Abstract

X-ray phase contrast imaging (PCI) combined with phase retrieval has the potential to improve soft-material visibility and discrimination. This work examined the accuracy, image quality gains, and robustness of a spectral phase retrieval method proposed by our group. Spectroscopic PCI measurements of a physical phantom were obtained using state-of-the-art photon-counting detectors in combination with a polychromatic x-ray source. The phantom consisted of four poorly attenuating materials. Excellent accuracy was demonstrated in simultaneously retrieving the complete refractive properties (photoelectric absorption, attenuation, and phase) of these materials. Approximately 10 times higher SNR was achieved in retrieved images compared to the original PCI intensity image. These gains are also shown to be robust against increasing quantum noise, even for acquisition times as low as 1 s with a low-flux microfocus x-ray tube (average counts of 250 photons/pixels). We expect that this spectral phase retrieval method, adaptable to several PCI geometries, will allow significant dose reduction and improved material discrimination in clinical and industrial x-ray imaging applications.

© 2020 Optical Society of America

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    [Crossref]
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    [Crossref]
  46. M. Das, H. C. Gifford, J. M. O’Connor, and S. J. Glick, “Penalized maximum likelihood reconstruction for improved microcalcification detection in breast tomosynthesis,” IEEE Trans. Med. Imaging 30, 904–914 (2010).
    [Crossref]
  47. H. C. Gifford, Z. Liang, and M. Das, “Visual-search observers for assessing tomographic x-ray image quality,” Med. Phys. 43, 1563–1575 (2016).
    [Crossref]

2019 (4)

I. Vazquez, N. R. Fredette, and M. Das, “Quantitative phase retrieval of heterogeneous samples from spectral x-ray measurements,” Proc. SPIE 10948, 1157–1164 (2019).
[Crossref]

N. R. Fredette, A. Kavuri, and M. Das, “Multi-step material decomposition for spectral computed tomography,” Phys. Med. Biol. 64, 145001 (2019).
[Crossref]

C. E. Lewis and M. Das, “Spectral signatures of X-ray scatter using energy-resolving photon-counting detectors,” Sensors 19, 5022 (2019).
[Crossref]

C. Lewis, I. Vazquez, S. Vespucci, and M. Das, “Contribution of scatter and beam hardening to phase contrast imaging,” Proc. SPIE 10948, 1274–1280 (2019).
[Crossref]

2018 (2)

S. Vespucci, C. S. Park, R. Torrico, and M. Das, “Robust energy calibration technique for photon counting spectral detectors,” IEEE Trans. Med. Imaging 38, 968–978 (2018).
[Crossref]

S. Vespucci, C. Lewis, C. S. Park, and M. Das, “Examining phase contrast sensitivity to signal location and tissue thickness in breast imaging,” Proc. SPIE 10573, 533–541 (2018).
[Crossref]

2017 (1)

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2017).
[Crossref]

2016 (2)

M. Das, C. S. Park, and N. R. Fredette, “Single-shot x-ray phase contrast imaging with an algorithmic approach using spectral detection,” Proc. SPIE 9783, 188–193 (2016).
[Crossref]

H. C. Gifford, Z. Liang, and M. Das, “Visual-search observers for assessing tomographic x-ray image quality,” Med. Phys. 43, 1563–1575 (2016).
[Crossref]

2015 (1)

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

2014 (4)

2013 (4)

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[Crossref]

D. Gürsoy and M. Das, “Single-step absorption and phase retrieval with polychromatic x rays using a spectral detector,” Opt. Lett. 38, 1461–1463 (2013).
[Crossref]

K. Taguchi and J. S. Iwanczyk, “Vision 20/20: single photon counting x-ray detectors in medical imaging,” Med. Phys. 40, 100901 (2013).
[Crossref]

Y. Sung and G. Barbastathis, “Rytov approximation for x-ray phase imaging,” Opt. Express 21, 2674–2682 (2013).
[Crossref]

2011 (1)

2010 (2)

K. S. Morgan, K. K. W. Siu, and D. M. Paganin, “The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge,” Opt. Express 18, 9865–9878 (2010).
[Crossref]

