Matching Shapes Using the Current Distance

The {current distance} was introduced by Vaillant and Glaunes as a way of comparing shapes (point sets, curves, surfaces) without having to rely on computing correspondences between features in each shape. This distance measure is defined by viewing a shape as a linear operator on a k-form field, and constructing a (dual) norm on the space of shapes. As formulated, it takes O(nm) time to compute the current distance between two shapes of size n and m, and there are no known algorithms to compute the current distance between shapes minimized under transformation groups. In this paper, we provide the first algorithmic analysis of the current distance. Our main results are (i) a method for computing the approximate current distance between two shapes in near-linear time, (ii) a coreset construction that allows us to approximate the current norm of a shape using a constant-sized sample, and (iii) an approximation algorithm for computing the current distance between two d-dimensional shapes under rigid transformations (rotations and translations). An interesting aspect of our work is that we can compute the current distance between curves, surfaces, and higher-order manifolds via a simple reduction to instances of weighted point sets, thus obviating the need for different kinds of algorithms for different kinds of shapes.