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 L₂ 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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20. J. P. Guigay, “Fourier transform analyis of Fresnel diffraction patterns in in-line holograms,” Optik 46, 12–125 (1977).
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32. C. T. Kelley, “Iterative methods for optimization,” Frontiers in Applied Mathematics (SIAM, 1999).

Appl. Opt.

• J.R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]

Eur. J. Radiol.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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

• 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.

• 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.

• 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.

• 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)

• 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

• 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.

• 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.

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

Opt. Lett.

• 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]

Optik

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

Phys. Rev. Lett.

• 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.

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

Proc. SPIE

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

Rev. Sci. Instrum

• 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.

• 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

• 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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• Dynamic sample imaging in coherent diffractive imaging (OL)
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