Abstract

Differential phase-contrast is a recent technique in the context of X-ray imaging. In order to reduce the specimen’s exposure time, we propose a new iterative algorithm that can achieve the same quality as FBP-type methods, while using substantially fewer angular views. Our approach is based on 1) a novel spline-based discretization of the forward model and 2) an iterative reconstruction algorithm using the alternating direction method of multipliers. Our experimental results on real data suggest that the method allows to reduce the number of required views by at least a factor of four.

© 2013 OSA

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  1. V. Ingal and E. Beliaevskaya, “X-ray plane-wave tomography observation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys 28, 2314–2317 (1995).
    [Crossref]
  2. T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nat. 373, 595–598 (1995).
    [Crossref]
  3. D. Chapman, S. Patel, and D. Fuhrman, “Diffraction enhanced X-ray imaging,” Phys., Med. and Bio. 42, 2015–2025 (1997).
    [Crossref]
  4. U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
    [Crossref]
  5. A. Momose, T. Takeda, Y. itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med 2, 473–475 (1996).
    [Crossref] [PubMed]
  6. 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, 6296–6304 (2005).
    [Crossref] [PubMed]
  7. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelekov, “On the possibilities of X-ray phase-contrast microimaging by coherent high-energy synchroton radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1997).
    [Crossref]
  8. 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, 2961–2964 (1996).
    [Crossref] [PubMed]
  9. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nat. 384, 335–338 (1996).
    [Crossref]
  10. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray talbot interferometry,” Jap. Jour. of Appl. Phys. 42, L866–L868 (2003).
    [Crossref]
  11. F. Pfieffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Inst. and Meth. in Phys. Res. 580.2, 925–928 (2007).
    [Crossref]
  12. M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
    [Crossref]
  13. M. Nilchian and M. Unser, “Differential phase-contrast X-ray computed tomography: From model discretization to image reconstruction,” Proc. of the Ninth IEEE Inter. Symp. on Biomed. Imag.: From Nano to Macro (ISBI’12), 90–93 (2012).
  14. M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Imag. Proc. 20, 681–695 (2011).
    [Crossref]
  15. S. Ramani and J. A. Fessler, “A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction,” IEEE Trans. Med. Imag. 31.3, 677–688 (2012).
    [Crossref]
  16. Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. of SPIE 7258, 4A1–8 (2009).
  17. T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. phys. 38, 4542–4545 (2011).
    [Crossref]
  18. Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography,” Opt. Express 20, 10724–10749 (2012).
    [Crossref] [PubMed]
  19. M. Unser, “Sampling–50 years after Shannon,” Proc. IEEE 88, 254104–1–3 (2000).
    [Crossref]
  20. E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
    [Crossref]
  21. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
    [Crossref]
  22. F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. 96, 215505-1–4 (2006).
    [Crossref]
  23. F. Natterer, The Mathematics of Computed Tomography (John Wiley and sons, 1986).
  24. H. Meijering, J. Niessen, and A. Viergever, “Quantitative evaluation of convolution-based methods for medical image interpolation,” Med. Imag. Anal. 5, 111–126 (2001).
    [Crossref]
  25. A. Entezari, M. Nilchian, and M. Unser, “A box spline calculus for the discretization of computed tomography reconstruction problems,” IEEE Trans. Med. Imag. 31, 1532 –1541 (2012).
    [Crossref]
  26. P. Thvenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imag. 19.7, 739–758 (2000).
    [Crossref]
  27. Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Jour. on Imag. Sci. 1, 248–272 (2008).
    [Crossref]
  28. T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM Jour. on Imag. Sci. 2, 323–343 (2009).
    [Crossref]
  29. M. Ng, P. Weiss, and X. Yuan, “Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods,” SIAM Jour. on Sci. Comp. 32, 2710–2736 (2010).
    [Crossref]
  30. B. Vandeghinste, B. Goossens, J. De Beenhouwer, A. Pizurica, W. Philips, S. Vandenberghe, and S. Staelens, “Split-bregman-based sparse-view CT reconstruction,” in “Fully 3D 2011 proc.,” 431–434 (2011).
  31. I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math. 57, 1413–1457 (2004).
    [Crossref]
  32. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Imag. Sci. 2, 183–202 (2009).
    [Crossref]
  33. Z. Wang and A. Bovik, “A universal image quality index,” IEEE Sig. Proc. Lett. 9, 81 –84 (2002).
    [Crossref]
  34. Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Imag. Proc. 13, 600–612 (2004).
    [Crossref]
  35. S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. 16, 562–572 (2009).
    [Crossref]

2012 (3)

S. Ramani and J. A. Fessler, “A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction,” IEEE Trans. Med. Imag. 31.3, 677–688 (2012).
[Crossref]

Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography,” Opt. Express 20, 10724–10749 (2012).
[Crossref] [PubMed]

A. Entezari, M. Nilchian, and M. Unser, “A box spline calculus for the discretization of computed tomography reconstruction problems,” IEEE Trans. Med. Imag. 31, 1532 –1541 (2012).
[Crossref]

2011 (4)

B. Vandeghinste, B. Goossens, J. De Beenhouwer, A. Pizurica, W. Philips, S. Vandenberghe, and S. Staelens, “Split-bregman-based sparse-view CT reconstruction,” in “Fully 3D 2011 proc.,” 431–434 (2011).

T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. phys. 38, 4542–4545 (2011).
[Crossref]

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
[Crossref]

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Imag. Proc. 20, 681–695 (2011).
[Crossref]

2010 (1)

M. Ng, P. Weiss, and X. Yuan, “Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods,” SIAM Jour. on Sci. Comp. 32, 2710–2736 (2010).
[Crossref]

2009 (4)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Imag. Sci. 2, 183–202 (2009).
[Crossref]

T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM Jour. on Imag. Sci. 2, 323–343 (2009).
[Crossref]

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. of SPIE 7258, 4A1–8 (2009).

S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. 16, 562–572 (2009).
[Crossref]

2008 (2)

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Jour. on Imag. Sci. 1, 248–272 (2008).
[Crossref]

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[Crossref]

2007 (1)

F. Pfieffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Inst. and Meth. in Phys. Res. 580.2, 925–928 (2007).
[Crossref]

2006 (2)

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. 96, 215505-1–4 (2006).
[Crossref]

2005 (1)

2004 (2)

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math. 57, 1413–1457 (2004).
[Crossref]

Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Imag. Proc. 13, 600–612 (2004).
[Crossref]

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray talbot interferometry,” Jap. Jour. of Appl. Phys. 42, L866–L868 (2003).
[Crossref]

2002 (1)

Z. Wang and A. Bovik, “A universal image quality index,” IEEE Sig. Proc. Lett. 9, 81 –84 (2002).
[Crossref]

2001 (1)

H. Meijering, J. Niessen, and A. Viergever, “Quantitative evaluation of convolution-based methods for medical image interpolation,” Med. Imag. Anal. 5, 111–126 (2001).
[Crossref]

2000 (2)

P. Thvenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imag. 19.7, 739–758 (2000).
[Crossref]

M. Unser, “Sampling–50 years after Shannon,” Proc. IEEE 88, 254104–1–3 (2000).
[Crossref]

1997 (2)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelekov, “On the possibilities of X-ray phase-contrast microimaging by coherent high-energy synchroton radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1997).
[Crossref]

D. Chapman, S. Patel, and D. Fuhrman, “Diffraction enhanced X-ray imaging,” Phys., Med. and Bio. 42, 2015–2025 (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, 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,” Nat. 384, 335–338 (1996).
[Crossref]

A. Momose, T. Takeda, Y. itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med 2, 473–475 (1996).
[Crossref] [PubMed]

1995 (2)

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

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nat. 373, 595–598 (1995).
[Crossref]

1965 (1)

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
[Crossref]

Afonso, M. V.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Imag. Proc. 20, 681–695 (2011).
[Crossref]

Anastasio, M. A.

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, 2961–2964 (1996).
[Crossref] [PubMed]

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Imag. Sci. 2, 183–202 (2009).
[Crossref]

Beliaevskaya, E.

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

Bevins, N.

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. of SPIE 7258, 4A1–8 (2009).

Bioucas-Dias, J. M.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Imag. Proc. 20, 681–695 (2011).
[Crossref]

Blu, T.

P. Thvenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imag. 19.7, 739–758 (2000).
[Crossref]

Bonse, U.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
[Crossref]

Bovik, A.

Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Imag. Proc. 13, 600–612 (2004).
[Crossref]

Z. Wang and A. Bovik, “A universal image quality index,” IEEE Sig. Proc. Lett. 9, 81 –84 (2002).
[Crossref]

Brendel, B.

T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. phys. 38, 4542–4545 (2011).
[Crossref]

Bunk, O.

F. Pfieffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Inst. and Meth. in Phys. Res. 580.2, 925–928 (2007).
[Crossref]

F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. 96, 215505-1–4 (2006).
[Crossref]

Chapman, D.

D. Chapman, S. Patel, and D. Fuhrman, “Diffraction enhanced X-ray imaging,” Phys., Med. and Bio. 42, 2015–2025 (1997).
[Crossref]

Chen, G.

