Abstract
Optimal hypercube algorithms are given for determining properties of labeled figures in a digitized black/white image stored one pixel per processor on a fine-grained hypercube. A figure (i.e., connected component) is a maximally connected set of black pixels in an image. The figures of an image are said to be labeled if every black pixel in the image has a label, with two black pixels having the same label if and only if they are in the same figure. We show that for input consisting of a labeled digitized image, a systematic use of divide-and-conquer into subimages of n c pixels, coupled with global operations such as parallel prefix and semigroup reduction over figures, can be used to rapidly determine many properties of the figures. Using this approach, we show that in Θ(log n) worst-case time the extreme points, area, perimeter, centroid, diameter, width and smallest enclosing rectangle of every figure can be determined. These times are optimal, and are superior to the best previously published times of Θ(log2 n).
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References
A. Aggarwal, B. Chazelle, L. Guibas, C. O'Dunlaing, and C. Yap, “Parallel computational geometry”, Algorithmica 3 (1988), pp. 293–327.
M.J. Atallah and M.T. Goodrich, “Efficient parallel solutions to some geometric problems”, J. Parallel and Distrib. Comput. 3 (1986), pp. 492–507.
K.E. Batcher, “Sorting networks and their applications”, Proc. AFIPS Spring Joint Comput. Conf. 32 (1968), pp. 307–314.
G. Blelloch, “Scans as primitive parallel operations” Proc. 1987 Int'l. Conf. Parallel Proc., pp. 355–362.
A. Borodin and J.E. Hopcroft, “Routing, merging and sorting on parallel models of computation”, J. Comp. and Sys. Sci. 30 (1985), pp. 130–145.
R. Cypher and J.L.C. Sanz, “Data reduction and fast routing: a strategy for efficient algorithms for message-passing parallel computers”, Algorithmica, to appear.
R. Cypher, J.L.C. Sanz, and L. Snyder, “Hypercube and shuffle-exchange algorithms for image component labeling”, Proc. Comp. Arch. Pat. Anal. and Mach. Intel. '87, pp. 5–10.
R. Cypher, J.L.C. Sanz, and L. Snyder, “EREW PRAM and Mesh Connected computer algorithms for image component labeling”, IEEE Trans. Pat. Anal. and Machine Intel., 11 (1989), pp. 258–262.
H. Freeman and R. Shapira, “Determining the minimal-area encasing rectangle for an arbitrary closed curve”, Comm. ACM 18 (1975), pp. 409–413.
A.E. Kayaalp and R. Jain, “Parallel implementation of an algorithm for three-dimensional reconstruction of integrated circuit pattern topography using the scanning electron microscope stereo technique on the NCUBE”, Hypercube Multiprocessors 1987, pp. 438–444.
C.P. Kruskal, L. Rudolf, and M. Snir, The power of parallel prefix, Proc. 1985 Intl. Conf. Parallel Proc., pp. 180–185.
W. Lim, A. Agrawal, and L. Nekludova, “A fast parallel algorithm for labeling connected components in image arrays”, Tech. report NA86-2, Thinking Machines Corp., 1986.
R. Miller and Q.F. Stout, “Some graph and image processing algorithms for the hypercube”, Hypercube Multiprocessors 1987, pp. 418–425.
R. Miller and Q.F. Stout, “Efficient parallel convex hull algorithms”, IEEE Trans. Computers 37 (1988), pp. 1605–1618.
R. Miller and Q.F. Stout, Parallel Algorithms for Regular Architectures, The MIT Press, 1989.
T.N. Mudge and T.S. Abdel-Rahman, “Vision algorithms for hypercube machines”, J. Parallel and Distrib. Comp. 4 (1987), pp. 79–94.
D. Nassimi and S. Sahni, “Parallel permutations and sorting algorithms and a new generalized connection network”, J. ACM 29 (1982), pp. 642–667.
F.P. Preparata, and M.I. Shamos, Computational Geometry, Springer-Verlag, 1985.
S. Ranka and S. Sahni, “Image template matching on SIMD hypercube multicomputers”, Proc. 1988 Intl. Conf. Parallel Proc., pp. 84–91.
E.M. Reingold, J. Nievergelt, and N. Deo, Combinatorial Algorithms, Prentice Hall, New York, 1977.
Y. Shiloach and U. Vishkin, “An O(logn) parallel connectivity algorithm”, J. Algorithms 3 (1982), pp. 57–67.
Q.F. Stout, “Hypercubes and pyramids”, Pyramidal Systems for Computer Vision, V.Cantoni and S. Levialdi, eds., Springer-Verlag, 1986, pp. 75–89.
L.G. Valiant, “Parallelism in comparison problems”, SIAM J. Comput. 4 (1975), pp. 151–162.
L.G. Valiant, “A scheme for fast parallel communication”, SIAM J. Comput. 11 (1982), pp. 350–361.
K. Voss and R. Klette, “On the maximum number of edges of convex digital polygons included into a square”, Friedrich-Schiller-Universitat Jena, Forschungsergegnisse, no. N/82/6.
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Miller, R., Stout, Q.F. (1989). Optimal hypercube algorithms for labeled images. In: Dehne, F., Sack, J.R., Santoro, N. (eds) Algorithms and Data Structures. WADS 1989. Lecture Notes in Computer Science, vol 382. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51542-9_43
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DOI: https://doi.org/10.1007/3-540-51542-9_43
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