Abstract

Several methods of phase retrieval for in-line phase tomography have already been investigated based on the linearization of the relation between the phase shift induced by the object and the diffracted intensity. In this work, we present a non-linear iterative approach using the Frechet derivative of the intensity recorded at a few number of propagation distances. A Landweber type iterative method with an analytic calculation of the Frechet derivative adjoint is proposed. The inverse problem is regularized with the smoothing L 2 norm of the phase gradient and evaluated for several different implementations. The evaluation of the method was performed using a simple phase map, both with and without noise. Our approach outperforms the linear methods on simulated noisy data up to high noise levels and thanks to the proposed analytical calculation is suited to the processing of large experimental image data sets.

© 2011 OSA

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    [CrossRef] [PubMed]
  2. M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. 26, 2194–2204 (1999).
    [CrossRef] [PubMed]
  3. S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel, and G. Boivin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Med. Phys. 29, 2672–2681 (2002).
    [CrossRef] [PubMed]
  4. S. Bayat, L. Apostol, E. Boller, T. Brochard, and F. Peyrin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Nucl. Instr. Meth. Phys. Res. A 548, 247–252 (2005).
    [CrossRef]
  5. C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. 33, 287–293 (2003).
  6. M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).
  7. G. J. Kazakia, A. J. Burghardt, S. Cheung, and S. Majumdar, “Assessment of bone tissue mineralization by conventional X-ray microcomputed tomography,” Med. Phys. 33, 3170–3179 (2008).
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    [CrossRef]
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    [CrossRef]
  12. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
    [CrossRef]
  13. P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997).
    [CrossRef]
  14. A. Momose, T. Takeda, Y. Itai, A. Yoneyama, and K. Hirano, “Phase-contrast tomographic imaging using an X-ray interferometer,” J. Synchrotron. Rad. 5, 309–314 (1998).
    [CrossRef]
  15. T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68, S13–S17 (2008).
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  18. T.E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
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  19. J.R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
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  20. J. P. Guigay, “Fourier transform analyis of Fresnel diffraction patterns in in-line holograms,” Optik 46, 12–125 (1977).
  21. S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X-rays,” Rev. Sci. Instrum 76, 1–7 (2005).
    [CrossRef]
  22. 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]
  23. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1997).
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    [CrossRef] [PubMed]
  25. M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in X-ray in-line phase tomography,” J. Opt. Soc. Am. A 28, 1877–1882 (2009).
    [CrossRef]
  26. M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2425–2436 (2010).
    [CrossRef]
  27. O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variationnal Methods in Imaging (Springer Verlag, 2008).
  28. M. Hanke, A. Neubauer, and O. Scherzer, “A convergence analysis of the landweber iteration for nonlinear ill-posed problems,” Numer. Math. 72, 21–37 (1995).
    [CrossRef]
  29. I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).
    [CrossRef]
  30. C. T. Kelley and P. Gilmore, “An implicit filtering algorithm for optimization of functions with many local minima,” SIAM J. Optm. 5, 269–285 (1985).
  31. D. Stoneking, G. L. Bilbro, R. Trew, P. Gilmore, and C. T. Kelley, “Yield optimization using a gaas process simulator coupled to a physical device model,” IEEE Trans. Microwave Theory Tech. 40, 1353–1363 (1992).
    [CrossRef]
  32. C. T. Kelley, “Iterative methods for optimization,” Frontiers in Applied Mathematics (SIAM, 1999).

