Journal article
Scattered manifold-valued data approximation
Numerische Mathematik, v 135(4), pp 987-1010
2017
PMID: 28615747
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Abstract
We consider the problem of approximating a function
f
from an Euclidean domain to a manifold
M
by scattered samples
\documentclass[12pt]{minimal}
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\begin{document}$$(f(\xi _i))_{i\in \mathcal {I}}$$\end{document}
(
f
(
ξ
i
)
)
i
∈
I
, where the data sites
\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\xi _i)_{i\in \mathcal {I}}$$\end{document}
(
ξ
i
)
i
∈
I
are assumed to be locally close but can otherwise be far apart points scattered throughout the domain. We introduce a natural approximant based on combining the moving least square method and the Karcher mean. We prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart. The analysis also tells us that the use of Karcher’s mean (dependent on a Riemannian metric and the associated exponential map) is inessential and one can replace it by a more general notion of ‘center of mass’ based on a general retraction on the manifold. Consequently, we can substitute the Karcher mean by a more computationally efficient mean. We illustrate our work with numerical results which confirm our theoretical findings.
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Details
- Title
- Scattered manifold-valued data approximation
- Creators
- Philipp Grohs - Oskar-Morgenstern Platz 1, 1090 Wien, AustriaMarkus Sprecher - Rämistrasse 101, 8092 Zürich, SwitzerlandThomas Yu - 3141 Chestnut Street, Korman 269, Philadelphia, PA 19104 USA
- Publication Details
- Numerische Mathematik, v 135(4), pp 987-1010
- Publisher
- Springer Berlin Heidelberg; Berlin/Heidelberg
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000398175500002
- Scopus ID
- 2-s2.0-84978058423
- Other Identifier
- 991014878160304721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics, Applied