An optimal <i>O</i>(<i>n</i> log <i>n</i>) algorithm for contour reconstruction from rays

Abstract

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 Ο(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 Ο(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 Ο(nlogn) time.