Manifold-valued subdivision schemes based on geodesic inductive averaging

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Abstract

Subdivision schemes have become an important tool for approximation of manifold-valued functions. In this paper, we describe a construction of manifold-valued subdivision schemes for geodesically complete manifolds. Our construction is based upon the adaptation of linear subdivision schemes using the notion of repeated binary averaging, where as a repeated binary average we propose to use the geodesic inductive mean. We derive conditions on the adapted schemes which guarantee convergence from any initial manifold-valued sequence. The definition and analysis of convergence are intrinsic to the manifold. The adaptation technique and the convergence analysis are demonstrated by several important examples of subdivision schemes. Two numerical examples visualizing manifold-valued curves generated by such schemes are given together with a link to the code that generated them.

Original languageEnglish
Pages (from-to)54-67
Number of pages14
JournalJournal of Computational and Applied Mathematics
Volume311
DOIs
StatePublished - 1 Feb 2017

Keywords

  • Contractivity
  • Convergence
  • Displacement-safe scheme
  • Inductive geodesic mean
  • Manifold-valued subdivision scheme

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