2012/08/09 by Sergey Kushnarev, Kushnarev, Sergey, Akil Narayan +1
Computer Science · Engineering · Mathematics · Social Sciences · #30F60 #65D19 #65K10 #Advanced Numerical Analysis Techniques #Advanced Vision and Imaging #Complex Variables (math.CV) #FOS: Mathematics #Historical Geography and Cartography #Morphological variations and asymmetry #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1208.2022
openalex publication_date 2012/08/09 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We propose and investigate a numerical shooting method for computing\ngeodesics in the Weil-Petersson (WP) metric on the universal Teichm "uller\nspace T(1). This space, or rather the coset subspace\n PSL2( R) backslash Diff(S1), has another realization as the space of\nsmooth, simple closed planar curves modulo translations and scalings. This\nalternate identification of T(1) is a convenient metrization of the space of\nshapes and provides an immediate application for our algorithm in computer\nvision. The geodesic equation on T(1) with the WP metric is EPDiff(S1),\nthe Euler-Poincare equation on the group of diffeomorphisms of the circle\nS1, and admits a class of soliton-like solutions named Teichons. Our method\nrelies on approximating the geodesic with these teichon solutions, which have\nmomenta given by a finite linear combination of delta functions. The geodesic\nequation for this simpler set of solutions is more tractable from the numerical\npoint of view. With a robust numerical integration of this equation, we\nformulate a shooting method utilizing a cross-ratio matching term. Several\nexamples of geodesics in the space of shapes are demonstrated.\n