2010 (1)

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2425–2436 (2010).
[CrossRef]

2009 (1)

M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in X-ray in-line phase tomography,” J. Opt. Soc. Am. A 28, 1877–1882 (2009).
[CrossRef]

2008 (4)

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef] [PubMed]

I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).
[CrossRef]

G. J. Kazakia, A. J. Burghardt, S. Cheung, and S. Majumdar, “Assessment of bone tissue mineralization by conventional X-ray microcomputed tomography,” Med. Phys. 33, 3170–3179 (2008).
[CrossRef]

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68, S13–S17 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

2005 (2)

S. Bayat, L. Apostol, E. Boller, T. Brochard, and F. Peyrin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Nucl. Instr. Meth. Phys. Res. A 548, 247–252 (2005).
[CrossRef]

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X-rays,” Rev. Sci. Instrum 76, 1–7 (2005).
[CrossRef]

2003 (2)

C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. 33, 287–293 (2003).

T.E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

2002 (1)

S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel, and G. Boivin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Med. Phys. 29, 2672–2681 (2002).
[CrossRef] [PubMed]

1999 (2)

M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. 26, 2194–2204 (1999).
[CrossRef] [PubMed]

U. Bonse, “Developments in X-ray tomography II,” Proc. SPIE 3775 (1999).

1998 (1)

A. Momose, T. Takeda, Y. Itai, A. Yoneyama, and K. Hirano, “Phase-contrast tomographic imaging using an X-ray interferometer,” J. Synchrotron. Rad. 5, 309–314 (1998).
[CrossRef]

1997 (1)

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997).
[CrossRef]

1996 (5)

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D.:Appl. Phys. 29, 133–146 (1996).
[CrossRef]

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

G. R. Davis and S. L. Wong, “X-ray microtomography of bones and teeth,” Physiol. Meas. 17, 121–146 (1996).
[CrossRef] [PubMed]

1995 (1)

M. Hanke, A. Neubauer, and O. Scherzer, “A convergence analysis of the landweber iteration for nonlinear ill-posed problems,” Numer. Math. 72, 21–37 (1995).
[CrossRef]

1992 (1)

D. Stoneking, G. L. Bilbro, R. Trew, P. Gilmore, and C. T. Kelley, “Yield optimization using a gaas process simulator coupled to a physical device model,” IEEE Trans. Microwave Theory Tech. 40, 1353–1363 (1992).
[CrossRef]

1985 (1)

C. T. Kelley and P. Gilmore, “An implicit filtering algorithm for optimization of functions with many local minima,” SIAM J. Optm. 5, 269–285 (1985).

1982 (1)

1977 (1)

J. P. Guigay, “Fourier transform analyis of Fresnel diffraction patterns in in-line holograms,” Optik 46, 12–125 (1977).

Apostol, L.

S. Bayat, L. Apostol, E. Boller, T. Brochard, and F. Peyrin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Nucl. Instr. Meth. Phys. Res. A 548, 247–252 (2005).
[CrossRef]

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D.:Appl. Phys. 29, 133–146 (1996).
[CrossRef]

Baruchel, J.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X-rays,” Rev. Sci. Instrum 76, 1–7 (2005).
[CrossRef]

S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel, and G. Boivin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Med. Phys. 29, 2672–2681 (2002).
[CrossRef] [PubMed]

M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. 26, 2194–2204 (1999).
[CrossRef] [PubMed]

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D.:Appl. Phys. 29, 133–146 (1996).
[CrossRef]

J. Baruchel, E. Marire, P. Merle, and G. Peix, X-ray Tomography in Material Science (Hermes Science Publications, 2000).

Basillais, A.

C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. 33, 287–293 (2003).

Bayat, S.

S. Bayat, L. Apostol, E. Boller, T. Brochard, and F. Peyrin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Nucl. Instr. Meth. Phys. Res. A 548, 247–252 (2005).
[CrossRef]

Benhamou, L.

C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. 33, 287–293 (2003).

Bilbro, G. L.

D. Stoneking, G. L. Bilbro, R. Trew, P. Gilmore, and C. T. Kelley, “Yield optimization using a gaas process simulator coupled to a physical device model,” IEEE Trans. Microwave Theory Tech. 40, 1353–1363 (1992).
[CrossRef]

Boistel, R.

Boivin, G.

