Abstract

Dark-field images are formed from x-ray small-angle scattering signals. The small-angle scattering signals are particularly sensitive to structural variation and density fluctuation on a length scale of several tens to hundreds of nanometers, offering a unique contrast mechanism to reveal subtle structural features of an object. In this study, based on the principle of energy conservation, we develop a physical model to describe the relationship between x-ray small-angle scattering coefficients of an object and dark-field intensity images. This model can be used to reconstruct volumetric x-ray small-angle scattering images of an object using classical tomographic algorithms. We also establish a relationship between the small-angle scattering intensity and the visibility function measured with x-ray grating imaging. The numerical simulations and phantom experiments have demonstrated the accuracy and practicability of the proposed model.

© 2012 Optical Society of America

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  1. A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
    [CrossRef]
  2. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
  3. D. I. Svergun and M. H. J. Koch, “Small-angle scattering studies of biological macromolecules in solution,” Rep. Prog. Phys. 66, 1735–1782 (2003).
    [CrossRef]
  4. V. Changizi, S. Wilkinson, C. J. Hall, and G. Grossmann, “A study of the effect of formalin preservation on normal and cancerous breast tissues using small angle X-ray scattering (SAXS),” Radiat. Phys. Chem. 75, 932–935 (2006).
    [CrossRef]
  5. M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
    [CrossRef]
  6. S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
    [CrossRef]
  7. C. G. Schroer, M. Kuhlmann, S. V. Roth, R. Gehrke, N. Stribeck, A. Almendarez-Camarillo, and B. Lengeler, “Mapping the local nanostructure inside a specimen by tomographic small-angle x-ray scattering,” Appl. Phys. Lett. 88, 164102 (2006).
    [CrossRef]
  8. J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
    [CrossRef]
  9. A. Harding, J.- P. Schlomka, and G. Harding, “Simulations and experimental feasibility study of fan-beam coherent scatter CT,” Proc. SPIE 4786, 202–209 (2002).
  10. G. Harding and J. Kosanetzky, “Status and outlook of coherent-x-ray scatter imaging,” J. Opt. Soc. Am. A 4, 933–944 (1987).
    [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. Mat. 7, 134–137 (2008).
    [CrossRef]
  12. G. Chen, J. Zambelli, K. Li, N. Bevins, and Z. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38, 584–588 (2011).
    [CrossRef]
  13. X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast CT compared with the conventional CT—noise power spectrum NPS(k),” Med. Phys. 38, 4386–4395 (2011).
    [CrossRef]
  14. Z. Wang, K. Kang, Z. Huang, and Z. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009).
    [CrossRef]
  15. G. Chen, N. Bevins, J. Zambelli, and Z. Qi, “Small-angle scattering computed tomography (SAS-CT) using a Talbot-Lau interferometer and a rotating anode x-ray tube: theory and experiments,” Opt. Express 18, 12960–12970(2010).
    [CrossRef]
  16. M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
    [CrossRef]
  17. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  18. W. Cong and G. Wang, “Analytic model for x-ray dark-field CT,” presented at The 11th International Meeting on Fully Three-dimensional Image Reconstruction in Radiology and Nuclear Medicine, Potsdam, Germany, 2011.
  19. A. C. Kak and M. Slane, Principles of Computerized Tomographic Imaging (IEEE, 1988).
  20. E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
    [CrossRef]
  21. J. Zambelli, N. Bevins, Z. Qi, and GH. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37, 2473–2479 (2010).
    [CrossRef]

2011 (3)

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[CrossRef]

G. Chen, J. Zambelli, K. Li, N. Bevins, and Z. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38, 584–588 (2011).
[CrossRef]

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast CT compared with the conventional CT—noise power spectrum NPS(k),” Med. Phys. 38, 4386–4395 (2011).
[CrossRef]

2010 (3)

G. Chen, N. Bevins, J. Zambelli, and Z. Qi, “Small-angle scattering computed tomography (SAS-CT) using a Talbot-Lau interferometer and a rotating anode x-ray tube: theory and experiments,” Opt. Express 18, 12960–12970(2010).
[CrossRef]

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[CrossRef]

J. Zambelli, N. Bevins, Z. Qi, and GH. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37, 2473–2479 (2010).
[CrossRef]

2009 (2)

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

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

2008 (1)

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. Mat. 7, 134–137 (2008).
[CrossRef]

2006 (3)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

C. G. Schroer, M. Kuhlmann, S. V. Roth, R. Gehrke, N. Stribeck, A. Almendarez-Camarillo, and B. Lengeler, “Mapping the local nanostructure inside a specimen by tomographic small-angle x-ray scattering,” Appl. Phys. Lett. 88, 164102 (2006).
[CrossRef]

V. Changizi, S. Wilkinson, C. J. Hall, and G. Grossmann, “A study of the effect of formalin preservation on normal and cancerous breast tissues using small angle X-ray scattering (SAXS),” Radiat. Phys. Chem. 75, 932–935 (2006).
[CrossRef]

2003 (1)

D. I. Svergun and M. H. J. Koch, “Small-angle scattering studies of biological macromolecules in solution,” Rep. Prog. Phys. 66, 1735–1782 (2003).
[CrossRef]

2002 (2)

M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
[CrossRef]

A. Harding, J.- P. Schlomka, and G. Harding, “Simulations and experimental feasibility study of fan-beam coherent scatter CT,” Proc. SPIE 4786, 202–209 (2002).

1997 (1)

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

1996 (1)

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef]

1987 (1)

Almendarez-Camarillo, A.

C. G. Schroer, M. Kuhlmann, S. V. Roth, R. Gehrke, N. Stribeck, A. Almendarez-Camarillo, and B. Lengeler, “Mapping the local nanostructure inside a specimen by tomographic small-angle x-ray scattering,” Appl. Phys. Lett. 88, 164102 (2006).
[CrossRef]

Arfelli, F.

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

Bech, M.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[CrossRef]

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. Mat. 7, 134–137 (2008).
[CrossRef]

Bevins, N.

G. Chen, J. Zambelli, K. Li, N. Bevins, and Z. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38, 584–588 (2011).
[CrossRef]

G. Chen, N. Bevins, J. Zambelli, and Z. Qi, “Small-angle scattering computed tomography (SAS-CT) using a Talbot-Lau interferometer and a rotating anode x-ray tube: theory and experiments,” Opt. Express 18, 12960–12970(2010).
[CrossRef]

J. Zambelli, N. Bevins, Z. Qi, and GH. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37, 2473–2479 (2010).
[CrossRef]

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. Mat. 7, 134–137 (2008).
[CrossRef]

Bubnova, R. S.

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

Bunk, O.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[CrossRef]

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. Mat. 7, 134–137 (2008).
[CrossRef]

Candès, E. J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Changizi, V.

V. Changizi, S. Wilkinson, C. J. Hall, and G. Grossmann, “A study of the effect of formalin preservation on normal and cancerous breast tissues using small angle X-ray scattering (SAXS),” Radiat. Phys. Chem. 75, 932–935 (2006).
[CrossRef]

Chapman, D.

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

Chen, G.

G. Chen, J. Zambelli, K. Li, N. Bevins, and Z. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38, 584–588 (2011).
[CrossRef]

G. Chen, N. Bevins, J. Zambelli, and Z. Qi, “Small-angle scattering computed tomography (SAS-CT) using a Talbot-Lau interferometer and a rotating anode x-ray tube: theory and experiments,” Opt. Express 18, 12960–12970(2010).
[CrossRef]

Chen, GH.

J. Zambelli, N. Bevins, Z. Qi, and GH. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37, 2473–2479 (2010).
[CrossRef]

Chen, Z.

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

Cong, W.

W. Cong and G. Wang, “Analytic model for x-ray dark-field CT,” presented at The 11th International Meeting on Fully Three-dimensional Image Reconstruction in Radiology and Nuclear Medicine, Potsdam, Germany, 2011.

David, C.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[CrossRef]

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. Mat. 7, 134–137 (2008).
[CrossRef]

Donath, T.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[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. Mat. 7, 134–137 (2008).
[CrossRef]

Falzon, G.

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[CrossRef]

Feidenhans’l, R.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[CrossRef]

Feldkamp, J. M.

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

Fernández, M.

M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
[CrossRef]

Filatov, S. K.

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

Fox, J. G.

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[CrossRef]

Gehrke, R.

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

C. G. Schroer, M. Kuhlmann, S. V. Roth, R. Gehrke, N. Stribeck, A. Almendarez-Camarillo, and B. Lengeler, “Mapping the local nanostructure inside a specimen by tomographic small-angle x-ray scattering,” Appl. Phys. Lett. 88, 164102 (2006).
[CrossRef]

Gmur, N.

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

Grossmann, G.

V. Changizi, S. Wilkinson, C. J. Hall, and G. Grossmann, “A study of the effect of formalin preservation on normal and cancerous breast tissues using small angle X-ray scattering (SAXS),” Radiat. Phys. Chem. 75, 932–935 (2006).
[CrossRef]

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. Mat. 7, 134–137 (2008).
[CrossRef]

Hall, C. J.

V. Changizi, S. Wilkinson, C. J. Hall, and G. Grossmann, “A study of the effect of formalin preservation on normal and cancerous breast tissues using small angle X-ray scattering (SAXS),” Radiat. Phys. Chem. 75, 932–935 (2006).
[CrossRef]

Harding, A.

A. Harding, J.- P. Schlomka, and G. Harding, “Simulations and experimental feasibility study of fan-beam coherent scatter CT,” Proc. SPIE 4786, 202–209 (2002).

Harding, G.

A. Harding, J.- P. Schlomka, and G. Harding, “Simulations and experimental feasibility study of fan-beam coherent scatter CT,” Proc. SPIE 4786, 202–209 (2002).

G. Harding and J. Kosanetzky, “Status and outlook of coherent-x-ray scatter imaging,” J. Opt. Soc. Am. A 4, 933–944 (1987).
[CrossRef]

Hart, S. A.

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[CrossRef]

Hirano, K.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef]

Huang, Z.

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

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Itai, Y.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef]

Johnston, R. E.

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

Kak, A. C.

A. C. Kak and M. Slane, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Kang, K.

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

Karjalainen-Lindsberg, M.-L.

M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
[CrossRef]

Keyriläinen, J.

M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
[CrossRef]

Koch, M. H. J.

D. I. Svergun and M. H. J. Koch, “Small-angle scattering studies of biological macromolecules in solution,” Rep. Prog. Phys. 66, 1735–1782 (2003).
[CrossRef]

Kosanetzky, J.

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. Mat. 7, 134–137 (2008).
[CrossRef]

Kuhlmann, M.

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

C. G. Schroer, M. Kuhlmann, S. V. Roth, R. Gehrke, N. Stribeck, A. Almendarez-Camarillo, and B. Lengeler, “Mapping the local nanostructure inside a specimen by tomographic small-angle x-ray scattering,” Appl. Phys. Lett. 88, 164102 (2006).
[CrossRef]

Lengeler, B.

C. G. Schroer, M. Kuhlmann, S. V. Roth, R. Gehrke, N. Stribeck, A. Almendarez-Camarillo, and B. Lengeler, “Mapping the local nanostructure inside a specimen by tomographic small-angle x-ray scattering,” Appl. Phys. Lett. 88, 164102 (2006).
[CrossRef]

Lewis, R. A.

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[CrossRef]

Li, K.

G. Chen, J. Zambelli, K. Li, N. Bevins, and Z. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38, 584–588 (2011).
[CrossRef]

Menk, R.

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

Momose, A.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef]

Paufler, P.

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

Pfeiffer, F.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[CrossRef]

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. Mat. 7, 134–137 (2008).
[CrossRef]

Pisano, E.

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

Qi, Z.

G. Chen, J. Zambelli, K. Li, N. Bevins, and Z. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38, 584–588 (2011).
[CrossRef]

G. Chen, N. Bevins, J. Zambelli, and Z. Qi, “Small-angle scattering computed tomography (SAS-CT) using a Talbot-Lau interferometer and a rotating anode x-ray tube: theory and experiments,” Opt. Express 18, 12960–12970(2010).
[CrossRef]

J. Zambelli, N. Bevins, Z. Qi, and GH. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37, 2473–2479 (2010).
[CrossRef]

Romberg, J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Roth, S. V.

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

C. G. Schroer, M. Kuhlmann, S. V. Roth, R. Gehrke, N. Stribeck, A. Almendarez-Camarillo, and B. Lengeler, “Mapping the local nanostructure inside a specimen by tomographic small-angle x-ray scattering,” Appl. Phys. Lett. 88, 164102 (2006).
[CrossRef]

Sayers, D.

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

Schlomka, J.- P.

A. Harding, J.- P. Schlomka, and G. Harding, “Simulations and experimental feasibility study of fan-beam coherent scatter CT,” Proc. SPIE 4786, 202–209 (2002).

Schroer, C. G.

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

C. G. Schroer, M. Kuhlmann, S. V. Roth, R. Gehrke, N. Stribeck, A. Almendarez-Camarillo, and B. Lengeler, “Mapping the local nanostructure inside a specimen by tomographic small-angle x-ray scattering,” Appl. Phys. Lett. 88, 164102 (2006).
[CrossRef]

Serimaa, R.

M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
[CrossRef]

Shakhverdova, I.

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

Sidhu, S.

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[CrossRef]

Siu, K. K. W.

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[CrossRef]

Slane, M.

A. C. Kak and M. Slane, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Stribeck, N.

C. G. Schroer, M. Kuhlmann, S. V. Roth, R. Gehrke, N. Stribeck, A. Almendarez-Camarillo, and B. Lengeler, “Mapping the local nanostructure inside a specimen by tomographic small-angle x-ray scattering,” Appl. Phys. Lett. 88, 164102 (2006).
[CrossRef]

Suortti, P.

M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
[CrossRef]

Svergun, D. I.

D. I. Svergun and M. H. J. Koch, “Small-angle scattering studies of biological macromolecules in solution,” Rep. Prog. Phys. 66, 1735–1782 (2003).
[CrossRef]

Takeda, T.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef]

Tang, S.

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast CT compared with the conventional CT—noise power spectrum NPS(k),” Med. Phys. 38, 4386–4395 (2011).
[CrossRef]

Tang, X.

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast CT compared with the conventional CT—noise power spectrum NPS(k),” Med. Phys. 38, 4386–4395 (2011).
[CrossRef]

Tao, T.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Tenhunen, M.

M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
[CrossRef]

Thomlinson, W.

M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
[CrossRef]

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

Timmann, A.

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

Torkkeli, M.

M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
[CrossRef]

Urban, V.

M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
[CrossRef]

Wang, G.

W. Cong and G. Wang, “Analytic model for x-ray dark-field CT,” presented at The 11th International Meeting on Fully Three-dimensional Image Reconstruction in Radiology and Nuclear Medicine, Potsdam, Germany, 2011.

Wang, Z.

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

Washburn, D.

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

Wilkinson, S.

V. Changizi, S. Wilkinson, C. J. Hall, and G. Grossmann, “A study of the effect of formalin preservation on normal and cancerous breast tissues using small angle X-ray scattering (SAXS),” Radiat. Phys. Chem. 75, 932–935 (2006).
[CrossRef]

Yang, Y.

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast CT compared with the conventional CT—noise power spectrum NPS(k),” Med. Phys. 38, 4386–4395 (2011).
[CrossRef]

Zambelli, J.

G. Chen, J. Zambelli, K. Li, N. Bevins, and Z. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38, 584–588 (2011).
[CrossRef]

G. Chen, N. Bevins, J. Zambelli, and Z. Qi, “Small-angle scattering computed tomography (SAS-CT) using a Talbot-Lau interferometer and a rotating anode x-ray tube: theory and experiments,” Opt. Express 18, 12960–12970(2010).
[CrossRef]

J. Zambelli, N. Bevins, Z. Qi, and GH. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37, 2473–2479 (2010).
[CrossRef]

Zhong, Z.

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

Appl. Phys. Lett. (2)

C. G. Schroer, M. Kuhlmann, S. V. Roth, R. Gehrke, N. Stribeck, A. Almendarez-Camarillo, and B. Lengeler, “Mapping the local nanostructure inside a specimen by tomographic small-angle x-ray scattering,” Appl. Phys. Lett. 88, 164102 (2006).
[CrossRef]

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

IEEE Trans. Inf. Theory (1)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

Med. Phys. (3)

J. Zambelli, N. Bevins, Z. Qi, and GH. Chen, “Radiation dose efficiency comparison between differential phase contrast CT and conventional absorption CT,” Med. Phys. 37, 2473–2479 (2010).
[CrossRef]

G. Chen, J. Zambelli, K. Li, N. Bevins, and Z. Qi, “Scaling law for noise variance and spatial resolution in differential phase contrast computed tomography,” Med. Phys. 38, 584–588 (2011).
[CrossRef]

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast CT compared with the conventional CT—noise power spectrum NPS(k),” Med. Phys. 38, 4386–4395 (2011).
[CrossRef]

Nat. Mat. (1)

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. Mat. 7, 134–137 (2008).
[CrossRef]

Nat. Med. (1)

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef]

Opt. Express (1)

Phys. Med. Biol. (4)

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[CrossRef]

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

M. Fernández, J. Keyriläinen, R. Serimaa, M. Torkkeli, M.-L. Karjalainen-Lindsberg, M. Tenhunen, W. Thomlinson, V. Urban, and P. Suortti, “Small-angle X-ray scattering studies of human breast tissue samples,” Phys. Med. Biol. 47, 577–592 (2002).
[CrossRef]

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[CrossRef]

Phys. Status Solidi A (1)

J. M. Feldkamp, M. Kuhlmann, S. V. Roth, A. Timmann, R. Gehrke, I. Shakhverdova, P. Paufler, S. K. Filatov, R. S. Bubnova, and C. G. Schroer, “Recent developments in tomographic small-angle X-ray scattering,” Phys. Status Solidi A 206, 1723–1726 (2009).
[CrossRef]

Proc. SPIE (1)

A. Harding, J.- P. Schlomka, and G. Harding, “Simulations and experimental feasibility study of fan-beam coherent scatter CT,” Proc. SPIE 4786, 202–209 (2002).

Radiat. Phys. Chem. (1)

V. Changizi, S. Wilkinson, C. J. Hall, and G. Grossmann, “A study of the effect of formalin preservation on normal and cancerous breast tissues using small angle X-ray scattering (SAXS),” Radiat. Phys. Chem. 75, 932–935 (2006).
[CrossRef]

Rep. Prog. Phys. (1)

D. I. Svergun and M. H. J. Koch, “Small-angle scattering studies of biological macromolecules in solution,” Rep. Prog. Phys. 66, 1735–1782 (2003).
[CrossRef]

Other (3)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

W. Cong and G. Wang, “Analytic model for x-ray dark-field CT,” presented at The 11th International Meeting on Fully Three-dimensional Image Reconstruction in Radiology and Nuclear Medicine, Potsdam, Germany, 2011.

A. C. Kak and M. Slane, Principles of Computerized Tomographic Imaging (IEEE, 1988).

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Figures (3)

Fig. 1.
Fig. 1.

Reconstruction of numerical phantom. (a) The original small-angle scattering image. (b) The reconstructed small-angle scattering image based on the proposed physical model.

Fig. 2.
Fig. 2.

Comparison of profiles. (a) Profiles of horizontal midlines in phantom and reconstructed images. (b) Profiles of vertical midlines in phantom and reconstructed images.

Fig. 3.
Fig. 3.

Reconstruction of physical phantom. The reconstructed image at the 15th slice (a), 45th slice (b), and the 100th slice (c) based on the proposed physical model.

Equations (14)

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Φ a ( r + R θ ) = Φ a ( r + t θ ) exp ( t R μ t ( r + s θ ) d s ) ,
ln [ Φ a ( r + t θ ) Φ a ( r + R θ ) ] = t R μ t ( r + s θ ) d s ,
d Φ s ( r ) d A = μ s d h Φ a ( r ) d A ( μ a + μ w ) d h Φ s ( r ) d A .
θ · Φ s ( r ) + ( μ a + μ w ) Φ s ( r ) = μ s Φ a ( r ) ,
Φ s ( r + R θ ) = 0 R μ s ( r + t θ ) Φ a ( r + t θ ) exp ( t R [ μ a ( r + s θ ) + μ w ( r + s θ ) ] d s ) d t .
Φ s ( r + R θ ) = Φ a ( r + R θ ) 0 R μ s ( r + t θ ) exp ( t R μ s ( r + s θ ) d s ) d t .
ln [ 1 + Φ s ( r + R θ ) Φ a ( r + R θ ) ] = 0 R μ s ( r + s θ ) d s ,
ln [ 1 + T s ( r + R θ ) T a ( r + R θ ) ] = 0 R μ s ( r + s θ ) d s ,
Φ s ( r 0 + R θ ) = Φ a ( r + R θ ) 0 R μ s ( r 0 + s θ ) d s .
I ( m , n , x g ) a 0 ( m , n ) + a 1 ( m , n ) cos ( k x g + ϕ ( m , n ) ) .
a 0 ( x , y ) = 1 M i = 1 M I i ( x , y )
a 1 ( x , y ) = ( 1 M i = 1 M I i ( x , y ) sin ( k · x g ) ) 2 + ( 1 M i = 1 M I i ( x , y ) cos ( k · x g ) ) 2 ,
a 1 h ( m , n ) a 0 s ( m , n ) a 0 r ( m , n ) a 1 r ( m , n ) = 1.
T s ( m , n ) T a ( m , n ) = [ a 1 h ( m , n ) a 1 s ( m , n ) a 1 r ( m , n ) ] / [ a 0 s ( m , n ) a 0 r ( m , n ) ] = 1 a 0 r ( m , n ) a 0 s ( m , n ) a 1 s ( m , n ) a 1 r ( m , n ) = 1 V ( m , n ) ,

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