M. Das, H. C. Gifford, J. M. O’Connor, and S. J. Glick, “Penalized maximum likelihood reconstruction for improved microcalcification detection in breast tomosynthesis,” IEEE Trans. Med. Imaging 30, 904–914 (2010).
[Crossref]

2009 (2)

T. G. Schmidt, “Optimal ‘image-based’ weighting for energy-resolved CT,” Med. Phys. 36, 3018–3027 (2009).
[Crossref]

J. Jakubek, “Semiconductor pixel detectors and their applications in life sciences,” J. Instrum. 4, P03013 (2009).
[Crossref]

2007 (1)

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
[Crossref]

2005 (1)

2004 (1)

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

2002 (1)

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

1998 (1)

E. Samei, M. J. Flynn, and D. A. Reimann, “A method for measuring the presampled MTF of digital radiographic systems using an edge test device,” Med. Phys. 25, 102–113 (1998).
[Crossref]

1997 (1)

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]

1996 (1)

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]

1986 (1)

C. R. Crawford, “Reprojection using a parallel backprojector,” Med. Phys. 13, 480–483 (1986).
[Crossref]

1983 (1)

1981 (1)

P. C. Shrimpton, “Electron density values of various human tissues: in vitro Compton scatter measurements and calculated ranges,” Phys. Med. Biol. 26, 907–911 (1981).
[Crossref]

1976 (2)

R. E. Alvarez and A. Macovski, “Energy-selective reconstructions in x-ray computerised tomography,” Phys. Med. Biol. 21, 733–744 (1976).
[Crossref]

R. A. Rutherford, B. R. Pullan, and I. Isherwood, “Measurement of effective atomic number and electron density using an EMI scanner,” Neuroradiology 11, 15–21 (1976).
[Crossref]

1975 (1)

M. E. Phelps, E. J. 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).
[Crossref]

1938 (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

Alozy, J.

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[Crossref]

Als-Nielsen, J.

J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics (Wiley, 2011).

Alvarez, R. E.

R. E. Alvarez and A. Macovski, “Energy-selective reconstructions in x-ray computerised tomography,” Phys. Med. Biol. 21, 733–744 (1976).
[Crossref]

Andres-Thio, N.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2017).
[Crossref]

Attix, F. H.

F. H. Attix, Introduction to Radiological Physics and Radiation Dosimetry (Wiley, 2008).

Ballabriga, R.

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[Crossref]

Barbastathis, G.

Barber, W. C.

J. Cammin, J. Xu, W. C. Barber, J. S. Iwanczyk, N. E. Hartsough, and K. Taguchi, “A cascaded model of spectral distortions due to spectral response effects and pulse pileup effects in a photon-counting x-ray detector for CT,” Med. Phys. 41, 041905 (2014).
[Crossref]

Blaj, G.

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[Crossref]

Buckley, G. A.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2017).
[Crossref]

Burvall, A.

Cammin, J.

J. Cammin, J. Xu, W. C. Barber, J. S. Iwanczyk, N. E. Hartsough, and K. Taguchi, “A cascaded model of spectral distortions due to spectral response effects and pulse pileup effects in a photon-counting x-ray detector for CT,” Med. Phys. 41, 041905 (2014).
[Crossref]

Campbell, M.

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[Crossref]

Chen, H.

Crawford, C. R.

C. R. Crawford, “Reprojection using a parallel backprojector,” Med. Phys. 13, 480–483 (1986).
[Crossref]

Das, M.

C. Lewis, I. Vazquez, S. Vespucci, and M. Das, “Contribution of scatter and beam hardening to phase contrast imaging,” Proc. SPIE 10948, 1274–1280 (2019).
[Crossref]

I. Vazquez, N. R. Fredette, and M. Das, “Quantitative phase retrieval of heterogeneous samples from spectral x-ray measurements,” Proc. SPIE 10948, 1157–1164 (2019).
[Crossref]

N. R. Fredette, A. Kavuri, and M. Das, “Multi-step material decomposition for spectral computed tomography,” Phys. Med. Biol. 64, 145001 (2019).
[Crossref]

C. E. Lewis and M. Das, “Spectral signatures of X-ray scatter using energy-resolving photon-counting detectors,” Sensors 19, 5022 (2019).
[Crossref]

