COMPUTING THE HAUSDORFF DISTANCE BETWEEN TWO SETS OF PARAMETRIC CURVES

Abstract

Abstract. We present an algorithm for computing the Hausdorff dis-tance between two parametric curves in R n , or more generally betweentwo sets of parametric curves in R n . During repeated subdivision of theparameter space, we prune subintervals that cannot contain an optimalpoint. Typically, our algorithm costs O(logM) operations, compared withO(M) operations for a direct, brute-force method, to achieve an accuracyof O(M −1 ). 1. IntroductionLet (X,d) be a metric space, so that d(x,y) is the distance between twopoints x and y ∈X. We denote the distance from a point x ∈X to a nonemptysubset B ⊆X by(1) d(x,B) = inf y∈B d(x,y).Given a second, nonempty subset A ⊆X, the directed Hausdorff distance fromA to B is defined byh(A,B) = sup x∈A d(x,B),and the Hausdorff distance between A and B byH(A,B) = maxh(A,B),h(B,A).If x ∗ ∈A is such that h(A,B) = d(x ∗ ,B), then we call x ∗ an optimal point.We remark that the Hausdorff distance defines a metric on the set of closed,bounded subsets of X.In pattern recognition and computer vision, it is very important to compareshapes and patterns, and to give a numerical value indicating their similar-ity. The Hausdorff distance is a well known similarity measure: the smallerthe Hausdorff distance between two shapes the greater is their degree of re-semblance. In practice, one often wants to find min