TY - JOUR
T1 - Manifold-valued subdivision schemes based on geodesic inductive averaging
AU - Dyn, Nira
AU - Sharon, Nir
N1 - Publisher Copyright:
© 2016 Elsevier B.V.
PY - 2017/2/1
Y1 - 2017/2/1
N2 - 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.
AB - 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.
KW - Contractivity
KW - Convergence
KW - Displacement-safe scheme
KW - Inductive geodesic mean
KW - Manifold-valued subdivision scheme
UR - http://www.scopus.com/inward/record.url?scp=84979924758&partnerID=8YFLogxK
U2 - 10.1016/j.cam.2016.07.008
DO - 10.1016/j.cam.2016.07.008
M3 - מאמר
AN - SCOPUS:84979924758
VL - 311
SP - 54
EP - 67
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
SN - 0377-0427
ER -