Non-convex contour reconstruction

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We present algorithms to reconstruct the planar cross-section of a simply connected 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. This paper shows that the information provided by the rays is crucial (though generally neglected) when solving 2-dimensional reconstruction problems. The main property of the rays is that they induce a total order on the measured points. This order is shown to be computable in optimal time O ( n log n ). The algorithm is fully dynamic and allows the insertion or the deletion of a point in O (log n ) time. From this order a polygonal approximation of the object can be deduced in a straightforward manner. However, if insufficient data are available or if the points belong to several connected objects, this polygonal approximation may not be a simple polygon or may intersect the rays. This can be checked in O ( n log n ) time. The order induced by the rays can also be used to find a strategy for discovering the exact shape of a simple (but not necessarily convex) polygon by means of a minimal number of probes. When each probe outcome consists of a contact point, a ray measuring that point and the normal to the object at the point, we have shown that 3 n -3 probes are necessary and sufficient if the object has n non-colinear edges. Each probe can be determined in O (log n ) time yielding an O ( n log n )-time 0 ( n )-space algorithm. When each probe outcome consists of a contact point and a ray measuring that point but not the normal, the same strategy can still be applied. Under a mild condition, 8 n -4 probes are sufficient to discover a shape that is almost surely the actual shape of the object.