S. Vespucci, C. Lewis, C. S. Park, and M. Das, “Examining phase contrast sensitivity to signal location and tissue thickness in breast imaging,” Proc. SPIE 10573, 533–541 (2018).
[Crossref]

S. Vespucci, C. S. Park, R. Torrico, and M. Das, “Robust energy calibration technique for photon counting spectral detectors,” IEEE Trans. Med. Imaging 38, 968–978 (2018).
[Crossref]

M. Das, C. S. Park, and N. R. Fredette, “Single-shot x-ray phase contrast imaging with an algorithmic approach using spectral detection,” Proc. SPIE 9783, 188–193 (2016).
[Crossref]

H. C. Gifford, Z. Liang, and M. Das, “Visual-search observers for assessing tomographic x-ray image quality,” Med. Phys. 43, 1563–1575 (2016).
[Crossref]

M. Das and Z. Liang, “Spectral x-ray phase contrast imaging for single-shot retrieval of absorption, phase, and differential-phase imagery,” Opt. Lett. 39, 6343–6346 (2014).
[Crossref]

M. Das and Z. Liang, “Approximated transport-of-intensity equation for coded-aperture x-ray phase-contrast imaging,” Opt. Lett. 39, 5395–5398 (2014).
[Crossref]

D. Gürsoy and M. Das, “Single-step absorption and phase retrieval with polychromatic x rays using a spectral detector,” Opt. Lett. 38, 1461–1463 (2013).
[Crossref]

M. Das, H. C. Gifford, J. M. O’Connor, and S. J. Glick, “Penalized maximum likelihood reconstruction for improved microcalcification detection in breast tomosynthesis,” IEEE Trans. Med. Imaging 30, 904–914 (2010).
[Crossref]

M. Das, “Single step differential phase contrast x-ray imaging,” U.S. Patent9,445,775 (20September2016).

M. Das and D. Gürsoy, “Single step X-ray phase imaging,” U.S. Patent9,237,876 (19January2016).

Dean, A.

X. Wu, A. Dean, and H. Liu, “X-ray diagnostic techniques,” in Biomedical Photonics Handbook (2003), Vol. 2, pp. 415–451.

Dolbnya, I.

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

Fiederle, M.

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[Crossref]

Flynn, M. J.

E. Samei, M. J. Flynn, and D. A. Reimann, “A method for measuring the presampled MTF of digital radiographic systems using an edge test device,” Med. Phys. 25, 102–113 (1998).
[Crossref]

Fredette, N. R.

N. R. Fredette, A. Kavuri, and M. Das, “Multi-step material decomposition for spectral computed tomography,” Phys. Med. Biol. 64, 145001 (2019).
[Crossref]

I. Vazquez, N. R. Fredette, and M. Das, “Quantitative phase retrieval of heterogeneous samples from spectral x-ray measurements,” Proc. SPIE 10948, 1157–1164 (2019).
[Crossref]

M. Das, C. S. Park, and N. R. Fredette, “Single-shot x-ray phase contrast imaging with an algorithmic approach using spectral detection,” Proc. SPIE 9783, 188–193 (2016).
[Crossref]

Frodjh, E.

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

Frojdh, E.

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[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]

Gao, K.

Gifford, H. C.

H. C. Gifford, Z. Liang, and M. Das, “Visual-search observers for assessing tomographic x-ray image quality,” Med. Phys. 43, 1563–1575 (2016).
[Crossref]

M. Das, H. C. Gifford, J. M. O’Connor, and S. J. Glick, “Penalized maximum likelihood reconstruction for improved microcalcification detection in breast tomosynthesis,” IEEE Trans. Med. Imaging 30, 904–914 (2010).
[Crossref]

Gimenez, E. N.

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

Glick, S. J.

M. Das, H. C. Gifford, J. M. O’Connor, and S. J. Glick, “Penalized maximum likelihood reconstruction for improved microcalcification detection in breast tomosynthesis,” IEEE Trans. Med. Imaging 30, 904–914 (2010).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

Gureyev, T. E.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2017).
[Crossref]

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

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]

Gürsoy, D.

Hartsough, N. E.

J. Cammin, J. Xu, W. C. Barber, J. S. Iwanczyk, N. E. Hartsough, and K. Taguchi, “A cascaded model of spectral distortions due to spectral response effects and pulse pileup effects in a photon-counting x-ray detector for CT,” Med. Phys. 41, 041905 (2014).
[Crossref]

Hatch, H. W.

