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Binary Matrices Under the Microscope: A Tomographical Problem

  • Andrea Frosini
  • Maurice Nivat
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3322)

Abstract

A binary matrix can be scanned by moving a fixed rectangular window (sub-matrix) across it, rather like examining it closely under a microscope. With each viewing, a convenient measurement is the number of 1s visible in the window, which might be thought of as the luminosity of the window. The rectangular scan of the binary matrix is then the collection of these luminosities presented in matrix form. We show that, at least in the technical case of a smoothm × n binary matrix, it can be reconstructed from its rectangular scan in polynomial time in the parameters m and n, where the degree of the polynomial depends on the size of the window of inspection. For an arbitrary binary matrix, we then extend this result by determining the entries in its rectangular scan that preclude the smoothness of the matrix.

Keywords

Discrete Tomography Reconstruction algorithm Computational complexity Projection Rectangular scan 

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References

  1. 1.
    Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations Algorithms and Applications. Birkhauser, Boston (1999)zbMATHGoogle Scholar
  2. 2.
    Nivat, M.: Sous-ensembles homogénes de ℤ2 et pavages du plan, vol. Ser. I 335, pp. 83–86. C. R. Acad. Sci., Paris (2002)Google Scholar
  3. 3.
    Ryser, H.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math. 9, 371–377 (1957)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Tijdeman, R., Hadju, L.: An algorithm for discrete tomography. Linear Algebra and Its Applications 339, 147–169 (2001)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Andrea Frosini
    • 1
  • Maurice Nivat
    • 2
  1. 1.Dipartimento di Scienze Matematiche ed Informatiche “Roberto Magari”Università degli Studi di SienaSienaItaly
  2. 2.Laboratoire d’Informatique, Algorithmique, Fondements et Applications (LIAFA)Université Denis DiderotParis 5Cedex 05France

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