A distance metric for multidimensional histograms

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Abstract

A metric is defined on the space of multidimensional histograms. Such histograms store in thexth location the number of events with feature vectorx; examples are gray level histograms and co-occurrence matrices of digital images. Given two multidimensional histograms, each is “unfolded” and a minimum distance pairing is performed using a distance metric on the feature vectorsx. The sum of the distances in the minimal pairing is used as the “match distance” between the histograms. This distance is shown to be a metric, and in the one-dimensional case is equal to the absolute difference of the two cumulative distribution functions. Among other applications, it facilitates direct computation of the distance between co-occurrence matrices or between point patterns. The problem of finding a translation to minimize the distance between point patterns is also discussed.

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Permanent Address: Dept. of Computer Science, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel. These authors were supported in Israel by a grant from the Israel Academy of Sciences.

The support of the National Science Foundation under grant DCR-82-18408 is gratefully acknowledged, as is the help of Janet Salzman in preparing this paper.

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