V. K. Shen, D. W. Siderius, W. P. Krekelberg, and H. W. Hatch, “NIST standard reference simulation website,” NIST Standard Reference Database (2017), pp. 2014–2017.

Heijne, E. H. M.

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[Crossref]

Hertz, H. M.

Hoffman, E. J.

M. E. Phelps, E. J. 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).
[Crossref]

Hooper, S. B.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2017).
[Crossref]

Horswell, I.

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

Hubbell, J. H.

J. H. Hubbell, “Photon cross sections, attenuation coefficients and energy absorption coefficients,” National Bureau of Standards Report NSRDS-NBS29 (1969).

Isherwood, I.

R. A. Rutherford, B. R. Pullan, and I. Isherwood, “Measurement of effective atomic number and electron density using an EMI scanner,” Neuroradiology 11, 15–21 (1976).
[Crossref]

Iwanczyk, J. S.

J. Cammin, J. Xu, W. C. Barber, J. S. Iwanczyk, N. E. Hartsough, and K. Taguchi, “A cascaded model of spectral distortions due to spectral response effects and pulse pileup effects in a photon-counting x-ray detector for CT,” Med. Phys. 41, 041905 (2014).
[Crossref]

K. Taguchi and J. S. Iwanczyk, “Vision 20/20: single photon counting x-ray detectors in medical imaging,” Med. Phys. 40, 100901 (2013).
[Crossref]

Jakubek, J.

J. Jakubek, “Semiconductor pixel detectors and their applications in life sciences,” J. Instrum. 4, P03013 (2009).
[Crossref]

Kavuri, A.

N. R. Fredette, A. Kavuri, and M. Das, “Multi-step material decomposition for spectral computed tomography,” Phys. Med. Biol. 64, 145001 (2019).
[Crossref]

Kitchen, M. J.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2017).
[Crossref]

Krekelberg, W. P.

V. K. Shen, D. W. Siderius, W. P. Krekelberg, and H. W. Hatch, “NIST standard reference simulation website,” NIST Standard Reference Database (2017), pp. 2014–2017.

Larsson, D. H.

Lewis, C.

C. Lewis, I. Vazquez, S. Vespucci, and M. Das, “Contribution of scatter and beam hardening to phase contrast imaging,” Proc. SPIE 10948, 1274–1280 (2019).
[Crossref]

S. Vespucci, C. Lewis, C. S. Park, and M. Das, “Examining phase contrast sensitivity to signal location and tissue thickness in breast imaging,” Proc. SPIE 10573, 533–541 (2018).
[Crossref]

Lewis, C. E.

C. E. Lewis and M. Das, “Spectral signatures of X-ray scatter using energy-resolving photon-counting detectors,” Sensors 19, 5022 (2019).
[Crossref]

Lewis, R. A.

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

Liang, Z.

Links, J. M.

J. L. Prince and J. M. Links, Medical Imaging Signals and Systems (Pearson Prentice Hall, 2006).

Liu, H.

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

X. Wu, A. Dean, and H. Liu, “X-ray diagnostic techniques,” in Biomedical Photonics Handbook (2003), Vol. 2, pp. 415–451.

Llopart, X.

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[Crossref]

Lundström, U.

Macovski, A.

R. E. Alvarez and A. Macovski, “Energy-selective reconstructions in x-ray computerised tomography,” Phys. Med. Biol. 21, 733–744 (1976).
[Crossref]

Marchal, J.

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

Mayo, S. C.

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

McGrath, J.

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

McMorrow, D.

J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics (Wiley, 2011).

Miller, P. R.

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

Morgan, K. S.

O’Connor, J. M.

M. Das, H. C. Gifford, J. M. O’Connor, and S. J. Glick, “Penalized maximum likelihood reconstruction for improved microcalcification detection in breast tomosynthesis,” IEEE Trans. Med. Imaging 30, 904–914 (2010).
[Crossref]

Olivo, A.

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
[Crossref]

Paganin, D.

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

D. Paganin, Coherent X-Ray Optics (Oxford University, 2006), Vol. 6.

Paganin, D. M.

