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Methods Inf Med 2004; 43(04): 320-326
DOI: 10.1055/s-0038-1633875
Original Article
Schattauer GmbH

A Linear Programming Approach to Limited Angle 3D Reconstruction from DSA Projections

S. Weber
1   University of Mannheim, Dept. M&CS, CVGPR-Group, Mannheim, Germany
3   Siemens Medical Solutions, Inc., Forchheim, Germany
,
T. Schüle
1   University of Mannheim, Dept. M&CS, CVGPR-Group, Mannheim, Germany
3   Siemens Medical Solutions, Inc., Forchheim, Germany
,
C. Schnörr
1   University of Mannheim, Dept. M&CS, CVGPR-Group, Mannheim, Germany
,
J. Hornegger
2   Friedrich-Alexander University Erlangen-Nürnberg, Dept. CS, Chair of Pattern Recognition, Nürnberg, Germany
› Author Affiliations
Further Information

Publication History

Publication Date:
05 February 2018 (online)

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Summary

Objectives: We investigate the feasibility of binary-valued 3D tomographic reconstruction using only a small number of projections acquired over a limited range of angles.

Methods: Regularization of this strongly ill-posed problem is achieved by (i) confining the reconstruction to binary vessel/non-vessel decisions, and (ii) by minimizing a global functional involving a smoothness prior.

Results: Our approach successfully reconstructs volumetric vessel structures from three projections taken within 90°. The percentage of reconstructed voxels differing from ground truth is below 1%.

Conclusion: We demonstrate that for particular applications – like Digital Subtraction Angiography – 3D reconstructions are possible where conventional methods must fail, due to a severely limited imaging geometry. This could play an important role for dose reduction and 3D reconstruction using non-conventional technical setups.

Keywords

X-ray tomography - computer-assisted image analysis - image reconstruction - computer-generated 3D imaging - digital subtraction angiography
 

  • References

  • 1 Natterer F, Wübbeling F. Mathematical methods in image reconstruction. SIAM, Philadelphia. 2001
    Google Scholar
  • 2 Herman GT, Kuba A. (eds) Discrete tomography: Foundations, algorithms, and applications. Birkhäuser; 1999
    Google Scholar
  • 3 Kuba A, Herman GT. Discrete tomography: A historical overview. In Herman GT, Kuba A. editors Discrete Tomography. Birkhäuser; 1999: 3-34.
    Google Scholar
  • 4 Gardner RJ, Gritzmann P. Discrete tomography: Determination of finite sets by x-rays. Transactions of the American Mathematical Society 1997; 349 (06) 2271-95.
    CrossrefPubMedGoogle Scholar
  • 5 Gritzmann P, Prangenberg D, Vries S, Wiegelmann M. Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography. International Journal of Imaging Systems and Technology 1998; 9: 101-9.
    CrossrefPubMedGoogle Scholar
  • 6 Fishburn P, Schwander P, Shepp L, Vanderbei R. The discrete radon transform and its approximate inversion via linear programming. Discrete Applied Mathematics 1997; 75: 39-61.
    CrossrefPubMedGoogle Scholar
  • 7 Gritzmann P, Vries S, Wiegelmann M. Approximating binary images from discrete x-rays. SIAM Journal on Optimization 2000; 11 (02) 522-46.
    CrossrefPubMedGoogle Scholar
  • 8 Bertero M, Poggio T, Torre V. Ill-posed problems in early vision. Proceedings of the IEEE 1998; 76: 869-89.
    PubMedGoogle Scholar
  • 9 Special issue on variational and level set methods in computer vision. International Journal on Computer Vision 2002 (Dec.).
    PubMedGoogle Scholar
  • 10 Kleinberg JM, Tardos E. Approximation algorithms for classification problems with pairwise relationships: Metric labeling and Markov random fields. IEEE Symposium on Foundations of Computer Science. 1999: 14-23.
    Google Scholar
  • 11 Matej S, Herman GT, Vardi A. Binary tomography on the hexagonal grid using gibbs priors. International Journal of Imaging Systems and Technology 1998; 9: 126-31.
    CrossrefPubMedGoogle Scholar
  • 12 Chan MT, Herman GT, Levitan E. Bayesian image reconstruction using image-modeling gibbs priors. International Journal of Imaging Systems and Technology 1998; 9: 85-98.
    CrossrefPubMedGoogle Scholar
  • 13 Frese T, Bouman CA, Sauer K. Multiscale bayesian methods for discrete tomography. In: Discrete tomography. Herman GT, Kuba A. (eds) Birkhäuser; 1999: 237-264.
    Google Scholar
  • 14 Censor Y. Binary steering in discrete tomography reconstruction with sequential and simultaneous iterative algorithms. Linear Algebra and its Applications 2001; 339: 111-24.
    CrossrefPubMedGoogle Scholar
  • 15 Censor Y, Zenios SA. Parallel optimization: Theory, algorithms, and applications. New York: Oxford University Press; 1998
    Google Scholar
  • 16 Lauritsch G, Haerer W. Theoretical framework for filtered back projection in tomosynthesis. Proceedings SPIE Conference In Image Processing 3338, San Diego. 1998: 1127
    Google Scholar