We present an optimal algorithm to reconstruct the planar cross section of a simple object from data points measured by rays. The rays are semi-infinite curves representing, for example, the laser beam or the articulated arms of a robot moving around the object. The object is assumed to be a unique simply connected object, and the contour to be reconstructed is a simple polygon having the data points as vertices and intersecting none of the measuring rays. Such a contour does not exist for any given sets of points and rays but only for legal data. In this paper, we prove that the solution to the contour problem is unique whenever such a solution exists. For a set of n points and n rays, the algorithm presented here provides in &Ogr;(nlogn) time, a polygon which is the solution to the contour problem when the data are legal. Updating this contour if a new measure is available can be done in &Ogr;(logn) time. Both results are asymptotically optimal in the worst-case. Moreover, once the solution has been found, we can check if the data are legal in &Ogr;(nlogn) time.
我们提出了一种最佳算法,可以根据射线测量的数据点重建简单对象的平面横截面。射线是半无限曲线,代表例如激光束或围绕对象移动的机器人的关节臂。假定该对象是唯一的简单连接的对象,要重建的轮廓是一个简单的多边形,其数据点为顶点,并且不与任何测量射线相交。对于任何给定的点和射线集,这种轮廓都不存在,而仅对于合法数据而言。在本文中,我们证明了只要存在这样的解决方案,轮廓问题的解决方案都是唯一的。对于一组n点和n射线,此处介绍的算法在(nlogn)时间中提供了一个多边形,当数据合法时,该多边形是轮廓问题的解决方案。如果有新的度量可用,则可以在(登录)时间内更新此轮廓。在最坏的情况下,两个结果都是渐近最优的。而且,一旦找到解决方案,我们就可以在  nlogn时间内检查数据是否合法。 P>
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