Skip to main content
SpringerLink
Log in

On the Number of hv-Convex Discrete Sets

  • Conference paper
Combinatorial Image Analysis (IWCIA 2008)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 4958))

Included in the following conference series:

Abstract

One of the basic problems in discrete tomography is the reconstruction of discrete sets from few projections. Assuming that the set to be reconstructed fulfills some geometrical properties is a commonly used technique to reduce the number of possibly many different solutions of the same reconstruction problem. The class of hv-convex discrete sets and its subclasses have a well-developed theory. Several reconstruction algorithms as well as some complexity results are known for those classes. The key to achieve polynomial-time reconstruction of an hv-convex discrete set is to have the additional assumption that the set is connected as well. This paper collects several statistics on hv-convex discrete sets, which are of great importance in the analysis of algorithms for reconstructing such kind of discrete sets.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Balázs, P.: A framework for generating some discrete sets with disjoint components using uniform distributions. Theor. Comput. Sci. (submitted)

    Google Scholar 

  2. Balázs, P.: On the ambiguity of reconstructing hv-convex binary matrices with decomposable configurations. Acta Cybernetica (accepted)

    Google Scholar 

  3. Balázs, P., Balogh, E., Kuba, A.: Reconstruction of 8-connected but not 4-connected hv-convex discrete sets. Disc. Appl. Math. 147, 149–168 (2005)

    Article  MATH  Google Scholar 

  4. Balogh, E., Kuba, A., Dévényi, C., Del Lungo, A.: Comparison of algorithms for reconstructing hv-convex discrete sets. Lin. Alg. Appl. 339, 23–35 (2001)

    Article  MATH  Google Scholar 

  5. Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Medians of polyominoes: A property for the reconstruction. Int. J. Imaging Systems and Techn. 9, 69–77 (1998)

    Article  Google Scholar 

  6. Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Reconstructing convex polyominoes from horizontal and vertical projections. Theor. Comput. Sci. 155, 321–347 (1996)

    Article  MATH  Google Scholar 

  7. Barcucci, E., Frosini, A., Rinaldi, S.: On directed-convex polyominoes in a rectangle. Discrete Math. 298, 62–78 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Batenburg, K.J.: An evolutionary algorithm for discrete tomography. Disc. Appl. Math. 151, 36–54 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brunetti, S., Del Lungo, A., Del Ristoro, F., Kuba, A., Nivat, M.: Reconstruction of 4- and 8-connected convex discrete sets from row and column projections. Lin. Alg. Appl. 339, 37–57 (2001)

    Article  MATH  Google Scholar 

  10. Chrobak, M., Dürr, C.: Reconstructing hv-convex polyominoes from orthogonal projections. Inform. Process. Lett. 69(6), 283–289 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dahl, G., Flatberg, T.: Optimization and reconstruction of hv-convex (0,1)-matrices. Disc. Appl. Math. 151, 93–105 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Delest, M.P., Viennot, G.: Algebraic languages and polyominoes enumeration. Theor. Comput. Sci. 34, 169–206 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gessel, I.: On the number of convex polyominoes. Ann. Sci. Math. Québec 24, 63–66 (2000)

    MATH  MathSciNet  Google Scholar 

  14. Golomb, S.W.: Polyominoes. Charles Scriber’s Sons, New York (1965)

    Google Scholar 

  15. Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms and Applications. Birkhäuser, Boston (1999)

    MATH  Google Scholar 

  16. Herman, G.T., Kuba, A. (eds.): Advances in Discrete Tomography and its Applications. Birkhäuser, Boston (2007)

    MATH  Google Scholar 

  17. Kuba, A.: The reconstruction of two-directionally connected binary patterns from their two orthogonal projections. Comp. Vision, Graphics, and Image Proc. 27, 249–265 (1984)

    Article  Google Scholar 

  18. Kuba, A.: Reconstruction in different classes of 2D discrete sets, Lecture Notes in Comput. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 153–163. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  19. Sloane, N.J.A.: The on-line encyclopedia of integer sequences, http://www.research.att.com/~njas/sequences/

  20. Woeginger, G.W.: The reconstruction of polyominoes from their orthogonal projections. Inform. Process. Lett. 77, 225–229 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Image Processing and Computer Graphics, University of Szeged, Árpád tér 2, H-6720, Szeged, Hungary

    Péter Balázs

Authors
  1. Péter Balázs
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Valentin E. Brimkov Reneta P. Barneva Herbert A. Hauptman

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Balázs, P. (2008). On the Number of hv-Convex Discrete Sets. In: Brimkov, V.E., Barneva, R.P., Hauptman, H.A. (eds) Combinatorial Image Analysis. IWCIA 2008. Lecture Notes in Computer Science, vol 4958. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78275-9_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-78275-9_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-78274-2

  • Online ISBN: 978-3-540-78275-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation