Computing digitized voronoi diagrams on a systolic screen and applications to clustering

  • Frank Dehne
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 401)

Abstract

A systolic screen of size M is a √M × √M mesh-of-processors where each processing element Pij represents the pixel (i,j) of a digitized plane П of √M × √M pixels. In this paper we study the computation of the Voronoi diagram of a set of n planar objects represented by disjoint images contained in П. We present O(√M) time algorithms to compute the Voronoi diagram for a large class of object types (e.g., points, line segments, circles, ellipses, and polygons of constant size) and distance functions (e.g., all Lp metrices).

Since the Voronoi diagram is used in many geometric applications, the above result has numerous consequences for the design of efficient image processing algorithms on a systolic screen. We obtain, e.g., an O(√M) time systolic screen algorithm for "optical clustering"; i.e., identifying those groups of objects in a digitized picture that are "close" in the sense of human perception.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AH86]
    M.J. Atallah, S.E. Hambrusch, "Solving tree problems on a mesh-connected processor array", Information and Control, Vol.69, Nos. 1–3, 1986, pp. 168–186.CrossRefGoogle Scholar
  2. [AK84]
    M.J. Atallah, S.R. Kosaraju, "Graph problems on a mesh-connected processor array", J. of the ACM, Vol.31:3, 1984, pp.649–667.CrossRefGoogle Scholar
  3. [De86]
    F. Dehne, "Optical clustering", The Visual Computer 2:1, 1986, pp. 39–43.CrossRefGoogle Scholar
  4. [DHSS87]
    F. Dehne, A. Hassenklover, J.-R. Sack, and N. Santoro, "Parallel visibility on a mesh-connected parallel computer", in Proc. International Conference on Parallel Processing and Applications, L'Aquila (Italy), 1987, North Holland 1988, pp. 203–210.Google Scholar
  5. [DSS87]
    F. Dehne, J.-R. Sack and N. Santoro, "Computing on a systolic screen: hulls, contours and applications", in Proc. Conference on Parallel Architectures and Languages Europe, Eindhoven (The Netherland), 1987, Vol. 1, Lecture Notes in Computer Science 258, Springer Verlag, pp. 121–133.Google Scholar
  6. [Ki82]
    C.E. Kim, "Digital disks", Report CS-82-104, Computer Science Dept., Washington State University, Dec. 1982.Google Scholar
  7. [Mi84]
    P.L. Mills, "The systolic pixel: A visible surface algorithm for VLSI", Computer Graphics Forum 3, 1984, pp.47–60.Google Scholar
  8. [Mo70]
    G.U. Montanari, "On limit properties of digitization schemes", J. ACM 17, 1970, pp 348–360.CrossRefGoogle Scholar
  9. [MS85]
    R. Miller and Q.F. Stout, "Geometric algorithms for digitised pictures on a mesh-connected computer", IEEE Trans. on PAMI 7:2, 1985, pp.216–228.Google Scholar
  10. [NS80]
    D. Nassimi and S. Sahni, "Finding connected components and connected ones on a mesh-connected parallel computer", SIAM J. Computing 9:4, 1980, pp.744–757.CrossRefGoogle Scholar
  11. [Re84]
    A.P. Reeves, "Survey parallel computer architectures for image processing", Computer Vision, Graphics, and Image Processing 25, 1984, pp.68–88.Google Scholar
  12. [Ro79]
    A. Rosenfeld, "Digital topology", Amer. Math. Monthly 86, 1979, pp 621–630.Google Scholar
  13. [S88]
    O.Schwarzkopf, "Parallel computation of discrete Voronoi diagrams", Tech. Rep., Fachbereich Mathematik, Freie Universität Berlin (W.-Germany), 1988.Google Scholar
  14. [SH75]
    M.I.Shamos, D.Hoey, "Closest Point Problems", Proc. 7th Ann. IEEE Symp. on Found. of Comp. Sci., 1975.Google Scholar
  15. [SM84]
    Q.F.Stout, R.Miller, "Mesh-connected computer algorithms for determining geometric properties of figures", in Proc. 7th Int. Conf. on Pattern Recognition, Montreal, 1984, pp.475–477.Google Scholar
  16. [Un58]
    S.H. Unger, "A computer oriented towards spatial problems", Proc. IRE 46, 1958, pp.1744–1750.Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Frank Dehne
    • 1
  1. 1.Center for Parallel and Distributed Computing School of Computer ScienceCarleton UniversityOttawaCanada

Personalised recommendations