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. of SPIE 7258, 4A1–8 (2009).

Cloetens, P.

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, 2961–2964 (1996).
[Crossref] [PubMed]

Daubechies, I.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math. 57, 1413–1457 (2004).
[Crossref]

David, C.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
[Crossref]

S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. 16, 562–572 (2009).
[Crossref]

F. Pfieffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Inst. and Meth. in Phys. Res. 580.2, 925–928 (2007).
[Crossref]

F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. 96, 215505-1–4 (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, 6296–6304 (2005).
[Crossref] [PubMed]

Davis, T.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nat. 373, 595–598 (1995).
[Crossref]

De Beenhouwer, J.

B. Vandeghinste, B. Goossens, J. De Beenhouwer, A. Pizurica, W. Philips, S. Vandenberghe, and S. Staelens, “Split-bregman-based sparse-view CT reconstruction,” in “Fully 3D 2011 proc.,” 431–434 (2011).

De Mol, C.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math. 57, 1413–1457 (2004).
[Crossref]

Defrise, M.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math. 57, 1413–1457 (2004).
[Crossref]

Diaz, A.

Entezari, A.

A. Entezari, M. Nilchian, and M. Unser, “A box spline calculus for the discretization of computed tomography reconstruction problems,” IEEE Trans. Med. Imag. 31, 1532 –1541 (2012).
[Crossref]

Fessler, J. A.

S. Ramani and J. A. Fessler, “A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction,” IEEE Trans. Med. Imag. 31.3, 677–688 (2012).
[Crossref]

Figueiredo, M. A. T.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Imag. Proc. 20, 681–695 (2011).
[Crossref]

Frei, G.

F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. 96, 215505-1–4 (2006).
[Crossref]

Fuhrman, D.

D. Chapman, S. Patel, and D. Fuhrman, “Diffraction enhanced X-ray imaging,” Phys., Med. and Bio. 42, 2015–2025 (1997).
[Crossref]

Gao, D.

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

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nat. 373, 595–598 (1995).
[Crossref]

Goldstein, T.

T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM Jour. on Imag. Sci. 2, 323–343 (2009).
[Crossref]

Goossens, B.

B. Vandeghinste, B. Goossens, J. De Beenhouwer, A. Pizurica, W. Philips, S. Vandenberghe, and S. Staelens, “Split-bregman-based sparse-view CT reconstruction,” in “Fully 3D 2011 proc.,” 431–434 (2011).

Grünzweig, C.

F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. 96, 215505-1–4 (2006).
[Crossref]

Gureyev, T.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nat. 373, 595–598 (1995).
[Crossref]

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,” Nat. 384, 335–338 (1996).
[Crossref]

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, 2961–2964 (1996).
[Crossref] [PubMed]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray talbot interferometry,” Jap. Jour. of Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Hart, M.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
[Crossref]

Hattori, T.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

Hauser, N.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
[Crossref]

Hintermuller, C.

S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. 16, 562–572 (2009).
[Crossref]

Hirano, K.

A. Momose, T. Takeda, Y. itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med 2, 473–475 (1996).
[Crossref] [PubMed]

Hohl, M.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
[Crossref]

Ingal, V.

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

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A. Momose, T. Takeda, Y. itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med 2, 473–475 (1996).
[Crossref] [PubMed]

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray talbot interferometry,” Jap. Jour. of Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Köhler, T.

T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. phys. 38, 4542–4545 (2011).
[Crossref]

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelekov, “On the possibilities of X-ray phase-contrast microimaging by coherent high-energy synchroton radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1997).
[Crossref]

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F. Pfieffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Inst. and Meth. in Phys. Res. 580.2, 925–928 (2007).
[Crossref]

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A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray talbot interferometry,” Jap. Jour. of Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Kubik-Huch, R.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
[Crossref]

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A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelekov, “On the possibilities of X-ray phase-contrast microimaging by coherent high-energy synchroton radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1997).
[Crossref]

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F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. 96, 215505-1–4 (2006).
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S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. 16, 562–572 (2009).
[Crossref]

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S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. 16, 562–572 (2009).
[Crossref]

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H. Meijering, J. Niessen, and A. Viergever, “Quantitative evaluation of convolution-based methods for medical image interpolation,” Med. Imag. Anal. 5, 111–126 (2001).
[Crossref]

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S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. 16, 562–572 (2009).
[Crossref]

Modregger, P.

Momose, A.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray talbot interferometry,” Jap. Jour. of Appl. Phys. 42, L866–L868 (2003).
[Crossref]

A. Momose, T. Takeda, Y. itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med 2, 473–475 (1996).
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F. Natterer, The Mathematics of Computed Tomography (John Wiley and sons, 1986).

