OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 20 — Oct. 7, 2013
  • pp: 23671–23679
« Show journal navigation

X-ray phase tomography of a moving object

Yongjin Sung and Rajiv Gupta  »View Author Affiliations


Optics Express, Vol. 21, Issue 20, pp. 23671-23679 (2013)
http://dx.doi.org/10.1364/OE.21.023671


View Full Text Article

Acrobat PDF (1066 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We propose an algorithm for tomographic reconstruction of the refractive index map of an object translated across a fan-shaped X-ray beam. We adopt a forward image model valid under the non-paraxial condition, and use a unique mapping of the acquired projection images to reduce the computational cost. Even though the imaging setup affords only a limited angular coverage, our algorithm provides accurate refractive index values by employing the positivity and piecewise-smoothness constraints.

© 2013 OSA

1. Introduction

The highly penetrating nature of X-rays [1

1. A. Stanton, “Wilhelm Conrad Röntgen on a new kind of rays: translation of a paper read before the Würzburg Physical and Medical Society, 1895,” Nature 53, 274–276 (1896).

] has been utilized in a variety of applications, including non-destructive evaluation, medical imaging [2

2. C. A. Helms, Fundamentals of Skeletal Radiology (Saunders, 2005).

, 3

3. E. D. Pisano, M. J. Yaffe, and C. M. Kuzmiak, Digital Mammography (Lippincott Williams & Wilkins, 2004).

] and baggage inspection [4

4. H. Vogel and D. Haller, “Luggage and shipped goods,” Eur. J. Radiol. 63(2), 242–253 (2007). [CrossRef] [PubMed]

]. The major source of contrast in the existing X-ray imaging techniques is derived from incoherent processes such as Compton scattering and photoelectric absorption. This type of absorption or attenuation contrast is sensitive to the atomic number of the sample under interrogation [5

5. J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data 4(3), 471–538 (1975). [CrossRef]

]. On the other hand, X-ray phase imaging (XPI) relies on the contrast due to non-homogeneous electron density distribution [6

6. D. M. Paganin, Coherent X-ray Optics (Oxford University, 2006).

]. Therefore, it can differentiate soft tissues (e.g., tumor and healthy tissue) and detect improvised homemade explosives with high sensitivity. Several XPI techniques have been demonstrated using synchrotron [7

7. A. Momose, T. Takeda, and Y. Itai, “Blood vessels: depiction at phase-contrast X-ray imaging without contrast agents in the mouse and rat-feasibility study,” Radiology 217(2), 593–596 (2000). [PubMed]

10

10. M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys. 41(Part 2, No. 9A/B), L1016–L1018 (2002). [CrossRef]

] and, more recently on a lab scale, using micro-focus X-ray sources [11

11. Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot. 66(8), 1083–1090 (2008). [CrossRef] [PubMed]

, 12

12. Z. Zaprazny, D. Korytar, V. Ac, P. Konopka, and J. Bielecki, “Phase contrast imaging of lightweight objects using microfocus X-ray source and high resolution CCD camera,” J. Instrum. 7(03), C03005 (2012). [CrossRef]

] or traditional X-ray tubes [13

13. 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(3Suppl), S13–S17 (2008). [CrossRef] [PubMed]

16

16. H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of X-ray scattering--initial results,” IEEE Trans. Med. Imaging 27(8), 997–1002 (2008). [CrossRef] [PubMed]

]. For a recent review of XPI techniques, the readers are referred to [17

17. A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58(1), R1–R35 (2013). [CrossRef] [PubMed]

].

X-ray phase tomography requires varying the angle of illumination that is incident on the specimen [18

18. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 1988).

