Abstract

The phase retrieval problem can be reduced to the second order partial differential equation. In order to retrieve the absolute values of the X-ray phase and to minimize the reconstruction artifacts we defined the mixed inhomogeneous boundary condition using available a priori information about the sample. Finite element technique was used to solve the boundary value problem. The approach is validated on numerical and experimental phantoms. In order to demonstrate a possible application of the method, we have processed an entire tomographic set of differential phase images and estimated the magnitude of the refractive index decrement for some tissues inside complex biomedical samples.

© 2015 Optical Society of America

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References

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2015 (1)

2014 (3)

J. Sperl, D. Bequ, G. Kudielka, K. Mahdi, P. Edic, and C. Cozzini, “A Fourier-domain algorithm for total-variation regularized phase retrieval in differential X-ray phase contrast imaging,” Opt. Express 22, 450–462 (2014).
[Crossref] [PubMed]

S. Gasilov, A. Mittone, E. Brun, A. Bravin, S. Grandl, A. Mirone, and P. Coan, “Tomographic reconstruction of the refractive index with hard X-rays: an efficient method based on the gradient vector-field approach,” Opt. Express 22, 5216–5227 (2014).
[Crossref] [PubMed]

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

2013 (3)

2012 (2)

2011 (1)

2007 (2)

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

A. Maksimenko, “Nonlinear extension of the X-ray diffraction enhanced imaging,” Appl. Phys. Lett. 90, 154106 (2007).
[Crossref]

2006 (1)

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

2005 (2)

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

M. O. Hasnah, C. Parham, E. D. Pisano, Z. Zhong, O. Oltulu, and D. Chapman, “Mass density images from the diffraction enhanced imaging technique,” Med. Phys. 32, 549–552 (2005).
[Crossref] [PubMed]

2000 (3)

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53, 23–27 (2000).
[Crossref]

C. T. Chantler, “Detailed tabulation of atomic form factors, photoelectric absorption and scattering cross section, and mass attenuation coefficients in the vicinity of absorption edges in the soft X-ray (Z=3036, Z=6089, E=0.1 keV10 keV),” J. Phys. Chem. Ref. Data 29, 597–1048 (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, 335–337 (1996).
[Crossref]

1995 (1)

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2318 (1995).
[Crossref]

1988 (1)

1926 (1)

C. M. Slack, “The refraction of X-Rays in prisms of various materials,” Phys. Rev. 27, 691–695 (1926).
[Crossref]

Adam-Neumair, S.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

Auweter, S. D.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

Ayubi, G. A.

Beliaevskaya, E. A.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2318 (1995).
[Crossref]

Bequ, D.

Berujon, S.

S. Berujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional X-Ray beam phase sensing,” Phys. Rev. Lett. 108, 158102 (2012).
[Crossref] [PubMed]

Birkhoff, G.

G. Birkhoff and R. E. Lynch, Numerical Solution of Elliptic Problems (SIAM Studies in Applied Mathematics1984, pp. 187).

Braun, C.

Bravin, A.

S. Gasilov, A. Mittone, E. Brun, A. Bravin, S. Grandl, A. Mirone, and P. Coan, “Tomographic reconstruction of the refractive index with hard X-rays: an efficient method based on the gradient vector-field approach,” Opt. Express 22, 5216–5227 (2014).
[Crossref] [PubMed]

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol 58, R1–R35 (2013).
[Crossref]

S. Gasilov, A. Mittone, E. Brun, A. Bravin, S. Grandl, and P. Coan, “On the possibility of quantitative refractive-index tomography of large biomedical samples with hard X-rays,” Biomed. Opt. Express 4, 1512–1518 (2013).
[Crossref] [PubMed]

Brun, E.

S. Gasilov, A. Mittone, E. Brun, A. Bravin, S. Grandl, A. Mirone, and P. Coan, “Tomographic reconstruction of the refractive index with hard X-rays: an efficient method based on the gradient vector-field approach,” Opt. Express 22, 5216–5227 (2014).
[Crossref] [PubMed]

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

S. Gasilov, A. Mittone, E. Brun, A. Bravin, S. Grandl, and P. Coan, “On the possibility of quantitative refractive-index tomography of large biomedical samples with hard X-rays,” Biomed. Opt. Express 4, 1512–1518 (2013).
[Crossref] [PubMed]

Bunk, O.

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

Byer, R. L.

Cerbino, R.

S. Berujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional X-Ray beam phase sensing,” Phys. Rev. Lett. 108, 158102 (2012).
[Crossref] [PubMed]

Chantler, C. T.

C. T. Chantler, “Detailed tabulation of atomic form factors, photoelectric absorption and scattering cross section, and mass attenuation coefficients in the vicinity of absorption edges in the soft X-ray (Z=3036, Z=6089, E=0.1 keV10 keV),” J. Phys. Chem. Ref. Data 29, 597–1048 (2000).
[Crossref]

Chapman, D.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

M. O. Hasnah, C. Parham, E. D. Pisano, Z. Zhong, O. Oltulu, and D. Chapman, “Mass density images from the diffraction enhanced imaging technique,” Med. Phys. 32, 549–552 (2005).
[Crossref] [PubMed]

Chapman, L. D.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Cloetens, P.

Coan, P.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

S. Gasilov, A. Mittone, E. Brun, A. Bravin, S. Grandl, A. Mirone, and P. Coan, “Tomographic reconstruction of the refractive index with hard X-rays: an efficient method based on the gradient vector-field approach,” Opt. Express 22, 5216–5227 (2014).
[Crossref] [PubMed]

S. Gasilov, A. Mittone, E. Brun, A. Bravin, S. Grandl, and P. Coan, “On the possibility of quantitative refractive-index tomography of large biomedical samples with hard X-rays,” Biomed. Opt. Express 4, 1512–1518 (2013).
[Crossref] [PubMed]

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol 58, R1–R35 (2013).
[Crossref]

Cozzini, C.

David, C.

Davis, C.

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

Di Martino, J. M.

Diaz, A.

Dilmanian, F. A.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Dreissi, D.

Duttenhofer, T.

Edic, P.

Faris, G. W.

Ferrari, J. A.

Fingerle, A.

Fitzgerald, R.

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53, 23–27 (2000).
[Crossref]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2. (Cambridge University, 1992, pp. 810).

Flores, J. L.

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–337 (1996).
[Crossref]

Gasilov, S.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

S. Gasilov, A. Mittone, E. Brun, A. Bravin, S. Grandl, A. Mirone, and P. Coan, “Tomographic reconstruction of the refractive index with hard X-rays: an efficient method based on the gradient vector-field approach,” Opt. Express 22, 5216–5227 (2014).
[Crossref] [PubMed]

S. Gasilov, A. Mittone, E. Brun, A. Bravin, S. Grandl, and P. Coan, “On the possibility of quantitative refractive-index tomography of large biomedical samples with hard X-rays,” Biomed. Opt. Express 4, 1512–1518 (2013).
[Crossref] [PubMed]

Geith, T.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

Grandl, S.

Gureyev, T. E.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–337 (1996).
[Crossref]

Hasnah, M.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

Hasnah, M. O.

M. O. Hasnah, C. Parham, E. D. Pisano, Z. Zhong, O. Oltulu, and D. Chapman, “Mass density images from the diffraction enhanced imaging technique,” Med. Phys. 32, 549–552 (2005).
[Crossref] [PubMed]

Herzen, J.

Horng, A.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

Ignatyev, K.

Ingal, V. N.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2318 (1995).
[Crossref]

Kaiser, K.

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988, Chap. 5).

Kottler, C.

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

Kudielka, G.

Lopez, F. C. M.

Lynch, R. E.

G. Birkhoff and R. E. Lynch, Numerical Solution of Elliptic Problems (SIAM Studies in Applied Mathematics1984, pp. 187).

Mahdi, K.

Maksimenko, A.

A. Maksimenko, “Nonlinear extension of the X-ray diffraction enhanced imaging,” Appl. Phys. Lett. 90, 154106 (2007).
[Crossref]

Meyer, P.

Mirone, A.

Mittone, A.

S. Gasilov, A. Mittone, E. Brun, A. Bravin, S. Grandl, A. Mirone, and P. Coan, “Tomographic reconstruction of the refractive index with hard X-rays: an efficient method based on the gradient vector-field approach,” Opt. Express 22, 5216–5227 (2014).
[Crossref] [PubMed]

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

S. Gasilov, A. Mittone, E. Brun, A. Bravin, S. Grandl, and P. Coan, “On the possibility of quantitative refractive-index tomography of large biomedical samples with hard X-rays,” Biomed. Opt. Express 4, 1512–1518 (2013).
[Crossref] [PubMed]

Modregger, P.

Mondal, I.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

Munro, P. R. T.

Nooel, P. B.

Olivo, A.

Oltulu, O.

M. O. Hasnah, C. Parham, E. D. Pisano, Z. Zhong, O. Oltulu, and D. Chapman, “Mass density images from the diffraction enhanced imaging technique,” Med. Phys. 32, 549–552 (2005).
[Crossref] [PubMed]

Orion, I.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Parham, C.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

M. O. Hasnah, C. Parham, E. D. Pisano, Z. Zhong, O. Oltulu, and D. Chapman, “Mass density images from the diffraction enhanced imaging technique,” Med. Phys. 32, 549–552 (2005).
[Crossref] [PubMed]

Peverini, L.

S. Berujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional X-Ray beam phase sensing,” Phys. Rev. Lett. 108, 158102 (2012).
[Crossref] [PubMed]

Pfeiffer, F.

Pinzer, B.

Pisano, E.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

Pisano, E. D.

M. O. Hasnah, C. Parham, E. D. Pisano, Z. Zhong, O. Oltulu, and D. Chapman, “Mass density images from the diffraction enhanced imaging technique,” Med. Phys. 32, 549–552 (2005).
[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, 335–337 (1996).
[Crossref]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2. (Cambridge University, 1992, pp. 810).

Reiser, M. F.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

Ren, B.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Rigon, L.

Sarapata, A.

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C. M. Slack, “The refraction of X-Rays in prisms of various materials,” Phys. Rev. 27, 691–695 (1926).
[Crossref]

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988, Chap. 5).

Speller, R. D.

Sperl, J.

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, 335–337 (1996).
[Crossref]

Suortti, P.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol 58, R1–R35 (2013).
[Crossref]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2. (Cambridge University, 1992, pp. 810).

Thomlinson, W. C.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Thuering, T.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2. (Cambridge University, 1992, pp. 810).

Walter, M.

Wang, Z.

Weber, L.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
[Crossref]

Weitkamp, T.

Wernick, M.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

Wilkins, S. 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, 335–337 (1996).
[Crossref]

Willner, M.

Wu, X. Y.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

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M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

Zhong, Z.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
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[Crossref] [PubMed]

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
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Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Maksimenko, “Nonlinear extension of the X-ray diffraction enhanced imaging,” Appl. Phys. Lett. 90, 154106 (2007).
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Biomed. Opt. Express (1)

Investigative Radiology (1)

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-Neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using X-rays propagation-based phase-contrast computed tomography of the human knee in comparison with clinical imaging techniques and histology,” Investigative Radiology 49, 627–634 (2014).
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J. Phys. Chem. Ref. Data (1)

C. T. Chantler, “Detailed tabulation of atomic form factors, photoelectric absorption and scattering cross section, and mass attenuation coefficients in the vicinity of absorption edges in the soft X-ray (Z=3036, Z=6089, E=0.1 keV10 keV),” J. Phys. Chem. Ref. Data 29, 597–1048 (2000).
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Med. Phys. (1)

M. O. Hasnah, C. Parham, E. D. Pisano, Z. Zhong, O. Oltulu, and D. Chapman, “Mass density images from the diffraction enhanced imaging technique,” Med. Phys. 32, 549–552 (2005).
[Crossref] [PubMed]

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, 335–337 (1996).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Phys. Med. Biol (1)

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol 58, R1–R35 (2013).
[Crossref]

Phys. Med. Biol. (2)

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006).
[Crossref] [PubMed]

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

Phys. Rev. (1)

C. M. Slack, “The refraction of X-Rays in prisms of various materials,” Phys. Rev. 27, 691–695 (1926).
[Crossref]

Phys. Rev. Lett. (2)

F. Pfeiffer, C. Kottler, O. Bunk, and C. Davis, “Hard X-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
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S. Berujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional X-Ray beam phase sensing,” Phys. Rev. Lett. 108, 158102 (2012).
[Crossref] [PubMed]

Phys. Today (1)

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53, 23–27 (2000).
[Crossref]

Other (5)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2. (Cambridge University, 1992, pp. 810).

G. Birkhoff and R. E. Lynch, Numerical Solution of Elliptic Problems (SIAM Studies in Applied Mathematics1984, pp. 187).

http://www.ansys.com/Products/ANSYS+15.0+Release+Highlights/ANSYS+Workbench

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988, Chap. 5).

Tissue Substitutes in Radiation Dosimetry and Measurement (ICRU Report 44, 1989).

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Figures (7)

Fig. 1
Fig. 1

Sketch of the PCI experiment with the definition of the coordinate system and parameters used in derivations. Two possible PCI experiments are distinguished depending on the orientation of the object rotation axis with respect to the phase-contrast sensitivity axis: panel (a) shows the in-plane geometry, panel (b) is the out-of-plane acquisition geometry.

