Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces

Authors:
Johannes Wallner;Esfandiar Nava Yazdani;Andreas Weinmann
Affiliations:
Institut für Geometrie, Technische Universität Graz, Graz, Austria 8010;Drexel University, Philadelphia, USA;Institut für Geometrie, Technische Universität Graz, Graz, Austria 8010
Venue:
Advances in Computational Mathematics
Year:
2011

Citing 5
Cited 2

A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices

SIAM Journal on Matrix Analysis and Applications
Smoothness Equivalence Properties of Manifold-Valued Data Subdivision Schemes Based on the Projection Approach

SIAM Journal on Numerical Analysis
Smoothness equivalence properties of univariate subdivision schemes and their projection analogues

Numerische Mathematik
Convergence and C1 analysis of subdivision schemes on manifolds by proximity

Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
Approximation order of interpolatory nonlinear subdivision schemes

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Approximation order of interpolatory nonlinear subdivision schemes

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Full length article: Invariance property of proximity conditions in nonlinear subdivision

Journal of Approximation Theory

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Abstract

After a discussion on definability of invariant subdivision rules we discuss rules for sequential data living in Riemannian manifolds and in symmetric spaces, having in mind the space of positive definite matrices as a major example. We show that subdivision rules defined with intrinsic means in Cartan-Hadamard manifolds converge for all input data, which is a much stronger result than those usually available for manifold subdivision rules. We also show weaker convergence results which are true in general but apply only to dense enough input data. Finally we discuss C 1 and C 2 smoothness of limit curves.