An efficient proximity probing algorithm for metrology

Abstract

Metrology, the theoretical and practical study of measurement, has applications in automated manufacturing, inspection, robotics, surveying, and healthcare. An important problem within metrology is how to interactively use a measuring device, or probe, to determine some geometric property of an unknown object; this problem is known as geometric probing. In this paper, we study a type of proximity probe which, given a point, returns the distance to the boundary of the object in question. We consider the case where the object is a convex polygon P in the plane, and the goal of the algorithm is to minimize the upper bound on the number of measurements necessary to exactly determine P . We show an algorithm which has an upper bound of 3.5n + k + 2 measurements necessary, where n is the number of vertices and k ≤ 3 is the number of acute angles of P . Furthermore, we show that our algorithm requires O(1) computations per probe, and hence O(n) time to determine P .

Active Learning a Convex Body in Low Dimensions

“…Various models of computation utilizing oracles have been previously studied within the community. Examples of other models include nearest-neighbor oracles (i.e., black-box access to nearest neighbor queries over a point set P ) [HKMR16], and proximity probes (which given a convex polygon C and a query q, returns the distance from q to C) [PAvdSG13]. It is reasonable to ask what classification-type problems can be solved with few oracle queries when using separation oracles.…”

Space Exploration via Proximity Search

“…Proximity queries have been applied before. Panahi et al [PASG13] use proximity probes on a convex polygon in the plane to reconstruct it exactly. Goel et al [GIV01], reduce the approximation versions of several problems like diameter, farthest neighbors, discrete center, metric facility location, bottleneck matching and minimum weight matching to nearest neighbor queries.…”

Efficient Proximity Probing Algorithms for Metrology

“…We now present two geometric lemmas which are necessary for our bound on the number of undesirable confirmations; we omit the proofs here, but interested readers can find the full arguments in our technical report [21].…”

“…This is an expanded and updated version of our conference paper [20], where we used proximity probes to determine the shape of an unknown convex polygon. We refer to this as Problem 1 and present Algorithm 1 to solve it, as illustrated in Fig.…”