A General Proximity Analysis of Nonlinear Subdivision Schemes

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 least as smooth as those of the linear scheme it has been derived from. If the manifold under consideration is a Cartan-Hadamard manifold, this result for the first time yields a manifold-valued subdivision scheme which converges for all input data and produces arbitrarily smooth limit curves. We also generalize our results to the case of multivariate subdivision schemes with arbitrary dilation matrix.