Fig. 2
Fig. 2

Definition of the phase and its derivative at object boundaries. Case (a) differs from case (b) in direction of the phase contrast sensitivity axis. It is assumed that the object (the gray area) exceeds the field of view (which is shown as a black rectangle) in the vertical direction. For the sake of compactness (x,y) coordinates of lines that define object boundaries are indicated with subscripts left, right, top, bottom. The Dirichlet boundary condition can be then set as following: ϕleft = ϕright = 0, since in the horizontal direction a part of the unperturbed wavefront is taken in the image; the phase can be constrained at the top and bottom boundaries assuming that object is homogeneous (ϕ : hom. obj. approx.) and its thickness is approximately known. Measured DPC values can be used in both cases as the Neumann condition for the phase derivative along the axis parallel to the DPC axis (x,yϕ : exp.). The phase derivative in the orthogonal direction is unknown, as, for instance, yϕ at the top and bottom edges in case (a). The periodic boundary condition can be used in this case. In case (b) xϕ is unknown, but a reasonable approximation (xϕ : approx.) can be made in order to reduce the artifacts.

Fig. 3
Fig. 3

Tests with the Shepp-Logan phantom: (a) ground truth image; (b) simulated noisy DPC image; (c) direct 1D integration of DPC image displays characteristic stripe artifacts. Phase retrieved using Eq. (6) with (d) γ = 0.001, own software; (e) γ = 0.05, own software; (f) γ = 0.05, ANSYS software. Insets (g) and (h) show 1D phase profiles over row 118 and column 118 respectively. Profiles were extracted from images (a,c,e) and there location is indicated by dashed red lines in Fig. 3(e).

Fig. 4
Fig. 4

Experimental images of a phantom: (a) DPC projection, (b) phase projection obtained with Eq. 6, (c) 1D phase profiles taken over line 280 (indicated with the dashed red line in panel (b)) together the expected (theoretical) phase profile.

Fig. 5
Fig. 5

Reconstruction of the refractive index distribution inside the rabbit knee leg. Results obtainable with the presented technique (shown in the bottom row) are compared with data reconstructed using FBP algorithm for gradient projections (FBP with Hilbert filter [9]). Images present the following results: one of the DPC projections obtained in the experiment (a), the colorbar on its left-hand side is proportional to the X-ray deflection angles in radians; axial and sagittal images of refractive index distribution obtained using the FBP algorithm for gradient projections (c) and (d); the phase projection (e) retrieved from the DCP image (a), colorbar on its left-hand side is proportional to the X-ray phase delay in the sample exit plane in radians; axial and sagittal views obtained using FBP algorithm for ordinary projections (g) and (h), the colorbar on the right-hand side shows the magnitude of the refractive index decrement. Insets (b) and (f) show magnified fragments of images (a) and (e) with automatic contrast adjustment made in ImageJ software.

Fig. 6
Fig. 6

Reconstructed distributions of the index of refraction in the breast sample. Axial and sagittal views obtained in the in-plane geometry are shown in panels (a,c) correspondingly. Panels (b,d) show axial and sagittal views reconstructed in the out-of-plane geometry. Dashed red lines in Figs. 6(a) and (b) show the location of sagittal sections. The right hand side and left hand side gray bars show tissue density in g/mm3 (derived from δ values) for images (a,c) and (b,d) correspondingly.

Fig. 7
Fig. 7

Workspace in the ANSYS mechanical. Note that: two Edge Sizing properties control the mesh sampling; Heat flow and Temperature are defined as boundary conditions. Imported phase derivative plays a role of the Heat Generation (note contours of the Shepp-Logan phantom in the right panel). Temperature is added in the Solution list.

Tables (3)

Tables Icon

Table 1 Root mean square error and contrast to noise ration for several γ values.

Tables Icon

Table 2 Measured and expected index of refraction decrement of materials and tissues inside the biological sample.

Tables Icon

Table 3 Quantitative results obtained from sagittal slices shown in Figs. 6(c) and 6(d). The values are measured in the regions marked by a rectangle (adipose tissue) and triangle (skin layer).

Equations (7)

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

d d s [ δ ( r ) t ( r ) ] = δ ( r ) ,
α ( x , y ) δ ( x , y , z ) y d z y δ ( x , y , z ) d z .
ϕ ( x , y ) = k δ ( x , y , z ) d z ,
ϕ ( x , y ) y = k α ( x , y ) ,
2 ϕ ( x , y ) 2 y = k α ( x , y ) y .
2 ϕ ( x , y ) 2 y + γ 2 ϕ ( x , y ) 2 x = k α ( x , y ) y .
2 ϕ ( x , y ) 2 y + γ 1 ( x , y ) 2 ϕ ( x , y ) 2 x + γ 2 ( x , y ) ϕ ( x , y ) x = k α ( x , y ) y .

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