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.
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A. Aggarwal, B. Chazelle, L. Guibas, C. O'Dunlaing, C. Yap, Parallel computational geometry,Algorithmica,3(3), 1988, 293–327.
M. J. Atallah, S. R. Kosaraju, Graph problems on a mesh-connected processor array,Journal of the ACM,31(3), 1984, 649–667.
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.
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.
L. P. Chew, R. L. Drysdale, Voronoi diagrams based on convex distance functions, inProc. Symp. on Computational Geometry, Baltimore, 1985, 235–244.
J. A. Dean, A. Lingas, J.-R. Sack, Recognizing Polygons: or How To Eavesdrop,The Visual Computer,3(6), 1988, 344–355.
F. Dehne, Optical clustering,The Visual Computer,21(1), 1986, 39–43.
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.
F. Dehne, Solving visibility and separability problems on a mesh-of-processors,The Visual Computer,4(6), 1988, 356–370.
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.
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.
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.
R. Dubes, A. K. Jain, Clustering methodologies in exploratory data analysis, inAdvances in Computers, Vol. 19 (M. C. Yovits, ed.), 1980, 113–228.
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.
F. Dehne, J.-R. Sack, Translation separability of sets of polygons,The Visual Computer,3(4), 1987, 227–235.
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.
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.
M. J. Duffet al., A cellular logic array for image processing,Pattern Recognition,5, 1973, 229–247.
R. H. Güting, O. Nurmi, T. Ottmann, The direct dominance problem, inProc. ACM Symp. on Computational Geometry, Baltimore, 1985, 81–88.
D. Hillis,The Connection Machine, MIT Press, Cambridge, Mass., 1985.
D. S. Hirschberg, Algorithms for the longest common subsequence problem,Journal of the ACM,24(4), 1977, 664–675.
P. J. Huber, Robust statistics: a review,Annals of Mathematical Statistics,43(3), 1972, 1041–1067.
J. W. Hunt, T. G. Szymanski, A fast algorithm for computing longest common subsequences,Communications of the ACM,20, 1977, 350–353.
C. E. Kim, Digital Disks, Report CS-82-104, Computer Science Department, Washington State University, Dec. 1982.
D. T. Lee, F. P. Preparata, Computational geometry—a survey,IEEE Transactions on Computers,33(12), 1984, 1072–1101.
B. H. McCormick, The Illinois pattern recognition computer—ILLIAC III,IEEE Transactions on Electronic Computers,12, 1963, 791–813.
G. U. Montanari, On limit properties of digitzation schemes,Journal of the ACM,17, 1970, 348–360.
M. F. Montuno, A. Fournier, Finding thex−y Convex Hull of a Set ofx−y Polygons, Technical Report CSRG-148, University of Toronto, Toronto, Nov. 1982.
R. Miller, Q. F. Stout, Computational geometry on a mesh-connected computer, inProc. Int. Conf. on Parallel Processing, 1984, 66–73.
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.
J. I. Munro, M. H. Overmars, D. Wood, Variations on visibility, inProc. ACM Symp. on Computational Geometry, Waterloo, Canada, 1987, 291–299.
N. Nakatsu, Y. Kambayashi, S. Yajima, A longest common subsequence algorithm suitable for similar text strings,Acta Informatica,18, 1982, 171–179.
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.
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.
M. H. Overmars, J. van Leeuwen, Maintenance of configurations in the plane,Journal of Computer and System Sciences,23, 1981, 166–204.
F. P. Preparata, M. I. Shamos,Computational Geometry, An Introduction, Springer-Verlag, Berlin, 1985.
A. P. Reeves, Survey parallel computer architectures for image processing,Computer Vision, Graphics and Image Processing,25, 1984, 68–88.
Y. Robert, M. Tchuente, A systolic array for the longest common subsequence problem,Information Processing Letters,21, 1985, 191–198.
A. Rosenfeld, Digital topology,The American Mathematical Monthly,86, 1979, 621–630.
A. Rosenfeld, Parallel image processing using cellular arrays,IEEE Transactions on Computers,16(1), 1983, 15–20.
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.
J.-R. Sack, Rectilinear Computational Geometry, Ph.D. thesis, McGill University, Montréal, 1984.
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.
M. I. Shamos, Computational Geometry, Ph.D. thesis, Department of Computer Science, Yale University, 1978.
M. I. Shamos, D. Hoey, Closest point problems, inProc. 7th Ann. IEEE Symp. on Foundations of Computer Science, 1975.
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.
C. D. Thompson, H. T. Kung, Sorting on a mesh-connected parallel computer,Communications of the ACM,20(4), 1977, 263–271.
G. T. Toussaint, Movable separability of sets, inComputational Geometry, (G. T. Toussaint, ed.), North-Holland, Amsterdam, 1985, 335–376.
S. H. Unger, A computer oriented towards spatial problems,Proceedings of the IRE,46, 1958, 1744–1750.
D. Wood, An isothetic view on computational geometry, inComputational Geometry, (G. T. Toussaint, ed.), North-Holland, Amsterdam, 1985, 429–459.
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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.
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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
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DOI: https://doi.org/10.1007/BF01759069