Approximation algorithms for the Earth mover's distance under transformations using reference points

Reference points have been introduced in [2] and [1] to construct approximation algorithms for matching compact subsets of R under a given class of transformations. Also a general discussion of reference point methods for matching according to the Hausdorffdistance has been given in [1]. Another distance measure used for shape matching is the Earth Mover’s Distance (EMD) for weighted point sets ([7]). Here we will extend the definition of reference points to weighted point sets and get fast constant factor approximation algorithms for matching weighted point sets under translations, rigid motions and similarity operations with respect to the Earth Mover’s Distance. A first iterative algorithm to solve this problem has been given by Cohen ([3]). Thus we want to find algorithms where EMD(A,B) ≤ εEMD(A,B). Under this assumption ε is called the loss factor of the approximation algorithm. Unless stated otherwise, the results given in this paper are independent of the distance measure on the ground set, therefore the results are widely applicable. Additionally, all theorems hold in arbitrary dimension d. For a full version of this extended abstract see [5].