Abstract

Iron oxide nano-particles have very different x-ray diffraction properties from tissue. They can be clearly visualized against suppressed tissue background in a single-shot x-ray diffraction imaging technique. This technique is able to acquire both diffraction and absorption images from a single grating-modulated projection image through analysis in the spatial frequency domain. We describe the use of two orthogonal transmission gratings to selectively retain diffraction signal from iron oxide particles that are larger than a threshold size, while eliminating the background signal from soft tissue and bone. This approach should help the tracking of functionalized particles in cell labeling and targeted therapy.

© 2010 OSA

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  1. S. Yokozeki and T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opti. 10(7), 1575-1580 (1971).
  2. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Japanese J. Appl. Phys. Part 2-Letters 42, L866–L868 (2003).
  3. 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]
  4. 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]
  5. 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]
  6. Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Japanese J. Appl. Phys. Part 2-Letters & Express Letters 46, L89–L91 (2007).
  7. 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]
  8. H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier X-ray scattering radiography yields bone structural information,” Radiology 251(3), 910–918 (2009).
    [CrossRef] [PubMed]
  9. L. E. Levine and G. G. Long, “X-ray imaging with ultra-small-angle x-ray scattering as a contrast mechanism,” J. Appl. Cryst. 37(5), 757–765 (2004).
    [CrossRef]
  10. J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Cryst. 42(3), 469–479 (2009).
    [CrossRef]
  11. P. C. Johns and M. J. Yaffe, “Coherent scatter in diagnostic radiology,” Med. Phys. 10(1), 40–50 (1983).
    [CrossRef] [PubMed]
  12. Q. A. Pankhurst, J. Connolly, S. K. Jones, and J. Dobson, “Applications of magnetic nanoparticles in biomedicine,” J. Phys. D Appl. Phys. 36(13), R167–R181 (2003).
    [CrossRef]

2009

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier X-ray scattering radiography yields bone structural information,” Radiology 251(3), 910–918 (2009).
[CrossRef] [PubMed]

J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Cryst. 42(3), 469–479 (2009).
[CrossRef]

2008

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]

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]

2007

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Japanese J. Appl. Phys. Part 2-Letters & Express Letters 46, L89–L91 (2007).

2005

2004

L. E. Levine and G. G. Long, “X-ray imaging with ultra-small-angle x-ray scattering as a contrast mechanism,” J. Appl. Cryst. 37(5), 757–765 (2004).
[CrossRef]

2003

Q. A. Pankhurst, J. Connolly, S. K. Jones, and J. Dobson, “Applications of magnetic nanoparticles in biomedicine,” J. Phys. D Appl. Phys. 36(13), R167–R181 (2003).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Japanese J. Appl. Phys. Part 2-Letters 42, L866–L868 (2003).

2002

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]

1983

P. C. Johns and M. J. Yaffe, “Coherent scatter in diagnostic radiology,” Med. Phys. 10(1), 40–50 (1983).
[CrossRef] [PubMed]

1971

S. Yokozeki and T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opti. 10(7), 1575-1580 (1971).

Allen, A. J.

J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Cryst. 42(3), 469–479 (2009).
[CrossRef]

Aoki, S.

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Japanese J. Appl. Phys. Part 2-Letters & Express Letters 46, L89–L91 (2007).

Bech, M.

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]

Bennett, E. E.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier X-ray scattering radiography yields bone structural information,” Radiology 251(3), 910–918 (2009).
[CrossRef] [PubMed]

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]

Brönnimann, Ch.

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]

Bunk, O.

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]

Carroll, S. C.

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]

Cloetens, P.

Connolly, J.

Q. A. Pankhurst, J. Connolly, S. K. Jones, and J. Dobson, “Applications of magnetic nanoparticles in biomedicine,” J. Phys. D Appl. Phys. 36(13), R167–R181 (2003).
[CrossRef]

David, C.

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]

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]

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]

Diaz, A.

Dobson, J.

Q. A. Pankhurst, J. Connolly, S. K. Jones, and J. Dobson, “Applications of magnetic nanoparticles in biomedicine,” J. Phys. D Appl. Phys. 36(13), R167–R181 (2003).
[CrossRef]

Eikenberry, E. F.

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]

Grünzweig, C.