21

21. S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11(19), 2289–2302 (2003). [CrossRef] [PubMed]

]. This may be achieved by rotating the imaging chain around the object, as in medical X-ray computed tomography, or by rotating the object in the case where the source is hard to move (for example, in synchrotron imaging). In either case, a rotational gantry or stage must be installed and the imaging chain must be carefully aligned with the object being imaged. For high-throughput applications such as luggage inspection at the airport, moving-object geometry is preferred [22

22. G. Donges and R. Dietrich, “Baggage inspection system,” U. S. patent 4,759,047 (1988).

, 23

23. R. Dietrich, “Baggage inspection system,” U. S. patent 4,783,794 (1988).

]. There are medical applications as well, e.g., breast tomosynthesis [24

24. J. M. Park, E. A. Franken Jr, M. Garg, L. L. Fajardo, and L. T. Niklason, “Breast tomosynthesis: present considerations and future applications,” Radiographics 27(Suppl 1), S231–S240 (2007). [CrossRef] [PubMed]

], where projections are available only from a linear track on one side of the object. Phase tomography of a moving object has been demonstrated in the ultrasound [25

25. D. Nahamoo, S. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason. 31(4), 218–229 (1984). [CrossRef]

], terahertz [26

26. T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun. 267(1), 128–136 (2006). [CrossRef]

], and optical regimes [27

27. N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Synthetic aperture tomographic phase microscopy for 3D imaging of live cells in translational motion,” Opt. Express 16(20), 16240–16246 (2008). [CrossRef] [PubMed]

], but not in the X-ray regime. We recently reported a forward image model for X-ray phase imaging of thick and large objects [28

28. Y. Sung, C. J. R. Sheppard, G. Barbastathis, M. Ando, and R. Gupta, “Full-wave approach for X-ray phase imaging,” Opt. Express 21(15), 17547–17557 (2013). [CrossRef] [PubMed]

]. It does not rely on the projection or paraxial approximation, and thus can be applied to the object at a far off-axis location. In this study, adopting the model, we propose an iterative algorithm for X-ray phase tomography of the object translated across an extended X-ray beam. We demonstrate that an artifact-free, depth-resolved refractive index map of a moving object can be obtained in an X-ray setup. We show that there exists an interesting analogy between our problem and plane-wave diffraction tomography (PW-DT).

2. Forward model for an object translated in a fan-beam X-ray system

Consider an object being translated across a fan-shaped X-ray beam in the geometry shown in Fig. 1
Fig. 1 Schematic diagram of X-ray phase tomography set-up for a moving sample.
. Suppose that the shape and material property of the object do not vary along the y-axis (i.e., perpendicular to the cross-sectional plane that is being imaged). In this 2-D model, we assume that a fan-shaped X-ray bean is emitted from an ideal line source. Let us place the source on the optical axis at z = -R1 and the 2-D object be translated along the line z = 0. The detector, located at z = R2, records the intensity of transmitted X-ray beam after the object. In this setup, the detector is translated with the object so as to capture the diffracted beam from the object at varying locations. The detector shift xi is determined by the magnification factor M = (R2 + R1)⁄R1 and the object shift xo; xi = Mxo, where xo = a(t) is the location of the object at time t. For a weakly absorbing object, the phase profile can be reasonably inferred from a single intensity image [29

29. A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in X-ray phase-contrast imaging suitable for tomography,” Opt. Express 19(11), 10359–10376 (2011). [CrossRef] [PubMed]

]. If the object absorption is non-negligible, one can derive the phase profile from a sequence of projection images acquired for different object-to-detector distance [30

30. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]

, 31

31. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984). [CrossRef]

]. Once the phase profile at the detector plane in Fig. 1 is derived, the iterative algorithm presented in this paper can be applied irrespective of whether the object is weakly or heavily absorbing. The same algorithm can be applied to the phase profiles acquired with other modalities, e.g. grating-based [13

13. 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(3Suppl), S13–S17 (2008). [CrossRef] [PubMed]

, 14

14. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008). [CrossRef] [PubMed]

], coded-aperture [15

15. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with laboratory sources,” Appl. Phys. Lett. 91(7), 074106 (2007). [CrossRef]

], or wire mesh-based [16

16. H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of X-ray scattering--initial results,” IEEE Trans. Med. Imaging 27(8), 997–1002 (2008). [CrossRef] [PubMed]

] system. An exception has to be made for thick metal or thick cortical bone that completely stops the X-rays. In that case, no information about phase or absorption can be derived from the projection data in the region where there is no X-ray penetration.

In a real experiment, the function u¯(s)(x;a) can be obtained from two measurements, one usamp(x;a) with the sample in the field-of-view and the other ubg(x;a) without the sample: u¯(s)(x;a)=log[usamp(x;a)/ubg(x;a)], where log is the natural logarithm. The background images are acquired only once and used for all the subsequent measurements. In this study, we numerically generate the phase profiles using Eqs. (2) and (3). Consider a line source emitting a fan-shaped, mono-energetic beam at 30 keV and a detector with a pixel size of 5 μm. R1 and R2 are fixed at 0.5 m. The object is a cylinder with a diameter of 1 mm and complex refractive index 1 - 0.56 × 10−7 + i × 0.00 (i.e., δ = 2.56 × 10−7 and β = 0.00). Figure 2(a)
Fig. 2 (a) Example of a numerically generated phase profile for a cylinder (1mm dia.) with negligible absorption (δ = 2.56 × 10−7, β = 0.00), and located at a = 0.289 m. In this simulation, the source energy is 30 keV, detector pitch is 5 μm, and R1 = R2 = 0.5 m. (b) 2-D mapping of the data acquired for different object locations along the x-axis.
shows an example phase profile, the argument of the complex function u¯(s)(x;a), for a = 0.289 m. Figure 2(b) shows a 2-D mapping of the phase profiles for varying locations of the cylinder. The width and height of each box in Fig. 2(b) represent object translation step and detector field-of-view, respectively. The dotted box in the figure corresponds to the phase profile in Fig. 2(a). For tomographic reconstruction of an object, multiple projection images are typically recorded for varying viewing angles on the object. In the geometry considered in this study [Fig. 1], the viewing angle is varied by translating the object across the fan-shaped X-ray beam. Given the phase profiles [Fig. 2(b)] and a forward image model [Eqs. (2) and (3)], one can retrieve the refractive index map in an iterative manner. We note, however, that direct evaluation of the integral in Eq. (3) is computationally expensive.

3. Iterative reconstruction algorithm for X-ray phase tomography of a moving object

Applying the mapping to Eq. (3), we obtain the following expression for Us(s;α):
Us(s;α)=(i4πw)1f˜(U,W)exp{i2π[U(s+αα')+WR2]}dU,
(5)
where the variables w and W are defined as w=(1/λ)2(U+α/λr)2, W=wm1/λ, respectively. The variables m1, r, and α are defined in terms of α as m1=1(α/r)2, r=α2+(R1+R2)2and α=α/M, respectively. Noteworthy, the variables w and W in Eq. (5) do not depend on s; thus, the integral in the equation can be efficiently evaluated for each α using a fast Fourier transform algorithm. We note that Eq. (5) has a similar form to the complex scattered phase in plane-wave diffraction tomography (PW-DT) [34

34. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969). [CrossRef]

, 35

35. Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009). [CrossRef] [PubMed]

]. From the analogy with PW-DT, Eq. (6) can be derived, which simply relates Us(s;α) to f(X,Z) in the 2-D spatial frequency plane:
f˜(U,W)=i4π(W+m1/λ)exp{i2π[U(αα')+WR2]}U˜s(U;α),
(6)
where U˜s(U;α) is the 1-D Fourier transform of Us(s;α) with respect to the coordinate s. Mapping the light field according to the method described above and using Eq. (6), we can obtain a 2-D map of f˜, and thus f. We call this method Fourier mapping and direct inversion as in PW-DT [35

35. Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009). [CrossRef] [PubMed]

].

The accuracy of tomographic reconstruction heavily depends on the angular coverage of the incident beam onto the object [36

36. P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6(2), 298–311 (1997). [CrossRef] [PubMed]

]. For the geometry considered in this study, the angular coverage is determined by the divergence angle of X-rays and the angle subtended by the detector trajectory at the source. The divergence angle of X-rays can be simply increased by rotating the source, but increasing the detector trajectory requires a larger system footprint. Therefore, the angular coverage cannot be increased too much in the geometry as shown in Fig. 1. In PW-DT, we have shown that an iterative reconstruction algorithm using the positivity and piecewise-smoothness constraints can effectively suppress the artifacts due to the limited angular coverage [37

37. Y. Sung and R. R. Dasari, “Deterministic regularization of three-dimensional optical diffraction tomography,” J. Opt. Soc. Am. A 28(8), 1554–1561 (2011). [CrossRef] [PubMed]

]. Here, we take a similar approach for X-ray phase tomography of a moving sample. The problem of retrieving the complex refractive index of an object from N scattered field measurements can be cast into the following form:
Anf=Us(s;αn),n=1,2,,N,
(7)
where the forward operator An is given from Eq. (5), in which α is replaced with αn. The scattered fields acquired experimentally contain noise, e.g., photon-shot or electronic noise. In the noisy case, an objective function that minimizes Eq. (8) is known to provide a robust solution for Eq. (7) [38

38. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).

]:
Φμ(f;g1,g2,,gN)=12n=1NAnfUs(s;αn)2+μJ(f),
(8)
where μ is the regularization parameter representing the trade-off between the data fidelity and penalty terms. J(f) is the penalty functional incorporating a priori information about the specimen. The objective function that minimizes Eq. (8) can be iteratively found using Eq. (9):
f(k+1)=f(k)+τn(AngnAnAnf(k))τμJ(f(k)),
(9)
where An is an adjoint operator of An. The variable τ is the relaxation parameter determining the speed of convergence. The superscripts k and k + 1 indicate the iteration number. In this study, the following constraints are used: (i) the refractive index profile is piecewise smooth; and (ii) the electron density and the attenuation coefficient are positive. The piecewise-smoothness constraint (i) can be imposed by using the penalty functional of the following form: J(f)=(1/2)|f|2+β2dV, where the parameter β is an arbitrary small number preventing blow-up of J in the case f = 0. The positivity constraint (ii) is separately imposed to the real and imaginary parts of the refractive index map at every step of iteration. The iteration is terminated when the following convergence criterion is fulfilled:
n(k+1)n(k)2<ε,
(10)
where ϵ is a number that decides the iteration error.

4. Numerical tests of the proposed algorithm

Next, we tested our algorithm using a more realistic phantom. The phantom is a water cylinder containing four cylinders of different materials: polypropylene, Mylar, Teflon, and PMMA (polymethyl methacrylate). The diameter of the water cylinder is 200 mm, and that of the four inner cylinders is 40 mm. The refractive index value of each material is listed in Table 1

Table 1. List of materials and refractive index values (at 30 keV) for the phantom considered in this study.

table-icon
View This Table
[32

32. B. Henke, E. Gullikson, and J. C. Davis, “X-ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables 54(2), 181–342 (1993). [CrossRef]

]. Figures 5(a)
Fig. 5 (a) Original refractive index map of the phantom considered in this study. The cylinder of diameter 200 mm is filled with water; it contains four cylinders (diameter 40 mm) of different materials (polypropylene, Mylar, Teflon, and PMMA). The refractive index values of the materials at the energy used in this study (30 keV) are listed in Table 1. (b) Reconstructed map of the refractive index after 1000 iterations. (c) Reconstruction using the Fourier mapping and direct inversion (without regularization). (d) Mean refractive index value within each cylinder region plotted at each iteration step. The value was normalized by the true value for each material.
and 5(b) show the refractive index map of original phantom and that after 1000 iterations, respectively. Compared to the result without regularization [Fig. 5(c)], Fig. 5(b) clearly shows the boundaries of outer and inner cylinders. Figure 5(d) shows the refractive index value (δ) averaged over each inner cylinder region at different iteration steps. Here, the refractive value was normalized by the true δ value for each material. The four materials have different speeds of convergence, the origin of which is not clear yet. Nonetheless, after 1000 iterations, the δ values of the four materials match with the true δ values within 10% accuracy. Considering that the data were collected only for the small detector movement that corresponds to the angular coverage of (−30° to 30°), this accuracy is remarkable. The accuracy may be further improved by utilizing more information about the specimen [39

39. L. Ritschl, F. Bergner, and M. Kachelrieß, “A new approach to limited angle tomography using the compressed sensing framework,” Proc. SPIE 7622, 76222H, 76222H-9 (2010). [CrossRef]

].

5. Conclusion

In this study, we derived an iterative algorithm for X-ray phase tomography of an object translated across a fan-shaped X-ray beam. The developed algorithm will be useful in a high-throughput X-ray phase tomography system continually interrogating objects on a conveyor belt or using multiple off-axis sources.

Acknowledgments

This work was supported by the Department of Homeland Security’s Science and Technology Directorate through contract HSHQDC-11-C-0083, the National Research Foundation of Singapore through the Singapore-MIT Alliance for Research and Technology Centre, and the DARPA AXiS program (Grant No. N66001-11-4204, P.R. No. 1300217190). The authors thank Prof. George Barbastathis and Dr. Niyom Lue for useful discussion.

References and links

1.

A. Stanton, “Wilhelm Conrad Röntgen on a new kind of rays: translation of a paper read before the Würzburg Physical and Medical Society, 1895,” Nature 53, 274–276 (1896).

2.

C. A. Helms, Fundamentals of Skeletal Radiology (Saunders, 2005).

3.

E. D. Pisano, M. J. Yaffe, and C. M. Kuzmiak, Digital Mammography (Lippincott Williams & Wilkins, 2004).

4.

H. Vogel and D. Haller, “Luggage and shipped goods,” Eur. J. Radiol. 63(2), 242–253 (2007). [CrossRef] [PubMed]

5.

J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data 4(3), 471–538 (1975). [CrossRef]

6.

D. M. Paganin, Coherent X-ray Optics (Oxford University, 2006).

7.

A. Momose, T. Takeda, and Y. Itai, “Blood vessels: depiction at phase-contrast X-ray imaging without contrast agents in the mouse and rat-feasibility study,” Radiology 217(2), 593–596 (2000). [PubMed]

8.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996). [CrossRef]

9.

U. Bonse and F. Beckmann, “Multiple-beam X-ray interferometry for phase-contrast microtomography,” J. Synchrotron Radiat. 8(1), 1–5 (2001). [CrossRef] [PubMed]

10.

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys. 41(Part 2, No. 9A/B), L1016–L1018 (2002). [CrossRef]

11.

Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot. 66(8), 1083–1090 (2008). [CrossRef] [PubMed]

12.

Z. Zaprazny, D. Korytar, V. Ac, P. Konopka, and J. Bielecki, “Phase contrast imaging of lightweight objects using microfocus X-ray source and high resolution CCD camera,” J. Instrum. 7(03), C03005 (2012). [CrossRef]

13.

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(3Suppl), S13–S17 (2008). [CrossRef] [PubMed]

14.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008). [CrossRef] [PubMed]

15.

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with laboratory sources,” Appl. Phys. Lett. 91(7), 074106 (2007). [CrossRef]

16.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of X-ray scattering--initial results,” IEEE Trans. Med. Imaging 27(8), 997–1002 (2008). [CrossRef] [PubMed]

17.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58(1), R1–R35 (2013). [CrossRef] [PubMed]

18.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 1988).

19.

