Manifold-valued subdivision schemes based on geodesic inductive averaging

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• 作者：Nira Dyn^a ; ^{niradyn@post.tau.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Nir Sharon^b ; ^{nsharon@math.princeton.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：40A30 ; 65D99 ; 41A05
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：311
• 期：Complete
• 页码：54-67
• 全文大小：694 K

文摘

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.