Abstract

We present a comparison for high-resolution imaging with a laboratory source between grating-based (GBI) and propagation-based (PBI) x-ray phase-contrast imaging. The comparison is done through simulations and experiments using a liquid-metal-jet x-ray microfocus source. Radiation doses required for detection in projection images are simulated as a function of the diameter of a cylindrical sample. Using monochromatic radiation, simulations show a lower dose requirement for PBI for small object features and a lower dose for GBI for larger object features. Using polychromatic radiation, such as that from a laboratory microfocus source, experiments and simulations show a lower dose requirement for PBI for a large range of feature sizes. Tested on a biological sample, GBI shows higher noise levels than PBI, but its advantage of quantitative refractive index reconstruction for multi-material samples becomes apparent.

© 2013 Optical Society of America

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    [Crossref] [PubMed]
  22. K. Li, N. Bevins, J. Zambelli, and G. H. Chen, “Fundamental relationship between the noise properties of grating-based differential phase contrast CT and absorption CT: theoretical framework using a cascaded system model and experimental validation,” Med. Phys. 40(2), 021908 (2013).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  25. T. Weber, P. Bartl, F. Bayer, J. Durst, W. Haas, T. Michel, A. Ritter, and G. Anton, “Noise in x-ray grating-based phase-contrast imaging,” Med. Phys. 38(7), 4133–4140 (2011).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  28. A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in X-ray phase-contrast imaging suitable for tomography,” Opt. Express 19(11), 10359–10376 (2011).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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  38. U. Lundström, D. H. Larsson, A. Burvall, L. Scott, U. K. Westermark, M. Wilhelm, M. Arsenian Henriksson, and H. M. Hertz, “X-ray phase-contrast CO2 angiography for sub-10 μm vessel imaging,” Phys. Med. Biol. 57(22), 7431–7441 (2012).
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    [PubMed]

2013 (4)

T. Thüring, T. Zhou, U. Lundström, A. Burvall, S. Rutishauser, C. David, H. M. Hertz, and M. Stampanoni, “X-ray grating interferometry with a liquid-metal-jet source,” Appl. Phys. Lett. 103(9), 091105 (2013).
[Crossref]

D. H. Larsson, U. Lundström, U. K. Westermark, M. Arsenian Henriksson, A. Burvall, and H. M. Hertz, “First application of liquid-metal-jet sources for small-animal imaging: high-resolution CT and phase-contrast tumor demarcation,” Med. Phys. 40(2), 021909 (2013).
[Crossref] [PubMed]

K. Li, N. Bevins, J. Zambelli, and G. H. Chen, “Fundamental relationship between the noise properties of grating-based differential phase contrast CT and absorption CT: theoretical framework using a cascaded system model and experimental validation,” Med. Phys. 40(2), 021908 (2013).
[Crossref] [PubMed]

T. Zhou, U. Lundström, D. H. Larsson, H. M. Hertz, and A. Burvall, “Low-dose phase-contrast X-ray imaging: a comparison of two methods,” J. Phys. Conf. Ser. 463, 012041 (2013).
[Crossref]

2012 (7)

U. Lundström, D. H. Larsson, A. Burvall, L. Scott, U. K. Westermark, M. Wilhelm, M. Arsenian Henriksson, and H. M. Hertz, “X-ray phase-contrast CO2 angiography for sub-10 μm vessel imaging,” Phys. Med. Biol. 57(22), 7431–7441 (2012).
[Crossref] [PubMed]

D. Hahn, P. Thibault, M. Bech, M. Stockmar, S. Schleede, I. Zanette, A. Rack, T. Weitkamp, A. Sztrókay, T. Schlossbauer, F. Bamberg, M. Reiser, and F. Pfeiffer, “Numerical comparison of X-ray differential phase contrast and attenuation contrast,” Biomed. Opt. Express 3(6), 1141–1148 (2012).
[Crossref] [PubMed]

E. Fredenberg, M. Danielsson, J. W. Stayman, J. H. Siewerdsen, and M. Aslund, “Ideal-observer detectability in photon-counting differential phase-contrast imaging using a linear-systems approach,” Med. Phys. 39(9), 5317–5335 (2012).
[Crossref] [PubMed]

P. C. Diemoz, A. Bravin, and P. Coan, “Theoretical comparison of three X-ray phase-contrast imaging techniques: propagation-based imaging, analyzer-based imaging and grating interferometry,” Opt. Express 20(3), 2789–2805 (2012).
[Crossref] [PubMed]

P. C. Diemoz, A. Bravin, M. Langer, and P. Coan, “Analytical and experimental determination of signal-to-noise ratio and figure of merit in three phase-contrast imaging techniques,” Opt. Express 20(25), 27670–27690 (2012).
[Crossref] [PubMed]

R. Raupach and T. Flohr, “Performance evaluation of x-ray differential phase contrast computed tomography (PCT) with respect to medical imaging,” Med. Phys. 39(8), 4761–4774 (2012).
[Crossref] [PubMed]

U. Lundström, D. H. Larsson, A. Burvall, P. A. Takman, L. Scott, H. Brismar, and H. M. Hertz, “X-ray phase contrast for CO2 microangiography,” Phys. Med. Biol. 57(9), 2603–2617 (2012).
[Crossref] [PubMed]

2011 (8)

D. H. Larsson, P. A. C. Takman, U. Lundström, A. Burvall, and H. M. Hertz, “A 24 keV liquid-metal-jet x-ray source for biomedical applications,” Rev. Sci. Instrum. 82(12), 123701 (2011).
[Crossref] [PubMed]

P. Modregger, B. R. Pinzer, T. Thüring, S. Rutishauser, C. David, and M. Stampanoni, “Sensitivity of X-ray grating interferometry,” Opt. Express 19(19), 18324–18338 (2011).
[Crossref] [PubMed]

T. Köhler, K. Jürgen Engel, and E. Roessl, “Noise properties of grating-based x-ray phase contrast computed tomography,” Med. Phys. 38(S1), S106–S116 (2011).
[Crossref] [PubMed]

R. Raupach and T. G. Flohr, “Analytical evaluation of the signal and noise propagation in x-ray differential phase-contrast computed tomography,” Phys. Med. Biol. 56(7), 2219–2244 (2011).
[Crossref] [PubMed]

G. H. Chen, J. Zambelli, K. Li, N. Bevins, and Z. H. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38(2), 584–588 (2011).
[Crossref] [PubMed]

A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in X-ray phase-contrast imaging suitable for tomography,” Opt. Express 19(11), 10359–10376 (2011).
[Crossref] [PubMed]

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast CT compared with the conventional CT--noise power spectrum NPS(k),” Med. Phys. 38(7), 4386–4395 (2011).
[Crossref] [PubMed]

T. Weber, P. Bartl, F. Bayer, J. Durst, W. Haas, T. Michel, A. Ritter, and G. Anton, “Noise in x-ray grating-based phase-contrast imaging,” Med. Phys. 38(7), 4133–4140 (2011).
[Crossref] [PubMed]

2010 (2)

J. Zambelli, N. Bevins, Z. Qi, and G. H. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37(6), 2473–2479 (2010).
[Crossref] [PubMed]

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[Crossref] [PubMed]

2009 (2)

C. Parham, Z. Zhong, D. M. Connor, L. D. Chapman, and E. D. Pisano, “Design and implementation of a compact low-dose diffraction enhanced medical imaging system,” Acad. Radiol. 16(8), 911–917 (2009).
[Crossref] [PubMed]

I. Nesch, D. P. Fogarty, T. Tzvetkov, B. Reinhart, A. C. Walus, G. Khelashvili, C. Muehleman, and D. Chapman, “The design and application of an in-laboratory diffraction-enhanced x-ray imaging instrument,” Rev. Sci. Instrum. 80(9), 093702 (2009).
[Crossref] [PubMed]

2008 (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008).
[Crossref] [PubMed]

2007 (2)

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98(10), 108105 (2007).
[Crossref] [PubMed]

J. Vlassenbroeck, M. Dierick, B. Masschaele, V. Cnudde, L. van Hoorebeke, and P. Jacobs, “Software tools for quantification of X-ray microtomography,” Nucl. Instrum. Methods Phys. Res. A 580(1), 442–445 (2007).
[Crossref]

2006 (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(4), 258–261 (2006).
[Crossref]

2005 (2)

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[Crossref] [PubMed]

2003 (1)

O. Hemberg, M. Otendal, and H. M. Hertz, “Liquid-metal-jet anode electron-impact x-ray source,” Appl. Phys. Lett. 83(7), 1483–1485 (2003).
[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(1), 33–40 (2002).
[Crossref] [PubMed]

2000 (1)

L. Kissel, “RTAB: the Rayleigh scattering database,” Radiat. Phys. Chem. 59(2), 185–200 (2000).
[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(6607), 335–338 (1996).
[Crossref]

1995 (1)

J. Baró, J. Sempau, J. M. Fernandezvarea, and F. Salvat, “Penelope - an algorithm for Monte-Carlo simulation of the penetration and energy-loss of electrons and positrons in matter,” Nucl. Instrum. Methods Phys. Res. B 100(1), 31–46 (1995).
[Crossref]

1988 (2)

A. Fandos-Morera, M. Prats-Esteve, J. M. Tura-Soteras, and A. Traveria-Cros, “Breast tumors: composition of microcalcifications,” Radiology 169(2), 325–327 (1988).
[PubMed]

G. W. Faris and R. L. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. 27(24), 5202–5212 (1988).
[Crossref] [PubMed]

1948 (1)

Anton, G.

T. Weber, P. Bartl, F. Bayer, J. Durst, W. Haas, T. Michel, A. Ritter, and G. Anton, “Noise in x-ray grating-based phase-contrast imaging,” Med. Phys. 38(7), 4133–4140 (2011).
[Crossref] [PubMed]

Arsenian Henriksson, M.

