Abstract

Propagation-based phase-contrast X-ray imaging provides high-resolution, dose-efficient images of biological materials. A crucial challenge is quantitative reconstruction, referred to as phase retrieval, of multi-material samples from single-distance, and hence incomplete, data. In this work, the two most promising methods for multi-material samples, the parallel method, and the linear method, are analytically, numerically, and experimentally compared. Both methods are designed for computed tomography, as they rely on segmentation in the tomographic reconstruction. The methods are found to result in comparable image quality, but the linear method provides faster reconstruction. In addition, as already done for the parallel method, we show that the linear method provides quantitative reconstruction for monochromatic radiation.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2016 (1)

A. Ruhlandt and T. Salditt, “Three-dimensional propagation in near-field tomographic X-ray phase retrieval,” Acta Crystallographica Section A 72(2), 215–221 (2016).
[Crossref]

2015 (1)

2014 (1)

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M.-C. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112, 253903 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (1)

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

2011 (2)

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]

M. Beltran, D. Paganin, K. Siu, A. Fouras, S. Hooper, D. Reser, and M. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

2010 (1)

2008 (1)

2007 (1)

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[Crossref]

2006 (2)

S. Zabler, H. Riesemeier, P. Fratzl, and P. Zaslansky, “Fresnel-propagated imaging for the study of human tooth dentin by partially coherent x-ray tomography,” Opt. Express 14(19), 8584–8597 (2006).
[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, 258–261 (2006).
[Crossref]

2005 (2)

2004 (1)

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] [PubMed]

2000 (1)

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000).
[Crossref]

1999 (1)

A. V. Bronnikov, “Reconstruction formulas for phase-contrast imaging,” Opt. Commun. 171, 239–244 (1999).
[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(7), 2774–2782 (1997).
[Crossref]

1996 (3)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

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]

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]

1995 (1)

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]

1983 (1)

1917 (1)

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Berichte über die Verhandlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig 69, 262 (1917).

Arhatari, B. D.

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Balaur, E.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[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]

Baruchel, J.

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]

Beltran, M.

M. Beltran, D. Paganin, K. Siu, A. Fouras, S. Hooper, D. Reser, and M. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

M. Beltran, D. Paganin, K. Uesugi, and M. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18(7), 6423–6436 (2010).
[Crossref] [PubMed]

Berk, K. N.

J. L. Devore and K. N. Berk, Modern Mathematical Statistics with Applications, 2nd ed. (Springer, 2012), Chap. 13.
[Crossref]

Beutel, J.

J. Beutel, H. L. Kundel, and R. L. Van Metter, Handbook of Medical Imaging: Volume 1. Physics and Psychophysics (SPIE Press, 2000), Chap. 3.

Bronnikov, A. V.

A. V. Bronnikov, “Reconstruction formulas for phase-contrast imaging,” Opt. Commun. 171, 239–244 (1999).
[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]

Burvall, A.

Cloetens, P.

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]

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. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

David, C.

Davids, T. J.

Devore, J. L.

J. L. Devore and K. N. Berk, Modern Mathematical Statistics with Applications, 2nd ed. (Springer, 2012), Chap. 13.
[Crossref]

Diaz, A.

Fitzgerald, R.

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000).
[Crossref]

Fouras, A.

M. Beltran, D. Paganin, K. Siu, A. Fouras, S. Hooper, D. Reser, and M. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

Fratzl, P.

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(7), 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]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics, 3rd ed. (Roberts & Company Publishers, 2005), Chap. 8.

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. E.

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[Crossref]

T. E. Gureyev, T. J. Davids, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born-and Rytov-type approximations,” Appl. Opt. 43(12), 2418–2430 (2004).
[Crossref] [PubMed]

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] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 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]

Hannah, K.

