Skip to main content
Log in

Computational geometry algorithms for the systolic screen

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Adigitized plane Π of sizeM is a rectangular √M × √M array of integer lattice points called pixels. A √M × √M mesh-of-processors in which each processorP ij represents pixel (i,j) is a natural architecture to store and manipulate images in Π; such a parallel architecture is called asystolic screen. In this paper we consider a variety of computational-geometry problems on images in a digitized plane, and present optimal algorithms for solving these problems on a systolic screen. In particular, we presentO(√M)-time algorithms for determining all contours of an image; constructing all rectilinear convex hulls of an image (peeling); solving the parallel and perspective visibility problem forn disjoint digitized images; and constructing the Voronoi diagram ofn planar objects represented by disjoint images, for a large class of object types (e.g., points, line segments, circles, ellipses, and polygons of constant size) and distance functions (e.g., allL p metrics). These algorithms implyO(√M)-time solutions to a number of other geometric problems: e.g., rectangular visibility, separability, detection of pseudo-star-shapedness, and optical clustering. One of the proposed techniques also leads to a new parallel algorithm for determining all longest common subsequences of two words.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Aggarwal, B. Chazelle, L. Guibas, C. O'Dunlaing, C. Yap, Parallel computational geometry,Algorithmica,3(3), 1988, 293–327.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. J. Atallah, S. R. Kosaraju, Graph problems on a mesh-connected processor array,Journal of the ACM,31(3), 1984, 649–667.

    Article  MATH  MathSciNet  Google Scholar 

  3. B. M. Chazelle, Optimal algorithms for computing depths and layers, inProc. 21st Allerton Conf. on Communication, Control and Computing, Urbana-Champaign, Ill., 1983, 427–436.

    Google Scholar 

  4. B. Chazelle, L. J. Guibas, D. T. Lee, The power of geometric duality, inProc 24th IEEE Symp. on Foundations of Computor Science, Tucson, Ariz., 1983.

  5. L. P. Chew, R. L. Drysdale, Voronoi diagrams based on convex distance functions, inProc. Symp. on Computational Geometry, Baltimore, 1985, 235–244.

  6. J. A. Dean, A. Lingas, J.-R. Sack, Recognizing Polygons: or How To Eavesdrop,The Visual Computer,3(6), 1988, 344–355.

    Article  MATH  Google Scholar 

  7. F. Dehne, Optical clustering,The Visual Computer,21(1), 1986, 39–43.

    Article  Google Scholar 

  8. F. Dehne,O(n1/2) algorithms for the maximal elements and ECDF searching problem on a mesh-conneced parallel computer,Information Processing Letters,22(6), 1986, 303–306.

    Article  MathSciNet  Google Scholar 

  9. F. Dehne, Solving visibility and separability problems on a mesh-of-processors,The Visual Computer,4(6), 1988, 356–370.

    Article  Google Scholar 

  10. F. Dehne, Computing the largest empty rectangle on one- and two-dimensional processor arrays,Journal of Parallel and Distributed Computing,9(1), 1990, 63–68.

    Article  MathSciNet  Google Scholar 

  11. F. Dehne, A. Hassenklover, J.-R. Sack, Computing the configuration space for a robot on a mesh-of-processors,Parallel Computing,12, 1989, 221–231.

    Article  MATH  MathSciNet  Google Scholar 

  12. F. Dehne, A. Hassenklover, J.-R. Sack, N. Santoro, Parallel visibility on a mesh-connected parallel computer, inProc. Int. Conf. on Parallel Processing and Applications, L'Aquila, Italy, 1987, 173–180.

    Google Scholar 

  13. R. Dubes, A. K. Jain, Clustering methodologies in exploratory data analysis, inAdvances in Computers, Vol. 19 (M. C. Yovits, ed.), 1980, 113–228.

  14. F. Dehne, Q. T. Pham, Visibility algorithms for binary images on the hypercube and the perfect-shuffle computer, inProc. IFIP WG 10.3 Conf. on Parallel processing, Pisa, 1988, North Holland, Amsterdam, 117–124.

    Google Scholar 

  15. F. Dehne, J.-R. Sack, Translation separability of sets of polygons,The Visual Computer,3(4), 1987, 227–235.

    Article  MATH  Google Scholar 

  16. F. Dehne, J.-R. Sack, Parallel computational geometry: a survey, invited paper, inProc. Parcella '88, Berlin, Lecture Notes in Computer Science, Vol. 342, Springer-Verlag, Berlin, 1988, 73–89.

    Google Scholar 

  17. F. Dehne, J.-R. Sack, N. Santoro, Computing on a systolic screen: hulls, contours and applications, inProc. Conf. on Parallel Architectures and Languages, Eindhoven, The Netherland, 1987, Lecture Notes in Computer Science, Vol. 258, Springer-Verlag, Berlin, 121–133.

    Google Scholar 

  18. M. J. Duffet al., A cellular logic array for image processing,Pattern Recognition,5, 1973, 229–247.

    Article  Google Scholar 

  19. R. H. Güting, O. Nurmi, T. Ottmann, The direct dominance problem, inProc. ACM Symp. on Computational Geometry, Baltimore, 1985, 81–88.

  20. D. Hillis,The Connection Machine, MIT Press, Cambridge, Mass., 1985.

    Google Scholar 

  21. D. S. Hirschberg, Algorithms for the longest common subsequence problem,Journal of the ACM,24(4), 1977, 664–675.

    Article  MATH  MathSciNet  Google Scholar 

  22. P. J. Huber, Robust statistics: a review,Annals of Mathematical Statistics,43(3), 1972, 1041–1067.

    Article  MATH  MathSciNet  Google Scholar 

  23. J. W. Hunt, T. G. Szymanski, A fast algorithm for computing longest common subsequences,Communications of the ACM,20, 1977, 350–353.

    Article  MATH  MathSciNet  Google Scholar 

  24. C. E. Kim, Digital Disks, Report CS-82-104, Computer Science Department, Washington State University, Dec. 1982.

  25. D. T. Lee, F. P. Preparata, Computational geometry—a survey,IEEE Transactions on Computers,33(12), 1984, 1072–1101.

    Article  MathSciNet  Google Scholar 

  26. B. H. McCormick, The Illinois pattern recognition computer—ILLIAC III,IEEE Transactions on Electronic Computers,12, 1963, 791–813.

    Article  Google Scholar 

  27. G. U. Montanari, On limit properties of digitzation schemes,Journal of the ACM,17, 1970, 348–360.

    Article  MATH  MathSciNet  Google Scholar 

  28. M. F. Montuno, A. Fournier, Finding thexy Convex Hull of a Set ofxy Polygons, Technical Report CSRG-148, University of Toronto, Toronto, Nov. 1982.

    Google Scholar 

  29. R. Miller, Q. F. Stout, Computational geometry on a mesh-connected computer, inProc. Int. Conf. on Parallel Processing, 1984, 66–73.

  30. R. Miller, Q. F. Stout, Geometric algorithms for digitised pictures on a mesh-connected computer,IEEE Transactions on Pattern Analysis and Machine Intelligence,7(2), 1985, 216–228.

    Article  Google Scholar 

  31. J. I. Munro, M. H. Overmars, D. Wood, Variations on visibility, inProc. ACM Symp. on Computational Geometry, Waterloo, Canada, 1987, 291–299.

    Google Scholar 

  32. N. Nakatsu, Y. Kambayashi, S. Yajima, A longest common subsequence algorithm suitable for similar text strings,Acta Informatica,18, 1982, 171–179.

    Article  MATH  MathSciNet  Google Scholar 

  33. D. Nassimi, S. Sahni, Finding connected components and connected ones on a mesh-connected parallel computer,SIAM Journal on Computing,9(4), 1980, 744–757.

    Article  MATH  MathSciNet  Google Scholar 

  34. O. Nurmi, J.-R. Sack, Separating a polyhedron by one translation from a set of obstacles, inProc. Workshop on Graph-Theoretic Concepts in Computer Science, Amsterdam, The Netherlands, June 1988, (J. van Leeuwen, ed.), Lecture Notes in Computer Science, Vol. 344, Springer-Verlag, Berlin, 202–212.

    Google Scholar 

  35. M. H. Overmars, J. van Leeuwen, Maintenance of configurations in the plane,Journal of Computer and System Sciences,23, 1981, 166–204.

    Article  MATH  MathSciNet  Google Scholar 

  36. F. P. Preparata, M. I. Shamos,Computational Geometry, An Introduction, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  37. A. P. Reeves, Survey parallel computer architectures for image processing,Computer Vision, Graphics and Image Processing,25, 1984, 68–88.

    Article  Google Scholar 

  38. Y. Robert, M. Tchuente, A systolic array for the longest common subsequence problem,Information Processing Letters,21, 1985, 191–198.

    Article  MATH  MathSciNet  Google Scholar 

  39. A. Rosenfeld, Digital topology,The American Mathematical Monthly,86, 1979, 621–630.

    Article  MATH  MathSciNet  Google Scholar 

  40. A. Rosenfeld, Parallel image processing using cellular arrays,IEEE Transactions on Computers,16(1), 1983, 15–20.

    Google Scholar 

  41. J.-R. Sack, A simple hidden-line algorithm for rectilinear-polygons, inProc. 21st Allerton Conf. on Communication, Control and Computing, Urbana-Champaign, Ill., Oct. 1983, 437–446.

    Google Scholar 

  42. J.-R. Sack, Rectilinear Computational Geometry, Ph.D. thesis, McGill University, Montréal, 1984.

    Google Scholar 

  43. O. Schwarzkopf, Parallel computation of discrete Voronoi diagrams, inProc. Symp. on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Vol. 349, Springer-Verlag, Berlin, 1989, 193–204.

    Google Scholar 

  44. M. I. Shamos, Computational Geometry, Ph.D. thesis, Department of Computer Science, Yale University, 1978.

  45. M. I. Shamos, D. Hoey, Closest point problems, inProc. 7th Ann. IEEE Symp. on Foundations of Computer Science, 1975.

  46. Q. F. Stout, R. Miller, Mesh-connected computer algorithms for determining geometric properties of figures, inProc. 7th Int. Conf. on Pattern Recognition, Montréal, 1984, 475–477.

  47. C. D. Thompson, H. T. Kung, Sorting on a mesh-connected parallel computer,Communications of the ACM,20(4), 1977, 263–271.

    Article  MATH  MathSciNet  Google Scholar 

  48. G. T. Toussaint, Movable separability of sets, inComputational Geometry, (G. T. Toussaint, ed.), North-Holland, Amsterdam, 1985, 335–376.

    Google Scholar 

  49. S. H. Unger, A computer oriented towards spatial problems,Proceedings of the IRE,46, 1958, 1744–1750.

    Article  Google Scholar 

  50. D. Wood, An isothetic view on computational geometry, inComputational Geometry, (G. T. Toussaint, ed.), North-Holland, Amsterdam, 1985, 429–459.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by Frank Dehne.

Research supported by the Naural Sciences and Engineering Research Council of Canada. With the Editor-in-Chief's permission, this paper was sent to the referees in a form which kept them unaware of the fact that the Guest Editor is one of the co-authors.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dehne, F., Hassenklover, A.L., Sack, J.R. et al. Computational geometry algorithms for the systolic screen. Algorithmica 6, 734–761 (1991). https://doi.org/10.1007/BF01759069

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01759069

Key words

Navigation