D. H. Larsson, U. Lundström, U. K. Westermark, M. Arsenian Henriksson, A. Burvall, and H. M. Hertz, “First application of liquid-metal-jet sources for small-animal imaging: high-resolution CT and phase-contrast tumor demarcation,” Med. Phys. 40(2), 021909 (2013).
[Crossref] [PubMed]

U. Lundström, D. H. Larsson, A. Burvall, L. Scott, U. K. Westermark, M. Wilhelm, M. Arsenian Henriksson, and H. M. Hertz, “X-ray phase-contrast CO2 angiography for sub-10 μm vessel imaging,” Phys. Med. Biol. 57(22), 7431–7441 (2012).
[Crossref] [PubMed]

Aslund, M.

E. Fredenberg, M. Danielsson, J. W. Stayman, J. H. Siewerdsen, and M. Aslund, “Ideal-observer detectability in photon-counting differential phase-contrast imaging using a linear-systems approach,” Med. Phys. 39(9), 5317–5335 (2012).
[Crossref] [PubMed]

Bamberg, F.

Baró, J.

J. Baró, J. Sempau, J. M. Fernandezvarea, and F. Salvat, “Penelope - an algorithm for Monte-Carlo simulation of the penetration and energy-loss of electrons and positrons in matter,” Nucl. Instrum. Methods Phys. Res. B 100(1), 31–46 (1995).
[Crossref]

Bartl, P.

T. Weber, P. Bartl, F. Bayer, J. Durst, W. Haas, T. Michel, A. Ritter, and G. Anton, “Noise in x-ray grating-based phase-contrast imaging,” Med. Phys. 38(7), 4133–4140 (2011).
[Crossref] [PubMed]

Baruchel, J.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[Crossref] [PubMed]

Bayer, F.

T. Weber, P. Bartl, F. Bayer, J. Durst, W. Haas, T. Michel, A. Ritter, and G. Anton, “Noise in x-ray grating-based phase-contrast imaging,” Med. Phys. 38(7), 4133–4140 (2011).
[Crossref] [PubMed]

Bech, M.

Bevins, N.

K. Li, N. Bevins, J. Zambelli, and G. H. Chen, “Fundamental relationship between the noise properties of grating-based differential phase contrast CT and absorption CT: theoretical framework using a cascaded system model and experimental validation,” Med. Phys. 40(2), 021908 (2013).
[Crossref] [PubMed]

G. H. Chen, J. Zambelli, K. Li, N. Bevins, and Z. H. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38(2), 584–588 (2011).
[Crossref] [PubMed]

J. Zambelli, N. Bevins, Z. Qi, and G. H. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37(6), 2473–2479 (2010).
[Crossref] [PubMed]

Bravin, A.

Brismar, H.

U. Lundström, D. H. Larsson, A. Burvall, P. A. Takman, L. Scott, H. Brismar, and H. M. Hertz, “X-ray phase contrast for CO2 microangiography,” Phys. Med. Biol. 57(9), 2603–2617 (2012).
[Crossref] [PubMed]

Brönnimann, Ch.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008).
[Crossref] [PubMed]

Bunk, O.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98(10), 108105 (2007).
[Crossref] [PubMed]

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(4), 258–261 (2006).
[Crossref]

Burvall, A.

D. H. Larsson, U. Lundström, U. K. Westermark, M. Arsenian Henriksson, A. Burvall, and H. M. Hertz, “First application of liquid-metal-jet sources for small-animal imaging: high-resolution CT and phase-contrast tumor demarcation,” Med. Phys. 40(2), 021909 (2013).
[Crossref] [PubMed]

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Larsson, D. H.

T. Zhou, U. Lundström, D. H. Larsson, H. M. Hertz, and A. Burvall, “Low-dose phase-contrast X-ray imaging: a comparison of two methods,” J. Phys. Conf. Ser. 463, 012041 (2013).
[Crossref]

D. H. Larsson, U. Lundström, U. K. Westermark, M. Arsenian Henriksson, A. Burvall, and H. M. Hertz, “First application of liquid-metal-jet sources for small-animal imaging: high-resolution CT and phase-contrast tumor demarcation,” Med. Phys. 40(2), 021909 (2013).
[Crossref] [PubMed]

U. Lundström, D. H. Larsson, A. Burvall, P. A. Takman, L. Scott, H. Brismar, and H. M. Hertz, “X-ray phase contrast for CO2 microangiography,” Phys. Med. Biol. 57(9), 2603–2617 (2012).
[Crossref] [PubMed]

U. Lundström, D. H. Larsson, A. Burvall, L. Scott, U. K. Westermark, M. Wilhelm, M. Arsenian Henriksson, and H. M. Hertz, “X-ray phase-contrast CO2 angiography for sub-10 μm vessel imaging,” Phys. Med. Biol. 57(22), 7431–7441 (2012).
[Crossref] [PubMed]

A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in X-ray phase-contrast imaging suitable for tomography,” Opt. Express 19(11), 10359–10376 (2011).
[Crossref] [PubMed]

D. H. Larsson, P. A. C. Takman, U. Lundström, A. Burvall, and H. M. Hertz, “A 24 keV liquid-metal-jet x-ray source for biomedical applications,” Rev. Sci. Instrum. 82(12), 123701 (2011).
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K. Li, N. Bevins, J. Zambelli, and G. H. Chen, “Fundamental relationship between the noise properties of grating-based differential phase contrast CT and absorption CT: theoretical framework using a cascaded system model and experimental validation,” Med. Phys. 40(2), 021908 (2013).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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T. Zhou, U. Lundström, D. H. Larsson, H. M. Hertz, and A. Burvall, “Low-dose phase-contrast X-ray imaging: a comparison of two methods,” J. Phys. Conf. Ser. 463, 012041 (2013).
[Crossref]

T. Thüring, T. Zhou, U. Lundström, A. Burvall, S. Rutishauser, C. David, H. M. Hertz, and M. Stampanoni, “X-ray grating interferometry with a liquid-metal-jet source,” Appl. Phys. Lett. 103(9), 091105 (2013).
[Crossref]

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[Crossref] [PubMed]

U. Lundström, D. H. Larsson, A. Burvall, P. A. Takman, L. Scott, H. Brismar, and H. M. Hertz, “X-ray phase contrast for CO2 microangiography,” Phys. Med. Biol. 57(9), 2603–2617 (2012).
[Crossref] [PubMed]

U. Lundström, D. H. Larsson, A. Burvall, L. Scott, U. K. Westermark, M. Wilhelm, M. Arsenian Henriksson, and H. M. Hertz, “X-ray phase-contrast CO2 angiography for sub-10 μm vessel imaging,” Phys. Med. Biol. 57(22), 7431–7441 (2012).
[Crossref] [PubMed]

D. H. Larsson, P. A. C. Takman, U. Lundström, A. Burvall, and H. M. Hertz, “A 24 keV liquid-metal-jet x-ray source for biomedical applications,” Rev. Sci. Instrum. 82(12), 123701 (2011).
[Crossref] [PubMed]

A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in X-ray phase-contrast imaging suitable for tomography,” Opt. Express 19(11), 10359–10376 (2011).
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J. Vlassenbroeck, M. Dierick, B. Masschaele, V. Cnudde, L. van Hoorebeke, and P. Jacobs, “Software tools for quantification of X-ray microtomography,” Nucl. Instrum. Methods Phys. Res. A 580(1), 442–445 (2007).
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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(1), 33–40 (2002).
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Muehleman, C.

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O. Hemberg, M. Otendal, and H. M. Hertz, “Liquid-metal-jet anode electron-impact x-ray source,” Appl. Phys. Lett. 83(7), 1483–1485 (2003).
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C. Parham, Z. Zhong, D. M. Connor, L. D. Chapman, and E. D. Pisano, “Design and implementation of a compact low-dose diffraction enhanced medical imaging system,” Acad. Radiol. 16(8), 911–917 (2009).
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[Crossref] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98(10), 108105 (2007).
[Crossref] [PubMed]

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(4), 258–261 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

Pinzer, B. R.

Pisano, E. D.

C. Parham, Z. Zhong, D. M. Connor, L. D. Chapman, and E. D. Pisano, “Design and implementation of a compact low-dose diffraction enhanced medical imaging system,” Acad. Radiol. 16(8), 911–917 (2009).
[Crossref] [PubMed]

Pogany, A.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Prats-Esteve, M.

A. Fandos-Morera, M. Prats-Esteve, J. M. Tura-Soteras, and A. Traveria-Cros, “Breast tumors: composition of microcalcifications,” Radiology 169(2), 325–327 (1988).
[PubMed]

Qi, Z.

J. Zambelli, N. Bevins, Z. Qi, and G. H. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37(6), 2473–2479 (2010).
[Crossref] [PubMed]

Qi, Z. H.

G. H. Chen, J. Zambelli, K. Li, N. Bevins, and Z. H. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38(2), 584–588 (2011).
[Crossref] [PubMed]

Rack, A.

Raupach, R.

R. Raupach and T. Flohr, “Performance evaluation of x-ray differential phase contrast computed tomography (PCT) with respect to medical imaging,” Med. Phys. 39(8), 4761–4774 (2012).
[Crossref] [PubMed]

R. Raupach and T. G. Flohr, “Analytical evaluation of the signal and noise propagation in x-ray differential phase-contrast computed tomography,” Phys. Med. Biol. 56(7), 2219–2244 (2011).
[Crossref] [PubMed]

Reinhart, B.

I. Nesch, D. P. Fogarty, T. Tzvetkov, B. Reinhart, A. C. Walus, G. Khelashvili, C. Muehleman, and D. Chapman, “The design and application of an in-laboratory diffraction-enhanced x-ray imaging instrument,” Rev. Sci. Instrum. 80(9), 093702 (2009).
[Crossref] [PubMed]

Reiser, M.

Revol, V.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[Crossref] [PubMed]

Ritter, A.

T. Weber, P. Bartl, F. Bayer, J. Durst, W. Haas, T. Michel, A. Ritter, and G. Anton, “Noise in x-ray grating-based phase-contrast imaging,” Med. Phys. 38(7), 4133–4140 (2011).
[Crossref] [PubMed]

Roessl, E.

