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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 5, Iss. 9 — Jul. 6, 2010
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Large phase-stepping approach for high-resolution hard X-ray grating-based multiple-information imaging

Zhifeng Huang, Zhiqiang Chen, Li Zhang, Kejun Kang, Fei Ding, Zhentian Wang, and Haozhi Ma  »View Author Affiliations


Optics Express, Vol. 18, Issue 10, pp. 10222-10229 (2010)
http://dx.doi.org/10.1364/OE.18.010222


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Abstract

High-resolution hard X-ray grating-based imaging method with conventional X-ray sources provides attenuation, refraction and scattering information synchronously, and it is regarded as the next-generation X-ray imaging technology for medical and industrial applications. In this letter, a large phase-stepping approach with at least one order of magnitude lower resolution of the movement is presented to equivalently substitute the current high-positioning-resolution phase-stepping approach. Both the theoretical deduction and actual experiment prove that the new approach is available to relax the requirement of high positioning resolution and strict circumstances so as to benefit the future commercial applications of the grating-based multiple-information imaging technology.

© 2010 OSA

1. Introduction

High-resolution hard X-ray grating-based multiple-information imaging technology is capable of providing attenuation, refraction (i.e., phase-contrast) and scattering (i.e., dark-field) information synchronously by a phase-stepping approach, and reconstructing the distributions of the linear attenuation coefficient, refractive index gradient and generalized scattering parameter of a tested sample in one computed tomographic scanning process, so it has been the most promising tool for clinical diagnosis, material examination, etc..

Above methods mostly adopted the phase-stepping (or phase-shifting) approach by moving either of the last two gratings relatively step by step over one grating period to retrieve multiple information by measuring intensity oscillation curves of each pixel on the detector. Nesterets and Wilkins presented another type of the phase-stepping approach by moving both gratings together [15

15. Y. I. Nesterets and S. W. Wilkins, “Phase-contrast imaging using a scanning-double-grating configuration,” Opt. Express 16(8), 5849–5867 (2008). [CrossRef] [PubMed]

]. Both gratings have very small periods, for example, 2 or 4 um in Refs [10

10. 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(4), 258–261 (2006). [CrossRef]

,11

11. 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]

], so the step length in the phase-stepping approach is usually less than 1 um. That is to say, the relative movement of two gratings requires very high positioning resolution on the order of sub-micrometers. So in the current phase-stepping approach, the positioning resolution and correlative circumstances (i.e., temperature and pressure, etc.) for the grating-based imaging system should have a very high level of stability. The requirements maybe mostly limit actual applications of the X-ray grating-based imaging technology. In this paper, we present an equivalent phase-stepping approach by moving the source grating with lower positioning resolution to solve this problem.

2. Theory

The X-ray grating-based multiple-information imaging method based on either Talbot-Lau interferometry or classic optics theory can be uniformly illustrated by Fig. 1
Fig. 1 The schematic diagram of the X-ray grating-based multiple-information imaging method.
. It consists of a source grating (G0) and two gratings (G1, G2). In general, the source grating has a large period on the order of tens of micrometers, even over one hundred micrometers, while the last two gratings have very small periods on the order of several micrometers. In the large phase-stepping approach, we fix the last two gratings and only relatively move the source grating over one grating period along the x-axis to achieve the measurement of the intensity oscillation curve of each pixel on the detector.

Suppose p0,p1 and p2 are the periods of G0, G1 and G2, respectively. L is the distance between G0 and G1. D is the distance between G1 and G2. Their relationship can be expressed by
p0p2=LD,
(1)
p1p2=LL+D
(2)
in classic optics theory [12

12. Z. Huang, K. Kang, L. Zhang, Z. Chen, F. Ding, Z. Wang, and Q. Fang, “Alternative method for differential phase-contrast imaging with weakly coherent hard x rays,” Phys. Rev. A 79(1), 013815 (2009). [CrossRef]

],
p2p1/2=LLDT
(3)
in Talbot–Lau interferometry [16

16. M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. 232(1), 145–157 (2008). [CrossRef] [PubMed]

], where DT is the Talbot distance.

Without loss of generality, the imaging system is supposed to be under the following ideal conditions in the x-z plane in order to simplify the computational complexity:

1) Suppose that the distance between G0 and the X-ray source is zero. The distance between G2 and the detector is zero. The detective efficiency of the detector is 100%. The influence of the noises in the whole imaging system doesn’t be considered.

2) Suppose that the sizes of three gratings along the x-axis are infinite. The transmission functions of G0, G1, and G2 can be expressed by T0(x), T1(x) and T2(x), respectively, that is,
T0(x)=nfne2πinp0x,
(4)
T1(x)=nane2πinp1x,
(5)
T2(x)=ntne2πinp2x,
(6)
where an, tn and fn denote the nth order Fourier coefficients.

Let the intensity distribution function of the X-ray source be S0(x), then the intensity distribution function of the X-ray downstream after G0, ST(x), is expressed by
ST(x)=S0(x)T0(x).
(7)
It can be regarded as an array of sub-sources from the source grating.

Consider the intensity distribution functions of the X-ray upstream before G2 in two cases: Is(x) without the sample and Js(x) with the sample present in the beam path, respectively,:
Is(x)=I1(x)S(x),
(8)
Js(x)=J1(x)S(x),
(9)
where I1(x) denotes the projection image or self-image of G1 illuminated by a point source from the position z0 to the position z2, J1(x) denotes the projection image or self-image of both the sample and G1 illuminated by a point source from the position z0 to the position z2.S(x) denotes the geometrical projection of ST(x) in the position z2, that is,
I1(x)=T1(LL+Dx),
(10)
J1(x)=T1'(LL+Dx),
(11)
S(x)=ST(LDx),
(12)
where T1'(x) denotes the distribution function of the X-rays pass through both the sample and G1 in the position z1. Note that I1(x), J1(x) and S(x) have the same period as T2(x)in the position z2, so p is utilized to denote their periods for convenience, then Eqs. (10)-(12) can be rewritten to
I1(x)=T1(LL+Dx)=nane2πinpx,
(13)
J1(x)=T1'(LL+Dx)=nan'e2πinp[x+Dϕ(x)],
(14)
S(x)=ST(LDx)=S0'(x)T0'(x)=S0(LDx)T0(LDx),
(15)
where S0'(x)=S0(LDx), T0'(x)=T0(LDx)=nfne2πinpx, ϕ(x) denotes the beam deflection angle at the sample caused by refraction and an' contains the attenuation effect of the sample. To simplify the computation, Eq. (15) can also be approximated to
S(x)=S0'(x)T0'(x)=S0'(x)[nfne2πinpx]nfn'e2πinpx,
(16)
where fn' contains the influence of the distribution of the X-ray source.

Finally, the X-ray intensities are captured by the detector in two cases: ID(x) without the sample and JD(x) with the sample present, respectively, that is,

ID(x)=Is(x)T2(x),
(17)
JD(x)=Js(x)T2(x).
(18)

In the large phase-stepping approach, G0 is relatively moved along the x-axis and the last two gratings G1 and G2 are fixed, then the intensity oscillation curve functions can be measured in two cases: ID(x,χ) without the sample and JD(x,χ) with the sample present, respectively, that is,
ID(x,χ)=Is(x,χ)T2(x)=[I1(x)S(x+χ)]T2(x)=[I1(x)(S0'(x)T0'(x+χ))]T2(x)=[(nane2πinpx)(S0'(x)mfme2πimp(x+χ))](ktke2πikpx)nanfn'tne2πinpχ,
(19)
JD(x,χ)=Js(x,χ)T2(x)=[J1(x)S(x+χ)]T2(x)=[J1(x)(S0'(x)T0'(x+χ))]T2(x)=[(nan'e2πinp[x+Dϕ(x)])(S0'(x)mfme2πimp(x+χ))](ktke2πikpx)nan'fn'tne2πinp[χ+Dϕ(x)],
(20)
where χ is the displacement of G0.

In the current high-positioning-resolution phase-stepping approach, G0 is fixed and either of the last two gratings G1 or G2 is relatively moved along the x-axis, then the intensity oscillation curve functions can be measured in two cases: ID'(x,χ) without the sample and JD'(x,χ) with the sample [17

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(6A6A), 5254–5262 (2006). [CrossRef]

20

20. Z. Wang, K. Kang, Z. Huang, and Z. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95(9), 094105 (2009). [CrossRef]

], respectively, that is,

ID'(x,χ)=Is(x)T2(x,χ)=[I1(x)S(x)]T2(x+χ)[(nane2πinpx)(mfm'e2πimpx)](ktke2πikp(x+χ))nanfn'tne2πinpχ,
(21)
JD'(x,χ)=Js(x)T2(x,χ)=[J1(x)S(x)]T2(x+χ)[(nan'e2πinp[x+Dϕ(x)])(mfm'e2πimpx)](ktke2πikp(x+χ))nan'fn'tne2πinp[χ+Dϕ(x)],
(22)

Note that Eq. (19) is equivalent to Eq. (21) and Eq. (20) is equivalent to Eq. (22) under the ideal conditions. It means that the large phase-stepping approach by moving the source grating has almost the same intensity oscillation curve functions as the current high-positioning-resolution phase-stepping approach by moving either of the last two gratings. The similar conclusion can also be drawn under actual conditions (for example, the finite sizes of the gratings, the systemic noises, the actual detective efficiency of the detector, unideal grating fabrication, etc), where actual intensity oscillation curve functions are approximately sinusoidal functions [7

7. A. Momose, S. Kawamoto, I. Koyama, and Y. Suzuki, “Phase tomography using an x-ray Talbot interferometer,” Proc. SPIE 5535, 352–360 (2004). [CrossRef]

,8

8. T. Weitkamp, A. Diaz, B. Nohammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137–142 (2004). [CrossRef]

,12

12. Z. Huang, K. Kang, L. Zhang, Z. Chen, F. Ding, Z. Wang, and Q. Fang, “Alternative method for differential phase-contrast imaging with weakly coherent hard x rays,” Phys. Rev. A 79(1), 013815 (2009). [CrossRef]

].

3. Experiment

An actual experiment on a phantom was carried out at the grating-based multiple-information imaging experimental setup at Tsinghua University in order to validate above deduction. The detailed description of the system can be found in Ref. 8

8. T. Weitkamp, A. Diaz, B. Nohammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137–142 (2004). [CrossRef]

and 9

9. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005). [CrossRef] [PubMed]

. In the experiment, the spot size of the X-ray tube was measured about 600 umwhen the tube was set around 30kV. The periods of G0, G1 and G2 were 110, 10 and 11 um, respectively, The distance between G0 and G1 was 1500 mm and the distance between G1 and G2 was 150 mm. The phantom consisted of two cylinders: the outer cylinder made of polymethylmethacrylate (PMMA) and the inner cylinder made of plastic. Both new and current phase-stepping approaches adopted 11 steps, while their step lengths were 10 um and 1 um, respectively. Their corresponding multiple images (that is, attenuation, refraction and scattering images) are shown in Fig. 2
Fig. 2 The comparisons of experimental results by use of two phase-stepping approaches. (a), (d) and (g) are the attenuation, refraction and scattering images, respectively, retrieved by the current high-positioning-resolution phase-stepping approach (Approach I). (b), (e) and (h) are the attenuation, refraction and scattering images, respectively, retrieved by the large phase-stepping approach (Approach II). (c) is the comparison of profile values in two attenuation images. (f) is the comparison of profile values in two refraction images. (i) is the comparison of profile values in two scattering images.
. The profile values of each image are plotted to compare the differences between the new and current phase-stepping approaches. It is found that their results are almost the same on the whole, though they aren’t identical completely because of the influences of systemic noises.

4. Conclusion

In conclusion, both the theoretical deduction and actual experiments prove that the large phase-stepping approach is equivalent to the current high-positioning-resolution phase-stepping approach. The period of the source grating is always much larger than the periods of last two gratings, so the large phase-stepping approach requires at least one order of magnitude lower positioning resolution than the current one. The large phase-stepping approach is applicable to both Talbot-Lau interferometric-based and classic-optics-based grating-based multiple-information imaging methods. It would relax the requirements of high positioning resolution and strict circumstances for future commercial applications of the X-ray grating-based multiple-information imaging technology in practice.

Acknowledgments

The work was supported by a grant from the National Natural Science Foundation of China (NNSFC) grants 10905031 and 10875066 and the Specialized Research Fund for the Doctoral Program of Higher Education grant 20090002120016.

References and links

1.

A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2(9), 413–415 (1971). [CrossRef]

2.

S. Yokozeki and T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10(7), 1575–1580 (1971). [CrossRef] [PubMed]

3.

O. Kafri, “Noncoherent method for mapping phase objects,” Opt. Lett. 5(12), 555–557 (1980). [CrossRef] [PubMed]

4.

J. F. Clauser and S. Li, ““Heisenberg microscope” decoherence atom interferometry,” Phys. Rev. A 50(3), 2430–2433 (1994). [CrossRef] [PubMed]

5.

C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002). [CrossRef]

6.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003). [CrossRef]

7.

A. Momose, S. Kawamoto, I. Koyama, and Y. Suzuki, “Phase tomography using an x-ray Talbot interferometer,” Proc. SPIE 5535, 352–360 (2004). [CrossRef]

8.

T. Weitkamp, A. Diaz, B. Nohammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137–142 (2004). [CrossRef]

9.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005). [CrossRef] [PubMed]

10.

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(4), 258–261 (2006). [CrossRef]

11.

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]

12.

Z. Huang, K. Kang, L. Zhang, Z. Chen, F. Ding, Z. Wang, and Q. Fang, “Alternative method for differential phase-contrast imaging with weakly coherent hard x rays,” Phys. Rev. A 79(1), 013815 (2009). [CrossRef]

13.

Z.Huang, Z.Wang, L.Zhang, Z.Chen, K.Kang, “Attenuation-Refraction-Scattering Computed Tomographic Experimental System with a Conventional X-Ray Tube: System Optimization & Image Fusion,” IEEE NSS/MIC Record, 2347–2350 (2009)

14.

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106(5), 054703 (2009). [CrossRef]

15.

Y. I. Nesterets and S. W. Wilkins, “Phase-contrast imaging using a scanning-double-grating configuration,” Opt. Express 16(8), 5849–5867 (2008). [CrossRef] [PubMed]

16.

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. 232(1), 145–157 (2008). [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(6A6A), 5254–5262 (2006). [CrossRef]

18.

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006). [CrossRef]

19.

W. Yashiro, Y. Takeda, and A. Momose, “Efficiency of capturing a phase image using cone-beam x-ray Talbot interferometry,” J. Opt. Soc. Am. A 25(8), 2025–2039 (2008). [CrossRef]

20.

Z. Wang, K. Kang, Z. Huang, and Z. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95(9), 094105 (2009). [CrossRef]

OCIS Codes
(110.7440) Imaging systems : X-ray imaging
(120.4820) Instrumentation, measurement, and metrology : Optical systems

ToC Category:
X-ray Optics

History
Original Manuscript: March 16, 2010
Revised Manuscript: April 21, 2010
Manuscript Accepted: April 21, 2010
Published: April 30, 2010

Virtual Issues
Vol. 5, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Zhifeng Huang, Zhiqiang Chen, Li Zhang, Kejun Kang, Fei Ding, Zhentian Wang, and Haozhi Ma, "Large phase-stepping approach for high-resolution hard X-ray grating-based multiple-information imaging," Opt. Express 18, 10222-10229 (2010)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-10-10222


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References

  1. A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2(9), 413–415 (1971). [CrossRef]
  2. S. Yokozeki and T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10(7), 1575–1580 (1971). [CrossRef] [PubMed]
  3. O. Kafri, “Noncoherent method for mapping phase objects,” Opt. Lett. 5(12), 555–557 (1980). [CrossRef] [PubMed]
  4. J. F. Clauser and S. Li, ““Heisenberg microscope” decoherence atom interferometry,” Phys. Rev. A 50(3), 2430–2433 (1994). [CrossRef] [PubMed]
  5. C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002). [CrossRef]
  6. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003). [CrossRef]
  7. A. Momose, S. Kawamoto, I. Koyama, and Y. Suzuki, “Phase tomography using an x-ray Talbot interferometer,” Proc. SPIE 5535, 352–360 (2004). [CrossRef]
  8. T. Weitkamp, A. Diaz, B. Nohammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137–142 (2004). [CrossRef]
  9. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005). [CrossRef] [PubMed]
  10. 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(4), 258–261 (2006). [CrossRef]
  11. 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]
  12. Z. Huang, K. Kang, L. Zhang, Z. Chen, F. Ding, Z. Wang, and Q. Fang, “Alternative method for differential phase-contrast imaging with weakly coherent hard x rays,” Phys. Rev. A 79(1), 013815 (2009). [CrossRef]
  13. Z.Huang, Z.Wang, L.Zhang, Z.Chen, K.Kang, “Attenuation-Refraction-Scattering Computed Tomographic Experimental System with a Conventional X-Ray Tube: System Optimization & Image Fusion,” IEEE NSS/MIC Record, 2347–2350 (2009)
  14. T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106(5), 054703 (2009). [CrossRef]
  15. Y. I. Nesterets and S. W. Wilkins, “Phase-contrast imaging using a scanning-double-grating configuration,” Opt. Express 16(8), 5849–5867 (2008). [CrossRef] [PubMed]
  16. M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. 232(1), 145–157 (2008). [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(6A6A), 5254–5262 (2006). [CrossRef]
  18. A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006). [CrossRef]
  19. W. Yashiro, Y. Takeda, and A. Momose, “Efficiency of capturing a phase image using cone-beam x-ray Talbot interferometry,” J. Opt. Soc. Am. A 25(8), 2025–2039 (2008). [CrossRef]
  20. Z. Wang, K. Kang, Z. Huang, and Z. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95(9), 094105 (2009). [CrossRef]

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