Abstract

The reconstruction problem in in-line X-ray Phase-Contrast Tomography is usually approached by solving two independent linearized sub-problems: phase retrieval and tomographic reconstruction. Both problems are often ill-posed and require the use of regularization techniques that lead to artifacts in the reconstructed image. We present a novel reconstruction approach that solves two coupled linear problems algebraically. Our approach is based on the assumption that the frequency space of the tomogram can be divided into bands that are accurately recovered and bands that are undefined by the observations. This results in an underdetermined linear system of equations. We investigate how this system can be solved using three different algebraic reconstruction algorithms based on Total Variation minimization. These algorithms are compared using both simulated and experimental data. Our results demonstrate that in many cases the proposed algebraic algorithms yield a significantly improved accuracy over the conventional L2-regularized closed-form solution. This work demonstrates that algebraic algorithms may become an important tool in applications where the acquisition time and the delivered radiation dose must be minimized.

© 2013 OSA

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References

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  1. R. C. Chen, L. Rigon, and R. Longo, “Quantitative 3D refractive index decrement reconstruction using single-distance phase-contrast tomography data,” J. Phys. D Appl. Phys.44, 9 (2011).
    [CrossRef]
  2. A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr.21, 710–723 (2013).
    [CrossRef]
  3. F. Natterer, The Mathematics of Computerized Tomography( New York: Wiley, 1986).
  4. E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52, 489–509 (2006).
    [CrossRef]
  5. M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. Signal Proc.50, 1417–1428 (2003).
    [CrossRef]
  6. X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inv. Probl. and Imaging2, 455–484 (2008).
    [CrossRef]
  7. L. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: demonstration of extended conditions with homogeneous objects,” Opt. Express12, 2960–2965 (2004).
    [CrossRef] [PubMed]
  8. D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc.206, 33–40 (2002).
    [CrossRef] [PubMed]
  9. X. Wu and A. Yan, “Phase retrieval from one single phase-contrast x-ray image,” Opt. Express17, 11187 (2009).
    [CrossRef] [PubMed]
  10. R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express19, 25881–25890 (2011).
    [CrossRef]
  11. A. C. Kak and M. Slaney, Principles of computerized tomographic imaging (IEEE Press, 1988).
  12. L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Math.16, 1–3 (1966).
    [CrossRef]
  13. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision20, 89–97 (2004).
    [CrossRef]
  14. J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
    [CrossRef]
  15. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Im. Sci.2, 183–202 (2009).
    [CrossRef]
  16. G. M. P. van Kemplen and L. J. van Vliet, “The influence of the regularization parameter and the first estimate on the performance of Tikhonov regularized non-linear image restoration algorithms,” J. of Microscopy198, 63–75 (2000).
    [CrossRef]
  17. S.-R. Zhao and H. Halling, “A new Fourier method for fan beam reconstruction,” IEEE Nucl. Sci. Symp. Med. and Imaging Conf.2, 1287–1291 (1995).
  18. G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys.32, 654–665 (2005).
    [CrossRef] [PubMed]
  19. P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
    [CrossRef]
  20. J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett.32, 1617–1619 (2007).
    [CrossRef] [PubMed]

2013 (1)

A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr.21, 710–723 (2013).
[CrossRef]

2011 (2)

R. C. Chen, L. Rigon, and R. Longo, “Quantitative 3D refractive index decrement reconstruction using single-distance phase-contrast tomography data,” J. Phys. D Appl. Phys.44, 9 (2011).
[CrossRef]

R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express19, 25881–25890 (2011).
[CrossRef]

2010 (1)

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

2009 (2)

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

X. Wu and A. Yan, “Phase retrieval from one single phase-contrast x-ray image,” Opt. Express17, 11187 (2009).
[CrossRef] [PubMed]

2008 (1)

X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inv. Probl. and Imaging2, 455–484 (2008).
[CrossRef]

2007 (1)

2006 (1)

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52, 489–509 (2006).
[CrossRef]

2005 (1)

G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys.32, 654–665 (2005).
[CrossRef] [PubMed]

2004 (2)

2003 (1)

M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. Signal Proc.50, 1417–1428 (2003).
[CrossRef]

2002 (1)

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. 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)

G. M. P. van Kemplen and L. J. van Vliet, “The influence of the regularization parameter and the first estimate on the performance of Tikhonov regularized non-linear image restoration algorithms,” J. of Microscopy198, 63–75 (2000).
[CrossRef]

1999 (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

1995 (1)

S.-R. Zhao and H. Halling, “A new Fourier method for fan beam reconstruction,” IEEE Nucl. Sci. Symp. Med. and Imaging Conf.2, 1287–1291 (1995).

1966 (1)

L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Math.16, 1–3 (1966).
[CrossRef]

Armijo, L.

L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Math.16, 1–3 (1966).
[CrossRef]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Batenburg, K.J.

A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr.21, 710–723 (2013).
[CrossRef]

Baumbach, T.

Beck, A.

