OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 23 — Nov. 7, 2011
  • pp: 22809–22819

Non-linear iterative phase retrieval based on Frechet derivative

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin  »View Author Affiliations

Optics Express, Vol. 19, Issue 23, pp. 22809-22819 (2011)

View Full Text Article

Enhanced HTML    Acrobat PDF (4308 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(100.3190) Image processing : Inverse problems
(100.5070) Image processing : Phase retrieval
(110.7440) Imaging systems : X-ray imaging
(180.7460) Microscopy : X-ray microscopy

ToC Category:
Image Processing

Original Manuscript: May 4, 2011
Revised Manuscript: June 22, 2011
Manuscript Accepted: June 24, 2011
Published: October 26, 2011

Virtual Issues
Vol. 7, Iss. 1 Virtual Journal for Biomedical Optics

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, "Non-linear iterative phase retrieval based on Frechet derivative," Opt. Express 19, 22809-22819 (2011)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. G. R. Davis and S. L. Wong, “X-ray microtomography of bones and teeth,” Physiol. Meas. 17, 121–146 (1996). [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). [CrossRef]
  8. J. Baruchel, E. Marire, P. Merle, and G. Peix, X-ray Tomography in Material Science (Hermes Science Publications, 2000).
  9. U. Bonse, “Developments in X-ray tomography II,” Proc. SPIE 3775 (1999).
  10. 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]
  11. 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]
  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). [CrossRef] [PubMed]
  16. D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006). [CrossRef]
  17. 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]
  18. T.E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003). [CrossRef]
  19. J.R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
  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).
  24. 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]
  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).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3
Fig. 4

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited