Abstract

Tomographic reconstruction requires precise knowledge of the position of the center of rotation in the sinogram data; otherwise, artifacts are introduced into the reconstruction. In parallel-beam microtomography, where resolution in the 1μm range is reached, the center of rotation is often only known with insufficient accuracy. We present three image metrics for the scoring of tomographic reconstructions and an iterative procedure for the determination of the position of the optimum center of rotation. The metrics are applied to model systems as well as to microtomography data from a synchrotron radiation source. The center of rotation is determined using the image metrics and compared with the results obtained by the center-of-mass method and by image registration. It is found that the image metrics make it possible to determine the axis position reliably at well below the resolution of one detector bin in an automated procedure.

© 2006 Optical Society of America

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References

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  1. U. Bonse and F. Busch, "X-ray computed microtomography (μCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
    [CrossRef] [PubMed]
  2. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).
  3. L. A. Shepp, S. K. Hilal, and R. A. Schulz, "Tuning fork artifact in computerized tomography," Comput. Graph. Image Process. 10, 246-255 (1979).
    [CrossRef]
  4. W. Lu and T. R. Mackie, "Tomographic motion detection and correction directly in sinogram space," Phys. Med. Biol. 47, 1267-1284 (2002).
    [CrossRef] [PubMed]
  5. S. G. Azevedo, D. J. Schneberk, J. P. Fitch, and H. E. Martz, "Calculation of the rotational centers in computed tomography sinograms," IEEE Trans. Nucl. Sci. 37, 1525-1540 (1990).
    [CrossRef]
  6. J. P. Hogan, R. A. Gonsalves, and A. S. Krieger, "Micro computed tomography: removal of translational stage backlash," IEEE Trans. Nucl. Sci. 40, 1238-1241 (1993).
  7. J. L. Prince and A. S. Willsky, "Hierarchical reconstruction using geometry and sinogram restoration," IEEE Trans. Image Process. 2, 401-416 (1993).
    [CrossRef] [PubMed]
  8. B. Jähne, Digital Image Processing, 5th ed. (Springer, 2002).
  9. J. Modersitzki, Numerical Methods for Image Registration, Numerical Mathematics and Scientific Computation (Oxford U. Press, 2004).
  10. L. Ibáñez, W. Schroeder, L. Ng, and J. Cates, The ITK Software Guide 1.4 (Kitware, 2003).
  11. D. Atkinson, D. L. G. Hill, P. N. R. Stoyle, P. E. Summers, S. Clare, R. Bowtell, and S. F. Keevil, "Automatic compensation of motion artifacts in MRI," Magn. Reson. Med. 41, 163-170 (1999).
    [CrossRef] [PubMed]
  12. K. P. McGee, A. Manduca, J. P. Felmlee, S. J. Riederer, and R. L. Ehman, "Image metric-based correction (autocorrection) of motion effects: analysis of image metrics," J. Magn. Reson Imaging 11, 174-181 (2000).
    [CrossRef] [PubMed]
  13. A. Brunetti and F. de Carlo, "A robust procedure for determination of center of rotation in tomography," in Proc. SPIE 5535, 652-659 (2004).
    [CrossRef]
  14. P. A. Jansson, Deconvolution: With Applications in Spectroscopy (Academic, 1984).
  15. A. K. Jain, Fundamentals of Digital Image Processing, 1st ed. (Prentice Hall, 1988).
  16. C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).
  17. R. H. Huesman, G. T. Gullberg, W. L. Greenberg, and T. F. Budinger, RECLBL Library Users Manual: Donner Algorithms for Reconstruction Tomography (Lawrence Berkeley Laboratory, University of California, 1977).
  18. F. Beckmann, T. Donath, T. Dose, T. Lippmann, R. V. Martins, J. Metge, and A. Schreyer, "Microtomography using synchrotron radiation at DESY: current status and future developments," in Proc. SPIE 5535, 1-10 (2004).
    [CrossRef]
  19. T. Donath, F. Beckmann, R. G. J. C. Heijkants, O. Brunke, and A. Schreyer, "Characterization of polyurethane scaffolds using synchrotron radiation computed microtomography," in Proc. SPIE 5535, 775-782 (2004).
    [CrossRef]
  20. L. Grodzins, "Optimum energies for x-ray transmission tomography of small samples," Nucl. Instrum. Methods Phys. Res. 206, 541-545 (1983).
    [CrossRef]
  21. L. Grodzins, "Critical absorption tomography of small samples," Nucl. Instrum. Methods Phys. Res. 206, 547-552 (1983).
    [CrossRef]

2004 (4)

J. Modersitzki, Numerical Methods for Image Registration, Numerical Mathematics and Scientific Computation (Oxford U. Press, 2004).

A. Brunetti and F. de Carlo, "A robust procedure for determination of center of rotation in tomography," in Proc. SPIE 5535, 652-659 (2004).
[CrossRef]

F. Beckmann, T. Donath, T. Dose, T. Lippmann, R. V. Martins, J. Metge, and A. Schreyer, "Microtomography using synchrotron radiation at DESY: current status and future developments," in Proc. SPIE 5535, 1-10 (2004).
[CrossRef]

T. Donath, F. Beckmann, R. G. J. C. Heijkants, O. Brunke, and A. Schreyer, "Characterization of polyurethane scaffolds using synchrotron radiation computed microtomography," in Proc. SPIE 5535, 775-782 (2004).
[CrossRef]

2003 (1)

L. Ibáñez, W. Schroeder, L. Ng, and J. Cates, The ITK Software Guide 1.4 (Kitware, 2003).

2002 (2)

W. Lu and T. R. Mackie, "Tomographic motion detection and correction directly in sinogram space," Phys. Med. Biol. 47, 1267-1284 (2002).
[CrossRef] [PubMed]

B. Jähne, Digital Image Processing, 5th ed. (Springer, 2002).

2000 (1)

K. P. McGee, A. Manduca, J. P. Felmlee, S. J. Riederer, and R. L. Ehman, "Image metric-based correction (autocorrection) of motion effects: analysis of image metrics," J. Magn. Reson Imaging 11, 174-181 (2000).
[CrossRef] [PubMed]

1999 (1)

D. Atkinson, D. L. G. Hill, P. N. R. Stoyle, P. E. Summers, S. Clare, R. Bowtell, and S. F. Keevil, "Automatic compensation of motion artifacts in MRI," Magn. Reson. Med. 41, 163-170 (1999).
[CrossRef] [PubMed]

1996 (1)

U. Bonse and F. Busch, "X-ray computed microtomography (μCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
[CrossRef] [PubMed]

1993 (2)

J. P. Hogan, R. A. Gonsalves, and A. S. Krieger, "Micro computed tomography: removal of translational stage backlash," IEEE Trans. Nucl. Sci. 40, 1238-1241 (1993).

J. L. Prince and A. S. Willsky, "Hierarchical reconstruction using geometry and sinogram restoration," IEEE Trans. Image Process. 2, 401-416 (1993).
[CrossRef] [PubMed]

1990 (1)

S. G. Azevedo, D. J. Schneberk, J. P. Fitch, and H. E. Martz, "Calculation of the rotational centers in computed tomography sinograms," IEEE Trans. Nucl. Sci. 37, 1525-1540 (1990).
[CrossRef]

1988 (2)

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

A. K. Jain, Fundamentals of Digital Image Processing, 1st ed. (Prentice Hall, 1988).

1984 (1)

P. A. Jansson, Deconvolution: With Applications in Spectroscopy (Academic, 1984).

1983 (2)

L. Grodzins, "Optimum energies for x-ray transmission tomography of small samples," Nucl. Instrum. Methods Phys. Res. 206, 541-545 (1983).
[CrossRef]

L. Grodzins, "Critical absorption tomography of small samples," Nucl. Instrum. Methods Phys. Res. 206, 547-552 (1983).
[CrossRef]

1979 (1)

L. A. Shepp, S. K. Hilal, and R. A. Schulz, "Tuning fork artifact in computerized tomography," Comput. Graph. Image Process. 10, 246-255 (1979).
[CrossRef]

1977 (1)

R. H. Huesman, G. T. Gullberg, W. L. Greenberg, and T. F. Budinger, RECLBL Library Users Manual: Donner Algorithms for Reconstruction Tomography (Lawrence Berkeley Laboratory, University of California, 1977).

1948 (1)

C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Atkinson, D.

D. Atkinson, D. L. G. Hill, P. N. R. Stoyle, P. E. Summers, S. Clare, R. Bowtell, and S. F. Keevil, "Automatic compensation of motion artifacts in MRI," Magn. Reson. Med. 41, 163-170 (1999).
[CrossRef] [PubMed]

Azevedo, S. G.

S. G. Azevedo, D. J. Schneberk, J. P. Fitch, and H. E. Martz, "Calculation of the rotational centers in computed tomography sinograms," IEEE Trans. Nucl. Sci. 37, 1525-1540 (1990).
[CrossRef]

Beckmann, F.

F. Beckmann, T. Donath, T. Dose, T. Lippmann, R. V. Martins, J. Metge, and A. Schreyer, "Microtomography using synchrotron radiation at DESY: current status and future developments," in Proc. SPIE 5535, 1-10 (2004).
[CrossRef]

T. Donath, F. Beckmann, R. G. J. C. Heijkants, O. Brunke, and A. Schreyer, "Characterization of polyurethane scaffolds using synchrotron radiation computed microtomography," in Proc. SPIE 5535, 775-782 (2004).
[CrossRef]

Bonse, U.

U. Bonse and F. Busch, "X-ray computed microtomography (μCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
[CrossRef] [PubMed]

Bowtell, R.

D. Atkinson, D. L. G. Hill, P. N. R. Stoyle, P. E. Summers, S. Clare, R. Bowtell, and S. F. Keevil, "Automatic compensation of motion artifacts in MRI," Magn. Reson. Med. 41, 163-170 (1999).
[CrossRef] [PubMed]

Brunetti, A.

A. Brunetti and F. de Carlo, "A robust procedure for determination of center of rotation in tomography," in Proc. SPIE 5535, 652-659 (2004).
[CrossRef]

Brunke, O.

T. Donath, F. Beckmann, R. G. J. C. Heijkants, O. Brunke, and A. Schreyer, "Characterization of polyurethane scaffolds using synchrotron radiation computed microtomography," in Proc. SPIE 5535, 775-782 (2004).
[CrossRef]

Budinger, T. F.

R. H. Huesman, G. T. Gullberg, W. L. Greenberg, and T. F. Budinger, RECLBL Library Users Manual: Donner Algorithms for Reconstruction Tomography (Lawrence Berkeley Laboratory, University of California, 1977).

Busch, F.

U. Bonse and F. Busch, "X-ray computed microtomography (μCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
[CrossRef] [PubMed]

Cates, J.

L. Ibáñez, W. Schroeder, L. Ng, and J. Cates, The ITK Software Guide 1.4 (Kitware, 2003).

Clare, S.

D. Atkinson, D. L. G. Hill, P. N. R. Stoyle, P. E. Summers, S. Clare, R. Bowtell, and S. F. Keevil, "Automatic compensation of motion artifacts in MRI," Magn. Reson. Med. 41, 163-170 (1999).
[CrossRef] [PubMed]

de Carlo, F.

A. Brunetti and F. de Carlo, "A robust procedure for determination of center of rotation in tomography," in Proc. SPIE 5535, 652-659 (2004).
[CrossRef]

Donath, T.

T. Donath, F. Beckmann, R. G. J. C. Heijkants, O. Brunke, and A. Schreyer, "Characterization of polyurethane scaffolds using synchrotron radiation computed microtomography," in Proc. SPIE 5535, 775-782 (2004).
[CrossRef]

F. Beckmann, T. Donath, T. Dose, T. Lippmann, R. V. Martins, J. Metge, and A. Schreyer, "Microtomography using synchrotron radiation at DESY: current status and future developments," in Proc. SPIE 5535, 1-10 (2004).
[CrossRef]

Dose, T.

F. Beckmann, T. Donath, T. Dose, T. Lippmann, R. V. Martins, J. Metge, and A. Schreyer, "Microtomography using synchrotron radiation at DESY: current status and future developments," in Proc. SPIE 5535, 1-10 (2004).
[CrossRef]

Ehman, R. L.

K. P. McGee, A. Manduca, J. P. Felmlee, S. J. Riederer, and R. L. Ehman, "Image metric-based correction (autocorrection) of motion effects: analysis of image metrics," J. Magn. Reson Imaging 11, 174-181 (2000).
[CrossRef] [PubMed]

Felmlee, J. P.

K. P. McGee, A. Manduca, J. P. Felmlee, S. J. Riederer, and R. L. Ehman, "Image metric-based correction (autocorrection) of motion effects: analysis of image metrics," J. Magn. Reson Imaging 11, 174-181 (2000).
[CrossRef] [PubMed]

Fitch, J. P.

S. G. Azevedo, D. J. Schneberk, J. P. Fitch, and H. E. Martz, "Calculation of the rotational centers in computed tomography sinograms," IEEE Trans. Nucl. Sci. 37, 1525-1540 (1990).
[CrossRef]

Gonsalves, R. A.

J. P. Hogan, R. A. Gonsalves, and A. S. Krieger, "Micro computed tomography: removal of translational stage backlash," IEEE Trans. Nucl. Sci. 40, 1238-1241 (1993).

Greenberg, W. L.

R. H. Huesman, G. T. Gullberg, W. L. Greenberg, and T. F. Budinger, RECLBL Library Users Manual: Donner Algorithms for Reconstruction Tomography (Lawrence Berkeley Laboratory, University of California, 1977).

Grodzins, L.

L. Grodzins, "Optimum energies for x-ray transmission tomography of small samples," Nucl. Instrum. Methods Phys. Res. 206, 541-545 (1983).
[CrossRef]

L. Grodzins, "Critical absorption tomography of small samples," Nucl. Instrum. Methods Phys. Res. 206, 547-552 (1983).
[CrossRef]

Gullberg, G. T.

R. H. Huesman, G. T. Gullberg, W. L. Greenberg, and T. F. Budinger, RECLBL Library Users Manual: Donner Algorithms for Reconstruction Tomography (Lawrence Berkeley Laboratory, University of California, 1977).

Heijkants, R. G. J. C.

T. Donath, F. Beckmann, R. G. J. C. Heijkants, O. Brunke, and A. Schreyer, "Characterization of polyurethane scaffolds using synchrotron radiation computed microtomography," in Proc. SPIE 5535, 775-782 (2004).
[CrossRef]

Hilal, S. K.

L. A. Shepp, S. K. Hilal, and R. A. Schulz, "Tuning fork artifact in computerized tomography," Comput. Graph. Image Process. 10, 246-255 (1979).
[CrossRef]

Hill, D. L. G.

D. Atkinson, D. L. G. Hill, P. N. R. Stoyle, P. E. Summers, S. Clare, R. Bowtell, and S. F. Keevil, "Automatic compensation of motion artifacts in MRI," Magn. Reson. Med. 41, 163-170 (1999).
[CrossRef] [PubMed]

Hogan, J. P.

J. P. Hogan, R. A. Gonsalves, and A. S. Krieger, "Micro computed tomography: removal of translational stage backlash," IEEE Trans. Nucl. Sci. 40, 1238-1241 (1993).

Huesman, R. H.

R. H. Huesman, G. T. Gullberg, W. L. Greenberg, and T. F. Budinger, RECLBL Library Users Manual: Donner Algorithms for Reconstruction Tomography (Lawrence Berkeley Laboratory, University of California, 1977).

Ibáñez, L.

L. Ibáñez, W. Schroeder, L. Ng, and J. Cates, The ITK Software Guide 1.4 (Kitware, 2003).

Jähne, B.

B. Jähne, Digital Image Processing, 5th ed. (Springer, 2002).

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing, 1st ed. (Prentice Hall, 1988).

Jansson, P. A.

P. A. Jansson, Deconvolution: With Applications in Spectroscopy (Academic, 1984).

Kak, A. C.

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

Keevil, S. F.

D. Atkinson, D. L. G. Hill, P. N. R. Stoyle, P. E. Summers, S. Clare, R. Bowtell, and S. F. Keevil, "Automatic compensation of motion artifacts in MRI," Magn. Reson. Med. 41, 163-170 (1999).
[CrossRef] [PubMed]

Krieger, A. S.

J. P. Hogan, R. A. Gonsalves, and A. S. Krieger, "Micro computed tomography: removal of translational stage backlash," IEEE Trans. Nucl. Sci. 40, 1238-1241 (1993).

Lippmann, T.

F. Beckmann, T. Donath, T. Dose, T. Lippmann, R. V. Martins, J. Metge, and A. Schreyer, "Microtomography using synchrotron radiation at DESY: current status and future developments," in Proc. SPIE 5535, 1-10 (2004).
[CrossRef]

Lu, W.

W. Lu and T. R. Mackie, "Tomographic motion detection and correction directly in sinogram space," Phys. Med. Biol. 47, 1267-1284 (2002).
[CrossRef] [PubMed]

Mackie, T. R.

W. Lu and T. R. Mackie, "Tomographic motion detection and correction directly in sinogram space," Phys. Med. Biol. 47, 1267-1284 (2002).
[CrossRef] [PubMed]

Manduca, A.

K. P. McGee, A. Manduca, J. P. Felmlee, S. J. Riederer, and R. L. Ehman, "Image metric-based correction (autocorrection) of motion effects: analysis of image metrics," J. Magn. Reson Imaging 11, 174-181 (2000).
[CrossRef] [PubMed]

Martins, R. V.

F. Beckmann, T. Donath, T. Dose, T. Lippmann, R. V. Martins, J. Metge, and A. Schreyer, "Microtomography using synchrotron radiation at DESY: current status and future developments," in Proc. SPIE 5535, 1-10 (2004).
[CrossRef]

Martz, H. E.

S. G. Azevedo, D. J. Schneberk, J. P. Fitch, and H. E. Martz, "Calculation of the rotational centers in computed tomography sinograms," IEEE Trans. Nucl. Sci. 37, 1525-1540 (1990).
[CrossRef]

McGee, K. P.

K. P. McGee, A. Manduca, J. P. Felmlee, S. J. Riederer, and R. L. Ehman, "Image metric-based correction (autocorrection) of motion effects: analysis of image metrics," J. Magn. Reson Imaging 11, 174-181 (2000).
[CrossRef] [PubMed]

Metge, J.

F. Beckmann, T. Donath, T. Dose, T. Lippmann, R. V. Martins, J. Metge, and A. Schreyer, "Microtomography using synchrotron radiation at DESY: current status and future developments," in Proc. SPIE 5535, 1-10 (2004).
[CrossRef]

Modersitzki, J.

J. Modersitzki, Numerical Methods for Image Registration, Numerical Mathematics and Scientific Computation (Oxford U. Press, 2004).

Ng, L.

L. Ibáñez, W. Schroeder, L. Ng, and J. Cates, The ITK Software Guide 1.4 (Kitware, 2003).

Prince, J. L.

J. L. Prince and A. S. Willsky, "Hierarchical reconstruction using geometry and sinogram restoration," IEEE Trans. Image Process. 2, 401-416 (1993).
[CrossRef] [PubMed]

Riederer, S. J.

K. P. McGee, A. Manduca, J. P. Felmlee, S. J. Riederer, and R. L. Ehman, "Image metric-based correction (autocorrection) of motion effects: analysis of image metrics," J. Magn. Reson Imaging 11, 174-181 (2000).
[CrossRef] [PubMed]

Schneberk, D. J.

S. G. Azevedo, D. J. Schneberk, J. P. Fitch, and H. E. Martz, "Calculation of the rotational centers in computed tomography sinograms," IEEE Trans. Nucl. Sci. 37, 1525-1540 (1990).
[CrossRef]

Schreyer, A.

F. Beckmann, T. Donath, T. Dose, T. Lippmann, R. V. Martins, J. Metge, and A. Schreyer, "Microtomography using synchrotron radiation at DESY: current status and future developments," in Proc. SPIE 5535, 1-10 (2004).
[CrossRef]

T. Donath, F. Beckmann, R. G. J. C. Heijkants, O. Brunke, and A. Schreyer, "Characterization of polyurethane scaffolds using synchrotron radiation computed microtomography," in Proc. SPIE 5535, 775-782 (2004).
[CrossRef]

Schroeder, W.

L. Ibáñez, W. Schroeder, L. Ng, and J. Cates, The ITK Software Guide 1.4 (Kitware, 2003).

Schulz, R. A.

L. A. Shepp, S. K. Hilal, and R. A. Schulz, "Tuning fork artifact in computerized tomography," Comput. Graph. Image Process. 10, 246-255 (1979).
[CrossRef]

Shannon, C. E.

C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Shepp, L. A.

L. A. Shepp, S. K. Hilal, and R. A. Schulz, "Tuning fork artifact in computerized tomography," Comput. Graph. Image Process. 10, 246-255 (1979).
[CrossRef]

Slaney, M.

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

Stoyle, P. N. R.

D. Atkinson, D. L. G. Hill, P. N. R. Stoyle, P. E. Summers, S. Clare, R. Bowtell, and S. F. Keevil, "Automatic compensation of motion artifacts in MRI," Magn. Reson. Med. 41, 163-170 (1999).
[CrossRef] [PubMed]

Summers, P. E.

D. Atkinson, D. L. G. Hill, P. N. R. Stoyle, P. E. Summers, S. Clare, R. Bowtell, and S. F. Keevil, "Automatic compensation of motion artifacts in MRI," Magn. Reson. Med. 41, 163-170 (1999).
[CrossRef] [PubMed]

Willsky, A. S.

J. L. Prince and A. S. Willsky, "Hierarchical reconstruction using geometry and sinogram restoration," IEEE Trans. Image Process. 2, 401-416 (1993).
[CrossRef] [PubMed]

Bell Syst. Tech. J. (1)

C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Comput. Graph. Image Process. (1)

L. A. Shepp, S. K. Hilal, and R. A. Schulz, "Tuning fork artifact in computerized tomography," Comput. Graph. Image Process. 10, 246-255 (1979).
[CrossRef]

IEEE Trans. Image Process. (1)

J. L. Prince and A. S. Willsky, "Hierarchical reconstruction using geometry and sinogram restoration," IEEE Trans. Image Process. 2, 401-416 (1993).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci. (2)

S. G. Azevedo, D. J. Schneberk, J. P. Fitch, and H. E. Martz, "Calculation of the rotational centers in computed tomography sinograms," IEEE Trans. Nucl. Sci. 37, 1525-1540 (1990).
[CrossRef]

J. P. Hogan, R. A. Gonsalves, and A. S. Krieger, "Micro computed tomography: removal of translational stage backlash," IEEE Trans. Nucl. Sci. 40, 1238-1241 (1993).

J. Magn. Reson Imaging (1)

K. P. McGee, A. Manduca, J. P. Felmlee, S. J. Riederer, and R. L. Ehman, "Image metric-based correction (autocorrection) of motion effects: analysis of image metrics," J. Magn. Reson Imaging 11, 174-181 (2000).
[CrossRef] [PubMed]

Magn. Reson. Med. (1)

D. Atkinson, D. L. G. Hill, P. N. R. Stoyle, P. E. Summers, S. Clare, R. Bowtell, and S. F. Keevil, "Automatic compensation of motion artifacts in MRI," Magn. Reson. Med. 41, 163-170 (1999).
[CrossRef] [PubMed]

Nucl. Instrum. Methods Phys. Res. (2)

L. Grodzins, "Optimum energies for x-ray transmission tomography of small samples," Nucl. Instrum. Methods Phys. Res. 206, 541-545 (1983).
[CrossRef]

L. Grodzins, "Critical absorption tomography of small samples," Nucl. Instrum. Methods Phys. Res. 206, 547-552 (1983).
[CrossRef]

Phys. Med. Biol. (1)

W. Lu and T. R. Mackie, "Tomographic motion detection and correction directly in sinogram space," Phys. Med. Biol. 47, 1267-1284 (2002).
[CrossRef] [PubMed]

Proc. SPIE (3)

A. Brunetti and F. de Carlo, "A robust procedure for determination of center of rotation in tomography," in Proc. SPIE 5535, 652-659 (2004).
[CrossRef]

F. Beckmann, T. Donath, T. Dose, T. Lippmann, R. V. Martins, J. Metge, and A. Schreyer, "Microtomography using synchrotron radiation at DESY: current status and future developments," in Proc. SPIE 5535, 1-10 (2004).
[CrossRef]

T. Donath, F. Beckmann, R. G. J. C. Heijkants, O. Brunke, and A. Schreyer, "Characterization of polyurethane scaffolds using synchrotron radiation computed microtomography," in Proc. SPIE 5535, 775-782 (2004).
[CrossRef]

Prog. Biophys. Mol. Biol. (1)

U. Bonse and F. Busch, "X-ray computed microtomography (μCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
[CrossRef] [PubMed]

Other (7)

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

B. Jähne, Digital Image Processing, 5th ed. (Springer, 2002).

J. Modersitzki, Numerical Methods for Image Registration, Numerical Mathematics and Scientific Computation (Oxford U. Press, 2004).

L. Ibáñez, W. Schroeder, L. Ng, and J. Cates, The ITK Software Guide 1.4 (Kitware, 2003).

P. A. Jansson, Deconvolution: With Applications in Spectroscopy (Academic, 1984).

A. K. Jain, Fundamentals of Digital Image Processing, 1st ed. (Prentice Hall, 1988).

R. H. Huesman, G. T. Gullberg, W. L. Greenberg, and T. F. Budinger, RECLBL Library Users Manual: Donner Algorithms for Reconstruction Tomography (Lawrence Berkeley Laboratory, University of California, 1977).

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Figures (9)

Fig. 1
Fig. 1

(Color online) Scheme of the microtomography setup at a synchrotron radiation source. The rotation axis is aligned parallel to the z axis ( ϑ 0 ) prior to the measurement, such that the rotation axis is perpendicular to the incoming almost parallel x-ray beam and perpendicular to the detector rows simultaneously. In this situation each detector row records the sinogram data of one tomographic slice. The position of the projection of the rotational axis onto the detector (dashed z axis) is described by the parameter t r .

Fig. 2
Fig. 2

(Color online) Definition of the parallel projection of a slice f ( x , y ) under angle θ (left). The corresponding sinogram consists of many projections (right). The position of the projected center of rotation t r corresponds to the vertical line in the sinogram.

Fig. 3
Fig. 3

Reconstructions from the sinogram data of Fig. 2 (left). The slices have been calculated on a 128 × 128 grid using centers of rotation t ̃ r = t r + Δ t r with Δ t r in the range from 4.0 to + 4.0 . Tuning-fork artifacts become visible for Δ t r 0 . They add a systematic error to the noisy background of the reconstruction. Profiles along the dashed lines in the reconstructions are plotted (right). The profiles have been shifted for better illustration. For comparison, the value of f ( x , y ) is 0.2 inside the circle. Even offsets | Δ t r | of only a fraction of a bin give rise to artifacts.

Fig. 4
Fig. 4

(Color online) Two-dimensional center of mass in f ( x , y ) is projected onto the center of mass t c in the one-dimensional projection p θ ( t ) . The center of mass t c as a function of θ describes a sinusoidal movement in the sinogram. For the typical case of a sinogram recorded over the interval θ [ 0 , π [ , this results in half a sinusoidal oscillation of the center of mass ( x c , y c ) .

Fig. 5
Fig. 5

Sinograms p θ i ( t j ) of the four model systems defined in Table 1 (top) and their reconstructions (bottom). Model system: (A) circle, (B) ellipse with gradient, (C) small circles, and (D) two circles of opposite attenuation coefficients. The sinograms have been calculated for 111 projection bins t j and 100 projection angles θ i , equally stepped over the interval [ 0 , π [ . The color scale shows the attenuation in the sinogram linearly ranging from 2.2 to + 2.2 . The reconstructions have been calculated on a 120 × 120 grid. The color scale in the reconstructions has been adapted to the individual systems. In the sinograms, the center of rotation is located five pixels to the left of the sinogram center. In the reconstructions, the center of rotation corresponds to the center of the reconstruction.

Fig. 6
Fig. 6

Value of the image metrics Q I A , Q I N , and Q H plotted as a function of the center of rotation t ̃ r for the model systems A, B, C, and D. The calculation was performed at 0.1 bin resolution around the true center of rotation at t r = 50 . For model system D, which contains negative attenuation, only the metric Q H is defined. In all cases the minimum is found at the true center of rotation.

Fig. 7
Fig. 7

Ideal sinogram of sample system A in Fig. 5 and the same sinogram corrupted by noise and random gradients (top). The bottom line of the sinograms is plotted (bottom).

Fig. 8
Fig. 8

Reconstruction of data sets recorded with projection width of 1536 bins at 720 projection angles at HASYLAB/DESY. (a) Cylindrical bone sample, (b) soil aggregate, (c) polymer foam, (d) bone sample with titanium implant. Data sets c and d have been recorded under nonoptimum conditions, i.e., under low x-ray absorption in c and with strong absorption caused by the implant in d. The minimum and maximum attenuation coefficients per edge length of a pixel and the edge length of a pixel are given for each reconstruction.

Fig. 9
Fig. 9

Value of the image metric Q H plotted as a function of the center of rotation t ̃ r for the four sample systems a, b, c, and d shown in Fig. 8 at 0.1 bin resolution. Metrics Q I A and Q I N return qualitatively the same results (not shown).

Tables (3)

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Table 1 Definition of the Model Systems as Ellipses in Relative Units

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Table 2 Optimum Value of t ̃ r as Determined by the Different Methods for the Model Systems a

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Table 3 Optimum Value of t ̃ r as Determined by the Different Methods for the Sample Systems Shown in Fig. 8

Equations (27)

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p θ ( t ) = f ( x , y ) δ ( x cos θ + y sin θ t ) d x d y ,
x c = f ( x , y ) x d x d y f ( x , y ) d x d y , y c = f ( x , y ) y d x d y f ( x , y ) d x d y .
t c ( θ ) = p θ ( t ) t d t p θ ( t ) d t .
t c ( θ ) = x c cos ( θ ) + y c sin ( θ ) + t r
= r c cos ( θ ϕ c ) + t r .
t c ( θ i ) = 1 m ¯ 0 j p θ i ( t j ) j .
t = t ̃ r t .
Q N C ( A , B ) = i ( A i B i ) [ ( i A i 2 ) ( i B i 2 ) ] 1 2 ,
Q I A ( f ̃ ) = 1 m 0 f ̃ ( x , y ) d x d y ,
m 0 = f ( x , y ) d x d y = p θ ( t ) d t .
f ( x , y ) d x d y = f ( x , y ) d x d y = f ̃ ( x , y ) d x d y f ̃ ( x , y ) d x d y .
Q I A ( f ) Q I A ( f ̃ ) .
Q I A ( f ̃ ) = 1 m ¯ 0 i , j f ̃ ( x i , y i ) ,
m ¯ 0 = Mean i ( j p θ i ( t j ) ) ,
Q I N ( f ̃ ) = 1 m 0 u [ f ̃ ( x , y ) ] f ̃ ( x , y ) d x d y ,
u ( α ) = { 1 : α 0 0 : else } .
Q I N ( f ) Q I N ( f ̃ ) .
Q I N ( f ̃ ) = 1 m ¯ 0 i , j u [ f ̃ ( x i , y j ) ] f ̃ ( x i , y j ) ,
H discrete = k = 1 G p k log 2 ( p k ) ,
p ( g ) = 1 h N i = 1 N K ( g g i h ) ,
K ( u ) = { 1 : u < 1 2 0 : else } .
H = d g p ( g ) log 2 [ h p ( g ) ]
= j = 1 2 N 1 Δ g j p j log 2 ( h p j ) .
Q H ( f ̃ ) = H H max ,
H = j = 1 2 N 1 Δ g j n j h N log 2 ( n j N )
= 1 h N j = 1 2 N 1 Δ g j n j [ log 2 ( n j ) log 2 ( N ) ] .
h = 0.01 1 M i , j p θ i ( t j ) .

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