Ng, M.

M. Ng, P. Weiss, and X. Yuan, “Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods,” SIAM Jour. on Sci. Comp. 32, 2710–2736 (2010).
[Crossref]

Niessen, J.

H. Meijering, J. Niessen, and A. Viergever, “Quantitative evaluation of convolution-based methods for medical image interpolation,” Med. Imag. Anal. 5, 111–126 (2001).
[Crossref]

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A. Entezari, M. Nilchian, and M. Unser, “A box spline calculus for the discretization of computed tomography reconstruction problems,” IEEE Trans. Med. Imag. 31, 1532 –1541 (2012).
[Crossref]

M. Nilchian and M. Unser, “Differential phase-contrast X-ray computed tomography: From model discretization to image reconstruction,” Proc. of the Ninth IEEE Inter. Symp. on Biomed. Imag.: From Nano to Macro (ISBI’12), 90–93 (2012).

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, 2961–2964 (1996).
[Crossref] [PubMed]

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T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM Jour. on Imag. Sci. 2, 323–343 (2009).
[Crossref]

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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, 2961–2964 (1996).
[Crossref] [PubMed]

Pan, X.

Patel, S.

D. Chapman, S. Patel, and D. Fuhrman, “Diffraction enhanced X-ray imaging,” Phys., Med. and Bio. 42, 2015–2025 (1997).
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Pfeiffer, F.

S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. 16, 562–572 (2009).
[Crossref]

F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. 96, 215505-1–4 (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, 6296–6304 (2005).
[Crossref] [PubMed]

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F. Pfieffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Inst. and Meth. in Phys. Res. 580.2, 925–928 (2007).
[Crossref]

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B. Vandeghinste, B. Goossens, J. De Beenhouwer, A. Pizurica, W. Philips, S. Vandenberghe, and S. Staelens, “Split-bregman-based sparse-view CT reconstruction,” in “Fully 3D 2011 proc.,” 431–434 (2011).

Pizurica, A.

B. Vandeghinste, B. Goossens, J. De Beenhouwer, A. Pizurica, W. Philips, S. Vandenberghe, and S. Staelens, “Split-bregman-based sparse-view CT reconstruction,” in “Fully 3D 2011 proc.,” 431–434 (2011).

Pogany, A.

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

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Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. of SPIE 7258, 4A1–8 (2009).

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S. Ramani and J. A. Fessler, “A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction,” IEEE Trans. Med. Imag. 31.3, 677–688 (2012).
[Crossref]

Roessl, E.

T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. phys. 38, 4542–4545 (2011).
[Crossref]

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
[Crossref]

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A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelekov, “On the possibilities of X-ray phase-contrast microimaging by coherent high-energy synchroton radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1997).
[Crossref]

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Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Imag. Proc. 13, 600–612 (2004).
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Simoncelli, E.

Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Imag. Proc. 13, 600–612 (2004).
[Crossref]

Singer, G.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
[Crossref]

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A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelekov, “On the possibilities of X-ray phase-contrast microimaging by coherent high-energy synchroton radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1997).
[Crossref]

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A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelekov, “On the possibilities of X-ray phase-contrast microimaging by coherent high-energy synchroton radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1997).
[Crossref]

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B. Vandeghinste, B. Goossens, J. De Beenhouwer, A. Pizurica, W. Philips, S. Vandenberghe, and S. Staelens, “Split-bregman-based sparse-view CT reconstruction,” in “Fully 3D 2011 proc.,” 431–434 (2011).

Stampanoni, M.

Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography,” Opt. Express 20, 10724–10749 (2012).
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M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
[Crossref]

S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. 16, 562–572 (2009).
[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, 6296–6304 (2005).
[Crossref] [PubMed]

Stevenson, A.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nat. 373, 595–598 (1995).
[Crossref]

Stevenson, A. W.

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

Suzuki, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray talbot interferometry,” Jap. Jour. of Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray talbot interferometry,” Jap. Jour. of Appl. Phys. 42, L866–L868 (2003).
[Crossref]

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A. Momose, T. Takeda, Y. itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med 2, 473–475 (1996).
[Crossref] [PubMed]

Takeda, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

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A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Imag. Sci. 2, 183–202 (2009).
[Crossref]

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M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
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P. Thvenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imag. 19.7, 739–758 (2000).
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M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
[Crossref]

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A. Entezari, M. Nilchian, and M. Unser, “A box spline calculus for the discretization of computed tomography reconstruction problems,” IEEE Trans. Med. Imag. 31, 1532 –1541 (2012).
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P. Thvenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imag. 19.7, 739–758 (2000).
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M. Unser, “Sampling–50 years after Shannon,” Proc. IEEE 88, 254104–1–3 (2000).
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M. Nilchian and M. Unser, “Differential phase-contrast X-ray computed tomography: From model discretization to image reconstruction,” Proc. of the Ninth IEEE Inter. Symp. on Biomed. Imag.: From Nano to Macro (ISBI’12), 90–93 (2012).

