Embedding and similarity search for point sets under translation

Pattern matching in point sets is a well studied problem with numerous applications. We assume that the point sets may contain outliers (missing or spurious points) and are subject to an unknown translation. We define the distance between any two point sets to be the minimum size of their symmetric difference over all translations of one set relative to the other. We consider the problem in the context of similarity search. We assume that a large database of point sets is to be preprocessed so that given any query point set, the closest matches in the database can be computed efficiently. Our approach is based on showing that there is a randomized algorithm that computes a translation-invariant embedding of any point set of size at most n into the L_1 metric, so that with high probability, distances are subject to a distortion that is O(log2 n).