Park, C. S.

S. Vespucci, C. Lewis, C. S. Park, and M. Das, “Examining phase contrast sensitivity to signal location and tissue thickness in breast imaging,” Proc. SPIE 10573, 533–541 (2018).
[Crossref]

S. Vespucci, C. S. Park, R. Torrico, and M. Das, “Robust energy calibration technique for photon counting spectral detectors,” IEEE Trans. Med. Imaging 38, 968–978 (2018).
[Crossref]

M. Das, C. S. Park, and N. R. Fredette, “Single-shot x-ray phase contrast imaging with an algorithmic approach using spectral detection,” Proc. SPIE 9783, 188–193 (2016).
[Crossref]

Phelps, M. E.

M. E. Phelps, E. J. 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).
[Crossref]

Pichotka, M.

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[Crossref]

Pogany, A.

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]

Prince, J. L.

J. L. Prince and J. M. Links, Medical Imaging Signals and Systems (Pearson Prentice Hall, 2006).

Procz, S.

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[Crossref]

Pullan, B. R.

R. A. Rutherford, B. R. Pullan, and I. Isherwood, “Measurement of effective atomic number and electron density using an EMI scanner,” Neuroradiology 11, 15–21 (1976).
[Crossref]

Reimann, D. A.

E. Samei, M. J. Flynn, and D. A. Reimann, “A method for measuring the presampled MTF of digital radiographic systems using an edge test device,” Med. Phys. 25, 102–113 (1998).
[Crossref]

Rutherford, R. A.

R. A. Rutherford, B. R. Pullan, and I. Isherwood, “Measurement of effective atomic number and electron density using an EMI scanner,” Neuroradiology 11, 15–21 (1976).
[Crossref]

Samei, E.

E. Samei, M. J. Flynn, and D. A. Reimann, “A method for measuring the presampled MTF of digital radiographic systems using an edge test device,” Med. Phys. 25, 102–113 (1998).
[Crossref]

Schmidt, T. G.

T. G. Schmidt, “Optimal ‘image-based’ weighting for energy-resolved CT,” Med. Phys. 36, 3018–3027 (2009).
[Crossref]

Shen, V. K.

V. K. Shen, D. W. Siderius, W. P. Krekelberg, and H. W. Hatch, “NIST standard reference simulation website,” NIST Standard Reference Database (2017), pp. 2014–2017.

Shrimpton, P. C.

P. C. Shrimpton, “Electron density values of various human tissues: in vitro Compton scatter measurements and calculated ranges,” Phys. Med. Biol. 26, 907–911 (1981).
[Crossref]

Siderius, D. W.

V. K. Shen, D. W. Siderius, W. P. Krekelberg, and H. W. Hatch, “NIST standard reference simulation website,” NIST Standard Reference Database (2017), pp. 2014–2017.

Siu, K. K. W.

Speller, R.

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
[Crossref]

Stevenson, A. W.

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]

Sung, Y.

Taguchi, K.

J. Cammin, J. Xu, W. C. Barber, J. S. Iwanczyk, N. E. Hartsough, and K. Taguchi, “A cascaded model of spectral distortions due to spectral response effects and pulse pileup effects in a photon-counting x-ray detector for CT,” Med. Phys. 41, 041905 (2014).
[Crossref]

K. Taguchi and J. S. Iwanczyk, “Vision 20/20: single photon counting x-ray detectors in medical imaging,” Med. Phys. 40, 100901 (2013).
[Crossref]

Takman, P. A.

Teague, M. R.

Ter-Pogossian, M. M.

M. E. Phelps, E. J. 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).
[Crossref]

Torrico, R.

S. Vespucci, C. S. Park, R. Torrico, and M. Das, “Robust energy calibration technique for photon counting spectral detectors,” IEEE Trans. Med. Imaging 38, 968–978 (2018).
[Crossref]

Uesugi, K.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2017).
[Crossref]

Vazquez, I.

I. Vazquez, N. R. Fredette, and M. Das, “Quantitative phase retrieval of heterogeneous samples from spectral x-ray measurements,” Proc. SPIE 10948, 1157–1164 (2019).
[Crossref]

C. Lewis, I. Vazquez, S. Vespucci, and M. Das, “Contribution of scatter and beam hardening to phase contrast imaging,” Proc. SPIE 10948, 1274–1280 (2019).
[Crossref]

Vespucci, S.

