Approximate Nearest Neighbor Search under Translation Invariant Hausdorff Distance

Abstract

The Hausdorff distance is a measure for the resemblance of two geometric objects. Given a set of n point patterns and a query point pattern Q , the nearest neighbor of Q under the Hausdorff distance is the point pattern which minimizes this distance to Q . An extension of the Hausdorff distance is the translation invariant Hausdorff distance which additionally allows the translation of the point patterns in order to minimize the distance. This paper introduces the first data structure which allows to solve the nearest neighbor problem for the directed Hausdorff distance under translation in sublinear query time in a non-heuristic manner, in the sense that the quality of the results, the performance, and the space bounds are guaranteed. The data structure answers queries for both directions of the directed Hausdorff distance with a $ \sqrt{d(s-1.5)}(1+\epsilon) $-approximation factor in $ O(\log \frac{n}{\epsilon}) $ query time for the nearest neighbor and O(k + logn) query time for the k -th nearest neighbor for any e> 0 . (The O -notation of the latter runtime contains terms that are quadratic in e -1 .) Furthermore it is shown how to find the exact nearest neighbor under the directed Hausdorff distance without transformation of the point sets within some weaker time and storage bounds.