Minimum Hausdorff Distance Research Articles

Geometric pattern matching under Euclidean motion

Given two planar sets A and B, we examine the problem of determining the smallest ϵ such that there is a Euclidean motion (rotation and translation) of A that brings each member of A within distance ϵ of some member of B. We establish upper bounds on the combinatorial complexity of this subproblem in model-based computer vision, when the sets A and B contain points, line segments, or (filled-in) polygons. We also show how to use our methods to substantially improve on existing algorithms for finding the minimum Hausdorff distance under Euclidean motion.

Applications of Parametric Searching in Geometric Optimization

We present several applications in computational geometry of Megiddo′s parametric searching technique. These applications include: (1) Finding the minimum Hausdorff distance in the Euclidean metric between two polygonal regions under translation; (2) Computing the biggest line segment that can be placed inside a simple polygon; (3) Computing the smallest width annulus that contains a given set of given points in the plane; (4) Given a set of n points in 3-space, finding the largest radius r such that if we place a ball of radius r around each point, no segment connecting a pair of points is intersected by a third ball. Besides obtaining efficient solutions to all these problems (which, in every case, either improve considerably previous solutions or are the first nontrivial solutions to these problems), our goal is to demonstrate the versatility of the parametric searching technique.

The upper envelope of voronoi surfaces and its applications

Given a setS ofsources (points or segments) in ?211C;d, we consider the surface in ?211C;d+1 that is the graph of the functiond(x)=minp?S?(x, p) for some metric?. This surface is closely related to the Voronoi diagram, Vor(S), ofS under the metric?. The upper envelope of a set of theseVoronoi surfaces, each defined for a different set of sources, can be used to solve the problem of finding the minimum Hausdorff distance between two sets of points or line segments under translation. We derive bounds on the number of vertices on the upper envelope of a collection of Voronoi surfaces, and provide efficient algorithms to calculate these vertices. We then discuss applications of the methods to the problems of finding the minimum Hausdorff distance under translation, between sets of points and segments.

Computing the minimum Hausdorff distance between two point sets on a line under translation

Given two sets of points on a line, we want to translate one of them so that their Hausdorff distance (the maximum of the distances from a point in any of the sets to the nearest point in the other set) is as small as possible. We present an optimal O( n log n) algorithm for this problem.