Applied Mathematics and Mechanics

, Volume 29, Issue 1, pp 101–112 | Cite as

Algorithm for phase contrast X-ray tomography based on nonlinear phase retrieval

  • Ni Wen-lei  (倪文磊)Email author
  • Zhou Tie  (周铁)


A new algorithm for phase contrast X-ray tomography under holographic measurement was proposed in this paper. The main idea of the algorithm was to solve the nonlinear phase retrieval problem using the Newton iterative method. The linear equations for the Newton directions were proved to be ill-posed and the regularized solutions were obtained by the conjugate gradient method. Some numerical experiments with computer simulated data were presented. The efficiency, feasibility and the numerical stability of the algorithm were illustrated by the numerical experiments. Compared with the results produced by the linearized phase retrieval algorithm, we can see that the new algorithm is not limited to be only efficient for the data measured in the near-field of the Fresnel region and thus it has a broader validity range.

Key words

phase contrast tomography holographic measurement phase retrieval 

Chinese Library Classification

O29 O434.19 

2000 Mathematics Subject Classification

65R32 78A46 92C55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Lewis R A. Medical phase contrast x-ray imaging: Current status and future prospects[J]. Physics in Medicine and Biology, 2004, 49:3573–3583.CrossRefGoogle Scholar
  2. [2]
    Suzuki Y, Yagi N, Uesugi K. X-ray refraction-enhanced imaging and a method for phase retrieval for a simple object[J]. Journal of Synchrotron Radiation, 2002, 9:160–165.CrossRefGoogle Scholar
  3. [3]
    Spanne P, Raven C, Snigireva I, et al. In-line holography and phase-contrast microtomography with high energy x-rays[J]. Physics in Medicines and Biology, 1999, 44:741–749.CrossRefGoogle Scholar
  4. [4]
    Arfelli F, Assante M, V Bonvicini, et al. Low-dose phase contrast x-ray medical imaging[J]. Physics in Medicine and Biology, 1998, 43:2845–2852.CrossRefGoogle Scholar
  5. [5]
    Ingal V N, Beliaevskaya E A, Brianskaya A P, et al. Phase mammography-a new technique for breast investigation[J]. Physics in Medicine and Biology, 1998, 43:2555–2567.CrossRefGoogle Scholar
  6. [6]
    Ando M, Hosoya S. An attempt at x-ray phase-contrast microscopy[C]. In: Shinoda G, Kohra K, and Ichinokawa T (eds). Proceedings of the 6th International Conference of X Ray Optics and Microanalysis, Tokyo: Univerisity of Tokyo Press, 1972, 63–68.Google Scholar
  7. [7]
    Momose A. Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer[J]. Nuclear Instruments and Methods in Physics Research A, 1995, 352:622–628.CrossRefGoogle Scholar
  8. [8]
    Chapman D, Thomlinson W, Johnston R E, et al. Diffraction enhanced x-ray imaging[J]. Physics in Medicine and Biology, 1997, 42:2015–2025.CrossRefGoogle Scholar
  9. [9]
    Dilmanian F A, Zhong Z, Ren B, et al. Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method[J]. Physics in Medicine and Biology, 2000, 45:933–946.CrossRefGoogle Scholar
  10. [10]
    Pfeiffer F, Weitkamp T, Bunk O, et al. Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources[J]. Nature Physics, 2006, 2:258–261.CrossRefGoogle Scholar
  11. [11]
    Momose A, Yashiro W, Takeda Y, et al. Phase tomography by x-ray talbot interferometry for biological imaging[J]. Japanese of Applied Physics, 2006, 45(6A):5254–5262.CrossRefGoogle Scholar
  12. [12]
    Gureyev T E, Nugent K A. Rapid quantitative phase imaging using the transport of intensity equation[J]. Optics Communications, 1997, 133:339–346.CrossRefGoogle Scholar
  13. [13]
    Barty A, Nugent K A, Roberts A, et al. Quantitative phase tomography[J]. Optics Communication, 2000, 175:329–336.CrossRefGoogle Scholar
  14. [14]
    Gureyev T E, Raven C, Snigirev A, Snigireva I, Wilkins S W. Hard x-ray quantitative non-interferometric phase-contrast microscopy[J]. Journal of Physics D: Applied Physics, 1999, 32:563–567.CrossRefGoogle Scholar
  15. [15]
    Jonas P, Louis A K. Phase contrast tomography using holographic measurements[J]. Inverse Problems, 2004, 20(1):75–102.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Bronnikov A V. Theory of quantitative phase-contrast computed tomography[J]. Journal of the Optical Society of America A, 2002, 19(3):472–480.CrossRefGoogle Scholar
  17. [17]
    Born M, Wolf E. Principles of optics: electromagnetic theory of propagation, interference and diffraction of light[M]. Cambridge: Cambridge University Press, 2001.Google Scholar
  18. [18]
    Gureyev T E, Wilkins S W. On x-ray phase imaging with a point source[J]. Journal of the Optical Society of America A, 1998, 15(3):579–585.CrossRefGoogle Scholar
  19. [19]
    Als-Nielsen J, McMorrow D. Elements of modern X-ray physics[M]. New York: Wiley, 2001.Google Scholar
  20. [20]
    Wu X, Deans A E, Liu H. X-ray diagnostic techniques[M]. In: T Vo-Dinh (ed). Biomedical Photonics Handbook, Tampa: CRC Press, 2003, 26.1–26.34.Google Scholar
  21. [21]
    Gureyev T E, Pogany A, Paganin D M, et al. Linear algorithms for phase retrieval in the Fresnel region[J]. Optics Communications, 2004, 231:53–70.CrossRefGoogle Scholar
  22. [22]
    Huntley J M. Noise-immune phase unwrapping algorithm[J]. Applied Optics, 1989, 28(15):3268–3270.CrossRefGoogle Scholar
  23. [23]
    Kak A C, Slaney M. Principles of computerized tomographic imaging[M]. New York: IEEE Press, 1988.Google Scholar
  24. [24]
    Kaipio J, Somersalo E. Statistical and computational inverse problems[M]. New York: Springer, 2005.Google Scholar

Copyright information

© Editorial Committee of Appl. Math. Mech. and Springer-Verlag 2008

Authors and Affiliations

  1. 1.LMAM, School of Mathematical SciencesPeking UniversityBeijingP. R. China

Personalised recommendations