Approximating the Weil-Petersson Metric Geodesics on the Universal Teichmüller 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 clo...

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Bibliographic Details
Published inarXiv.org
Main Authors Kushnarev, Sergey, Narayan, Akil
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 19.10.2012
Subjects
Algorithms
Approximation
Computer vision
Delta function
Geodesy
Isomorphism
Numerical integration
Robustness (mathematics)
Shape recognition
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Summary: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.
ISSN:2331-8422