Ambiguity Results in the Characterization of hv-convex Polyominoes from Projections

  • Elena Barcucci
  • Paolo Dulio
  • Andrea Frosini
  • Simone Rinaldi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10502)


In 1997 R. Gardner and P. Gritzmann proved a milestone result for uniqueness in Discrete Tomography: a finite convex discrete set can be uniquely determined by projections taken in any set of seven planar directions. The number of required directions can be reduced to 4, providing their cross-ratio, arranged in order of increasing angle with the positive x-axis, does not belong to the set \(\{4/3, 3/2, 2, 3, 4\}\).

Later studies, supported by experimental evidence, allow us to conjecture that a similar result may also hold for the wider class of hv-convex polyominoes.

In this paper we shed some light on the differences between these two classes, providing new 4-tuples of discrete directions that do not lead to a unique reconstruction of hv-convex polyominoes. We reach our main result by a constructive process. This generates switching components along four directions by a recursive composition of only three of them, and then by shifting the obtained structure along the fourth one.

Furthermore, we stress the role that the horizontal and the vertical directions have in preserving the hv-convexity property. This is pointed out by showing that these often appear in the 4-tuples of directions that allow uniqueness.

A final characterization theorem for hv-convex polyominoes is still left as open question.


Discrete geometry Discrete tomography hv-convex set Uniqueness problem Switching component 



This study has been partially supported by INDAM - GNCS Project 2016.


  1. 1.
    Alpers, A., Larman, D.G.: The smallest sets of points not determined by their X-rays. Bull. London Math. Soc. 47, 171–176 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alpers, A., Tijdeman, R.: The two-dimensional Prouhet-Tarry-Escott problem. J. Number Theor. 123, 403–412 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brunetti, S., Dulio, P., Hajdu, L., Peri, C.: Ghosts in discrete tomography. J. Math. Imaging Vis. 53(2), 210–224 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: X-rays characterizing some classes of discrete sets. Linear Algebra Appl. 339, 3–21 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cipolla, M., Lo Bosco, G., Millonzi, F., Valenti, C.: An island strategy for memetic discrete tomography reconstruction. Inform. Sci. 257, 357–368 (2104)Google Scholar
  6. 6.
    Dulio, P.: Convex decomposition of \(U\)-polygons. Theor. Comput. Sci. 406, 80–89 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dulio, P., Peri, C.: On the geometric structure of lattice U-polygons. Discrete Math. 307, 2330–2340 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fishburn, P., Lagarias, J., Reeds, J., Shepp, L.: Sets uniquely determined by projections on axes II. Discrete Appl. Math. 91, 149–159 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Castiglione, G., Frosini, A., Restivo, A., Rinaldi, S.: Enumeration of L-convex polyominoes by rows and columns. Theor. Comput. Sci. 347(1-2), 336–352 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gardner, R.J., Gritzmann, P.: Discrete tomography: determination of finite sets by X-rays. Trans. Amer. Math. Soc. 349, 2271–2295 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gardner, R., Gritzmann, P.: Uniqueness and complexity in discrete tomography. In: Herman, G., Kuba, A. (eds.) Discrete Tomography: Foundations, Algorithms, and Applications, pp. 85–113. Birkhäuser, Basel (1999)CrossRefGoogle Scholar
  12. 12.
    Herman, G.T., Kuba, A. (eds.): Discrete tomography: Foundations algorithms and applications. Birkhauser, Boston (1999)zbMATHGoogle Scholar
  13. 13.
    Kuba, A.: Reconstruction of unique binary matrices with prescribed elements. Acta Cybern. 12, 57–70 (1995)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ryser, H.: Combinatorial Mathematics, The Carus Mathematical Monographs No. 14, The Mathematical Association of America, Rahway (1963)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Elena Barcucci
    • 1
  • Paolo Dulio
    • 2
  • Andrea Frosini
    • 1
  • Simone Rinaldi
    • 3
  1. 1.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFlorenceItaly
  2. 2.Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoMilanItaly
  3. 3.Dipartimento di Ingegneria dell’Informazione e Scienze MatematicheUniversità di SienaSienaItaly

Personalised recommendations