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(6A), 5254–5262 (2006). [CrossRef]

20.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98(10), 108105 (2007). [CrossRef] [PubMed]

21.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11(19), 2289–2302 (2003). [CrossRef] [PubMed]

22.

G. Donges and R. Dietrich, “Baggage inspection system,” U. S. patent 4,759,047 (1988).

23.

R. Dietrich, “Baggage inspection system,” U. S. patent 4,783,794 (1988).

24.

J. M. Park, E. A. Franken Jr, M. Garg, L. L. Fajardo, and L. T. Niklason, “Breast tomosynthesis: present considerations and future applications,” Radiographics 27(Suppl 1), S231–S240 (2007). [CrossRef] [PubMed]

25.

D. Nahamoo, S. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason. 31(4), 218–229 (1984). [CrossRef]

26.

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun. 267(1), 128–136 (2006). [CrossRef]

27.

N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Synthetic aperture tomographic phase microscopy for 3D imaging of live cells in translational motion,” Opt. Express 16(20), 16240–16246 (2008). [CrossRef] [PubMed]

28.

Y. Sung, C. J. R. Sheppard, G. Barbastathis, M. Ando, and R. Gupta, “Full-wave approach for X-ray phase imaging,” Opt. Express 21(15), 17547–17557 (2013). [CrossRef] [PubMed]

29.

A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in X-ray phase-contrast imaging suitable for tomography,” Opt. Express 19(11), 10359–10376 (2011). [CrossRef] [PubMed]

30.

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]

31.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984). [CrossRef]

32.

B. Henke, E. Gullikson, and J. C. Davis, “X-ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables 54(2), 181–342 (1993). [CrossRef]

33.

Y. Sung and G. Barbastathis, “Rytov approximation for x-ray phase imaging,” Opt. Express 21(3), 2674–2682 (2013). [CrossRef] [PubMed]

34.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969). [CrossRef]

35.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009). [CrossRef] [PubMed]

36.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6(2), 298–311 (1997). [CrossRef] [PubMed]

37.

Y. Sung and R. R. Dasari, “Deterministic regularization of three-dimensional optical diffraction tomography,” J. Opt. Soc. Am. A 28(8), 1554–1561 (2011). [CrossRef] [PubMed]

38.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).

39.

L. Ritschl, F. Bergner, and M. Kachelrieß, “A new approach to limited angle tomography using the compressed sensing framework,” Proc. SPIE 7622, 76222H, 76222H-9 (2010). [CrossRef]

OCIS Codes
(100.6950) Image processing : Tomographic image processing
(340.7440) X-ray optics : X-ray imaging
(290.5825) Scattering : Scattering theory

ToC Category:
X-ray Optics

History
Original Manuscript: July 25, 2013
Revised Manuscript: September 2, 2013
Manuscript Accepted: September 13, 2013
Published: September 27, 2013

Citation
Yongjin Sung and Rajiv Gupta, "X-ray phase tomography of a moving object," Opt. Express 21, 23671-23679 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23671


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. A. Stanton, “Wilhelm Conrad Röntgen on a new kind of rays: translation of a paper read before the Würzburg Physical and Medical Society, 1895,” Nature53, 274–276 (1896).
  2. C. A. Helms, Fundamentals of Skeletal Radiology (Saunders, 2005).
  3. E. D. Pisano, M. J. Yaffe, and C. M. Kuzmiak, Digital Mammography (Lippincott Williams & Wilkins, 2004).
  4. H. Vogel and D. Haller, “Luggage and shipped goods,” Eur. J. Radiol.63(2), 242–253 (2007). [CrossRef] [PubMed]
  5. J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data4(3), 471–538 (1975). [CrossRef]
  6. D. M. Paganin, Coherent X-ray Optics (Oxford University, 2006).
  7. A. Momose, T. Takeda, and Y. Itai, “Blood vessels: depiction at phase-contrast X-ray imaging without contrast agents in the mouse and rat-feasibility study,” Radiology217(2), 593–596 (2000). [PubMed]
  8. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996). [CrossRef]
  9. U. Bonse and F. Beckmann, “Multiple-beam X-ray interferometry for phase-contrast microtomography,” J. Synchrotron Radiat.8(1), 1–5 (2001). [CrossRef] [PubMed]
  10. M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys.41(Part 2, No. 9A/B), L1016–L1018 (2002). [CrossRef]
  11. Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot.66(8), 1083–1090 (2008). [CrossRef] [PubMed]
  12. Z. Zaprazny, D. Korytar, V. Ac, P. Konopka, and J. Bielecki, “Phase contrast imaging of lightweight objects using microfocus X-ray source and high resolution CCD camera,” J. Instrum.7(03), C03005 (2012). [CrossRef]
  13. 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(3Suppl), S13–S17 (2008). [CrossRef] [PubMed]
  14. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008). [CrossRef] [PubMed]
  15. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with laboratory sources,” Appl. Phys. Lett.91(7), 074106 (2007). [CrossRef]
  16. H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of X-ray scattering--initial results,” IEEE Trans. Med. Imaging27(8), 997–1002 (2008). [CrossRef] [PubMed]
  17. A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol.58(1), R1–R35 (2013). [CrossRef] [PubMed]
  18. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 1988).
  19. 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(6A), 5254–5262 (2006). [CrossRef]
  20. F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett.98(10), 108105 (2007). [CrossRef] [PubMed]
  21. S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express11(19), 2289–2302 (2003). [CrossRef] [PubMed]
  22. G. Donges and R. Dietrich, “Baggage inspection system,” U. S. patent 4,759,047 (1988).
  23. R. Dietrich, “Baggage inspection system,” U. S. patent 4,783,794 (1988).
  24. J. M. Park, E. A. Franken, M. Garg, L. L. Fajardo, and L. T. Niklason, “Breast tomosynthesis: present considerations and future applications,” Radiographics27(Suppl 1), S231–S240 (2007). [CrossRef] [PubMed]
  25. D. Nahamoo, S. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason.31(4), 218–229 (1984). [CrossRef]
  26. T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun.267(1), 128–136 (2006). [CrossRef]
  27. N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Synthetic aperture tomographic phase microscopy for 3D imaging of live cells in translational motion,” Opt. Express16(20), 16240–16246 (2008). [CrossRef] [PubMed]
  28. Y. Sung, C. J. R. Sheppard, G. Barbastathis, M. Ando, and R. Gupta, “Full-wave approach for X-ray phase imaging,” Opt. Express21(15), 17547–17557 (2013). [CrossRef] [PubMed]
  29. A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in X-ray phase-contrast imaging suitable for tomography,” Opt. Express19(11), 10359–10376 (2011). [CrossRef] [PubMed]
  30. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am.73(11), 1434–1441 (1983). [CrossRef]
  31. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49(1), 6–10 (1984). [CrossRef]
  32. B. Henke, E. Gullikson, and J. C. Davis, “X-ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables54(2), 181–342 (1993). [CrossRef]
  33. Y. Sung and G. Barbastathis, “Rytov approximation for x-ray phase imaging,” Opt. Express21(3), 2674–2682 (2013). [CrossRef] [PubMed]
  34. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun.1(4), 153–156 (1969). [CrossRef]
  35. Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express17(1), 266–277 (2009). [CrossRef] [PubMed]
  36. P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process.6(2), 298–311 (1997). [CrossRef] [PubMed]
  37. Y. Sung and R. R. Dasari, “Deterministic regularization of three-dimensional optical diffraction tomography,” J. Opt. Soc. Am. A28(8), 1554–1561 (2011). [CrossRef] [PubMed]
  38. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).
  39. L. Ritschl, F. Bergner, and M. Kachelrieß, “A new approach to limited angle tomography using the compressed sensing framework,” Proc. SPIE7622, 76222H, 76222H-9 (2010). [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.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited