Efficient Proximity Probing Algorithms for Metrology

Abstract

Abstract-Metrology, the theoretical and practical study of measurement, has applications in automated manufacturing, inspection, robotics, surveying, and healthcare. The geometric probing problem considers how to optimally use a probe to measure geometric properties. In this paper, we consider a proximity probe which, given a point, returns the distance to the boundary of the nearest object. When there is an unknown convex polygon P in the plane, the goal is to minimize the number of probe measurement needed to exactly determine the shape and location of P . We present an algorithm with upper bound of 3.5n + k + 2 probes, where n is the number of vertices and k ≤ 3 is the number of acute angles of P . The algorithm requires constant time per probe, and hence O(n) time to determine P . We also address the related problem where the unknown polygon is a member of a known finite set Γ and the goal is to efficiently determine which polygon is present. When m is the size of Γ and n is the maximum number of vertices of any member of Γ, we present an O(n m) algorithm with an upper bound of 2n + 2 probes.Note to Practitioners: Abstract-This paper was inspired by the problem of using a sensor with low-dimensional output, such as a range sensor, to determine the shape of an object, which is typically too complex for a single measurement to characterize. Existing approaches to shape measurement generally use sensors with high-dimensional output, such as cameras, or use lowdimensional sensors to measure fixed points, such as with Scanning Probe Microscopy (SPM). It is shown here that by employing an algorithm that uses the results of previous measurements to determine how the next measurement is taken, a non-directional range sensor can be used to efficiently and exactly determine the shape of a convex polygon. This suggests an alternative approach to obtaining information on the shape of an object in cases where low-dimensional sensors are more accurate, faster, or cheaper than their counterparts.