C. Lewis, I. Vazquez, S. Vespucci, and M. Das, “Contribution of scatter and beam hardening to phase contrast imaging,” Proc. SPIE 10948, 1274–1280 (2019).
[Crossref]

S. Vespucci, C. Lewis, C. S. Park, and M. Das, “Examining phase contrast sensitivity to signal location and tissue thickness in breast imaging,” Proc. SPIE 10573, 533–541 (2018).
[Crossref]

S. Vespucci, C. S. Park, R. Torrico, and M. Das, “Robust energy calibration technique for photon counting spectral detectors,” IEEE Trans. Med. Imaging 38, 968–978 (2018).
[Crossref]

Wallace, M. J.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2017).
[Crossref]

Wang, D.

Wang, S.

Wang, Z.

Wilkins, S. W.

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

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]

Wu, X.

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

X. Wu, A. Dean, and H. Liu, “X-ray diagnostic techniques,” in Biomedical Photonics Handbook (2003), Vol. 2, pp. 415–451.

Wu, Z.

Xu, J.

J. Cammin, J. Xu, W. C. Barber, J. S. Iwanczyk, N. E. Hartsough, and K. Taguchi, “A cascaded model of spectral distortions due to spectral response effects and pulse pileup effects in a photon-counting x-ray detector for CT,” Med. Phys. 41, 041905 (2014).
[Crossref]

Yagi, N.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2017).
[Crossref]

Yan, A.

Zernike, F.

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

Appl. Phys. Lett. (1)

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
[Crossref]

IEEE Trans. Med. Imaging (2)

S. Vespucci, C. S. Park, R. Torrico, and M. Das, “Robust energy calibration technique for photon counting spectral detectors,” IEEE Trans. Med. Imaging 38, 968–978 (2018).
[Crossref]

M. Das, H. C. Gifford, J. M. O’Connor, and S. J. Glick, “Penalized maximum likelihood reconstruction for improved microcalcification detection in breast tomosynthesis,” IEEE Trans. Med. Imaging 30, 904–914 (2010).
[Crossref]

IEEE Trans. Nucl. Sci. (1)

E. N. Gimenez, R. Ballabriga, G. Blaj, M. Campbell, I. Dolbnya, E. Frodjh, I. Horswell, X. Llopart, J. Marchal, and J. McGrath, “Medipix3RX: characterizing the Medipix3 redesign with synchrotron radiation,” IEEE Trans. Nucl. Sci. 62, 1413–1421 (2015).
[Crossref]

J. Instrum. (2)

R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, and S. Procz, “The Medipix3RX: a high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrum. 8, C02016 (2013).
[Crossref]

J. Jakubek, “Semiconductor pixel detectors and their applications in life sciences,” J. Instrum. 4, P03013 (2009).
[Crossref]

J. Microsc. (1)

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

J. Opt. Soc. Am. (1)

Med. Phys. (6)

T. G. Schmidt, “Optimal ‘image-based’ weighting for energy-resolved CT,” Med. Phys. 36, 3018–3027 (2009).
[Crossref]

K. Taguchi and J. S. Iwanczyk, “Vision 20/20: single photon counting x-ray detectors in medical imaging,” Med. Phys. 40, 100901 (2013).
[Crossref]

J. Cammin, J. Xu, W. C. Barber, J. S. Iwanczyk, N. E. Hartsough, and K. Taguchi, “A cascaded model of spectral distortions due to spectral response effects and pulse pileup effects in a photon-counting x-ray detector for CT,” Med. Phys. 41, 041905 (2014).
[Crossref]

H. C. Gifford, Z. Liang, and M. Das, “Visual-search observers for assessing tomographic x-ray image quality,” Med. Phys. 43, 1563–1575 (2016).
[Crossref]

C. R. Crawford, “Reprojection using a parallel backprojector,” Med. Phys. 13, 480–483 (1986).
[Crossref]

E. Samei, M. J. Flynn, and D. A. Reimann, “A method for measuring the presampled MTF of digital radiographic systems using an edge test device,” Med. Phys. 25, 102–113 (1998).
[Crossref]

Nature (1)

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]

Neuroradiology (1)

R. A. Rutherford, B. R. Pullan, and I. Isherwood, “Measurement of effective atomic number and electron density using an EMI scanner,” Neuroradiology 11, 15–21 (1976).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Phys. Med. Biol. (4)

R. E. Alvarez and A. Macovski, “Energy-selective reconstructions in x-ray computerised tomography,” Phys. Med. Biol. 21, 733–744 (1976).
[Crossref]

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

N. R. Fredette, A. Kavuri, and M. Das, “Multi-step material decomposition for spectral computed tomography,” Phys. Med. Biol. 64, 145001 (2019).
[Crossref]

P. C. Shrimpton, “Electron density values of various human tissues: in vitro Compton scatter measurements and calculated ranges,” Phys. Med. Biol. 26, 907–911 (1981).
[Crossref]

Physica (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

Proc. SPIE (4)

S. Vespucci, C. Lewis, C. S. Park, and M. Das, “Examining phase contrast sensitivity to signal location and tissue thickness in breast imaging,” Proc. SPIE 10573, 533–541 (2018).
[Crossref]

C. Lewis, I. Vazquez, S. Vespucci, and M. Das, “Contribution of scatter and beam hardening to phase contrast imaging,” Proc. SPIE 10948, 1274–1280 (2019).
[Crossref]

I. Vazquez, N. R. Fredette, and M. Das, “Quantitative phase retrieval of heterogeneous samples from spectral x-ray measurements,” Proc. SPIE 10948, 1157–1164 (2019).
[Crossref]

M. Das, C. S. Park, and N. R. Fredette, “Single-shot x-ray phase contrast imaging with an algorithmic approach using spectral detection,” Proc. SPIE 9783, 188–193 (2016).
[Crossref]

Radiology (1)

M. E. Phelps, E. J. 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).
[Crossref]

Rev. Sci. Instrum. (1)

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]

Sci. Rep. (1)

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2017).
[Crossref]

Sensors (1)

C. E. Lewis and M. Das, “Spectral signatures of X-ray scatter using energy-resolving photon-counting detectors,” Sensors 19, 5022 (2019).
[Crossref]

Other (10)

M. Das, “Single step differential phase contrast x-ray imaging,” U.S. Patent9,445,775 (20September2016).

M. Das and D. Gürsoy, “Single step X-ray phase imaging,” U.S. Patent9,237,876 (19January2016).

D. Paganin, Coherent X-Ray Optics (Oxford University, 2006), Vol. 6.

X. Wu, A. Dean, and H. Liu, “X-ray diagnostic techniques,” in Biomedical Photonics Handbook (2003), Vol. 2, pp. 415–451.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

V. K. Shen, D. W. Siderius, W. P. Krekelberg, and H. W. Hatch, “NIST standard reference simulation website,” NIST Standard Reference Database (2017), pp. 2014–2017.

F. H. Attix, Introduction to Radiological Physics and Radiation Dosimetry (Wiley, 2008).

J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics (Wiley, 2011).

J. H. Hubbell, “Photon cross sections, attenuation coefficients and energy absorption coefficients,” National Bureau of Standards Report NSRDS-NBS29 (1969).

J. L. Prince and J. M. Links, Medical Imaging Signals and Systems (Pearson Prentice Hall, 2006).

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Figures (7)

Fig. 1.
Fig. 1. (a) Depiction of a propagation-based phase contrast experiment geometry. The small focal spot (high lateral coherence) x-ray source and the sample are separated by a distance ${R_1}$, while the distance from the sample plane to the detector is ${R_2}$. In conventional x-ray imaging, ${R_2} \approx 0$. (b) Two enlarged pixels illustrate the basic steps that enable spectroscopic imaging with PCDs. Photons deposit a charge proportional to their energy. The charge is converted to a voltage that is later amplified, compared to a user-defined threshold, and counted if it exceeds the threshold.

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Fig. 2.
Fig. 2. Illustration of the experimental setup used to image the multimaterial phantom. The process of energy binning is also depicted. Namely, the Medipix3RX detector captures specific sections of the incident spectrum and, after energy binning, a set of bins corresponding to narrow sections of the incident spectrum are formed. This allows for accurate spectral measurements without the need for monochromatic sources.

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Fig. 3.
Fig. 3. PCI measurement of the multimaterial sample showing the expected edge enhancement at material boundaries.

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Fig. 4.
Fig. 4. Phase retrieval results for the multimaterial phantom. (a) The figure shows the pLAC map with the average retrieved profile (solid) and expected (dashed) values superimposed. (b) The retrieved phase difference map is shown along with the retrieved (solid) and expected (dashed) average profiles. The retrieved quantities correspond to an energy of 40 keV. (c) Percent error between the retrieved and expected phase and pLAC of the four rods.

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Fig. 5.
Fig. 5. (a) Average profile of the retrieved and expected photoelectric term for the four material phantom. (b) Normalized contrast for the three retrieved properties at 40 keV. The contrast of each property was normalized by the maximum value between all four materials.

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Fig. 6.
Fig. 6. In all of the plots, ${{2\times}}$ indicates the results for the measurements taken at a two-times magnification, while the ${{3\times}}$ the configuration with a magnification of three. (a) Signal-difference-to-noise ratio (SNR) of the retrieved projected linear attenuation coefficient (pLAC), retrieved phase, and negative log of the measured phase contrast images (PCIs). An effective energy of 38 keV was estimated for the PCI measurement, and the retrieved quantities were computed for this energy. (b) Gain in SNR found as the ratio between the SNR of the pLAC and PCI images as well as the phase and PCI images. (c) Average percent error computed from the estimated and expected electron density of POM.

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Fig. 7.
Fig. 7. (a) Photograph of Emory oak (Quercus emoryi) leaf. A black box encloses the region imaged with conventional and phase-enhancing configurations. (b) Negative log of the normalized intensity, ${-}\ln [I/{I^{{\rm{in}}}}]$, from measurements taken at the contact plane, that is, $M \approx 1$. (c) The same quantity as (b) but using measurements taken in a phase-enhancing configuration, namely, $M = 3$. (d) Same quantity as (a) and (b) but with intensity values reconstructed from the retrieved material properties. Insets 1, 2, and 3 show the same region of interest (ROI) in images (b), (c), and (d), respectively.

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Tables (1)

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Table 1. Properties Listed Are Effective Atomic Number ( Z e f f ), Electron Density ( ρ e ), Mass Density ( ρ m ), Total Linear Attenuation (µ), Photoelectric Absorption ( τ p e ), Compton Scattering ( σ c s ), and Rod Diameter (ø)

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Equations (10)

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− 2 π λ ∂ I ( r , z ; E ) ∂ z = ∇ r ⋅ I ( r , z ; E ) ∇ r φ ( r , z ; E ) .
I ( r , R 2 ; E ) = I ( r , 0 ; E ) [ 1 − λ R 2 2 π ∇ r 2 φ ( r , 0 ; E ) ] .
ln ⁡ [ I i n ( M r ; E ) I ( M r , R 2 ; E ) ] = ∫ μ ( r , z ; E ) d z + λ 2 π R 2 M ∇ r 2 φ ( r , 0 ; E ) ,
∫ μ ( r , z ; E ) d z = ∫ [ k Z n ( r , z ) E m ρ e ( r , z ) + f K N ( E ) ρ e ( r , z ) ] d z = 1 E m a 1 ( r ) + f K N ( E ) a 2 ( r ) ,
φ ( r , 0 ; E ) = − r e λ a 2 ( r ) ,
D ( M r ; E ) = [ 1 E m ] a 1 ( r ) + [ f K N ( E ) − λ 2 r e R 2 2 π M ∇ r 2 ] a 2 ( r ) ,
F { D ( ξ ; E ) } = [ 1 E m ] F { a 1 ( ξ / M ) } + [ f K N ( E ) + M λ 2 r e R 2 2 π | u | 2 ] F { a 2 ( ξ / M ) } ,
C N V R i = ⟨ D ( ξ m ; E i ) ⟩ − ⟨ D ( ξ b ; E i ) ⟩ σ 2 ( ξ b ; E i ) ,
E e f f j = ∑ i = 1 M c i E i j / ∑ i = 1 M c i ,
ϵ = 100 % × 1 P ∑ i = 1 P | r i − e i e i | ,

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