On the Non-additive Sets of Uniqueness in a Finite Grid

  • Sara Brunetti
  • Paolo Dulio
  • Carla Peri
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7749)

Abstract

In Discrete Tomography there is a wide literature concerning (weakly) bad configurations. These occur in dealing with several questions concerning the important issues of uniqueness and additivity. Discrete lattice sets which are additive with respect to a given set S of lattice directions are uniquely determined by X-rays in the direction of S. These sets are characterized by the absence of weakly bad configurations for S. On the other side, if a set has a bad configuration with respect to S, then it is not uniquely determined by the X-rays in the directions of S, and consequently it is also non-additive. Between these two opposite situations there are also the non-additive sets of uniqueness, which deserve interest in Discrete Tomography, since their unique reconstruction cannot be derived via the additivity property. In this paper we wish to investigate possible interplays among such notions in a given lattice grid \(\mathcal{A}\), under X-rays taken in directions belonging to a set S of four lattice directions.

2000 Mathematics Subject Classification

Primary 05D05 Secondary 05A17 11P81 

Keywords

Additivity bad-configuration reconstruction uniqueness weakly bad configuration 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sara Brunetti
    • 1
  • Paolo Dulio
    • 2
  • Carla Peri
    • 3
  1. 1.Dipartimento di Scienze Matematiche e InformaticheUniversità di SienaSienaItaly
  2. 2.Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoMilanoItaly
  3. 3.Università Cattolica S.C.PiacenzaItaly

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