S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel, and G. Boivin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Med. Phys. 29, 2672–2681 (2002).
[CrossRef] [PubMed]

Boller, E.

S. Bayat, L. Apostol, E. Boller, T. Brochard, and F. Peyrin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Nucl. Instr. Meth. Phys. Res. A 548, 247–252 (2005).
[CrossRef]

Bonassie, A.

C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. 33, 287–293 (2003).

Bonnet, N.

C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. 33, 287–293 (2003).

Bonse, U.

U. Bonse, “Developments in X-ray tomography II,” Proc. SPIE 3775 (1999).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1997).

Brochard, T.

S. Bayat, L. Apostol, E. Boller, T. Brochard, and F. Peyrin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Nucl. Instr. Meth. Phys. Res. A 548, 247–252 (2005).
[CrossRef]

Bruder, J.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68, S13–S17 (2008).
[CrossRef] [PubMed]

Brunet-Imbault, B.

C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. 33, 287–293 (2003).

Buffiere, J. Y.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997).
[CrossRef]

Bunk, O.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68, S13–S17 (2008).
[CrossRef] [PubMed]

Burghardt, A. J.

G. J. Kazakia, A. J. Burghardt, S. Cheung, and S. Majumdar, “Assessment of bone tissue mineralization by conventional X-ray microcomputed tomography,” Med. Phys. 33, 3170–3179 (2008).
[CrossRef]

Chappard, C.

C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. 33, 287–293 (2003).

Cheung, S.

G. J. Kazakia, A. J. Burghardt, S. Cheung, and S. Majumdar, “Assessment of bone tissue mineralization by conventional X-ray microcomputed tomography,” Med. Phys. 33, 3170–3179 (2008).
[CrossRef]

Cloetens, P.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2425–2436 (2010).
[CrossRef]

M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in X-ray in-line phase tomography,” J. Opt. Soc. Am. A 28, 1877–1882 (2009).
[CrossRef]

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef] [PubMed]

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68, S13–S17 (2008).
[CrossRef] [PubMed]

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]

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X-rays,” Rev. Sci. Instrum 76, 1–7 (2005).
[CrossRef]

S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel, and G. Boivin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Med. Phys. 29, 2672–2681 (2002).
[CrossRef] [PubMed]

M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. 26, 2194–2204 (1999).
[CrossRef] [PubMed]

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D.:Appl. Phys. 29, 133–146 (1996).
[CrossRef]

Daubechies, I.

I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).
[CrossRef]

David, C.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68, S13–S17 (2008).
[CrossRef] [PubMed]

Davis, G. R.

G. R. Davis and S. L. Wong, “X-ray microtomography of bones and teeth,” Physiol. Meas. 17, 121–146 (1996).
[CrossRef] [PubMed]

Ejiri, S.

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

Fienup, J.R.

Fornasier, M.

I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).
[CrossRef]

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

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

Gilmore, P.

D. Stoneking, G. L. Bilbro, R. Trew, P. Gilmore, and C. T. Kelley, “Yield optimization using a gaas process simulator coupled to a physical device model,” IEEE Trans. Microwave Theory Tech. 40, 1353–1363 (1992).
[CrossRef]

C. T. Kelley and P. Gilmore, “An implicit filtering algorithm for optimization of functions with many local minima,” SIAM J. Optm. 5, 269–285 (1985).

Grasmair, M.

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variationnal Methods in Imaging (Springer Verlag, 2008).

Grossauer, H.

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variationnal Methods in Imaging (Springer Verlag, 2008).

Guigay, J. P.

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef] [PubMed]

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, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D.:Appl. Phys. 29, 133–146 (1996).
[CrossRef]

J. P. Guigay, “Fourier transform analyis of Fresnel diffraction patterns in in-line holograms,” Optik 46, 12–125 (1977).

Guigay, J.-P.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X-rays,” Rev. Sci. Instrum 76, 1–7 (2005).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

Gureyev, T.E.

T.E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

Haltmeier, M.

