Geometry, Imaging and Computing

Volume 1 (2014)

Number 2

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

Pages: 173 – 221



Matt Feiszli (New Haven, Connecticut, U.S.A.)

Sergey Kushnarev (Engineering Systems & Design, Singapore University of Technology & Design, Singapore)

Kathryn Leonard (Department of Mathematics, California State University, Channel Islands, Camarillo, Calif., U.S.A.)


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 $\epsilon$-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 $S^1 \to \mathbb{R}^2$, allows a low dimensional shape representation, an $N$-Teichon, whose initial conditions are closely linked to curvature.

Full Text (PDF format)