Approximating the Weil-Petersson Metric Geodesics on the Universal Teichmuller Space by Singular Solutions

作者：Kushnarev Sergey^*; Narayan Akil

来源：SIAM Journal on Imaging Sciences, 2014, 7(2): 900-923.

DOI：10.1137/120898565

• universal Teichmuller space
• Weil-Petersson metric
• teichons
• shooting
• shapes

We propose and investigate a numerical shooting method for computing geodesics in the Weil-Petersson (WP) metric on the universal Teichmuller space T (1). This space, or rather the coset subspace PSL2(R)\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 [S. Kushnarev, Experiment. Math., 18 (2009), pp. 325-336]. 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.

• 出版日期2014