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The Number of Line-Convex Directed Polyominoes Having the Same Orthogonal Projections

  • Péter Balázs
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4245)

Abstract

The number of line-convex directed polyominoes with given horizontal and vertical projections is studied. It is proven that diagonally convex directed polyominoes are uniquely determined by their orthogonal projections. The proof of this result is algorithmical. As a counterpart, we show that ambiguity can be exponential if antidiagonal convexity is assumed about the polyomino. Then, the results are generalised to polyominoes having convexity property along arbitrary lines.

Keywords

Discrete tomography line-convex directed polyomino reconstruction from projections 

References

  1. 1.
    Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms and Applications. Birkhäuser, Boston (1999)zbMATHGoogle Scholar
  2. 2.
    Crewe, A.V., Crewe, D.A.: Inexact Reconstruction: Some Improvements. Ultramicroscopy 16, 33–40 (1985)CrossRefGoogle Scholar
  3. 3.
    Gordon, R., Herman, G.T.: Reconstruction of Pictures from their Projections. Graphics Image Process 14, 759–768 (1971)zbMATHGoogle Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    Prause, G.M.P., Onnasch, D.G.W.: Binary Reconstruction of the Heart Chambers from Biplane Angiographic Image Sequences. IEEE Trans. Medical Imaging MI-15, 532–546 (1996)CrossRefGoogle Scholar
  6. 6.
    Schilferstein, A.R., Chien, Y.T.: Switching Components and the Ambiguity Problem in the Reconstruction of Pictures from their Projections. Pattern Recognition 10, 327–340 (1978)CrossRefGoogle Scholar
  7. 7.
    Balázs, P., Balogh, E., Kuba, A.: Reconstruction of 8-Connected but not 4-Connected hv-Convex Discrete Sets. Discrete App. Math. 147, 149–168 (2005)zbMATHCrossRefGoogle Scholar
  8. 8.
    Del Lungo, A.: Polyominoes Defined by Two Vectors. Theor. Comput. Sci. 127, 187–198 (1994)zbMATHCrossRefGoogle Scholar
  9. 9.
    Kuba, A., Balogh, E.: Reconstruction of Convex 2D Discrete Sets in Polynomial Time. Theor. Comput. Sci. 283, 223–242 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Reconstructing Convex Polyominoes from Horizontal and Vertical Projections. Theor. Comput. Sci. 155, 321–347 (1996)zbMATHCrossRefGoogle Scholar
  11. 11.
    Del Lungo, A., Nivat, M., Pinzani, R.: The Number of Convex Polyominoes Recostructible from their Orthogonal Projections. Discrete Math. 157, 65–78 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Feretic, S., Svrtan, D.: Combinatorics of Diagonally Convex Directed Polyominoes. Discrete Math. 157, 147–168 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Golomb, S.W.: Polyominoes. Charles Scriber’s Sons, New York (1965)Google Scholar
  14. 14.
    Balázs, P.: A Decomposition Technique for Reconstructing Discrete Sets from Four Projections. Image and Vision Computing (submitted)Google Scholar
  15. 15.
    Ryser, H.J.: Combinatorial Mathematics. The Carus Mathematical Monographs 14 (1963)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Péter Balázs
    • 1
  1. 1.Department of Computer Algorithms and Artificial IntelligenceUniversity of SzegedSzegedHungary

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