Abstract
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.
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Alpers, A.: Instability and Stability in Discrete Tomography, PhD thesis, Tech-nische Universität München, Shaker Verlag, ISBN 3-8322-2355-X (2003)
Alpers, A., Gritzmann, P.: On stability, error correction and noise compensation in discrete tomography (in preparation)
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)
Brualdi, R.A.: Matrices of zeros and ones with fixed row and column sum vectors. Linear Algebra Appl. 33, 159–231 (1980)
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)
Brunetti, S., Daurat, A.: Stability in discrete tomography: Some positive results. To appear in Discrete Appl. Math.
Haber, R.M.: Term rank of 0,1 matrices. Rend. Sem. Mat. Univ. Padova 30, 24–51 (1960)
Herman, G.T., Kuba, A.: Discrete tomography: Foundations, algorithms and applications. Birkhäuser, Basel (1999)
Kuba, A.: Determination of the structure class \(\cal{A}(R,S)\) of (0,1)-matrices. Acta Cybernet 9-2, 121–132 (1989)
Lorentz, G.G.: A problem of plane measure. Amer. J. Math. 71, 417–426 (1949)
Matej, S., Vardi, A., Herman, G.T., Vardi, E.: Binary tomography using Gibbs priors. In: Discrete tomography: Foundations, algorithms and applications, ch. 8 (1999)
Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Can. J. Mathematics 9, 371–377 (1957)
Ryser, H.J.: The term rank of a matrix. Canad. J. Math. 10, 57–65 (1958)
Ryser, H.J.: Matrices of zeros and ones. Bull. Amer. Math. 66, 442–464 (1960)
Valenti, C.: An experimental study of the stability problem in discrete tomography. Electron. Notes Discrete Math. 12 (2003)
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Alpers, A., Brunetti, S. (2005). On the Stability of Reconstructing Lattice Sets from X-rays Along Two Directions. In: Andres, E., Damiand, G., Lienhardt, P. (eds) Discrete Geometry for Computer Imagery. DGCI 2005. Lecture Notes in Computer Science, vol 3429. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31965-8_9
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DOI: https://doi.org/10.1007/978-3-540-31965-8_9
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