Hertz, H. M.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M.-C. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112, 253903 (2014).
[Crossref] [PubMed]

T. Zhou, U. Lundström, T. Thüring, S. Rutishauser, D. H. Larsson, M. Stampanoni, C. David, H. M. Hertz, and A. Burvall, “Comparison of two x-ray phase-contrast imaging methods with a microfocus source,” Opt. Express 21(25), 30183–30195 (2013).
[Crossref]

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]

W. Vågberg, J. Persson, L. Szekely, and H. M. Hertz, “Cellular-resolution 3D virtual histology of human coronary arteries using x-ray phase tomography,” Submitted.

Hooper, S.

M. Beltran, D. Paganin, K. Siu, A. Fouras, S. Hooper, D. Reser, and M. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

Kitchen, M.

M. Beltran, D. Paganin, K. Siu, A. Fouras, S. Hooper, D. Reser, and M. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

M. Beltran, D. Paganin, K. Uesugi, and M. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18(7), 6423–6436 (2010).
[Crossref] [PubMed]

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]

Kundel, H. L.

J. Beutel, H. L. Kundel, and R. L. Van Metter, Handbook of Medical Imaging: Volume 1. Physics and Psychophysics (SPIE Press, 2000), Chap. 3.

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]

Larsson, D. H.

Lundström, U.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M.-C. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112, 253903 (2014).
[Crossref] [PubMed]

T. Zhou, U. Lundström, T. Thüring, S. Rutishauser, D. H. Larsson, M. Stampanoni, C. David, H. M. Hertz, and A. Burvall, “Comparison of two x-ray phase-contrast imaging methods with a microfocus source,” Opt. Express 21(25), 30183–30195 (2013).
[Crossref]

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]

U. Lundström, Phase-Contrast X-ray Carbon Dioxide Angiography, Ph.D. thesis (KTH Royal Institute of Technology, 2014).

Mayo, S. C.

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[Crossref]

T. E. Gureyev, T. J. Davids, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born-and Rytov-type approximations,” Appl. Opt. 43(12), 2418–2430 (2004).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

Momose, A.

A. Momose, “Recent Advances in X-ray Phase Imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
[Crossref]

Morgan, K. S.

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

Myers, G. R.

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[Crossref]

Nugent, K. A.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Paganin, D.

M. Beltran, D. Paganin, K. Siu, A. Fouras, S. Hooper, D. Reser, and M. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

M. Beltran, D. Paganin, K. Uesugi, and M. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18(7), 6423–6436 (2010).
[Crossref] [PubMed]

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] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Paganin, D. M.

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[Crossref]

Peele, A. G.

Persson, J.

W. Vågberg, J. Persson, L. Szekely, and H. M. Hertz, “Cellular-resolution 3D virtual histology of human coronary arteries using x-ray phase tomography,” Submitted.

Pfeiffer, F.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M.-C. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112, 253903 (2014).
[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, 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]

Pogany, A.

T. E. Gureyev, T. J. Davids, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born-and Rytov-type approximations,” Appl. Opt. 43(12), 2418–2430 (2004).
[Crossref] [PubMed]

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 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]

Radon, J.

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Berichte über die Verhandlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig 69, 262 (1917).

Reser, D.

M. Beltran, D. Paganin, K. Siu, A. Fouras, S. Hooper, D. Reser, and M. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

Riesemeier, H.

Ruhlandt, A.

A. Ruhlandt and T. Salditt, “Three-dimensional propagation in near-field tomographic X-ray phase retrieval,” Acta Crystallographica Section A 72(2), 215–221 (2016).
[Crossref]

Rutishauser, S.

Salditt, T.

A. Ruhlandt and T. Salditt, “Three-dimensional propagation in near-field tomographic X-ray phase retrieval,” Acta Crystallographica Section A 72(2), 215–221 (2016).
[Crossref]

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, 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]

Siu, K.

M. Beltran, D. Paganin, K. Siu, A. Fouras, S. Hooper, D. Reser, and M. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

Siu, K. K. W.

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

Snigirev, A.

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

Snigireva, I.

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

Stampanoni, M.

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]

Szekely, L.

W. Vågberg, J. Persson, L. Szekely, and H. M. Hertz, “Cellular-resolution 3D virtual histology of human coronary arteries using x-ray phase tomography,” Submitted.

Takman, P. A. C.

Teague, M. R.

Thibault, P.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M.-C. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112, 253903 (2014).
[Crossref] [PubMed]

Thüring, T.

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Uesugi, K.

Ullherr, M.

Vågberg, W.

W. Vågberg, J. Persson, L. Szekely, and H. M. Hertz, “Cellular-resolution 3D virtual histology of human coronary arteries using x-ray phase tomography,” Submitted.

Van Metter, R. L.

J. Beutel, H. L. Kundel, and R. L. Van Metter, Handbook of Medical Imaging: Volume 1. Physics and Psychophysics (SPIE Press, 2000), Chap. 3.

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]

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]

Wilkins, S. W.

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[Crossref]

T. E. Gureyev, T. J. Davids, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born-and Rytov-type approximations,” Appl. Opt. 43(12), 2418–2430 (2004).
[Crossref] [PubMed]

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] [PubMed]

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 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]

Zabler, S.

Zanette, I.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M.-C. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112, 253903 (2014).
[Crossref] [PubMed]

Zaslansky, P.

Zdora, M.-C.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M.-C. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112, 253903 (2014).
[Crossref] [PubMed]

Zhou, T.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M.-C. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112, 253903 (2014).
[Crossref] [PubMed]

T. Zhou, U. Lundström, T. Thüring, S. Rutishauser, D. H. Larsson, M. Stampanoni, C. David, H. M. Hertz, and A. Burvall, “Comparison of two x-ray phase-contrast imaging methods with a microfocus source,” Opt. Express 21(25), 30183–30195 (2013).
[Crossref]

Ziegler, E.

Acta Crystallographica Section A (1)

A. Ruhlandt and T. Salditt, “Three-dimensional propagation in near-field tomographic X-ray phase retrieval,” Acta Crystallographica Section A 72(2), 215–221 (2016).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

Berichte über die Verhandlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig (1)

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Berichte über die Verhandlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig 69, 262 (1917).

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] [PubMed]

J. Opt. Soc. Am. (1)

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]

Jpn. J. Appl. Phys. (1)

A. Momose, “Recent Advances in X-ray Phase Imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
[Crossref]

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 (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]

Opt. Commun. (1)

A. V. Bronnikov, “Reconstruction formulas for phase-contrast imaging,” Opt. Commun. 171, 239–244 (1999).
[Crossref]

Opt. Express (7)

M. Ullherr and S. Zabler, “Correcting multi material artifacts from single material phase retrieved holo-tomograms with a simple 3D Fourier method,” Opt. Express 23(25), 32718–32727 (2015).
[Crossref] [PubMed]

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase Imaging Using A Polychromatic X-ray Laboratory Source,” Opt. Express 16(24), 19950–19956 (2008).
[Crossref] [PubMed]

S. Zabler, H. Riesemeier, P. Fratzl, and P. Zaslansky, “Fresnel-propagated imaging for the study of human tooth dentin by partially coherent x-ray tomography,” Opt. Express 14(19), 8584–8597 (2006).
[Crossref] [PubMed]

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]

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]

M. Beltran, D. Paganin, K. Uesugi, and M. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18(7), 6423–6436 (2010).
[Crossref] [PubMed]

T. Zhou, U. Lundström, T. Thüring, S. Rutishauser, D. H. Larsson, M. Stampanoni, C. David, H. M. Hertz, and A. Burvall, “Comparison of two x-ray phase-contrast imaging methods with a microfocus source,” Opt. Express 21(25), 30183–30195 (2013).
[Crossref]

Phys. Med. Biol. (1)

M. Beltran, D. Paganin, K. Siu, A. Fouras, S. Hooper, D. Reser, and M. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

Phys. Rev. A (1)

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[Crossref]

Phys. Rev. Lett. (2)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M.-C. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112, 253903 (2014).
[Crossref] [PubMed]

Phys. Today (1)

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000).
[Crossref]

Rev. Sci. Instrum. (2)

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 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]

Other (7)

W. Vågberg, J. Persson, L. Szekely, and H. M. Hertz, “Cellular-resolution 3D virtual histology of human coronary arteries using x-ray phase tomography,” Submitted.

