Advertisement

Incremental Algorithms Based on Discrete Green Theorem

  • Srečko Brlek
  • Gilbert Labelle
  • Annie Lacasse
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2886)

Abstract

By using the discrete version of Green’s theorem and bivariate difference calculus we provide incremental algorithms to compute various statistics about polyominoes given, as input, by 4-letter words describing their contour. These statistics include area, coordinates of the center of gravity, moment of inertia, higher order moments, size of projections, hook lengths, number of pixels in common with a given set of pixels and also q-statistics.

Keywords

Discrete Green Theorem statistics about polyominoes 

References

  1. 1.
    Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Reconstructing convex polyominoes from their vertical and horizontal projections. Theoret. Comput. Sci. 155, 321–347 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bousquet-Mélou, M.: New enumerative results on two-dimensional directed animals. Discrete Math 180(1-3), 73–106 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brlek, S., Labelle, G., Lacasse, A.: Incremental Algorithms for Polyominoes Coded by their Contour, Research Report, Lacim, Un. Quebec à, Montréal (2003)Google Scholar
  4. 4.
    Clarke, A.L.: Isometrical polyominoes. J. Recreational Math. 13, 18–25 (1980)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Comtet, L.: Advanced Combinatorics. Reidel, Dordrechtz (1974)zbMATHGoogle Scholar
  6. 6.
    Delest, M.P., Gouyou-Beauchamps, D., Vauquelin, B.: Enumeration of parallelogram polyominoes with given bound and site perimeter. Graphs Comb. 3, 325–339 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Del Lungo, A.: Polyominoes defined by two vectors. Theoret. Comput. Sci. 127(1), 187–198 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Freeman, H.: On the Encoding of Arbitrary Geometric Configurations, IRE Trans. Electronic Computer 10, 260–268 (1961)MathSciNetGoogle Scholar
  9. 9.
    Freeman, H.: Boundary encoding and processing. In: Lipkin, B.S., Rosenfeld, A. (eds.) Picture Processing and Psychopictorics, pp. 241–266. Academic Press, New York (1970)Google Scholar
  10. 10.
    Golomb, S.W.: Polyominoes: Puzzles, Patterns, Problems, and Packings. Princeton University Press, Princeton (1996)Google Scholar
  11. 11.
    Philips, W.: A new fast algorithm for moment computation. Pattern Recognition 26(11), 1619–1621 (1993)CrossRefGoogle Scholar
  12. 12.
    Tang, G.Y., Lien, B.: Region Filling With The Use Of The Discrete Green Theorem. Proc. CVGIP (42), 297–305 (1988)Google Scholar
  13. 13.
    Viennot, X.G.: A survey of polyomino enumeration, Proc. Séries formelles et combinatoire algébrique, Montréal, Juin 1992. Publications de LACIM 11, Université du Québec à Montréal (1996)Google Scholar
  14. 14.
    Yang, L., Albregtsen, F.: Fast computation of invariant geometric moments. A new method giving correct results. In: Proceeding of the International Conference on Pattern Recognition (ICPR 1994), pp. 201–204 (1994)Google Scholar
  15. 15.
    Yang, L., Albregtsen, F.: Fast and exact computation of Cartesian geometric moments using discrete Green’s theorem. Pattern Recognition 29(7), 1061–1073 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Srečko Brlek
    • 1
  • Gilbert Labelle
    • 1
  • Annie Lacasse
    • 1
  1. 1.Laboratoire de Combinatoire et d’Informatique MathématiqueUniversité du Québec à MontréalMontréalCanada

Personalised recommendations