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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 3339–3346
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Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval

Hidenosuke Itoh, Kentaro Nagai, Genta Sato, Kimiaki Yamaguchi, Takashi Nakamura, Takeshi Kondoh, Chidane Ouchi, Takayuki Teshima, Yutaka Setomoto, and Toru Den  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 3339-3346 (2011)
http://dx.doi.org/10.1364/OE.19.003339


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Abstract

We demonstrate a single shot two-dimensional grating-based X-ray phase-contrast imaging using a synchrotron radiation source. A checkerboard designed phase grating for π phase modulation at 17 keV and 35 keV, and a lattice-shaped amplitude grating with a high aspect ratio to shield X-rays up to 35 keV were fabricated. A Fourier analysis of Moiré fringe generated by the gratings was introduced to obtain the two-dimensional differential phase-contrast image with a single exposure. The results show that soft tissues and cartilages of a chicken wing sample are clearly seen with differential phase variation in two-dimensional directions. Using this method not only the whole of an object but also only an inner part of the object can be imaged.

© 2011 OSA

1. Introduction

Conventional X-ray imaging has been widely used for medical imaging and non-destructive imaging. Since Röntgen found X-ray in 1895, X-ray imaging has been developed based on the absorption contrast. However, due to a large amount of the dose and the difficulty of imaging soft tissues, X-ray imaging is a limited tool for medical imaging. Since 1990s, the studies of X-ray phase-contrast imaging have been active using synchrotron radiation sources. The Bonse-Hart interferometer [1

1. U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965). [CrossRef]

] using triple silicon single-crystal plates is able to image the phase of an object, but it requires parallel and monochromatic X-rays. The diffraction enhanced imaging (DEI) [2

2. V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D Appl. Phys. 28(11), 2314–2317 (1995). [CrossRef]

4

4. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997). [CrossRef] [PubMed]

] is used to obtain the differential phase-contrast image using double or triple silicon single-crystal monochromators, and as a consequence the X-ray photons are lost by the monochromators. On the other hand, the propagation-based imaging [5

5. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995). [CrossRef]

-6

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

] is a simple method that X-ray detector is kept as far away from an object as possible, but the obtained image is mainly governed by the absorption contrast with an edge enhancement. On the contrary to these three methods, the grating-based interferometer using the Talbot effect [7

7. H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).

] is one of the promising methods to achieve a novel X-ray medical imaging modality using a conventional X-ray tube.

Momose’s [8

8. 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]

-9

9. 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(No. 6A), 5254–5262 (2006). [CrossRef]

] and Pfeiffer’s [10

10. 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]

12

12. 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]

] groups have studied the grating-based X-ray phase-contrast imaging using a phase stepping technique. Their imaging method has an advantage of a high spatial resolution, but a disadvantage of requirement of multiple exposures at least three images. Recently, another phase retrieval approach was introduced to X-ray phase-contrast imaging by Momose et al. [13

13. A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009). [CrossRef] [PubMed]

] to obtain the time resolved phase-contrast images using a white beam of synchrotron radiation source. They applied the Fourier transform phase retrieval for their study, but only one-dimensional differential phase-contrast image was obtained. On the other hand, Kottler et al. [14

14. C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard X-rays,” Opt. Express 15(3), 1175–1181 (2007). [CrossRef] [PubMed]

] have retrieved two-dimensional phase-contrast image via an algorism to combine x- and y-directional differential phase-contrast images using one-dimensional gratings. In their imaging method, however, a double number of exposures are required to obtain two-dimensional phase information due to the use of one-dimensional gratings with rotation. As a consequence, a multiple shot imaging increases the radiation dose and is not resistant to the movement of the object for medical applications. Recently, differential phase-contrast images were obtained by the refraction effect of X-rays using two-dimensional transmission gratings, a stack of two Bucky grids, by Wen et al. [15

15. H. H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission gratings,” Opt. Lett. 35(12), 1932–1934 (2010). [CrossRef] [PubMed]

]. A focal spot size of X-ray source is expected to be limited due to the separation of absorption and phase components in the image.

In this study we demonstrate a single shot two-dimensional grating-based X-ray phase-contrast imaging using a synchrotron radiation source. A checkerboard designed phase grating for π phase modulation and a lattice-shaped amplitude grating were positioned with a distance of the first Talbot position. Differential phase-contrast image was obtained by the Fourier analysis of Moiré fringe generated by the gratings on X-ray detector. The present imaging method includes obtaining the phase-contrast image and the absorption image in two-dimensional directions with a single exposure.

2. Principles

Figure 1
Fig. 1 Experimental setup of two-dimensional grating-based (Talbot) interferometer. Z denotes the Talbot length which is expressed by the pitch of phase grating (p) and X-ray wavelength (λ). An amplitude grating is positioned close to X-ray detector to generate Moiré fringe.
shows our experimental set up. It consists of a phase grating, an amplitude grating, and an X-ray detector. The phase grating has a structure of checkerboard with the height equivalent to π phase modulation at the set X-ray energy. Silicon is chosen for the phase grating because of the fabrication properties and the high transparency of X-rays. The lattice-shaped amplitude grating is used to shield X-rays partially and generate Moiré fringe. The amplitude grating is a fabricated silicon structure with filling of gold in order to shield X-rays. The amplitude grating is positioned along X-ray beams with a distance from the phase grating, called Talbot distance defined by the period of the phase grating and X-ray wavelength.

The principle of the present imaging method is the following. The X-rays are modulated with π phase by the phase grating and generate a self-image at the position of Talbot distance where the amplitude grating is arranged. Following the Talbot effect the self-image is created by the interference of diffracted X-rays from the phase grating. The intensity of the self-image generated at the position (x, y) of the wavefront and the distance from the phase grating, z, can be written as

ISelf(x,y,z)|nx,nyψnxnyexp[i2πλz]exp[iπzλ(nx2+ny2)pG12]exp[i2π(nxx+nyy)pG1]×exp[iΦ(xzλnxpG1,yzλnypG1)]|2.
(1)

Here, ψ is a Fourier series expansion coefficient of the period of phase grating, nx and ny are the counting number, PG1 is the period of phase grating, Φ is the phase of object, and λ is the wavelength of X-ray.

The intensity of Moiré fringe generated by the amplitude grating at the position of (x, y, z) is written by
IMoire(x,y,z)=ISelf(x,y,z)×TAmp(x,y),
(2)
a(x,y)+mx,mybmx,my(x,y){exp(i2πmxxpMoireφxmx(x,y,z))×exp(i2πmyypMoireφymy(x,y,z)),
(3)
φxmx(x,y,z)=zλmxpG1Φ(x,y)x,
(4)
φymy(x,y,z)=zλmypG1Φ(x,y)y,
(5)
where TAmp is an absorption coefficient of amplitude grating, a is an absorption component, b is a Fourier series expansion coefficient of Moiré fringe, pMoire is the period of Moiré fringe, and ϕx and ϕy are the phase variation caused by an object.

It is necessary for the grating-based X-ray phase-contrast imaging that the self-image transforms to Moiré fringe by introducing the amplitude grating due to a low spatial resolution of conventional X-ray detectors compared to the self-image. Two-dimensional Moiré fringe is generated by the combination of the self-image and the slightly rotated amplitude grating with the same period of the self-image. The wavefront of X-rays through an object placed upstream of the phase grating is deformed by the phase shift of the object. This phase shift can be detected by the change of Moiré fringe on the X-ray detector.

The analysis of the Moiré fringe is carried out using Fourier transform method based on the paper by Takeda et al. [16

16. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computed-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

] shown in Fig. 2
Fig. 2 Fourier analysis of Moiré fringe. A simple sphere phantom is used in this simulation. The differential phase image is calculated via the inverse Fourier transform of the first Fourier spectrum of Moiré fringe.
.

The Fourier spectra are obtained by the Fourier transform of Moiré fringe. The spectrum at the center (the 0th spectrum) is attributed to an absorption component of X-rays, and the carrier frequency spectrum (the 1st spectrum) includes the phase information. To retrieve the phase information we compute the inverse Fourier transform of the carrier frequency spectrum with the surrounding region. The differential phase-contrast images with horizontal and vertical directions are obtained by the above-mentioned phase retrieval method. The Fourier transform of the intensity of Moiré fringe is written by
F[IMoire(x,y,z)]=A˜(kx,ky)+mx,my{B˜mxmy(kx,ky)δ(kx2πmxpMoire)δ(ky2πmypMoire)},
(6)
=A˜(kx,ky)+mx,my{B˜mx,my(kx2πmxpMoire,ky2πmypMoire)},
(7)
A˜(kx,ky)=F[a(x,y)],
(8)
B˜mx,my(kx,ky)=F[bmx,my(x,y)exp[iφxmx(x,y,z)iφymy(x,y,z)]],
(9)
Φ(x,y)x=pG1zλarg[F1[B˜1,0(kx,ky)]],
(10)
Φ(x,y)y=pG1zλarg[F1[B˜0,1(kx,ky)]],
(11)
where, kx and ky are the wave number, mx and my are the counting number, ⊗ is a convolution operator.

The technique has some advantages. This method uniquely determines the phase integrated from the differential phase-contrast images, whereas the phase stepping approach requires the origin of the integration. The present imaging method is able to obtain the phase-contrast image of an object bigger than the field of view, whereas the phase stepping method requires a space out of the object.

3. Experimental details

The experiments were performed with monochromatic X-rays of 17.5 and 35 keV at BL-20B2 of SPring-8, Japan. It is a nearly parallel X-ray beam reaching at the distance of 206 meters from the radiation-emitted point. We used an X-ray beam with a dimension of 16 mm by 24 mm shaped from the original beam size 50 mm by 300 mm using a double slits system. The X-rays were monochromatized by double single crystals of silicon (111).

The phase gratings were fabricated by a process including photolithography, and deep etching into silicon. The checkerboard designed phase grating [Fig. 3(a)
Fig. 3 SEM image of the cross-section of the checkerboard designed phase grating (a) with π modulation at 35 keV with the pitch of 11.4 μm and the depth 46 μm. The phase grating was fabricated by deep etching into silicon substrate. SEM image of the cross-section (b) and the expanded image of the region surrounded by the white line (c) of lattice-shaped amplitude grating with the pitch of 8.24 μm and the depth 100 μm. The amplitude grating was fabricated from silicon-based structure with gold electroplating. The amplitude grating is expected to shield about 97% of X-rays at 35 keV.
] with the period of 11.4 μm and the height of 23 μm for π phase modulation was prepared for the experiment with X-ray of 17.5 keV. The phase grating with the height of 46 μm was fabricated for the 35 keV use. The filling of gold by the electroplating was added to the fabrication process of the phase grating to fabricate the amplitude grating. The lattice-shaped amplitude grating with the period of 8.24 μm and the height of 100 μm was successfully fabricated as shown in Fig. 3(b) and (c). Many silicon pillars, whose cross section is a square 4μm on a side, are two-dimensionally arranged in the amplitude grating. The height of the gold is able to block X-rays up to 47 keV with a shielding rate of 80%.

The distance between the phase grating and the amplitude grating was set to 459 mm for 17.5 keV, and 921 mm for 35 keV corresponding to the 1st Talbot distance, respectively. The images were recorded with a fluorescent-screen lens-coupling system, Hamamatsu C4742-95HR combined with AA60 lens system. X-rays passing through the object are transformed into a visible light by the fluorescent screen. The images on the screen are read by a cooled CCD camera with a high numerical aperture lens. The module consists of an array of 4000 (H) × 2624 (V) pixels with a pixel size of 6.0 µm. We used this detector with binned pixels and the pixel size is 12 µm.

4. Results

The raw image of a Teflon rod and a polyamide rod was captured at 17.5 keV with an exposure time of 1.5 seconds. The differential phase-contrast image [Fig. 4(b)
Fig. 4 Comparison of differential phase-contrast image in x-direction of a Teflon rod (left) and a polyamide rod (right) with the absorption image taken at 17.5 keV. A double layered structure of the polyamide rod can be identified in the differential phase-contrast image (b), whereas the absorption image has a very low contrast (a). The dashed line in (b) represents the edge of the inner part of the double layered structure of the Teflon rod. The line profile of the differential phase of the polyamide rod (c) is obtained following the solid line in (b). The differential phase resolution about 0.05 rad/μm at the boundary of a double layered structure can be resolved by the present imaging system.
] was calculated from the inverse Fourier transform of the 1st Fourier spectrum of the raw image. Compared to the absorption image [Fig. 4(a)] retrieved from the 0th Fourier spectrum, the polyamide rod having a very low X-ray absorption can be clearly seen in the differential phase-contrast image [Fig. 4(b)]. We can also identify a double layered structure of the polyamide rod from Fig. 4(b). The polyamide rod is filled with the material and the boundary of the double layered structure in the rod is invisible. The line profile of the polyamide rod shows the double layered structure more clearly with double gaps equivalent to 0.05 rad/μm differential phase variation as shown in Fig. 4(c).

A chicken wing was chosen as a biological sample with a simple reason of easy handling. The field of view was limited to 12 mm by 12mm due to the active area of the amplitude grating. The chicken wing was fixed on an acrylic plate by polyimide tapes without any wrapping. The absorption image was taken with the sample as close as possible to the detector, that is called contact image. Figure 5
Fig. 5 Comparison of differential phase-contrast image in x-direction of a joint part of a chicken wing (a) with the contact image (b) taken at 17.5 keV.
shows the differential phase-contrast image and the contact image of a joint part of the chicken wing sample at 17.5 keV with an exposure time of 1.5 seconds. Some soft tissues between bones can be clearly seen in the differential phase-contrast image [Fig. 5(a)] compared with the contact image [Fig. 5(b)].

The present imaging method has an advantage in the scene of an object bigger than the field of view such as the image shown in Fig. 5. The grating-based X-ray phase-contrast imaging using one-dimensional gratings requires a space out of an object for the starting phase value in integration. The fringe analysis method used in the present study does not need any space out of an object, i.e. the phase-contrast image of an object filled in the field of view can be retrieved.

Cartilage is one of the most difficult parts of human bodies to be imaged in conventionalX-ray imaging. A cartilage of chicken wing can be seen in the differential phase-contrast image shown in Fig. 6
Fig. 6 Differential phase-contrast image in x-direction of a joint part of a chicken wing taken at 17.5 keV (a) and 35 keV (b). The inset figures represent the cartilage around the edge of the bone.
, whereas the contact image does not show any profile. The cartilage is more clearly identified at 17.5 keV [Fig. 6(a)] with comparison of the image at 35 keV [Fig. 6(b)]. It is considered that this is mainly caused by the difference of refractive index calculated from the composition of cartilage. The calculated refractive index of the cartilage is 7.98 x 10−6 at 17.5 keV and 2.04 x 10−6 at 35 keV. From these refractive indices the phase shift of cartilage is 0.716 rad/μm at 17.5 keV, 0.362 rad/μm at 35 keV, respectively.

A full image of the chicken wing sample was obtained from 99 individual differential phase-contrast images using an image stitching technology shown in Fig. 7(a)
Fig. 7 Full image of a chicken wing stitched from 99 images in size of 15mm by 15mm taken at 35 keV. Differential phase-contrast image in x-direction (a) and the square-root of sum of squares of the differential phase-contrast image in x- and y-directions (b).
. The detailed texture of bones and soft tissues were imaged with differential phase variation. Another way of showing the differential phase-contrast image was also examined. The image shown in Fig. 7(b) has information of two-dimensional differential phase of the sample. The square-root of sum of squares of the differential phase-contrast image in x- and y-directions was calculated. The texture of the chicken wing sample was clearly observed with an edge enhancement. This is one of the most important features of our two-dimensional grating-based X-ray phase-contrast imaging. In addition, the present imaging method can avoid the artifact and the phase mismatch that may be happened if the misalignment of the gratings or the movement of an object in one-dimensional phase stepping approach with a double exposures.

5. Conclusion

We have demonstrated a single shot two-dimensional grating-based X-ray phase-contrast imaging using a synchrotron radiation source. We used a checkerboard designed phase grating and a lattice-shaped amplitude grating in Talbot interferometer to obtain two-dimensional phase information of the object. The Fourier analysis of Moiré fringe formed by the gratings was introduced to obtain the differential phase-contrast images with a single exposure. The present imaging method has advantages of reducing exposure times and retrieving the phase-contrast image of an object bigger than the field of view. Some soft tissues and cartilages of a chicken wing can be clearly imaged with differential phase variation in two-dimensional directions. This demonstration results show a high potential application of the present imaging system for medical imaging.

Acknowledgments

The authors gratefully acknowledge Mr. Masanobu Hasegawa and Mr. Naoki Kohara (both Canon Inc.) for helpful discussions, and Dr. Peter Fletcher and Dr. Stephen Hardy for image processing (both Canon Information Systems Research Australia), and Mr. Kentaro Uesugi and Dr. Masato Hoshino (both SPring-8) for technical supports. The synchrotron radiation experiments were performed at the BL-20B2 in the SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2009A1952).

References and links

1.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965). [CrossRef]

2.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D Appl. Phys. 28(11), 2314–2317 (1995). [CrossRef]

3.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995). [CrossRef]

4.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997). [CrossRef] [PubMed]

5.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995). [CrossRef]

6.

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

7.

H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).

8.

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]

9.

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(No. 6A), 5254–5262 (2006). [CrossRef]

10.

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]

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(4), 258–261 (2006). [CrossRef]

12.

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]

13.

A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009). [CrossRef] [PubMed]

14.

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard X-rays,” Opt. Express 15(3), 1175–1181 (2007). [CrossRef] [PubMed]

15.

H. H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission gratings,” Opt. Lett. 35(12), 1932–1934 (2010). [CrossRef] [PubMed]

16.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computed-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(100.5070) Image processing : Phase retrieval
(340.7440) X-ray optics : X-ray imaging
(340.7450) X-ray optics : X-ray interferometry

ToC Category:
X-ray Optics

History
Original Manuscript: December 10, 2010
Revised Manuscript: January 23, 2011
Manuscript Accepted: January 28, 2011
Published: February 4, 2011

Citation
Hidenosuke Itoh, Kentaro Nagai, Genta Sato, Kimiaki Yamaguchi, Takashi Nakamura, Takeshi Kondoh, Chidane Ouchi, Takayuki Teshima, Yutaka Setomoto, and Toru Den, "Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval," Opt. Express 19, 3339-3346 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3339


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References

  1. U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965). [CrossRef]
  2. V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D Appl. Phys. 28(11), 2314–2317 (1995). [CrossRef]
  3. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995). [CrossRef]
  4. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997). [CrossRef] [PubMed]
  5. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995). [CrossRef]
  6. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996). [CrossRef]
  7. H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).
  8. 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]
  9. 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(No. 6A), 5254–5262 (2006). [CrossRef]
  10. 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]
  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(4), 258–261 (2006). [CrossRef]
  12. 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]
  13. A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009). [CrossRef] [PubMed]
  14. C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard X-rays,” Opt. Express 15(3), 1175–1181 (2007). [CrossRef] [PubMed]
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