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Discrete Tomography: Reconstruction under Periodicity Constraints

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Automata, Languages and Programming (ICALP 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2380))

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Abstract

This paper studies the problem of reconstructing binary matrices that are only accessible through few evaluations of their discrete X-rays. Such question is prominently motivated by the demand in material science for developing a tool for the reconstruction of crystalline structures from their images obtained by high-resolution transmission electron microscopy. Various approaches have been suggested for solving the general problem of reconstructing binary matrices that are given by their discrete X-rays in a number of directions, but more work have to be done to handle the ill-posedness of the problem. We can tackle this ill-posedness by limiting the set of possible solutions, by using appropriate a priori information, to only those which are reasonably typical of the class of matrices which contains the unknown matrix that we wish to reconstruct. Mathematically, this information is modelled in terms of a class of binary matrices to which the solution must belong. Several papers study the problem on classes of binary matrices on which some connectivity and convexity constraints are imposed.

We study the reconstruction problem on some new classes consisting of binary matrices with periodicity properties, and we propose a polynomial-time algorithm for reconstructing these binary matrices from their orthogonal discrete X-rays.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Siena, Via del Capitano 15, 53100, Siena, Italy

    Alberto Del Lungo & Andrea Frosini

  2. Laboratoire d’Informatique, Algorithmique, Fondements et Applications (LIAFA), Université Denis Diderot, 2, place Jussieu, 75251, Paris Cedex 05, France

    Maurice Nivat & Laurent Vuillon

Authors
  1. Alberto Del Lungo
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  2. Andrea Frosini
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  3. Maurice Nivat
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  4. Laurent Vuillon
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Editor information

Editors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zentrum, ETH Zürich, 8092, Zürich, Switzerland

    Peter Widmayer  & Stephan Eidenbenz  & 

  2. Department of Languages and Sciences of the Computation E.T.S. de Ingeniería Informática, University of Málaga, Campus de Teatinos, 29071, Málaga, Spain

    Francisco Triguero , Rafael Morales  & Ricardo Conejo ,  & 

  3. School of Cognitive and Computing Sciences, University of Sussex, Falmer, Brighton, BN1 9QN, UK

    Matthew Hennessy

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© 2002 Springer-Verlag Berlin Heidelberg

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Del Lungo, A., Frosini, A., Nivat, M., Vuillon, L. (2002). Discrete Tomography: Reconstruction under Periodicity Constraints. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds) Automata, Languages and Programming. ICALP 2002. Lecture Notes in Computer Science, vol 2380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45465-9_5

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  • DOI: https://doi.org/10.1007/3-540-45465-9_5

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  • Print ISBN: 978-3-540-43864-9

  • Online ISBN: 978-3-540-45465-6

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