OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 14 — Jul. 7, 2008
  • pp: 10736–10749

The binary dissector: phase contrast tomography of two- and three-material objects from few projections

G. R. Myers, T. E. Gureyev, D. M. Paganin, and S. C. Mayo  »View Author Affiliations

Optics Express, Vol. 16, Issue 14, pp. 10736-10749 (2008)

View Full Text Article

Enhanced HTML    Acrobat PDF (2309 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



In X-ray computed tomography (CT) increased information requirements (e.g. increased resolution) typically lead to a concurrent increase in the required number of viewing angles, scanning time and delivered dose. We demonstrate that using phase-contrast imaging it is possible to “dissect” two- and three-material objects into their component materials, which in combination with binary tomographic techniques allows us to satisfy increased information requirements without taking the usual images at additional viewing angles. This imaging scheme reduces the scanning time and dose delivered to samples by at least an order of magnitude when compared to conventional X-ray CT. The effects of noise on our reconstruction scheme are investigated for simulated data. Finally, a slice through a glass tube filled with silica and water is reconstructed from 18 projection images taken on an X-ray ultra Microscope (XuM).

© 2008 Optical Society of America

OCIS Codes
(100.5070) Image processing : Phase retrieval
(100.6950) Image processing : Tomographic image processing
(340.7440) X-ray optics : X-ray imaging
(090.1995) Holography : Digital holography
(110.3200) Imaging systems : Inverse scattering

ToC Category:
Image Processing

Original Manuscript: May 28, 2008
Revised Manuscript: June 22, 2008
Manuscript Accepted: June 23, 2008
Published: July 2, 2008