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]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Japanese J. Appl. Phys. Part 2-Letters 42, L866–L868 (2003).

Hattori, T.

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Japanese J. Appl. Phys. Part 2-Letters & Express Letters 46, L89–L91 (2007).

Hegedus, M. M.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier X-ray scattering radiography yields bone structural information,” Radiology 251(3), 910–918 (2009).
[CrossRef] [PubMed]

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]

Ilavsky, J.

J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Cryst. 42(3), 469–479 (2009).
[CrossRef]

Jemian, P. R.

J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Cryst. 42(3), 469–479 (2009).
[CrossRef]

Johns, P. C.

P. C. Johns and M. J. Yaffe, “Coherent scatter in diagnostic radiology,” Med. Phys. 10(1), 40–50 (1983).
[CrossRef] [PubMed]

Jones, S. K.

Q. A. Pankhurst, J. Connolly, S. K. Jones, and J. Dobson, “Applications of magnetic nanoparticles in biomedicine,” J. Phys. D Appl. Phys. 36(13), R167–R181 (2003).
[CrossRef]

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Japanese J. Appl. Phys. Part 2-Letters 42, L866–L868 (2003).

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Japanese J. Appl. Phys. Part 2-Letters 42, L866–L868 (2003).

Kraft, P.

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]

Levine, L. E.

J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Cryst. 42(3), 469–479 (2009).
[CrossRef]

L. E. Levine and G. G. Long, “X-ray imaging with ultra-small-angle x-ray scattering as a contrast mechanism,” J. Appl. Cryst. 37(5), 757–765 (2004).
[CrossRef]

Long, G. G.

J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Cryst. 42(3), 469–479 (2009).
[CrossRef]

L. E. Levine and G. G. Long, “X-ray imaging with ultra-small-angle x-ray scattering as a contrast mechanism,” J. Appl. Cryst. 37(5), 757–765 (2004).
[CrossRef]

Momose, A.

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Japanese J. Appl. Phys. Part 2-Letters & Express Letters 46, L89–L91 (2007).

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Japanese J. Appl. Phys. Part 2-Letters 42, L866–L868 (2003).

Nohammer, B.

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]

Pankhurst, Q. A.

Q. A. Pankhurst, J. Connolly, S. K. Jones, and J. Dobson, “Applications of magnetic nanoparticles in biomedicine,” J. Phys. D Appl. Phys. 36(13), R167–R181 (2003).
[CrossRef]

Pfeiffer, F.

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]

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]

Rapacchi, S.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier X-ray scattering radiography yields bone structural information,” Radiology 251(3), 910–918 (2009).
[CrossRef] [PubMed]

Solak, H. H.

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]

Stampanoni, M.

Suzuki, T.

S. Yokozeki and T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opti. 10(7), 1575-1580 (1971).

Suzuki, Y.

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Japanese J. Appl. Phys. Part 2-Letters & Express Letters 46, L89–L91 (2007).

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Japanese J. Appl. Phys. Part 2-Letters 42, L866–L868 (2003).

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Japanese J. Appl. Phys. Part 2-Letters 42, L866–L868 (2003).

Takeda, Y.

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Japanese J. Appl. Phys. Part 2-Letters & Express Letters 46, L89–L91 (2007).

Weitkamp, T.

Wen, H.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier X-ray scattering radiography yields bone structural information,” Radiology 251(3), 910–918 (2009).
[CrossRef] [PubMed]

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]

Yaffe, M. J.

P. C. Johns and M. J. Yaffe, “Coherent scatter in diagnostic radiology,” Med. Phys. 10(1), 40–50 (1983).
[CrossRef] [PubMed]

Yashiro, W.

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Japanese J. Appl. Phys. Part 2-Letters & Express Letters 46, L89–L91 (2007).

Yokozeki, S.

S. Yokozeki and T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opti. 10(7), 1575-1580 (1971).

Zhang, F.

J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Cryst. 42(3), 469–479 (2009).
[CrossRef]

Ziegler, E.

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]

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]

Appl. Phys. Lett.

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]

IEEE Trans. Med. Imaging

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]

J. Appl. Cryst.

L. E. Levine and G. G. Long, “X-ray imaging with ultra-small-angle x-ray scattering as a contrast mechanism,” J. Appl. Cryst. 37(5), 757–765 (2004).
[CrossRef]

J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Cryst. 42(3), 469–479 (2009).
[CrossRef]

J. Phys. D Appl. Phys.

Q. A. Pankhurst, J. Connolly, S. K. Jones, and J. Dobson, “Applications of magnetic nanoparticles in biomedicine,” J. Phys. D Appl. Phys. 36(13), R167–R181 (2003).
[CrossRef]

Med. Phys.

P. C. Johns and M. J. Yaffe, “Coherent scatter in diagnostic radiology,” Med. Phys. 10(1), 40–50 (1983).
[CrossRef] [PubMed]

Nat. Mater.

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]

Opt. Express

Radiology

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier X-ray scattering radiography yields bone structural information,” Radiology 251(3), 910–918 (2009).
[CrossRef] [PubMed]

Other

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Japanese J. Appl. Phys. Part 2-Letters & Express Letters 46, L89–L91 (2007).

S. Yokozeki and T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opti. 10(7), 1575-1580 (1971).

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Japanese J. Appl. Phys. Part 2-Letters 42, L866–L868 (2003).

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Figures (6)

Fig. 1
Fig. 1

The double-grid diffraction imaging setup. After passing through the sample, the x-ray cone beam is spatially modulated by a horizontal, object grid and a vertical, camera grid before detection by a camera.

Fig. 2
Fig. 2

Process to extract absorption and diffraction images from a single raw image. The sample includes a vial of black watercolor paint on the left which is a suspension of black iron oxide nano-particles, and a vial of 150 mg/ml KI solution on the right. (a) A grid-masked raw image is acquired and (b) 2D Fourier transformed into the spatial frequency domain. The spectrum contains a number of peaks corresponding to the frequencies of the grids. (c) An absorption image is obtained from the inverse Fourier transform of the segment around the zeroth order peak. (d,e) Harmonic images from the camera and object grids are obtained from the inverse Fourier transform of the segments surrounding the first order harmonic peaks on the X and Y axes. They are shown in log scale. (f) The final double-grid diffraction image is the ratio of the two single-grid harmonic images. Note that the stripe pattern in the raw image (a) is not the actually grid lines but a Moire effect from the low resolution of the graph.

Fig. 3
Fig. 3

(a) Calculated single-grid diffraction image intensities from a random suspension of cylindrical particles for the camera (blue) and object (red) grids. The scattering length scales of the grids are 4 nm and 87 nm. (b) Calculated double-grid diffraction image intensity. The vertical lines mark the scattering length scales of the two grids. This graph shows that the double-grid image is a highpass filter on the particle size.

Fig. 4
Fig. 4

Absorption and diffraction images of a rat leg. (a) Absorption, (b) object grid diffraction, (c) camera grid diffraction and (d) double grid diffraction images. Both bone and soft tissue are effectively nulled in the double grid diffraction image.

Fig. 5
Fig. 5

Absorption and diffraction images of a chicken wing injected with iron oxide particles and potassium iodide. (a) Absorption image showing the patch of iron oxide (single-line oval) and KI (double-line oval), (b) object grid diffraction image, (c) camera grid diffraction image and (d) double-grid diffraction image. The double-grid diffraction only retains the iron oxide particles.

Fig. 6
Fig. 6

The USAXS measurement from the cortical bone of a rat tibia. Based on Eq. (5) the double-grid diffraction signal mainly comes from the shaded band of q values, which is negligible in the bone sample.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

D n = ln ( | I n | | I 0 | ) .
D d 1 = ln ( | I 1 , o j e c t | | I 1 , c a m e r a | ) = ln ( | I 1 , o b j e c t | | I 0 | ) + ln ( | I 1 , c a m e r a | | I 0 | ) = D 1 , o b j e c t D 1 , c a m e r a .
D 1 = { ( s a m p l e t h i c k n e s s ) f ( 2 π λ Δ n ) 2 l [ 1 + 2 π L l 1 ( L l ) 2 2 π arccos ( L l ) ] f o r l > L , ( s a m p l e t h i c k n e s s ) f ( 2 π λ Δ n ) 2 l f o r l L ,
L = λ projected   grid   period ( grid   to   camera   distance ) .
D d I ( q ) [ cos ( q x L C ) cos ( q x L O ) ] d 2 q ,

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