Vandeghinste, B.

B. Vandeghinste, B. Goossens, J. De Beenhouwer, A. Pizurica, W. Philips, S. Vandenberghe, and S. Staelens, “Split-bregman-based sparse-view CT reconstruction,” in “Fully 3D 2011 proc.,” 431–434 (2011).

Vandenberghe, S.

B. Vandeghinste, B. Goossens, J. De Beenhouwer, A. Pizurica, W. Philips, S. Vandenberghe, and S. Staelens, “Split-bregman-based sparse-view CT reconstruction,” in “Fully 3D 2011 proc.,” 431–434 (2011).

Viergever, A.

H. Meijering, J. Niessen, and A. Viergever, “Quantitative evaluation of convolution-based methods for medical image interpolation,” Med. Imag. Anal. 5, 111–126 (2001).
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Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Jour. on Imag. Sci. 1, 248–272 (2008).
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Wang, Z.

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
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Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Imag. Proc. 13, 600–612 (2004).
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Z. Wang and A. Bovik, “A universal image quality index,” IEEE Sig. Proc. Lett. 9, 81 –84 (2002).
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M. Ng, P. Weiss, and X. Yuan, “Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods,” SIAM Jour. on Sci. Comp. 32, 2710–2736 (2010).
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Weitkamp, T.

Wilkins, S.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nat. 373, 595–598 (1995).
[Crossref]

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S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nat. 384, 335–338 (1996).
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Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Jour. on Imag. Sci. 1, 248–272 (2008).
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A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
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Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Jour. on Imag. Sci. 1, 248–272 (2008).
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Yuan, X.

M. Ng, P. Weiss, and X. Yuan, “Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods,” SIAM Jour. on Sci. Comp. 32, 2710–2736 (2010).
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Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. of SPIE 7258, 4A1–8 (2009).

Zhang, Y.

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Jour. on Imag. Sci. 1, 248–272 (2008).
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Appl. Phys. Lett. (1)

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
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Comm. Pure Appl. Math. (1)

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math. 57, 1413–1457 (2004).
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Fully 3D 2011 proc. (1)

B. Vandeghinste, B. Goossens, J. De Beenhouwer, A. Pizurica, W. Philips, S. Vandenberghe, and S. Staelens, “Split-bregman-based sparse-view CT reconstruction,” in “Fully 3D 2011 proc.,” 431–434 (2011).

IEEE Sig. Proc. Lett. (1)

Z. Wang and A. Bovik, “A universal image quality index,” IEEE Sig. Proc. Lett. 9, 81 –84 (2002).
[Crossref]

IEEE Trans. Imag. Proc. (2)

Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Imag. Proc. 13, 600–612 (2004).
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M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Imag. Proc. 20, 681–695 (2011).
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IEEE Trans. Med. Imag. (3)

S. Ramani and J. A. Fessler, “A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction,” IEEE Trans. Med. Imag. 31.3, 677–688 (2012).
[Crossref]

A. Entezari, M. Nilchian, and M. Unser, “A box spline calculus for the discretization of computed tomography reconstruction problems,” IEEE Trans. Med. Imag. 31, 1532 –1541 (2012).
[Crossref]

P. Thvenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imag. 19.7, 739–758 (2000).
[Crossref]

Inves. radio. (1)

M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. 46, 801–806 (2011).
[Crossref]

J. Phys. D: Appl. Phys (1)

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

Jap. Jour. of Appl. Phys. (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray talbot interferometry,” Jap. Jour. of Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Jpn. J. Appl. Phys. (1)

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

Med. Imag. Anal. (1)

H. Meijering, J. Niessen, and A. Viergever, “Quantitative evaluation of convolution-based methods for medical image interpolation,” Med. Imag. Anal. 5, 111–126 (2001).
[Crossref]

Med. phys. (1)

T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. phys. 38, 4542–4545 (2011).
[Crossref]

Nat. (2)

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nat. 373, 595–598 (1995).
[Crossref]

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

Nat. Med (1)

A. Momose, T. Takeda, Y. itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med 2, 473–475 (1996).
[Crossref] [PubMed]

