = ω m ϕ ̃ d ( u , v ) } ,
Δ I m ( x , y ) 2 I i R ( ω m ) exp ( 2 A b ) { ( 1 + b m 2 π x ) A d ( x , y ) + | 1 4 π R ( ω m ) d R ( u ) d u | u = ω m x ϕ d ( x , y ) } .
SNR A 2 = 4 I i 2 R 2 ( ω m ) exp ( 4 A b ) d u d v | A ̃ d ( u , v ) | 2 ( 1 + b m 2 u 2 ) NEQ ( u , v ) ,
SNR ϕ 2 = I i 2   exp ( 4 A b ) ( d R ( ω m ) d u ) 2 d u d v | ϕ ̃ d ( u , v ) | 2 u 2 NEQ ( u , v ) ,
SNR c c 2 = 4 I i 2 R ( ω m ) exp ( 4 A b ) | d R ( u ) d u | u = ω m d u d v   Im { A ̃ d ( u , v ) ϕ ̃ d ( u , v ) ( 1 + j b m u ) } u NEQ ( u , v ) .
SNR A 2 = 4 I i 2 R 2 ( ω m ) exp ( 4 A b ) d x d y | ( 1 + b m 2 π x ) A d ( x , y ) n e q 1 / 2 ( x , y ) | 2 ,
SNR ϕ 2 = I i 2   exp ( 4 A b ) 4 π 2 ( d R ( ω m ) d u ) 2 d x d y | x ϕ d ( x , y ) n e q 1 / 2 ( x , y ) | 2 ,
SNR c c 2 = 2 π I i 2 R ( ω m ) exp ( 4 A b ) | d R ( u ) d u | u = ω m d x d y   Re { ( ( 1 + b m 2 π x ) A d ( x , y ) n e q 1 / 2 ( x , y ) ) ( x ϕ d ( x , y ) n e q 1 / 2 ( x , y ) ) } .

Metrics