A Pseudo-Metric for Weighted Point Sets

We present a pseudo-metric for weighted point sets. There are numerous situations, for example in the shape description domain, where the individual points in a feature point set have an associated attribute, a weight. A distance function that incorporates this extra information apart from the points' position can be very useful for matching and retrieval purposes. There are two main approaches to do this. One approach is to interpret the point sets as fuzzy sets. However, a distance measure for fuzzy sets that is a metric, invariant under rigid motion and respects scaling of the underlying ground distance, does not exist. In addition, a Hausdor -like pseudo-metric fails to di erentiate between fuzzy sets with arbitrarily di erent maximum membership values. An alternative approach is the Earth Mover's Distance. However, for sets of unequal total weights, it gives zero distance for arbitrarily di erent sets, and does not obey the triangle inequality. In this paper we derive a distance measure, based on weight transportation, that is invariant under rigid motion, respects scaling, and obeys the triangle inequality, so that it can be used in eÆcient database searching. Moreover, our pseudo-metric identi es only weight-scaled versions of the same set. We demonstrate its potential use by testing it on two di erent collections, one of company logos and another one of sh contours. In addition, simple upper bounds on its value, related to incremental change of the point sets, are given. Finally, we address the diÆcult problem of partial matching in its most general form, giving useful insight and interesting research directions.