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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 25 — Dec. 6, 2010
  • pp: 25771–25785

Nonlinear phase retrieval from single-distance radiograph

Julian Moosmann, Ralf Hofmann, Andrei V. Bronnikov, and Tilo Baumbach  »View Author Affiliations

Optics Express, Vol. 18, Issue 25, pp. 25771-25785 (2010)

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Phase contrast in the object plane of a phase object is retrieved from intensity contrast at a single object-detector distance. Expanding intensity contrast and phase shift in the detector plane in powers of object-detector distance, phase retrieval is performed beyond the solution to the linearized transport-of-intensity equation. The expansion coefficients are determined by the entire paraxial wave equation. The Laplacian of the phase shift in the object plane thus is written as a local expression linear in the intensity contrast and nonlinear in the phase shift in the object plane. A perturbative approach to this expression is proposed and tested with simulated phantom data.

© 2010 Optical Society of America

OCIS Codes
(100.2960) Image processing : Image analysis
(100.3010) Image processing : Image reconstruction techniques
(100.5070) Image processing : Phase retrieval
(340.7440) X-ray optics : X-ray imaging

ToC Category:
Image Processing

Original Manuscript: September 16, 2010
Revised Manuscript: October 11, 2010
Manuscript Accepted: October 14, 2010
Published: November 23, 2010

Julian Moosmann, Ralf Hofmann, Andrei Bronnikov, and Tilo Baumbach, "Nonlinear phase retrieval from single-distance radiograph," Opt. Express 18, 25771-25785 (2010)

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  1. A. Papoulis, "Ambiguity function in Fourier optics," J. Opt. Soc. Am. 64, 779 (1974). [CrossRef]
  2. J.-P. Guigay, "Fourier transform analysis of Fresnel diffraction patterns and in-line holograms," Optik 49, 121 (1977).
  3. J.-P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136 (1978). [CrossRef]
  4. A. V. Bronnikov, "Theory of quantitative phase-contrast computed tomography," J. Opt. Soc. Am. A 19, 472 (2002). [CrossRef]
  5. M. Born, and E. Wolf, Principles of Optics, 6th ed., Pergamon Press, Oxford, New York (1980).
  6. A. Momose, "Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer," Nucl. Instrum. Methods Phys. Res. A 352, 622 (1995). [CrossRef]
  7. F. Beckmann, U. Bonse, F. Busch, O. Günnewig, and T. Biermann, "A novel system for X-ray phase-contrast microtomography," HASYLAB Annual Report II, 691 (1995).
  8. U. Bonse, and M. Hart, "An X-ray interferometer," Appl. Phys. Lett. 6, 155 (1965). [CrossRef]
  9. F. Zernike, "Phase-contrast, a new method for microscopic observation of transparent objects. Part I," Physica 9, 686 (1942). [CrossRef]
  10. C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287 (2002). [CrossRef]
  11. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nat. Phys. 2, 258 (2006). [CrossRef]
  12. E. Förster, K. Goetz, and P. Zaumseil, "Double crystal diffractometry for the characterization of targets for laser fusion experiments," Krist. Tech. 1, 937 (1980). [CrossRef]
  13. V. A. Bushuev, V. N. Ingal, and E. A. Belyaevskaya, "Dynamical Theory of Images Generated by Noncrystalline Objects for the Method of Phase-Dispersive Introscopy," Kristallografiya 41, 808 (1996).
  14. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and W. Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature 373, 595 (1995). [CrossRef]
  15. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of xray phase contrast microimaging by coherent high energy synchrotron radiation," Rev. Sci. Instrum. 66(12), 5486 (1995). [CrossRef]
  16. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335 (1996). [CrossRef]
  17. D. Paganin, Coherent X-Ray Optics, Oxford University Press (2006). [CrossRef]
  18. T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials," J. Opt. Soc. Am. A 12, 1942 (1995).
  19. D. Paganin, and K. A. Nugent, "Noninterferometric Phase Imaging with Partially Coherent Light," Phys. Rev. Lett. 80, 2586 (1998). [CrossRef]
  20. L. D. 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 for homogeneous objects," Opt. Express 12, 2960 (2004). [CrossRef] [PubMed]
  21. T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569 (2006). [CrossRef]
  22. T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006). [CrossRef]
  23. M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167 (1996) and refs. therein. [CrossRef]
  24. P. Cloetens, Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation, PhD thesis at Vrije Universiteit Brussel (1999) and references therein. [PubMed]
  25. X. Wu, and H. Liu, "A new theory of phase-contrast x-ray imaging based on Wigner distributions," Med. Phys. 31, 2378 (2004). [CrossRef] [PubMed]
  26. T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53 (2004). [CrossRef]
  27. M. Langer, F. Peyrin, P. Cloetens, and J.-P. Guigay, "Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography," Med. Phys. 35, 4556 (2008). [CrossRef] [PubMed]
  28. A. Groso, R. Abela, and M. Stampanoni, "Implementation of a fast method for high resolution phase contrast tomography," Opt. Express 14, 8103 (2006). [CrossRef] [PubMed]
  29. M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, "Practical use of the modified Bronnikov algorithm in micro-CT," Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009). [CrossRef]
  30. J.-P. Guigay, R. H. Wade, and C. Delpha, "Optical diffraction of Lorentz microscope images," Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W. C. Nixon ed. (The Institute of Physics, London, 1971), pp. 238-239.
  31. M. Langer, Phase Retrieval in the Fresnel Region for Hard X-ray Tomography, PhD thesis at L’Insitut National des Sciences Appliquees de Lyon (2008) and references therein.

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