T. Köhler, K. Jürgen Engel, and E. Roessl, “Noise properties of grating-based x-ray phase contrast computed tomography,” Med. Phys. 38(S1), S106–S116 (2011).
[Crossref] [PubMed]

Rose, A.

Rutishauser, S.

T. Thüring, T. Zhou, U. Lundström, A. Burvall, S. Rutishauser, C. David, H. M. Hertz, and M. Stampanoni, “X-ray grating interferometry with a liquid-metal-jet source,” Appl. Phys. Lett. 103(9), 091105 (2013).
[Crossref]

P. Modregger, B. R. Pinzer, T. Thüring, S. Rutishauser, C. David, and M. Stampanoni, “Sensitivity of X-ray grating interferometry,” Opt. Express 19(19), 18324–18338 (2011).
[Crossref] [PubMed]

Salvat, F.

J. Baró, J. Sempau, J. M. Fernandezvarea, and F. Salvat, “Penelope - an algorithm for Monte-Carlo simulation of the penetration and energy-loss of electrons and positrons in matter,” Nucl. Instrum. Methods Phys. Res. B 100(1), 31–46 (1995).
[Crossref]

Schleede, S.

Schlossbauer, T.

Scott, L.

U. Lundström, D. H. Larsson, A. Burvall, L. Scott, U. K. Westermark, M. Wilhelm, M. Arsenian Henriksson, and H. M. Hertz, “X-ray phase-contrast CO2 angiography for sub-10 μm vessel imaging,” Phys. Med. Biol. 57(22), 7431–7441 (2012).
[Crossref] [PubMed]

U. Lundström, D. H. Larsson, A. Burvall, P. A. Takman, L. Scott, H. Brismar, and H. M. Hertz, “X-ray phase contrast for CO2 microangiography,” Phys. Med. Biol. 57(9), 2603–2617 (2012).
[Crossref] [PubMed]

Sempau, J.

J. Baró, J. Sempau, J. M. Fernandezvarea, and F. Salvat, “Penelope - an algorithm for Monte-Carlo simulation of the penetration and energy-loss of electrons and positrons in matter,” Nucl. Instrum. Methods Phys. Res. B 100(1), 31–46 (1995).
[Crossref]

Siewerdsen, J. H.

E. Fredenberg, M. Danielsson, J. W. Stayman, J. H. Siewerdsen, and M. Aslund, “Ideal-observer detectability in photon-counting differential phase-contrast imaging using a linear-systems approach,” Med. Phys. 39(9), 5317–5335 (2012).
[Crossref] [PubMed]

Stampanoni, M.

Stayman, J. W.

E. Fredenberg, M. Danielsson, J. W. Stayman, J. H. Siewerdsen, and M. Aslund, “Ideal-observer detectability in photon-counting differential phase-contrast imaging using a linear-systems approach,” Med. Phys. 39(9), 5317–5335 (2012).
[Crossref] [PubMed]

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(6607), 335–338 (1996).
[Crossref]

Stockmar, M.

Straumann, U.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[Crossref] [PubMed]

Sztrókay, A.

Takman, P. A.

U. Lundström, D. H. Larsson, A. Burvall, P. A. Takman, L. Scott, H. Brismar, and H. M. Hertz, “X-ray phase contrast for CO2 microangiography,” Phys. Med. Biol. 57(9), 2603–2617 (2012).
[Crossref] [PubMed]

Takman, P. A. C.

D. H. Larsson, P. A. C. Takman, U. Lundström, A. Burvall, and H. M. Hertz, “A 24 keV liquid-metal-jet x-ray source for biomedical applications,” Rev. Sci. Instrum. 82(12), 123701 (2011).
[Crossref] [PubMed]

A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in X-ray phase-contrast imaging suitable for tomography,” Opt. Express 19(11), 10359–10376 (2011).
[Crossref] [PubMed]

Tang, S.

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast CT compared with the conventional CT--noise power spectrum NPS(k),” Med. Phys. 38(7), 4386–4395 (2011).
[Crossref] [PubMed]

Tang, X.

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast CT compared with the conventional CT--noise power spectrum NPS(k),” Med. Phys. 38(7), 4386–4395 (2011).
[Crossref] [PubMed]

Thibault, P.

Thüring, T.

T. Thüring, T. Zhou, U. Lundström, A. Burvall, S. Rutishauser, C. David, H. M. Hertz, and M. Stampanoni, “X-ray grating interferometry with a liquid-metal-jet source,” Appl. Phys. Lett. 103(9), 091105 (2013).
[Crossref]

P. Modregger, B. R. Pinzer, T. Thüring, S. Rutishauser, C. David, and M. Stampanoni, “Sensitivity of X-ray grating interferometry,” Opt. Express 19(19), 18324–18338 (2011).
[Crossref] [PubMed]

Traveria-Cros, A.

A. Fandos-Morera, M. Prats-Esteve, J. M. Tura-Soteras, and A. Traveria-Cros, “Breast tumors: composition of microcalcifications,” Radiology 169(2), 325–327 (1988).
[PubMed]

Tura-Soteras, J. M.

A. Fandos-Morera, M. Prats-Esteve, J. M. Tura-Soteras, and A. Traveria-Cros, “Breast tumors: composition of microcalcifications,” Radiology 169(2), 325–327 (1988).
[PubMed]

Tzvetkov, T.

I. Nesch, D. P. Fogarty, T. Tzvetkov, B. Reinhart, A. C. Walus, G. Khelashvili, C. Muehleman, and D. Chapman, “The design and application of an in-laboratory diffraction-enhanced x-ray imaging instrument,” Rev. Sci. Instrum. 80(9), 093702 (2009).
[Crossref] [PubMed]

Urban, C.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[Crossref] [PubMed]

van Hoorebeke, L.

J. Vlassenbroeck, M. Dierick, B. Masschaele, V. Cnudde, L. van Hoorebeke, and P. Jacobs, “Software tools for quantification of X-ray microtomography,” Nucl. Instrum. Methods Phys. Res. A 580(1), 442–445 (2007).
[Crossref]

Vlassenbroeck, J.

J. Vlassenbroeck, M. Dierick, B. Masschaele, V. Cnudde, L. van Hoorebeke, and P. Jacobs, “Software tools for quantification of X-ray microtomography,” Nucl. Instrum. Methods Phys. Res. A 580(1), 442–445 (2007).
[Crossref]

von Smitten, K.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[Crossref] [PubMed]

Walus, A. C.

I. Nesch, D. P. Fogarty, T. Tzvetkov, B. Reinhart, A. C. Walus, G. Khelashvili, C. Muehleman, and D. Chapman, “The design and application of an in-laboratory diffraction-enhanced x-ray imaging instrument,” Rev. Sci. Instrum. 80(9), 093702 (2009).
[Crossref] [PubMed]

Weber, T.

T. Weber, P. Bartl, F. Bayer, J. Durst, W. Haas, T. Michel, A. Ritter, and G. Anton, “Noise in x-ray grating-based phase-contrast imaging,” Med. Phys. 38(7), 4133–4140 (2011).
[Crossref] [PubMed]

Weitkamp, T.

Westermark, U. K.

D. H. Larsson, U. Lundström, U. K. Westermark, M. Arsenian Henriksson, A. Burvall, and H. M. Hertz, “First application of liquid-metal-jet sources for small-animal imaging: high-resolution CT and phase-contrast tumor demarcation,” Med. Phys. 40(2), 021909 (2013).
[Crossref] [PubMed]

U. Lundström, D. H. Larsson, A. Burvall, L. Scott, U. K. Westermark, M. Wilhelm, M. Arsenian Henriksson, and H. M. Hertz, “X-ray phase-contrast CO2 angiography for sub-10 μm vessel imaging,” Phys. Med. Biol. 57(22), 7431–7441 (2012).
[Crossref] [PubMed]

Wilhelm, M.

U. Lundström, D. H. Larsson, A. Burvall, L. Scott, U. K. Westermark, M. Wilhelm, M. Arsenian Henriksson, and H. M. Hertz, “X-ray phase-contrast CO2 angiography for sub-10 μm vessel imaging,” Phys. Med. Biol. 57(22), 7431–7441 (2012).
[Crossref] [PubMed]

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(1), 33–40 (2002).
[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(6607), 335–338 (1996).
[Crossref]

Yang, Y.

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast CT compared with the conventional CT--noise power spectrum NPS(k),” Med. Phys. 38(7), 4386–4395 (2011).
[Crossref] [PubMed]

Zambelli, J.

K. Li, N. Bevins, J. Zambelli, and G. H. Chen, “Fundamental relationship between the noise properties of grating-based differential phase contrast CT and absorption CT: theoretical framework using a cascaded system model and experimental validation,” Med. Phys. 40(2), 021908 (2013).
[Crossref] [PubMed]

G. H. Chen, J. Zambelli, K. Li, N. Bevins, and Z. H. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38(2), 584–588 (2011).
[Crossref] [PubMed]

J. Zambelli, N. Bevins, Z. Qi, and G. H. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37(6), 2473–2479 (2010).
[Crossref] [PubMed]

Zanette, I.

Zhong, Z.

C. Parham, Z. Zhong, D. M. Connor, L. D. Chapman, and E. D. Pisano, “Design and implementation of a compact low-dose diffraction enhanced medical imaging system,” Acad. Radiol. 16(8), 911–917 (2009).
[Crossref] [PubMed]

Zhou, T.

T. Thüring, T. Zhou, U. Lundström, A. Burvall, S. Rutishauser, C. David, H. M. Hertz, and M. Stampanoni, “X-ray grating interferometry with a liquid-metal-jet source,” Appl. Phys. Lett. 103(9), 091105 (2013).
[Crossref]

T. Zhou, U. Lundström, D. H. Larsson, H. M. Hertz, and A. Burvall, “Low-dose phase-contrast X-ray imaging: a comparison of two methods,” J. Phys. Conf. Ser. 463, 012041 (2013).
[Crossref]

Ziegler, E.