World Health Organization, Health in 2015: from MDGs, Millennium Development Goals to SDGs, Sustainable Development Goals (World Health OrganizationFrance, 2015).

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

J. W. Goodman, Introduction to Fourier optics, 3rd ed. (Roberts & Company Publishers, 2005), Chap. 8.

U. Lundström, Phase-Contrast X-ray Carbon Dioxide Angiography, Ph.D. thesis (KTH Royal Institute of Technology, 2014).

J. Beutel, H. L. Kundel, and R. L. Van Metter, Handbook of Medical Imaging: Volume 1. Physics and Psychophysics (SPIE Press, 2000), Chap. 3.

J. L. Devore and K. N. Berk, Modern Mathematical Statistics with Applications, 2nd ed. (Springer, 2012), Chap. 13.
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 The parallel algorithm: 1: The projection image (B) together with the projected thickness A(r) (A) are used in Beltran’s formula to retrieve (μ2μ1)T2 (C). 2: D is also retrieved from B using Eq. (1). 3 & 4: C and D are tomographically reconstructed. Slices are shown in E and F, respectively. 5 & 6: The segmentation is performed with an intensity threshold in F. This is used to obtain only material 2 from E, and to remove material 2 from F. 7: The cut out G is combined with H. The value of the encasing (μ1) must be added to G before merging to get μ in the final image I.
Fig. 2
Fig. 2 The linear method’s algorithm: 1: phase retrieval on projection images (A), 2: tomographic reconstruction, 3: segmentation (highly absorbing part D removed), 4: additional phase retrieval on volume image (E), 5: merge of D and F to form final result G.
Fig. 3
Fig. 3 Phase retrieval using the linear method on simulated cylindrical object. (a) Tomographic slice without phase retrieval. (b) Slice after partial phase retrieval. (c) Slice after second phase retrieval and merge. (d–f) Profiles of the images above.
Fig. 4
Fig. 4 Simulated slices after the linear method’s partial phase retrieval showing three noise levels: (a) no added noise, (b) SNRdiff = 1.4, and (c) SNRdiff = 0.9.
Fig. 5
Fig. 5 Phase retrieval using the parallel method on simulated cylindrical object. (a) Tomographic slice after phase retrieval with the parallel formula. (b) Slice after phase retrieval with Paganin’s formula. (c) Merge of (a) and (b) according to the parallel algorithm. (d–f) Profiles of the images above.
Fig. 6
Fig. 6 Phase retrieved images of coronary artery with (a) partial and (b) full extended Paganin phase retrieval as well as (c) the linear method, and (d) the parallel method.
Fig. 7
Fig. 7 Effect of a Wiener filter on a phase retrieval filter H as a function of frequency ω. The dashed dotted line in blue is a conventional filter obtained for positive Δδμ. The solid red line shows a filter with zeros in the denominator caused by negative Δδμ and the dashed black line the result of applying a Wiener filter to the red filter.
Fig. 8
Fig. 8 Sinusoidal Siemens star with negative Δδμ material combination. The diameter in the detector plane is 6.3 mm. (a) Object, (b) simulated phase contrast image, and (c) phase retrieved image.
Fig. 9
Fig. 9 (a) Tomographic slice of a sample containing two materials. The interface has negative edge enhancement (compare with Fig. 3). (b) Result of erroneous phase retrieval with positive Δδμ if applied to image (a). (c) Result of phase retrieval with negative Δδμ and a Wiener filter. Although the retrieval of the edge is improved, amplification of low frequencies close to w0 are visible.
Fig. 10
Fig. 10 For two materials there is a linear relationship between δ and μ at a given energy.

Tables (3)

Tables Icon

Table 1 Agreement between simulated and phase retrieved μ-values [m−1]. All values are given as mean ± standard deviation. Differential signal-to-noise ratio (SNRdiff) for simulated projections is given.

Tables Icon

Table 2 χ2 values for the multi-material reconstructions with different levels of noise.

Tables Icon

Table 3 Overshoot and rise distance at interfaces BD and AD for all images in Fig. 6. An advantage of multi-material methods as well as clear trade-off between overshoot and rise distance is evident from the numbers.

Equations (37)

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

μ T ( r ) = ln ( 𝔉 2 D 1 { 1 4 π 2 d δ μ w 2 + 1 × 𝔉 2 D { I I 0 } } ) ,
( μ 2 μ 1 ) T 2 ( r ) = ln ( 𝔉 2 D 1 { 1 4 π 2 d Δ δ Δ μ w 2 + 1 × 𝔉 2 D { I I 0 exp ( μ 1 A ( r ) ) } } ) ,
μ d z = ln ( 𝔉 2 D 1 { 1 4 π 2 d Δ δ Δ μ w 2 + 1 × 𝔉 2 D { I I 0 } } ) .
V B = M 1 V 1 + M 2 ( V 2 + μ 1 )
V U = M H V + M L 𝔉 2 D 1 { K L 𝔉 2 D { V M L } } 𝔉 2 D 1 { K L 𝔉 2 D { M L } } .
K L = 4 π 2 d Δ δ 1 Δ μ 1 w 2 + 1 4 π 2 d Δ δ 2 Δ μ 2 w 2 + 1 .
SNR diff = S ¯ A S ¯ B σ B
𝔉 { exp ( ( μ 2 μ 1 ) T 2 ( r ) ) } ( d ( δ 2 δ 1 ) ( μ 2 μ 1 ) 4 π 2 w 2 + 1 ) = 𝔉 { I ( r , z = d ) I 0 exp ( μ 1 A ( r ) ) } .
w 0 2 = μ 2 μ 1 δ 2 δ 1 1 4 π 2 d
[ I ( r , z ) ϕ ( r , z ) ] = k z I ( r , z ) ,
I ( r , z = 0 ) = I 0 exp ( μ 1 T 1 ( r ) μ 2 T 2 ( r ) ) ,
φ ( r , z = 0 ) = k [ δ 1 T 1 ( r ) + δ 2 T 2 ( r ) ] .
A ( r ) = T 1 ( r ) + T 2 ( r ) .
[ I 0 exp ( μ 1 T 1 ( r ) μ 2 T 2 ( r ) ) [ k ( δ 1 T 1 ( r ) + δ 2 T 2 ( r ) ) ] ] = k z I ( r , z ) ,
[ I 0 exp ( μ 1 T 1 ( r ) μ 2 T 2 ( r ) ) [ ( δ 1 T 1 ( r ) + δ 2 T 2 ( r ) ) ] ] = z I ( r , z ) .
δ 1 T 1 + δ 2 T 2 = δ 1 ( T 1 + T 2 ) + ( δ 2 δ 1 ) T 2 = δ 1 A + ( δ 2 δ 1 ) T 2
μ 1 T 1 μ 2 T 2 = μ 1 ( T 1 + T 2 ) ( μ 2 μ 1 ) T 2 = μ 1 A ( μ 2 μ 1 ) T 2 .
[ I 0 exp ( μ 1 A ( r ) ) exp ( ( μ 2 μ 1 ) T 2 ( r ) ) [ δ 1 A ( r ) + ( δ 2 δ 1 ) T 2 ( r ) ] ] = z I ( r , z ) .
I 0 exp ( μ 1 A ( r ) ) [ exp ( ( μ 2 μ 1 ) T 2 ( r ) ) [ ( δ 2 δ 1 ) T 2 ( r ) ] ] = z I ( r , z ) .
( δ 2 δ 1 ) I 0 exp ( μ 1 A ( r ) ) ( 1 μ 2 μ 1 2 exp ( ( μ 2 μ 1 ) T 2 ( r ) ) ) .
z I ( r , z ) I ( r , z = d ) I ( r , z = 0 ) d = I ( r , z = d ) I 0 exp ( ( μ 2 μ 1 ) T 2 ( r ) ) exp ( μ 1 A ( r ) ) d .
( δ 2 δ 1 ) I 0 exp ( μ 1 A ( r ) ) ( 1 μ 2 μ 1 2 exp ( ( μ 2 μ 1 ) T 2 ( r ) ) ) = I ( r , z = d ) I 0 exp ( ( μ 2 μ 1 ) T 2 ( r ) ) exp ( μ 1 A ( r ) ) d .
[ d ( δ 2 δ 1 ) ( μ 2 μ 1 ) 2 + 1 ] exp ( ( μ 2 μ 1 ) T 2 ( r ) ) = I ( r , z = d ) I 0 exp ( μ 1 A ( r ) ) .
𝔉 { exp ( ( μ 2 μ 1 ) T 2 ( r ) ) } ( d ( δ 2 δ 1 ) ( μ 2 μ 1 ) 4 π 2 w 2 + 1 ) = 𝔉 { I ( r , z = d ) I 0 exp ( μ 1 A ( r ) ) } .
( μ 2 μ 1 ) T 2 ( r ) = ln ( 𝔉 1 { 1 d ( δ 2 δ 1 ) ( μ 2 μ 1 ) 4 π 2 w 2 + 1 𝔉 { I ( r , z = d ) I 0 exp ( μ 1 A ( r ) ) } } ) .
I ( r , z = 0 ) = I 0 exp ( μ 2 T 2 ( r ) μ 1 T 1 ( r ) ) = I 0 exp ( μ ( r , z ) d z ) .
φ ( r , z = 0 ) = k [ δ 2 T 2 ( r ) + δ 1 T 1 ( r ) ] .
M ( r ) = μ 2 T 2 ( r ) + μ 1 T 1 ( r ) ,
δ 1 = δ 0 + Δ δ Δ μ μ 1
φ ( r , z = 0 ) k = δ 2 T 2 ( r ) + δ 1 T 1 ( r ) = ( δ 0 Δ δ Δ μ μ 2 ) T 2 ( r ) + ( δ 0 Δ δ Δ μ μ 1 ) T 1 ( r )
= δ 0 ( T 2 ( r ) + T 1 ( r ) ) + Δ δ Δ μ M ( r ) = δ 0 A ( r ) + Δ δ Δ μ M ( r )
[ I 0 exp ( M ( r ) ) [ k ( δ 0 A ( r ) + Δ δ Δ μ M ( r ) ) ] ] = k z I ( r , z ) .
Δ δ Δ μ [ I 0 exp ( M ( r ) ) M ( ) ] = z I ( r , z ) .
Δ δ Δ μ I 0 2 e M ( r ) = I ( r , z = d ) I 0 exp ( M ( r ) ) d
( Δ δ Δ μ d 2 + 1 ) I 0 exp ( M ( r ) ) = I ( r , z = d )
𝔉 { exp ( M ( r ) ) } ( Δ δ Δ μ 4 π 2 d w 2 + 1 ) = 𝔉 { I ( r , z = d ) I 0 }
μ ( r , z ) d z = M ( r ) = ln ( 𝔉 1 { 1 Δ δ Δ μ 4 π 2 d w 2 + 1 𝔉 { I ( r , z = d ) I 0 } } )

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