Abstract
Labelling the lines of a planar line drawing of a 3-D object in a way that reflects the geometric properties of the object is a much studied problem in computer vision, considered to be an important step towards understanding the object from its 2-D drawing. Combinatorially, the labellability problem is a Constraint Satisfaction Problem and has been shown to be NP-complete even for images of polyhedral scenes. In this paper, we examine scenes that consist of a set of objects each obtained by rotating a polygon around an arbitrary axis. The objects are allowed to arbitrarily intersect or overlay. We show that for these scenes, there is a sequential lineartime labelling algorithm. Moreover, we show that the algorithm has a fast parallel version that executes inO(log3 n) time on an Exclusive-Read-Exclusive-Write Parallel Random Access Machine withO(n 3/log3 n) processors. The algorithm not only answers the decision problem of labellability, but also produces a legal labelling, if there is one. This parallel algorithm should be contrasted with the techniques of dealing with special cases of the constraint satisfaction problem. These techniques employ an effective, but inherently sequential, relaxation procedure in order to restrict the domains of the variables.
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References
P. Alevizos, “A linear algorithm for labeling planar projections of polyhedra,” inProc. IEEE/RSJ Int. Workshop on Intelligent Robots and Systems, Osaka, Japan, 1991, pp. 595–601.
M.B. Clowes, “On seeing things,”Artificial Intelligence, vol. 2, pp. 79–116, 1971.
M. Goldberg and T. Spencer, “Constructing a maximal independent set in paprallel,”SIAM J. Discrete Mathematics, vol. 2, pp. 322–328, 1989.
D.A. Huffman, “Impossible objects as nonsense sentences,”Machine Intelligence, vol. 6, B. Meltzer and D. Michie eds., Edinburgh University Press: Edinburgh, 1971, pp. 295–323.
R.M. Karp and V. Ramachandran, “Parallel algorithms for shared-memory machines,”Handbook of Theoretical Computer Science, vol. A, J. van Leeuwen ed., Elsevier: Amsterdam, 1990, pp. 869–942.
R.M. Karp and A. Wigderson, “A fast parallel algorithm for the maximal independent set problem,”J. Association of Computing Machinery, vol. 32, pp. 762–773, 1985.
S. Kasif, “On the parallel complexity of discrete relaxation in constraint satisfaction networks,”Artificial Intelligence, vol. 45, pp. 275–286, 1990.
L.M. Kirousis, “Effectively labeling planar projections of polyhedra,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, pp. 123–130, 1990.
L.M. Kirousis, “Fast parallel constraint satisfaction,”Artificial Intelligence, vol. 64, pp. 147–160, 1993.
L.M. Kirousis and C.H. Papadimitriou, “The complexity of recognizing polyhedral scenes,”Journal of Computer and System Sciences, vol. 37, pp. 14–38, 1988.
M. Luby, “A simple parallel algorithm for the maximal independent set problem,”SIAM J. Computing, vol. 15, pp. 1036–1053, 1986.
J. Malik, “Interpreting line drawings of curved objects,”International J. of Computer Vision, vol. 1, pp. 73–103, 1987.
J. Malik and D. Maydan, “Recovering three-dimensional shape from a single image of curved objects,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, pp. 555–566, 1989.
A.K. Mackworth and E.C. Freuder, “The complexity of some polynomial network consistency algorithms for constraint satisfaction problems,”Artificial Intelligence, vol. 25, pp. 65–74, 1985.
U. Montanari and F. Rossi, “Constraint relaxation may be perfect,”Artificial Intelligence, vol. 48, pp. 143–170, 1991.
P. Parodi and V. Torre, “A linear complexity procedure for labelling line drawings of polyhedral scenes using vanishing points,” inProc. International Conference on Computer Vision ICCV-93, Berlin, Germany, 1993, pp. 291–295.
K. Sugihara, “A necessary and sufficient condition for a picture to represent a polyhedral scene,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 6, pp. 578–586, 1984.
D. Waltz, “Understanding line drawings of scenes with shadows,”The Psychology of Computer Vision, P.H. Winston ed., McGraw-Hill: New York, 1975, pp. 19–91.
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This research was partially supported by the European Community ESPRIT Basic Research Program under contracts 7141 (project ALCOM II) and 6019 (project Insight II).
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Dendris, N.D., Kalafatis, I.A. & Kirousis, L.M. An efficient parallel algorithm for geometrically characterising drawings of a class of 3-D objects. J Math Imaging Vis 4, 375–387 (1994). https://doi.org/10.1007/BF01262403
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DOI: https://doi.org/10.1007/BF01262403