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variationnal Methods in Imaging (Springer Verlag, 2008).

Hanke, M.

M. Hanke, A. Neubauer, and O. Scherzer, “A convergence analysis of the landweber iteration for nonlinear ill-posed problems,” Numer. Math. 72, 21–37 (1995).
[CrossRef]

Hayashi, K.

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

Hirano, K.

A. Momose, T. Takeda, Y. Itai, A. Yoneyama, and K. Hirano, “Phase-contrast tomographic imaging using an X-ray interferometer,” J. Synchrotron. Rad. 5, 309–314 (1998).
[CrossRef]

Ikeda, S.

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

Itai, Y.

A. Momose, T. Takeda, Y. Itai, A. Yoneyama, and K. Hirano, “Phase-contrast tomographic imaging using an X-ray interferometer,” J. Synchrotron. Rad. 5, 309–314 (1998).
[CrossRef]

Ito, M.

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

Jinnai, H.

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

Kazakia, G. J.

G. J. Kazakia, A. J. Burghardt, S. Cheung, and S. Majumdar, “Assessment of bone tissue mineralization by conventional X-ray microcomputed tomography,” Med. Phys. 33, 3170–3179 (2008).
[CrossRef]

Kelley, C. T.

D. Stoneking, G. L. Bilbro, R. Trew, P. Gilmore, and C. T. Kelley, “Yield optimization using a gaas process simulator coupled to a physical device model,” IEEE Trans. Microwave Theory Tech. 40, 1353–1363 (1992).
[CrossRef]

C. T. Kelley and P. Gilmore, “An implicit filtering algorithm for optimization of functions with many local minima,” SIAM J. Optm. 5, 269–285 (1985).

C. T. Kelley, “Iterative methods for optimization,” Frontiers in Applied Mathematics (SIAM, 1999).

Kono, J.

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

Langer, M.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2425–2436 (2010).
[CrossRef]

M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in X-ray in-line phase tomography,” J. Opt. Soc. Am. A 28, 1877–1882 (2009).
[CrossRef]

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef] [PubMed]

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]

Laval-Jeantet, A. M.

M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. 26, 2194–2204 (1999).
[CrossRef] [PubMed]

Lenzen, F.

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variationnal Methods in Imaging (Springer Verlag, 2008).

Loris, I.

I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).
[CrossRef]

Majumdar, S.

G. J. Kazakia, A. J. Burghardt, S. Cheung, and S. Majumdar, “Assessment of bone tissue mineralization by conventional X-ray microcomputed tomography,” Med. Phys. 33, 3170–3179 (2008).
[CrossRef]

Marire, E.

J. Baruchel, E. Marire, P. Merle, and G. Peix, X-ray Tomography in Material Science (Hermes Science Publications, 2000).

Merle, P.

J. Baruchel, E. Marire, P. Merle, and G. Peix, X-ray Tomography in Material Science (Hermes Science Publications, 2000).

Momose, A.

A. Momose, T. Takeda, Y. Itai, A. Yoneyama, and K. Hirano, “Phase-contrast tomographic imaging using an X-ray interferometer,” J. Synchrotron. Rad. 5, 309–314 (1998).
[CrossRef]

Neubauer, A.

M. Hanke, A. Neubauer, and O. Scherzer, “A convergence analysis of the landweber iteration for nonlinear ill-posed problems,” Numer. Math. 72, 21–37 (1995).
[CrossRef]

Nishida, A.

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

Nugent, K. A.

Nuzzo, S.

S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel, and G. Boivin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Med. Phys. 29, 2672–2681 (2002).
[CrossRef] [PubMed]

Odet, C.

M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. 26, 2194–2204 (1999).
[CrossRef] [PubMed]

Paganin, D. M.