Nucl. Inst. and Meth. in Phys. Res. (1)

F. Pfieffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Inst. and Meth. in Phys. Res. 580.2, 925–928 (2007).
[Crossref]

Opt. Express (2)

Phys. Med. Biol. (1)

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[Crossref]

Phys. Rev. Lett. (2)

F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. 96, 215505-1–4 (2006).
[Crossref]

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, 2961–2964 (1996).
[Crossref] [PubMed]

Phys., Med. and Bio. (1)

D. Chapman, S. Patel, and D. Fuhrman, “Diffraction enhanced X-ray imaging,” Phys., Med. and Bio. 42, 2015–2025 (1997).
[Crossref]

Proc. IEEE (1)

M. Unser, “Sampling–50 years after Shannon,” Proc. IEEE 88, 254104–1–3 (2000).
[Crossref]

Proc. of SPIE (1)

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. of SPIE 7258, 4A1–8 (2009).

Rev. Sci. Instrum. (1)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelekov, “On the possibilities of X-ray phase-contrast microimaging by coherent high-energy synchroton radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1997).
[Crossref]

SIAM Imag. Sci. (1)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Imag. Sci. 2, 183–202 (2009).
[Crossref]

SIAM Jour. on Imag. Sci. (2)

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Jour. on Imag. Sci. 1, 248–272 (2008).
[Crossref]

T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM Jour. on Imag. Sci. 2, 323–343 (2009).
[Crossref]

SIAM Jour. on Sci. Comp. (1)

M. Ng, P. Weiss, and X. Yuan, “Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods,” SIAM Jour. on Sci. Comp. 32, 2710–2736 (2010).
[Crossref]

Sync. Rad. (1)

S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. 16, 562–572 (2009).
[Crossref]

Other (2)

F. Natterer, The Mathematics of Computed Tomography (John Wiley and sons, 1986).

M. Nilchian and M. Unser, “Differential phase-contrast X-ray computed tomography: From model discretization to image reconstruction,” Proc. of the Ninth IEEE Inter. Symp. on Biomed. Imag.: From Nano to Macro (ISBI’12), 90–93 (2012).

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Figures (5)

Fig. 1
Fig. 1

Differential phase-contrast tomography is based on a grating interferometer setup. The phase grating introduces a phase shift of π in the transmitted wave. The absorption grating is necessary for measuring the received wave given the limited resolution of the detector. The diagram on the right shows the received intensity for one pixel of the CCD camera with respect to the position of the grating. The red curve corresponds to the situation where the object is in the illumination path. The black curve shows the received intensity when there is no object.

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Fig. 2
Fig. 2

(a) Analytic phantom. (b) Analytical values of the derivative of the Radon transform of analytic phantom.

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Fig. 3
Fig. 3

Convergence-speed comparison of different iterative techniques for solving our regularized problem.

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Fig. 4
Fig. 4

Comparison of the reconstruction results for 721 viewing angles (first column) and 181 viewing angles (second column). (a,d) GFBP, (b,e) the iterative ADMM, (c,f) GFBP with smoothing kernel. The sub-images correspond to the region between the thalamus and the hippocampus (top), a part of the talamus (middle) and the Fornix (bottom). Notice the oscillatory artifacts produced by GFBP. Applying a smoothing kernel reduces the artifacts but also blurs the reconstruction.

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Fig. 5
Fig. 5

The SNR and SSIM metrics for images reconstructed from a subset of projections.

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Tables (3)

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Algorithm 1 Generalized filtered back projection(GFBP) for the inverse problem ℛ θ ( n ) f ( y ) = g θ ( y )

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Table 1 Comparison of the projection and reconstruction accuracy using cubic B-splines and Kaiser-Bessel functions with the parameters proposed in [18].

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Table 3 Algorithm 2 ADMM-PCG with warm initialization reconstruction method

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Equations (55)

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ϕ ( y , θ ) = 2 π λ ℛ { f ( x ) } ( y , θ ) = 2 π λ ∫ ℝ 2 f ( x ) δ ( y − 〈 x , θ 〉 ) d x ,
ℛ { f ( x ) } ( y , θ ) = ∫ ℝ 2 f ( x ) δ ( y − 〈 x , θ 〉 ) d x = ∫ ℝ f ( y θ + t θ ⊥ ) d t ,
α ( y , θ ) = λ 2 π ∂ ϕ ( y , θ ) ∂ y ,
Δ x g ( y , θ ) = d α ( y , θ ) ,
g ( y , θ ) = 1 d Δ x g ( y , θ ) = ∂ ℛ { f } ∂ y ( y , θ ) .
ℛ ( n ) f ( y , θ ) = ∂ n ∂ y n ℛ f ( y , θ ) .
ℛ ( n ) { f ( α x ) } ( y , θ ) = α n + 1 ℛ ( n ) f ( α y , θ ) ,             α ∈ ℝ + .
ℛ ( n ) { ( f * g ) ( x ) } ( y , θ ) = ( ℛ ( n ) f ( ⋅ , θ ) * ℛ g ( ⋅ , θ ) ) ( y , θ ) = ( ℛ f ( ⋅ , θ ) * ℛ ( n ) g ( ⋅ , θ ) ) ( y , θ ) .
ℛ ( n ) { f ( ⋅ − x 0 ) } ( y , θ ) = ℛ ( n ) f ( y − 〈 x 0 , θ 〉 , θ ) .
ℋ 2 { f ( x ) } ^ ( ω 1 , ω 2 ) = − i ⋅ sgn ( ω 2 ) f ^ ( ω 1 , ω 2 ) ,
ℛ * { ∂ n ∂ y n ℛ { f } ( y , θ ) } ( x ) = 2 π ( − 1 ) n ℋ 2 n ( − Δ ) n − 1 2 { f } ( x ) ,
g ^ ( ω , θ ) = f ^ ( ω cos θ , ω sin θ ) .
( i ω ) n g ^ ( ω , θ ) = i n × sgn n ( ω ) | ω | n f ^ ( ω cos θ , ω sin θ ) .
∂ n ∂ y n ℛ f ( y , θ ) = ℛ { ( ( − 1 ) n ( ℋ 2 ) n ( − Δ ) n 2 { f } ) ( x ) } ( y , θ ) , ∀ θ ∈ ( 0 , π ) .
ℛ * { ∂ n ∂ y n ℛ { f } ( y , θ ) } ( x ) = ℛ * ℛ { ( ( − 1 ) n ( ℋ 2 ) n ( − Δ ) n 2 { f } ) ( x ) } ( y , θ ) , ∀ θ ∈ ( 0 , π ) .
ℛ ( n ) f ( y , θ ) = g ( y , θ ) ,
ℛ ( n ) * { ( q * ℛ ( n ) f ( ⋅ , θ ) ) ( y ) } ( x ) = f ( x ) ,
q ^ ( ω y ) = 1 2 π × 1 | ω y | 2 n − 1 .
q ^ ( ω y ) = 1 2 π × 1 | ω y | 2 n − 1
V ( φ ) = { f ( x ) = ∑ k ∈ ℤ 2 c k φ ( x − k ) : c ∈ l 2 ( ℤ 2 ) } .
f ( x ) = ∑ k ∈ ℤ 2 c k φ k ( x ) ,
ℛ { f } ( y , θ ) = ∑ k ∈ ℤ 2 c k ℛ { φ k } ( y , θ ) ,
ℛ { φ k } ( y , θ ) = ℛ { φ } ( y − 〈 θ , k 〉 , θ ) ,
ℛ ( n ) { f } ( y , θ ) = ∑ k ∈ ℤ 2 c k ℛ ( n ) { φ k } ( y , θ ) ,
ℛ ( n ) { φ k } ( y , θ ) = ℛ ( n ) { φ } ( y − 〈 θ , k 〉 , θ ) .
φ ( x ) = β ( x ) = β ( x 1 ) β ( x 2 ) ,
β ( x ) = Δ 1 m + 1 m ! x + m ↔ ℱ ( sin ( ω / 2 ) ω / 2 ) m + 1 ,
ℛ ( n ) { β ( x ) } ( y , θ ) = Δ cos θ m + 1 Δ sin θ m + 1 ( 2 m − n + 1 ) ! y + 2 m − n + 1 = ∑ k 1 = 0 m + 1 ∑ k 2 = 0 m + 1 ( − 1 ) k 1 + k 2 ( m + 1 k 1 ) ( m + 1 k 2 ) × ( y + ( m + 1 2 − k 1 ) cos θ + ( m + 1 2 − k 2 ) sin θ ) + 2 m − n + 1 ( 2 m − n + 1 ) ! cos θ m + 1 sin θ m + 1 .
g ( y , θ ) = ℱ − 1 { ( j ω ) n β ^ ( ω cos θ ) β ^ ( ω sin θ ) } .
