On the Stability of Reconstructing Lattice Sets from X-rays Along Two Directions

  • Andreas Alpers
  • Sara Brunetti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3429)


We consider the stability problem of reconstructing lattice sets from their noisy X-rays (i.e. line sums) taken along two directions. Stability is of major importance in discrete tomography because, in practice, these X-rays are affected by errors due to the nature of measurements. It has been shown that the reconstruction from noisy X-rays taken along more than two directions can lead to dramatically different reconstructions. In this paper we prove a stability result showing that the same instability result does not hold for the reconstruction from two directions. We also show that the derived stability result can be carried over by similar techniques to lattice sets with invariant points.


  1. 1.
    Alpers, A.: Instability and Stability in Discrete Tomography, PhD thesis, Tech-nische Universität München, Shaker Verlag, ISBN 3-8322-2355-X (2003)Google Scholar
  2. 2.
    Alpers, A., Gritzmann, P.: On stability, error correction and noise compensation in discrete tomography (in preparation)Google Scholar
  3. 3.
    Alpers, A., Gritzmann, P., Thorens, L.: Stability and instability in discrete tomography. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 175–186. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Brualdi, R.A.: Matrices of zeros and ones with fixed row and column sum vectors. Linear Algebra Appl. 33, 159–231 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Brunetti, S., Daurat, A.: Stability in discrete tomography: Linear programming, additivity and convexity. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 398–407. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Brunetti, S., Daurat, A.: Stability in discrete tomography: Some positive results. To appear in Discrete Appl. Math.Google Scholar
  7. 7.
    Haber, R.M.: Term rank of 0,1 matrices. Rend. Sem. Mat. Univ. Padova 30, 24–51 (1960)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Herman, G.T., Kuba, A.: Discrete tomography: Foundations, algorithms and applications. Birkhäuser, Basel (1999)zbMATHGoogle Scholar
  9. 9.
    Kuba, A.: Determination of the structure class \(\cal{A}(R,S)\) of (0,1)-matrices. Acta Cybernet 9-2, 121–132 (1989)MathSciNetGoogle Scholar
  10. 10.
    Lorentz, G.G.: A problem of plane measure. Amer. J. Math. 71, 417–426 (1949)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Matej, S., Vardi, A., Herman, G.T., Vardi, E.: Binary tomography using Gibbs priors. In: Discrete tomography: Foundations, algorithms and applications, ch. 8 (1999)Google Scholar
  12. 12.
    Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Can. J. Mathematics 9, 371–377 (1957)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ryser, H.J.: The term rank of a matrix. Canad. J. Math. 10, 57–65 (1958)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Ryser, H.J.: Matrices of zeros and ones. Bull. Amer. Math. 66, 442–464 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Valenti, C.: An experimental study of the stability problem in discrete tomography. Electron. Notes Discrete Math. 12 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Andreas Alpers
    • 1
  • Sara Brunetti
    • 2
  1. 1.Zentrum MathematikTechnische Universität MünchenGarching bei MünchenGermany
  2. 2.Dipartimento di Scienze Matematiche e InformaticheUniversitá degli Studi di SienaSienaItaly

Personalised recommendations