Matching Shapes Using the Current Distance

In this paper, we study shape matching under the current distance, a distance measure on shapes proposed by Vaillant and Glaunès [4]. This measure has four attractive properties. Firstly, it is global in nature, and thus does not require the determination of correspondences between features. Computing correspondences is often the most expensive part of computing distance between shapes. Secondly, although inspired by clever ideas from geometric measure theory, it can be expressed as a direct (but expensive) computation, and so is more tractable than many global shape distance measures (that typically require computation of geodesics on manifolds). Thirdly, it generalizes easily to higher dimensional structures: the current distance can be defined between pairs of point sets, curves, surfaces, and even higher-dimensional manifolds. Finally, it is defined in terms of a norm on the shape by the usual construction d(S, S′) = ‖S − S′‖. This norm acts as a signature of the shape, and is potentially useful for building data structures to answer more generalized queries about shapes (like near neighbors, clustering, etc). We present the first algorithmic analysis of the current distance. Our main contributions in this work are (1) a fast approximation for computing the current distance that runs in near-linear time (as opposed to the quadratic bound implicit in the definition) (2) A coreset-like construction for approximating the current norm of a point set by a small-sized sample and (3) an FPTAS for minimizing the current distance between two point sets under translations.