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[CrossRef]

Pateyron-Salome, M.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997).
[CrossRef]

Peix, G.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997).
[CrossRef]

J. Baruchel, E. Marire, P. Merle, and G. Peix, X-ray Tomography in Material Science (Hermes Science Publications, 2000).

Peyrin, F.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2425–2436 (2010).
[CrossRef]

M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in X-ray in-line phase tomography,” J. Opt. Soc. Am. A 28, 1877–1882 (2009).
[CrossRef]

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef] [PubMed]

S. Bayat, L. Apostol, E. Boller, T. Brochard, and F. Peyrin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Nucl. Instr. Meth. Phys. Res. A 548, 247–252 (2005).
[CrossRef]

C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. 33, 287–293 (2003).

S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel, and G. Boivin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Med. Phys. 29, 2672–2681 (2002).
[CrossRef] [PubMed]

M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. 26, 2194–2204 (1999).
[CrossRef] [PubMed]

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997).
[CrossRef]

Pfeiffer, F.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68, S13–S17 (2008).
[CrossRef] [PubMed]

Pogany, A.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

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

Salomé, M.

M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. 26, 2194–2204 (1999).
[CrossRef] [PubMed]

Scherzer, O.

M. Hanke, A. Neubauer, and O. Scherzer, “A convergence analysis of the landweber iteration for nonlinear ill-posed problems,” Numer. Math. 72, 21–37 (1995).
[CrossRef]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variationnal Methods in Imaging (Springer Verlag, 2008).

Schlenker, M.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X-rays,” Rev. Sci. Instrum 76, 1–7 (2005).
[CrossRef]

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D.:Appl. Phys. 29, 133–146 (1996).
[CrossRef]

Spanne, P.

M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. 26, 2194–2204 (1999).
[CrossRef] [PubMed]

Stevenson, A. W.

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

Stoneking, D.

D. Stoneking, G. L. Bilbro, R. Trew, P. Gilmore, and C. T. Kelley, “Yield optimization using a gaas process simulator coupled to a physical device model,” IEEE Trans. Microwave Theory Tech. 40, 1353–1363 (1992).
[CrossRef]

Takeda, T.

A. Momose, T. Takeda, Y. Itai, A. Yoneyama, and K. Hirano, “Phase-contrast tomographic imaging using an X-ray interferometer,” J. Synchrotron. Rad. 5, 309–314 (1998).
[CrossRef]

Tanaka, M.

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

Trew, R.

D. Stoneking, G. L. Bilbro, R. Trew, P. Gilmore, and C. T. Kelley, “Yield optimization using a gaas process simulator coupled to a physical device model,” IEEE Trans. Microwave Theory Tech. 40, 1353–1363 (1992).
[CrossRef]

Uesugi, K.

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

Weitkamp, T.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68, S13–S17 (2008).
[CrossRef] [PubMed]

Wilkins, S. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1997).

Wong, S. L.

G. R. Davis and S. L. Wong, “X-ray microtomography of bones and teeth,” Physiol. Meas. 17, 121–146 (1996).
[CrossRef] [PubMed]

Yagi, N.

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

Yoneyama, A.

A. Momose, T. Takeda, Y. Itai, A. Yoneyama, and K. Hirano, “Phase-contrast tomographic imaging using an X-ray interferometer,” J. Synchrotron. Rad. 5, 309–314 (1998).
[CrossRef]

Zabler, S.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X-rays,” Rev. Sci. Instrum 76, 1–7 (2005).
[CrossRef]

Appl. Opt. (1)

Eur. J. Radiol. (1)

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68, S13–S17 (2008).
[CrossRef] [PubMed]

IEEE Trans. Image Process. (1)

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2425–2436 (2010).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

D. Stoneking, G. L. Bilbro, R. Trew, P. Gilmore, and C. T. Kelley, “Yield optimization using a gaas process simulator coupled to a physical device model,” IEEE Trans. Microwave Theory Tech. 40, 1353–1363 (1992).
[CrossRef]

J. Appl. Phys. (1)

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997).
[CrossRef]

J. Bone Miner. Metab. (1)

M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).