Acad. Radiol. (1)

C. Parham, Z. Zhong, D. M. Connor, L. D. Chapman, and E. D. Pisano, “Design and implementation of a compact low-dose diffraction enhanced medical imaging system,” Acad. Radiol. 16(8), 911–917 (2009).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

O. Hemberg, M. Otendal, and H. M. Hertz, “Liquid-metal-jet anode electron-impact x-ray source,” Appl. Phys. Lett. 83(7), 1483–1485 (2003).
[Crossref]

T. Thüring, T. Zhou, U. Lundström, A. Burvall, S. Rutishauser, C. David, H. M. Hertz, and M. Stampanoni, “X-ray grating interferometry with a liquid-metal-jet source,” Appl. Phys. Lett. 103(9), 091105 (2013).
[Crossref]

Biomed. Opt. Express (1)

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(1), 33–40 (2002).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Phys. Conf. Ser. (1)

T. Zhou, U. Lundström, D. H. Larsson, H. M. Hertz, and A. Burvall, “Low-dose phase-contrast X-ray imaging: a comparison of two methods,” J. Phys. Conf. Ser. 463, 012041 (2013).
[Crossref]

Med. Phys. (9)

G. H. Chen, J. Zambelli, K. Li, N. Bevins, and Z. H. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38(2), 584–588 (2011).
[Crossref] [PubMed]

K. Li, N. Bevins, J. Zambelli, and G. H. Chen, “Fundamental relationship between the noise properties of grating-based differential phase contrast CT and absorption CT: theoretical framework using a cascaded system model and experimental validation,” Med. Phys. 40(2), 021908 (2013).
[Crossref] [PubMed]

J. Zambelli, N. Bevins, Z. Qi, and G. H. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37(6), 2473–2479 (2010).
[Crossref] [PubMed]

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast CT compared with the conventional CT--noise power spectrum NPS(k),” Med. Phys. 38(7), 4386–4395 (2011).
[Crossref] [PubMed]

T. Weber, P. Bartl, F. Bayer, J. Durst, W. Haas, T. Michel, A. Ritter, and G. Anton, “Noise in x-ray grating-based phase-contrast imaging,” Med. Phys. 38(7), 4133–4140 (2011).
[Crossref] [PubMed]

D. H. Larsson, U. Lundström, U. K. Westermark, M. Arsenian Henriksson, A. Burvall, and H. M. Hertz, “First application of liquid-metal-jet sources for small-animal imaging: high-resolution CT and phase-contrast tumor demarcation,” Med. Phys. 40(2), 021909 (2013).
[Crossref] [PubMed]

T. Köhler, K. Jürgen Engel, and E. Roessl, “Noise properties of grating-based x-ray phase contrast computed tomography,” Med. Phys. 38(S1), S106–S116 (2011).
[Crossref] [PubMed]

E. Fredenberg, M. Danielsson, J. W. Stayman, J. H. Siewerdsen, and M. Aslund, “Ideal-observer detectability in photon-counting differential phase-contrast imaging using a linear-systems approach,” Med. Phys. 39(9), 5317–5335 (2012).
[Crossref] [PubMed]

R. Raupach and T. Flohr, “Performance evaluation of x-ray differential phase contrast computed tomography (PCT) with respect to medical imaging,” Med. Phys. 39(8), 4761–4774 (2012).
[Crossref] [PubMed]

Nat. Mater. (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008).
[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(4), 258–261 (2006).
[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(6607), 335–338 (1996).
[Crossref]

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

J. Vlassenbroeck, M. Dierick, B. Masschaele, V. Cnudde, L. van Hoorebeke, and P. Jacobs, “Software tools for quantification of X-ray microtomography,” Nucl. Instrum. Methods Phys. Res. A 580(1), 442–445 (2007).
[Crossref]

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

J. Baró, J. Sempau, J. M. Fernandezvarea, and F. Salvat, “Penelope - an algorithm for Monte-Carlo simulation of the penetration and energy-loss of electrons and positrons in matter,” Nucl. Instrum. Methods Phys. Res. B 100(1), 31–46 (1995).
[Crossref]

Opt. Express (5)

Phys. Med. Biol. (4)

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[Crossref] [PubMed]

R. Raupach and T. G. Flohr, “Analytical evaluation of the signal and noise propagation in x-ray differential phase-contrast computed tomography,” Phys. Med. Biol. 56(7), 2219–2244 (2011).
[Crossref] [PubMed]

U. Lundström, D. H. Larsson, A. Burvall, P. A. Takman, L. Scott, H. Brismar, and H. M. Hertz, “X-ray phase contrast for CO2 microangiography,” Phys. Med. Biol. 57(9), 2603–2617 (2012).
[Crossref] [PubMed]

U. Lundström, D. H. Larsson, A. Burvall, L. Scott, U. K. Westermark, M. Wilhelm, M. Arsenian Henriksson, and H. M. Hertz, “X-ray phase-contrast CO2 angiography for sub-10 μm vessel imaging,” Phys. Med. Biol. 57(22), 7431–7441 (2012).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98(10), 108105 (2007).
[Crossref] [PubMed]

