Measuring Point Set Similarity with the Hausdorff Distance: Theory and Applications

We consider the problem of measuring the similarity between two nite point sets using the Hausdor distance H(A;B). This distance is small when each point in A is close to some point in B, and each point in B is close to some point in A. The de nition does not require that A and B have the same number of points. More importantly, H(A;B) is a metric. A point set is identical only to itself, the order of comparison is irrelevant, and the triangle inequality holds. If two point sets are similar to a third, then they are similar to each other. This agrees with our intuition of shape similarity. The Hausdor distance is insensitive to small perturbations of the point sets. This stability is important because it allows for small positional errors in point feature sets. Also, the Hausdor distance does not build one-to-one correspondences between the points in the two sets. Shape matchers which do so will require an unreasonable amount of time when the number of features detected becomes very large. For cases in which we do not want symmetry, there is a directed version of the Hausdor distance, denoted h(A;B). This distance from A to B is small whenever each point of A is close to some point in B. Consider, for example, an illustration retrieval system which compares point feature sets of a database illustration and a given query. We may decide that a query is similar to a database gure if features present in the query are also present in the database gure, but not necessarily vice versa. In this case we would use the directed Hausdor distance from the query set to the database gure set. This paper is intended to be a summary and synthesis of four papers: