On the estimation of a convex set with corners

In robotic vision using laser-radar measurements, noisy data on convex sets with corners are derived in terms of the set's support function. The corners represent abutting edges of manufactured items, and convey important information about the items' shape. However, simple methods for set estimation, for example based on fitting random polygons or smooth sets, either add additional corners as an artifact of the algorithm, or approximate corners by smooth curves. It might be argued, however, that corners have special significance in the interpretation of a set, and should not be introduced as an artifact of the estimation procedure. In this paper we suggest a corner-diagnostic approach, in the form of a three-step algorithm which (a) identifies the number and positions of corners, (b) fits smooth curves between corners, and (c) splices together the smooth curves and the corners, to produce an over-all estimate of the convex set. The corner-finding step is parametric in character, and although it is based on detecting change points in high-order derivatives of the support function, it produces root-n consistent estimators of the locations of corners. On the other hand, the smooth-curve fitting step is entirely nonparametric. The splicing step marries these two disparate approaches into a single, practical method.