Object Detection in Flatland

Given a rectangle with emitters and receivers on its perimeter, one can detect objects in it by determining which of the line segments between emitters and receivers are blocked by objects. The problem of object detection can be formulated as the problem of finding all non-empty n-wedge intersections, where a wedge is defined by a consecutive set of blocked line segments from the same emitter. We show that for a given set of wedges, one emanating from each emitter, we can determine the intersection (i.e., the convex polygon) in time linear in the number of wedges, assuming some given ordering of the wedges. We present two algorithms that efficiently determine all non-empty n-wedge intersections, assuming that objects are sufficiently large.