Virtual Issues
Vol. 3, Iss. 8 Virtual Journal for Biomedical Optics

G. R. Myers, T. E. Gureyev, D. M. Paganin, and S. C. Mayo, "The binary dissector: phase contrast tomography of two- and three-material objects from few projections," Opt. Express 16, 10736-10749 (2008)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).
  2. R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53, 23-26 (2000). [CrossRef]
  3. J. ?ehá?ek, Z. Hradil, J. Pe?ina, S. Pascazio, P. Facchi, and M. Zawisky, "Advanced neutron imaging and sensing," Adv. Imaging Electron. Phys. 142, 53-157 (2006). [CrossRef]
  4. E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969). [CrossRef]
  5. A. J. Devaney, "Reconstructive tomography with diffracting wave-fields," Inv.Problems 2, 161-183 (1986). [CrossRef]
  6. A. Momose, T. Takeda, and Y. Itai, "Phase-contrast X-ray computed tomography for observing biological specimens and organic materials," Rev. Sci. Instrum. 66, 1434-1436 (1995). [CrossRef]
  7. U. Bonse and F. Busch, "X-ray computed microtomography (µCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996). [CrossRef] [PubMed]
  8. C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x rays," Appl. Phys. Lett. 69, 1826-1828 (1996). [CrossRef]
  9. 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]
  10. A. V. Bronnikov, "Reconstruction formulas in phase-contrast tomography," Opt. Commun. 171, 239-244 (1999). [CrossRef]
  11. F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000). [CrossRef] [PubMed]
  12. K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D: Appl. Phys. 34, A168-A172 (2001). [CrossRef]
  13. Y. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "General reconstruction formulas for analyzer-based computed tomography," Appl. Phys. Lett. 89, 264103 (2006). [CrossRef]
  14. S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003). [CrossRef] [PubMed]
  15. M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Image reconstruction in spherical-wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 2651-2661 (2005). [CrossRef]
  16. X. Wu and H. Liu, "X-ray cone-beam phase tomography formulas based on phase-attenuation duality," Opt. Express 13, 6000-6014 (2005). [CrossRef] [PubMed]
  17. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, "Phase tomography by X-ray Talbot interferometry for biological imaging," Jpn. J. Appl. Phys. 45, 5254-5262 (2006). [CrossRef]
  18. 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]
  19. F. Pfeiffer, C. Kottler, O. Bunk, and C. David, "Hard x-ray phase tomography with low-brilliance sources," Phys. Rev. Lett. 98, 108105 (2007). [CrossRef] [PubMed]
  20. M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, "High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source," Appl. Phys. Lett. 90, 224101 (2007). [CrossRef]
  21. D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, New York, 2006). [CrossRef]
  22. G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys. Rev. A 76, 045804 (2007). [CrossRef]
  23. T. E. Gureyev, Y. I. Nesterets, K. M. Pavlov, and S. W. Wilkins, "Computed tomography with linear shift-invariant optical systems," J. Opt. Soc. Am. A 24, 2230-2241 (2007). [CrossRef]
  24. J. Miao, C.-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T.-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by X-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503 (2006). [CrossRef] [PubMed]
  25. A. V. Bronnikov, "Phase contrast CT: Fundamental theorem and fast image reconstruction algorithms," Proc. SPIE 6318, 63180Q (2006). [CrossRef]
  26. F. Pfeiffer, O. Bunk, C. Kottler, and C. David, "Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images," Nucl. Instrum. Methods Phys. Res. A 580, 925-928 (2007). [CrossRef]
  27. A. Groso, R. Abela, and M. Stampanoni, "Implementation of a fast method for high resolution phase contrast tomography," Opt. Express 14, 8103-8110 (2006). [CrossRef] [PubMed]
  28. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996). [CrossRef]
  29. M. R. Teague, "Deterministic phase retrieval: a Green??s function solution, " J. Opt. Soc. Am. 73, 1434-1441 (1983). [CrossRef]
  30. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002). [CrossRef] [PubMed]
  31. G. T. Herman and A. Kuba, eds., Discrete Tomography: Foundations, Algorithms and Applications (Birkhäuser, Boston, 1999).
  32. F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001). [CrossRef]
  33. R. J. Gardner and P. Gritzmann, "Discrete tomography: Determination of finite sets by x-rays," Trans. Amer. Math. Soc. 349, 2271-2295 (1997). [CrossRef]
  34. P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, "Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography," Int. J. Imaging Syst. Technol. 9, 101-109 (1998). [CrossRef]
  35. P. Fishburn, P. Schwander, L. Shepp, and R. J. Vanderbei, "The discrete Radon transform and its approximate inversion via linear programming," Discrete Appl. Math. 75, 39-61 (1997). [CrossRef]
  36. P. Gritzmann, S. de Vries, and M. Wiegelmann, "Approximating binary images from discrete x-rays," SIAM J. Optimization 11, 522-546 (2000). [CrossRef]
  37. S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, "Binary tomography by iterating linear programs from noisy projections," Lecture Notes in Comput.Science 3322, 38-51 (2004). [CrossRef]
  38. T. Schüle, C. Schnörr, S. Weber, and J. Hornegger, "Discrete tomography by convex-concave regularization and D.C. programming," Discrete Appl. Math. 151, 229-243 (2005). [CrossRef]
  39. K. J. Batenburg, "A network flow algorithm for binary image reconstruction from few projections," Lecture Notes in Comput.Science 4245, 86-97 (2006). [CrossRef]
  40. L. Hajdu and R. Tijdeman, "An algorithm for discrete tomography," Linear Algebra Appl. 339, 119-128 (2001). [CrossRef]
  41. A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, "A discrete tomography algorithm for improving the quality of 3DXRD grain maps," J. Appl. Cryst. 39, 582-588 (2006). [CrossRef]
  42. A. Alpers and P. Gritzmann, "On the degree of ill-posedness in discrete tomography," preprint, 2004.
  43. A. Alpers, P. Gritzmann, and L. Thorens, "Stability and instability in discrete tomography," Lecture Notes in Comput.Science 2243, 175-186 (2001). [CrossRef]
  44. A. Alpers and P. Gritzmann, "On stability, error correction, and noise compensation in discrete tomography," SIAM J. Discrete Math. 20, 227-239 (2006). [CrossRef]
  45. R. J. Gardner, P. Gritzmann, and D. Prangenberg, "On the computational complexity of reconstructing lattice sets from their X-rays," Discrete Math. 202, 45-71 (1999). [CrossRef]
  46. P. Gritzmann and S. de Vries, "Reconstructing crystalline structures from few images under high resolution transmission electron microscopy," in Mathematics: Key Technology for the Future, W. Jäger and H.-J. Krebs, eds. (Springer-Verlag, Berlin, 2003) pp. 441-459. [CrossRef]
  47. G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, "Phase-contrast tomography of single-material objects from few projections," Opt. Express 16, 908-919 (2008). [CrossRef] [PubMed]
  48. T. E. Gureyev, "Transport of intensity equation for beams in an arbitrary state of temporal and spatial coherence," Optik 110, 263-266 (1999).
  49. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C, The art of scientific computing, 2nd ed. (Cambridge University Press, Cambridge, 1995).
  50. D. M. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy. III. The effects of noise," J. Microsc. 214, 51-61 (2004). [CrossRef] [PubMed]
  51. D. M. Paganin, "Studies in Phase Retrieval," PhD Thesis, University of Melbourne (1999).
  52. E. Wolf and J. R. Shewell, "The inverse wave propagator," Phys. Lett. A 25, 417-418 (1967); see also E. Wolf and J. R. Shewell, "Errata," Phys. Lett. A 26, 104 (1967). [CrossRef]
  53. J. R. Shewell and E. Wolf, "Inverse diffraction and a new reciprocity theorem," J. Opt. Soc. Am. A 58, 1596-1603 (1968). [CrossRef]
  54. E. Lalor, "Inverse wave propagator," J. Math. Phys. 9, 2001-2006 (1968). [CrossRef]
  55. G. C. Sherman, "Diffracted wave fields expressible by plane-wave expansions containing only homogenous components," Phys. Rev. Lett. 21, 761-764 (1968). [CrossRef]
  56. P. Besl, "Active, optical range imaging sensors", Machine Vision and Applications 1, 127-152 (1988). [CrossRef]
  57. T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two component samples using in-line hard X-ray images," J. Synchrotron Rad. 9, 148-153 (2002). [CrossRef]
  58. W. R. Brody, G. Butt, A. Hall, and A. Macovski, "A method for selective tissue and bone visualization using dual energy scanned projection radiography," Med. Phys. 8, 353-357 (1980). [CrossRef]
  59. L. A. Shepp and B. F. Logan, "Reconstructing interior head tissue from X-ray transmissions," IEEE Trans. Nucl. Sci. 21, 228-236 (1974). [CrossRef]

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.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited