Fast Approximation of Distance Between Elastic Curves using Kernels

Elastic shape analysis on non-linear Riemannian manifolds provides an efficient and elegant way for simultaneous comparison and registration of non-rigid shapes. In such formulation, shapes become points on some high dimensional shape space. A geodesic between two points corresponds to the optimal deformation needed to register one shape onto another. The length of the geodesic provides a proper metric for shape comparison. However, the computation of geodesics, and therefore the metric, is computationally very expensive as it involves a search over the space of all possible rotations and reparameterization. This problem is even more important in shape retrieval scenarios where the query shape is compared to every element in the collection to search. In this paper, we propose a new procedure for metric approximation using the framework of kernel functions. We will demonstrate that this provides a fast approximation of the metric while preserving its invariance properties.