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

Blu, T.

M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. Signal Proc.50, 1417–1428 (2003).
[CrossRef]

Boistel, R.

Bresson, X.

X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inv. Probl. and Imaging2, 455–484 (2008).
[CrossRef]

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52, 489–509 (2006).
[CrossRef]

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision20, 89–97 (2004).
[CrossRef]

Chan, T. F.

X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inv. Probl. and Imaging2, 455–484 (2008).
[CrossRef]

Chen, G.-H.

G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys.32, 654–665 (2005).
[CrossRef] [PubMed]

Chen, R. C.

R. C. Chen, L. Rigon, and R. Longo, “Quantitative 3D refractive index decrement reconstruction using single-distance phase-contrast tomography data,” J. Phys. D Appl. Phys.44, 9 (2011).
[CrossRef]

Cloetens, P.

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett.32, 1617–1619 (2007).
[CrossRef] [PubMed]

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Dahl, J.

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

Dhal, B.

Guigay, J. P.

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett.32, 1617–1619 (2007).
[CrossRef] [PubMed]

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Gureyev, T.

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

Halling, H.

S.-R. Zhao and H. Halling, “A new Fourier method for fan beam reconstruction,” IEEE Nucl. Sci. Symp. Med. and Imaging Conf.2, 1287–1291 (1995).

Hansen, P. C.

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

Hayes, J.

Hofmann, R.

Jensen, S. H.

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

Jensen, T. L.

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of computerized tomographic imaging (IEEE Press, 1988).

Kostenko, A.

A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr.21, 710–723 (2013).
[CrossRef]

Langer, M.

Leng, S.

G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys.32, 654–665 (2005).
[CrossRef] [PubMed]

Longo, R.

R. C. Chen, L. Rigon, and R. Longo, “Quantitative 3D refractive index decrement reconstruction using single-distance phase-contrast tomography data,” J. Phys. D Appl. Phys.44, 9 (2011).
[CrossRef]

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Mancuso, A.

Marziliano, P.

M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. Signal Proc.50, 1417–1428 (2003).
[CrossRef]

Mayo, S.

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. 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.

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

Mistretta, C. A.

G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys.32, 654–665 (2005).
[CrossRef] [PubMed]

Moosmann, J.

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography( New York: Wiley, 1986).

Nugent, K.

Offerman, S.E.

A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr.21, 710–723 (2013).
[CrossRef]

Paganin, D.

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

Paterson, D.

Peele, A.

Rigon, L.

R. C. Chen, L. Rigon, and R. Longo, “Quantitative 3D refractive index decrement reconstruction using single-distance phase-contrast tomography data,” J. Phys. D Appl. Phys.44, 9 (2011).
[CrossRef]

Romberg, J.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52, 489–509 (2006).
[CrossRef]

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Scholten, R.

Slaney, M.

A. C. Kak and M. Slaney, Principles of computerized tomographic imaging (IEEE Press, 1988).

Suhonen, H.

A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr.21, 710–723 (2013).
[CrossRef]

Tao, T.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52, 489–509 (2006).
[CrossRef]

Teboulle, M.

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

Tran, C.

Turner, L.

van Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

van Kemplen, G. M. P.

G. M. P. van Kemplen and L. J. van Vliet, “The influence of the regularization parameter and the first estimate on the performance of Tikhonov regularized non-linear image restoration algorithms,” J. of Microscopy198, 63–75 (2000).
[CrossRef]

van Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

van Vliet, L. J.

G. M. P. van Kemplen and L. J. van Vliet, “The influence of the regularization parameter and the first estimate on the performance of Tikhonov regularized non-linear image restoration algorithms,” J. of Microscopy198, 63–75 (2000).
[CrossRef]

van Vliet, L.J.

A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr.21, 710–723 (2013).
[CrossRef]

Vetterli, M.

M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. Signal Proc.50, 1417–1428 (2003).
[CrossRef]

Wilkins, S.

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

Wu, X.

Yan, A.

Zhao, S.-R.

S.-R. Zhao and H. Halling, “A new Fourier method for fan beam reconstruction,” IEEE Nucl. Sci. Symp. Med. and Imaging Conf.2, 1287–1291 (1995).

Appl. Phys. Lett. (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

IEEE Nucl. Sci. Symp. Med. and Imaging Conf. (1)

S.-R. Zhao and H. Halling, “A new Fourier method for fan beam reconstruction,” IEEE Nucl. Sci. Symp. Med. and Imaging Conf.2, 1287–1291 (1995).