Radiat. Phys. Chem. (1)

L. Kissel, “RTAB: the Rayleigh scattering database,” Radiat. Phys. Chem. 59(2), 185–200 (2000).
[Crossref]

Radiology (1)

A. Fandos-Morera, M. Prats-Esteve, J. M. Tura-Soteras, and A. Traveria-Cros, “Breast tumors: composition of microcalcifications,” Radiology 169(2), 325–327 (1988).
[PubMed]

Rev. Sci. Instrum. (3)

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[Crossref] [PubMed]

D. H. Larsson, P. A. C. Takman, U. Lundström, A. Burvall, and H. M. Hertz, “A 24 keV liquid-metal-jet x-ray source for biomedical applications,” Rev. Sci. Instrum. 82(12), 123701 (2011).
[Crossref] [PubMed]

I. Nesch, D. P. Fogarty, T. Tzvetkov, B. Reinhart, A. C. Walus, G. Khelashvili, C. Muehleman, and D. Chapman, “The design and application of an in-laboratory diffraction-enhanced x-ray imaging instrument,” Rev. Sci. Instrum. 80(9), 093702 (2009).
[Crossref] [PubMed]

Other (3)

D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2009).

J. H. Hubbell and S. M. Seltzer, Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients from 1 keV to 20 MeV for Elements Z = 1 to 92 and 48 Additional Substances of Dosimetric Interest (Radiation and Biomolecular Physics Division, PML, NIST, 1996).

H. H. Barrett and K. J. Myers, Foundations of Image Science (John Wiley, 2003).

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Figures (6)

Fig. 1
Fig. 1

(a) Experimental arrangement for PBI consisting of the x-ray source, the object and the detector. (b) The same for GBI also including phase (G1) and absorption (G2) gratings. (c) The emission spectrum of the Ga/In/Sn liquid-metal-jet x-ray source after passage through the liquid-paraffin bath.

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Fig. 2
Fig. 2

Schematic illustration of the simulation process for GBI, where × indicates multiplication and ⊗ convolution. For PBI, the principle is the same but G1 and G2 are not included.

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Fig. 3
Fig. 3

Simulated and experimental PBI (a-e) and DPC images (f-j) of a phantom consisting of four PET monofilaments in a liquid-paraffin bath. Their diameters were found to be 494, 213, 100 and 23 µm respectively by matching the widths of the simulated images to the experimental ones [38]. Projections (c, h) have 4 h total exposure time. Profiles are integrated vertically over 33 pixels from the images above them. Slices from tomographic reconstructions (e, j) have an exposure time of 30 h.

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Fig. 4
Fig. 4

Simulated dose required for detecting a PET cylinder, in a liquid-paraffin bath, as a function of the cylinder diameter. Simulations for DPC and PBI were done both with a liquid-metal-jet source spectrum and for 25 keV monochromatic x-rays. For the DPC Polychromatic Adjusted line, the visibility was adjusted to agree with the experiments. Similarly, the signal of the PBI Polychromatic Adjusted line was modified to match experimental data.

Download Full Size | PPT Slide | PDF

Fig. 5
Fig. 5

Projection images of a breast biopsy sample using (a) PBI and (b) GBI absorption and (c) GBI DPC. The exposure time is 16 min per projection for PBI and 2 min times 8 steps for GBI. Inserts (d) and (e) are the enlarged views of a region of 0.5 × 0.4 mm2, containing calcifications, for PBI and GBI absorption respectively. (a, b, d, e) have a linear color scale from 0.5 to 1.05, where the background is normalized to 1, and (c) from –π to π. Tomography slices using (f) PBI with 5 min exposure time per projection, (g) absorption and (h) DPC from the same data set with 1 min exposure time per phase-scanning step and 5 steps per projection direction. The images were acquired over 360° with 0.5° increment for both methods.

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Fig. 6
Fig. 6

Magnified images of calcifications (upper row) and adipose tissue (lower row) of PBI reconstructed using parameters of breast tissue and calcifications (a, b) from CT slice Fig. 5(f), PBI reconstructed using parameters of breast tissue and adipose tissue (c, d), GBI DPC (e, f) from CT slice Fig. 5(h), and GBI absorption from slice Fig. 5(g).

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Tables (1)

Tables Icon

Table 1 Geometry parameters of the experiments given as distances from the source

View Table

Equations (7)

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I(n)= I 0 +acos( 2πn N +φ ), with n=0, 1, ..., N-1,
c(k)= 1 N ∑ n=0 N−1 I(n)⋅ e i2π n N k ,
I 0 =c(0), a=2| c(1) |, φ=arg[ c(1) ].
V= a I 0
SNR 2 = ∬ | ΔG(u) | 2 NPS(u) d 2 u ,
σ φ 2 = 2 V 2 (N I 0 ) 2 ∑ n=0 N−1 σ n 2 .
NPS φ = 2 V 2 (N I 0 ) 2 ∑ n=0 N−1 NPS n .

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