Approximate Earth Mover's Distance in Linear Time

Abstract

The earth moverpsilas distance (EMD) is an important perceptually meaningful metric for comparing histograms, but it suffers from high (O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> logN)) computational complexity. We present a novel linear time algorithm for approximating the EMD for low dimensional histograms using the sum of absolute values of the weighted wavelet coefficients of the difference histogram. EMD computation is a special case of the Kantorovich-Rubinstein transshipment problem, and we exploit the Holder continuity constraint in its dual form to convert it into a simple optimization problem with an explicit solution in the wavelet domain. We prove that the resulting wavelet EMD metric is equivalent to EMD, i.e. the ratio of the two is bounded. We also provide estimates for the bounds. The weighted wavelet transform can be computed in time linear in the number of histogram bins, while the comparison is about as fast as for normal Euclidean distance or chi <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> statistic. We experimentally show that wavelet EMD is a good approximation to EMD, has similar performance, but requires much less computation.