A General Proximity Analysis of Nonlinear Subdivision Schemes

Abstract

In recent work nonlinear subdivision schemes which operate on manifold-valued data have been successfully analyzed with the aid of so-called proximity conditions bounding the difference between a linear scheme and the nonlinear one. The main difficulty with this method is the verification of these conditions. In the present paper we obtain a very clear understanding of which properties a nonlinear scheme has to satisfy in order to fulfill proximity conditions. To this end we introduce a novel polynomial generation property for linear subdivision schemes and obtain a characterization of this property via simple multiplicativity properties of the moments of the mask coefficients. As a main application of our results we prove that the Riemannian analogue of a linear subdivision scheme which is defined by replacing linear averages by the Riemannian center of mass satisfies proximity conditions of arbitrary order. As a corollary we conclude that the Riemannian analogue always produces limit curves which are at...