Approximating the Weil-Petersson Metric Geodesics on the Universal Teichm\"uller space by Singular Solutions

TL;DR

This paper introduces a numerical shooting method to approximate geodesics in the Weil-Petersson metric on the universal Teichmüller space, with applications to shape analysis and computer vision.

Contribution

The authors develop a novel numerical approach using Teichon solutions to efficiently compute WP geodesics, bridging complex geometric theory with practical shape analysis.

Findings

Successfully computed geodesics in shape space

Demonstrated applications in computer vision tasks

Validated the robustness of the shooting method

Abstract

We propose and investigate a numerical shooting method for computing geodesics in the Weil-Petersson ($WP$) metric on the universal Teichm\"uller space T(1). This space, or rather the coset subspace $\PSL_2(\R)\backslash\Diff(S^1)$, has another realization as the space of smooth, simple closed planar curves modulo translations and scalings. This alternate identification of T(1) is a convenient metrization of the space of shapes and provides an immediate application for our algorithm in computer vision. The geodesic equation on T(1) with the $WP$ metric is EPDiff($S^1$), the Euler-Poincare equation on the group of diffeomorphisms of the circle $S^1$, and admits a class of soliton-like solutions named Teichons. Our method relies on approximating the geodesic with these teichon solutions, which have momenta given by a finite linear combination of delta functions. The geodesic equation for this simpler set of solutions is more tractable from the numerical point of view. With a robust numerical integration of this equation, we formulate a shooting method utilizing a cross-ratio matching term. Several examples of geodesics in the space of shapes are demonstrated.