A Pseudo-Metric for Weighted Point Sets

We derive 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 Hausdorff-like pseudo-metric fails to differentiate between fuzzy sets with arbitrarily different maximum memebership values. The other approach is the Earth Mover's Distance. However, for sets of unequal total weights, it gives zero distance for arbitrarily different 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 efficient database searching. Moreover, our pseudo-metric identifies only weight-scaled versions of the same set. We demonstrate its potential use by testing it on two different collections, one of company logos and another one of fish contours.