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

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.