Numerical Computation of Weil--Peterson Geodesics in the Universal Teichmüller Space

Abstract

We propose an optimization algorithm for computing geodesics on the universal Teichmuller space $T(1)$ in the Weil--Petersson (WP) metric. Another realization for T(1) is the space of planar shapes, modulo translation and scale, and thus our algorithm addresses a fundamental problem in computer vision: compute the distance between two given shapes. The identification of planar shapes with elements on T(1) allows us to represent a shape as a homeomorphism of $S^1$. Then given two homeomorphisms of $S^1$ (i.e., two shapes we want to connect with a flow), we formulate a discretized WP energy and the resulting problem is a boundary-value minimization problem. We numerically solve this problem, providing several examples of geodesic flow on the space of shapes, and verifying mathematical properties of $T(1)$. Our algorithm is more general than the application here in the sense that it can be used to compute geodesics on any other Riemannian manifold.