Metric spaces of shapes and applications: compression, curve matching and low-dimensional representation

In this paper we present three metrics on classes of 2D shapes whose outlines are simple closed planar curves. The first, a C 1 -type metric on classes of shapes with Lipschitz tangent angle, allows for estimates of massiveness such as -entropy. A Sobolev-type metric on piecewise C 2 curves allows for efficient curve matching based on a multiscale wavelet-like analysis. Finally, the Weil-Petersson metric, a Riemannian metric on the class of smooth diffeomorphisms of S 1 R 2 , allows a low dimensional shape representation, an N -Teichon, whose initial conditions are closely linked to curvature.