## 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 in O(log^{3}n) time on an Exclusive-Read-Exclusive-Write Parallel Random Access Machine with O(n^{3}/log^{3}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.

Original language | English (US) |
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Pages (from-to) | 375-387 |

Number of pages | 13 |

Journal | Journal of Mathematical Imaging and Vision |

Volume | 4 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1994 |

## Keywords

- efficient parallel algorithms
- image analysis
- labelling of lines of drawings
- line drawings
- planar projections of 3-D objects

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Condensed Matter Physics
- Computer Vision and Pattern Recognition
- Geometry and Topology
- Applied Mathematics