IEEE Trans. Inf. Theory (1)

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52, 489–509 (2006).
[CrossRef]

IEEE Trans. Signal Proc. (1)

M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. Signal Proc.50, 1417–1428 (2003).
[CrossRef]

Inv. Probl. and Imaging (1)

X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inv. Probl. and Imaging2, 455–484 (2008).
[CrossRef]

J. Math. Imaging Vision (1)

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision20, 89–97 (2004).
[CrossRef]

J. Microsc. (1)

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

J. of Microscopy (1)

G. M. P. van Kemplen and L. J. van Vliet, “The influence of the regularization parameter and the first estimate on the performance of Tikhonov regularized non-linear image restoration algorithms,” J. of Microscopy198, 63–75 (2000).
[CrossRef]

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

R. C. Chen, L. Rigon, and R. Longo, “Quantitative 3D refractive index decrement reconstruction using single-distance phase-contrast tomography data,” J. Phys. D Appl. Phys.44, 9 (2011).
[CrossRef]

Med. Phys. (1)

G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys.32, 654–665 (2005).
[CrossRef] [PubMed]

Num. Alg. (1)

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

Opt. Expr. (1)

A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr.21, 710–723 (2013).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Pacific J. Math. (1)

L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Math.16, 1–3 (1966).
[CrossRef]

SIAM J. Im. Sci. (1)

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

Other (2)

F. Natterer, The Mathematics of Computerized Tomography( New York: Wiley, 1986).

A. C. Kak and M. Slaney, Principles of computerized tomographic imaging (IEEE Press, 1988).

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Figures (5)

Fig. 1
Fig. 1

Central slice theorem for single-distance X-ray Phase Retrieval Tomography. Spatial domain representation: observations I(θ, s) can be modeled as the projection of the unknown image f(x, y) followed by the convolution with a linear propagator. Fourier domain representation: Fourier transform of the observed image Ĩ(θ, w) can be modeled as the slice of the unknown image f̃(wcosθ, wsinθ)) multiplied with P(w). The Fourier representation of the unknown image f (u, v) is irrecoverable at the frequency bands corresponding to zero-crossings of P(w).

Fig. 2
Fig. 2

Simulated phase-contrast tomography of the Shepp-Logan phantom. (a) Ground truth; (b) projected attenuation; (c) observed phase-contrast sinogram; (d) sinogram after phase-retrieval; (e) ’direct’ filtered-back projection; (f) filtered-back projection after phase retrieval.

Fig. 3
Fig. 3

Reconstructions of the simulated data (error magnitude). Columns correspond to the reconstruction methods: sequential, unconstrained algebraic, full algebraic, algebraic tomographic reconstruction and algebraic phase retrieval. Rows correspond to the different simulations: weak phase, few projections, blur, strong phase, noise, realistic.

Fig. 4
Fig. 4

X-ray PCT reconstruction of the glass capillary filled with copper spheres (slice from the middle of the volume). Reconstructions based on 650 projections: (a) - sequential approach based on L2-regularized phase retrieval, (c) - full algebraic reconstruction, (e) - algebraic phase retrieval. Reconstructions based on 65 projections: (b) - sequential approach, (d) - full algebraic reconstruction, (f) - algebraic phase retrieval. Red line shows position of the profiles depicted on the next figure.

Fig. 5
Fig. 5

Attenuation profiles of the copper sphere and the quartz wall for different reconstruction algorithms. (a) Reconstruction based on 650 projections; (b) reconstruction based on 65 projections.

Tables (2)

Tables Icon

Table 1 Simulation parameter sets

Tables Icon

Table 2 RMSE for six different simulations (rows) and five reconstruction algorithms (columns).

Equations (20)

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

I ˜ ( w ) = δ ( w ) ( 2 cos ( π λ D w 2 ) 2 σ sin ( π λ D w 2 ) ) μ ˜ ( w ) .
I = 𝒫 μ .
𝒫 = diag ( P ( w 1 ) , P ( w 2 ) , , P ( w k ) ) ,
μ = argmin μ 𝒫 μ I 2 2 .
𝒵 = diag ( Z ( w 1 ) , Z ( w 2 ) , , Z ( w k ) ) .
𝒵 I = 𝒵 𝒫 μ .
p ( θ , s ) = f ( t sin θ + s cos θ , t cos θ + s sin θ ) d t .
f ˜ ( u , v ) = f ( x , y ) e 2 π i ( x u + y v ) d x d y ,
f ˜ ( w cos θ , w sin θ ) = p ( θ , s ) e 2 π i s w d s .
f ˜ ( w cos θ , w sin θ ) = I ˜ ( θ , w ) δ ( w ) 2 σ sin ( π λ D w 2 ) 2 cos ( π λ D w 2 ) .
p = f .
𝒵 I = 𝒵 𝒫 f .
𝒵 μ = 𝒵 f ,
^ 1 I = 𝒫 ^ ^ f ,
arg min f : 𝒜 f I ˜ 2 ,
f j + 1 = f j 2 α 𝒜 T ( 𝒜 f j I ^ ) ,
argmin f : 𝒜 f I ^ 2 + λ T V f T V ,
T ( θ , s ) = exp ( 2 π λ ( β + i δ ) f ( x , y ) ) ,
I ( θ , s ) = 1 ( O T F | 1 ( P λ T ( θ , s ) ) | 2 ) + noise ,
f j + 1 f j 2 f j 2 < 10 5 .

Metrics