Approximate congruence in nearly linear time

Abstract

The problem of geometric point set matching has been studied extensively in the domain of computational geometry, and has many applications in areas such as computer vision, computational chemistry, and pattern recognition. One of the commonly used metrics is the bottleneck distance, which for two point sets P and Q is the minimum over all one-to-one mappings f:P→Q of max[subscript p∈Pd(p,f(p))], where d is the Euclidean distance. Much effort has gone into developing efficient algorithms for minimising the bottleneck distance between two point sets under groups of transformations. However, the algorithms that have thus far been developed suffer from running times that are large polynomials in the size of the input, even for approximate formulations of the problem. In this paper we define a point set similarity measure that includes both the bottleneck distance and the Hausdorff distance as special cases. This measure relaxes the condition that the mapping must be one-to-one, but guarantees that only a few points are mapped to any point. Using a novel application of Hall's Theorem to reduce the geometric matching problem to a combinatorial matching problem, we present near-linear time approximation schemes for minimising this distance between two point sets in the plane under isometries; we note here that the best known algorithms for congruence under the bottleneck measure run in time [~ over O](n[superscript 2.5]). We also obtain a combinatorial bound on the metric entropy of certain families of geometric objects. This result yields improved algorithms for approximate congruence, and may be of independent interest.