( j ω ) n β ^ m ( ω cos θ ) β ^ m ( ω sin θ ) = ( j ω ) n ( sin ( ω cos θ 2 ) ω cos θ 2 ) m + 1 ( sin ( ω sin θ 2 ) ω sin θ 2 ) m + 1 = ( j ω ) n ( e j ω cos θ 2 − e − j ω cos θ 2 j ω cos θ ) m + 1 ( e j ω sin θ 2 − e − j ω sin θ 2 j ω sin θ ) m + 1 = ( e j ω cos θ 2 − e − j ω cos θ 2 cos θ ) m + 1 ( e j ω sin θ 2 − e − j ω sin θ 2 sin θ ) m + 1 1 ( j ω ) 2 m − n + 2 .
ℛ ( n ) f ( y j , θ i ) = ∑ k ∈ ℤ 2 c k ℛ ( n ) φ k ( y j , θ i ) .
g = Hc ,
[ H ] ( i , j ) , k = ℛ ( 1 ) φ k ( y j , θ i ) .
H ( i , j ) , k = ℛ ( 1 ) φ ( y j − 〈 k , θ i 〉 , θ i ) .
I : ℝ × [ 0 , π ] → { 1 , 2 , … , K } × { 1 , 2 , … . P }                             ( y , θ ) ↦ ( j , i ) ,
[ H ] ( i , j ) , k ≃ L I ( y j − 〈 k , θ i 〉 , θ i ) .
p a ( x ) = { ( a 2 − ‖ x ‖ 2 ) 2 ‖ x ‖ ≤ a 0 Oth . ,
∂ ℛ p a ( y , θ ) ∂ y = − 16 3 y ( a 2 − y 2 ) 3 2 .
∂ ℛ p a ( y , θ ) ∂ y = 2 ∂ ∂ y ∫ 0 a 2 − y 2 ( a 2 − y 2 − x 2 2 ) 2 d y
f ( x ) = ∑ k ∈ ℤ α k p a k ( x − x k ) ,
J ( c ) = 1 2 ‖ Hc − g ‖ 2 ,
J ( c ) ≜ 1 2 ‖ Hc − g ‖ 2 + Ψ ( c ) ,
Ψ ( c ) = λ 1 2 ‖ c ‖ 2 + λ 2 ∑ k ‖ { Lc } k ‖ 1 ,
∂ f ∂ x 1 [ k 1 , k 2 ] = ( ( h 1 [ ⋅ , k 2 ] * c [ ⋅ , k 2 ] ) [ k 1 , ⋅ ] * b 2 [ k 1 , ⋅ ] ) [ k 1 , k 2 ] ∂ f ∂ x 2 [ k 1 , k 2 ] = ( ( h 2 [ k 1 , ⋅ ] * c [ k 1 , ⋅ ] ) [ ⋅ , k 2 ] * b 1 [ ⋅ , k 2 ] ) [ k 1 , k 2 ] ,
c = argmin c , u { 1 2 ‖ Hc − g ‖ 2 + λ 1 2 ‖ c ‖ 2 + λ 2 ∑ k ‖ u k ‖ 1 } .     subject   to   u = Lc
ℒ u ( c , u , α ) = 1 2 ‖ Hc − g ‖ 2 + λ 1 2 ‖ c ‖ 2 + λ 2 ∑ k ‖ u k ‖ 1 + α T ( Lc − u ) + μ 2 ‖ Lc − u ‖ 2 ,
{ c k + 1 ← argmin c ℒ μ ( c , u k , α k ) u k + 1 ← argmin c ℒ μ ( c k + 1 , u , α k ) α k + 1 ← α k + μ ( L c k + 1 − u k + 1 ) .
∇ ℒ μ ( c , u k , α k ) = ( H T H + μ L T L + λ 1 I ) ︸ A c − ( H T g + μ L T ( u k − α k μ ) ) ︸ b .
ℛ ( n ) * ℛ ( n ) { f } ( x ) = 2 π × ( − Δ ) 2 n − 1 2 { f } ( x ) .
u k + 1 = max { | L c k + 1 + α k μ | − λ 2 μ , 0 } sgn ( L c k + 1 + α k μ ) .
SSIM ( x , x ^ ) = ( 2 μ x μ x ^ + C 1 ) ( 2 σ x x ^ + C 2 ) ( μ x 2 + μ x ^ 2 + C 1 ) ( σ x 2 + σ x ^ 2 + C 2 ) ,
SNR ( x , x ^ ) = max a , b ∈ ℝ 20 log ‖ x ‖ 2 ‖ x − a x ^ + b ‖ 2
c ^ = argmin c { 1 2 ‖ Hc − g ‖ 2 − σ n 2 log p ( c ) } ,
− log p ( c ) ∝ 1 σ u ∑ k ‖ u k ‖ 1 ,
q ^ ( ω y ) = 1 2 π 1 | ω y | × h k ( ω y ) ,

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