J. Fourier Anal. Appl. (1)

I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).
[CrossRef]

J. Opt. Soc. Am. A (2)

M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in X-ray in-line phase tomography,” J. Opt. Soc. Am. A 28, 1877–1882 (2009).
[CrossRef]

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

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

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D.:Appl. Phys. 29, 133–146 (1996).
[CrossRef]

J. Synchrotron. Rad. (1)

A. Momose, T. Takeda, Y. Itai, A. Yoneyama, and K. Hirano, “Phase-contrast tomographic imaging using an X-ray interferometer,” J. Synchrotron. Rad. 5, 309–314 (1998).
[CrossRef]

Med. Phys. (5)

G. J. Kazakia, A. J. Burghardt, S. Cheung, and S. Majumdar, “Assessment of bone tissue mineralization by conventional X-ray microcomputed tomography,” Med. Phys. 33, 3170–3179 (2008).
[CrossRef]

M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. 26, 2194–2204 (1999).
[CrossRef] [PubMed]

S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel, and G. Boivin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Med. Phys. 29, 2672–2681 (2002).
[CrossRef] [PubMed]

C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. 33, 287–293 (2003).

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef] [PubMed]

Nature (London) (1)

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

Nucl. Instr. Meth. Phys. Res. A (1)

S. Bayat, L. Apostol, E. Boller, T. Brochard, and F. Peyrin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Nucl. Instr. Meth. Phys. Res. A 548, 247–252 (2005).
[CrossRef]

Numer. Math. (1)

M. Hanke, A. Neubauer, and O. Scherzer, “A convergence analysis of the landweber iteration for nonlinear ill-posed problems,” Numer. Math. 72, 21–37 (1995).
[CrossRef]

Opt. Commun. (1)

T.E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

Opt. Lett. (1)

Optik (1)

J. P. Guigay, “Fourier transform analyis of Fresnel diffraction patterns in in-line holograms,” Optik 46, 12–125 (1977).

Phys. Rev. Lett. (1)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

Physiol. Meas. (1)

G. R. Davis and S. L. Wong, “X-ray microtomography of bones and teeth,” Physiol. Meas. 17, 121–146 (1996).
[CrossRef] [PubMed]

Proc. SPIE (1)

U. Bonse, “Developments in X-ray tomography II,” Proc. SPIE 3775 (1999).

Rev. Sci. Instrum (1)

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X-rays,” Rev. Sci. Instrum 76, 1–7 (2005).
[CrossRef]

SIAM J. Optm. (1)

C. T. Kelley and P. Gilmore, “An implicit filtering algorithm for optimization of functions with many local minima,” SIAM J. Optm. 5, 269–285 (1985).

Other (5)

C. T. Kelley, “Iterative methods for optimization,” Frontiers in Applied Mathematics (SIAM, 1999).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1997).

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variationnal Methods in Imaging (Springer Verlag, 2008).

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[CrossRef]

J. Baruchel, E. Marire, P. Merle, and G. Peix, X-ray Tomography in Material Science (Hermes Science Publications, 2000).

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Figures (4)

Fig. 1
Fig. 1

(a) Original phase map to be retrieved, (b) Original absorption, (c) Fresnel diffraction pattern at propagation distance D=1.4 m and(d) Phase obtained with the mixed approximation.

Fig. 2
Fig. 2

(a) Error map for the phase retrieved with the mixed approach [22], (b) Error map for the phase retrieved with the algorithm A 1 (PPSNR=24 dB, α = 0.01, δ = 0.01), (c) Error map for the phase obtained with the algorithm A 2 (α = 0.01, δ = 0.01) and (d) Error map for the phase obtained with the algorithm A 3 (PPSNR=24 dB, α = 0.01, δ = 0.01).

Fig. 3
Fig. 3

Normalized mean square error for the phase versus iteration number with the edge fixed to zero: (a) for the noise-free data (α = 0.01, δ = 0.01) and (b) for the noisy data (α = 0.01, δ = 0.01, PPSNR= 24dB).

Fig. 4
Fig. 4

Reconstructed phase with A 3 algorithm for (a) noisy-data (PPSNR=24 dB, α = 0.01) and (b) for noise-free simulations (α = 0.01).

Equations (30)

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

n ( x , y , z ) = 1 δ r ( x , y , z ) + i β ( x , y , z )
T ( x ) = exp [ B ( x ) + i φ ( x ) ] = a ( x ) exp [ i φ ( x ) ]
B ( x ) = 2 π λ β ( x , y , z ) d z , φ ( x ) = 2 π λ δ r ( x , y , z ) d z .
I D ( x ) = | T ( x ) * P D ( x ) | 2
P D ( x ) = 1 i λ D exp ( i π λ D | x | 2 )
{ g } ( f ) = g ˜ ( f ) = g ( x ) exp ( 2 i π x f ) d x
I ˜ D ( f ) = I ˜ D φ = 0 ( f ) + 2 sin ( π λ D | f | 2 ) { I 0 φ } ( f ) + λ D 2 π cos ( π λ D | f | 2 ) { ( φ I 0 ) } ( f )
ψ ˜ ( n + 1 ) ( f ) = Σ D A D * ( f ) [ I ˜ D ( f ) I ˜ D φ = 0 ( f ) Δ D ( n ) ( f ) ] Σ D | A D 2 ( f ) | 2 + α
A D ( f ) = 2 sin ( π λ D | f | 2 )
Δ D ( f ) = λ D 2 π cos ( π λ D | f | 2 ) { [ ψ ( n ) ln ( I 0 ) ] } ( f )
H 2 , 2 ( Ω ) = { φ H 2 , 2 ( Ω ) , φ n = 0 }
J ( φ ) = | | I D ( φ ) I δ | | L 2 ( Ω ) 2
| | I D I δ | | L 2 ( Ω ) δ .
J α ( φ ) = 1 2 | | I D ( φ ) I δ | | L 2 ( Ω ) 2 + α 2 | | φ | | L 2 ( Ω ) 2
J α ( φ ) h = 0
2 I D ( φ ) I δ , I D ( φ ) h L 2 ( Ω ) + 2 α φ , h L 2 = 0
2 I ( φ ) * [ I D ( φ ) I δ ] , h L 2 ( Ω ) + 2 α φ , h L 2 = 0
I D ( φ ) * [ I D ( φ ) I δ ] α ( φ ) = 0
τ k = argmin J α k ( φ k τ δ k ) ,
φ k + 1 = φ k τ k J α k ( φ k ) .
φ k + 1 = φ k τ k { I D ( φ k ) * [ I D ( φ k ) I δ ] α φ k }
I D ( φ k + ɛ ) = I D ( φ k ) + G k ( ɛ ) + O ( ɛ 2 ) .
G k ( ɛ ) = { [ i a ɛ exp ( i φ k ) ] * P D } { [ a exp ( i φ k ) ] * P D ¯ } + { [ a exp ( i φ k ) ] * P D } { [ i a ɛ exp ( i φ k ) ] * P D ¯ }
G k ( ɛ ) = 2 Real ( { [ i a ɛ exp ( i φ k ) ] * P D } { [ a exp ( i φ k ) ] * P D ¯ } )
G k * ( ɛ ) = 2 Real [ ( { ɛ [ a exp ( i φ k ) * P D ] } * P D ¯ ) { i a exp ( i φ k ) } ] .
P P S N R = 20 log ( f m a x n m a x ) ,
| | φ k φ * | | / | | φ * | |
J α ( φ k + 1 ) J α ( φ k )
| | I ( φ k + 1 ) I δ | | L 2 ( Ω ) | | I ( φ k ) I δ | | L 2 ( Ω ) .
| | I ( φ k ) I δ | | δ

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