Numerical Computation of Weil-Peterson Geodesics in the Universal Teichm\"uller Space

TL;DR

This paper introduces a numerical algorithm to compute Weil-Petersson geodesics in the universal Teichmüller space, which corresponds to shape analysis in computer vision, enabling shape distance measurement.

Contribution

It presents a novel optimization method for geodesic computation in T(1), applicable to shape analysis and general Riemannian manifolds.

Findings

Successfully computes geodesics between shapes

Verifies mathematical properties of T(1)

Demonstrates applicability to shape analysis

Abstract

We propose an optimization algorithm for computing geodesics on the universal Teichm\"uller space T(1) in the Weil-Petersson ($W P$) 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 smooth shapes with elements on T(1) allows us to represent a shape as a diffeomorphism on $S^1$. Then given two diffeomorphisms on $S^1$ (i.e., two shapes we want connect with a flow), we formulate a discretized $W P$ 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.