Pattern Matching for sets of segments

Abstract

In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plane. Our work consists of two main parts. In the first, we address problems and measures that relate to collections of orthogonal line segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration. We propose a new measure of segment similarity called a \emph{coverage measure}, and present efficient algorithms for maximising this measure between sets of axis-parallel segments under translations. Our algorithms run in time $O(n^3\polylog n)$ in the general case, and run in time $O(n^2\polylog n)$ for the case when all segments are horizontal. In addition, we show that when restricted to translations that are only vertical, the Hausdorff distance between two sets of horizontal segments can be computed in time roughly $O(n^{3/2}{\sl polylog}n)$. These algorithms form significant improvements over the general algorithm of Chew et al. that takes time $O(n^4 \log^2 n)$. In the second part of this paper we address the problem of matching polygonal chains. We study the well known \Frd, and present the first algorithm for computing the \Frd under general translations. Our methods also yield algorithms for computing a generalization of the \Fr distance, and we also present a simple approximation algorithm for the \Frd that runs in time $O(n^2\polylog n)$.