Smoothness of Nonlinear Subdivision Process

Subdivision algorithms paly a prominent role in Computer Grahics, in Geometry Modeling, and in connection with wavelts. Their study poses a number of mathematical challenges. Convergence and smoothness analysis of linear subdivision schemes in the one-dimensional case can be considered complete, and also in the higher-dimensional setting (subdivision of polyhedra) the smoothness problem has been solved in 1995. In view of the wealth of applications it is not surprising that subdivision processes have been generalized to nonlinear geometries like surfaces, Riemannian manifolds, Euclidean space minus obstacles, and Lie groups. Also in the univariate case there are signal processing applications where some properties of wavelet transforms are incompatible with linearity. Recently a systematic theory of proximity of subdivison schemes has emerged, and meanwhile a rich class of nonlinear analogues of linear one-dimensional subdivision schemes has been investigated with regard to convergence, approximation properties, and smoothness. Despite these successes, great parts of the theory are still missing: the systematic development of the higher- dimensional case, especially smoothness analysis at extraordinary vertices; finer smoothness analysis (Hoelder regularity), and nonlinear energy minimizing subdivision algorithms. It is the aim of the proposed research project to continue research in these directions.

Subdivision means the refinement of discrete data with the purpose of generating a continuous or even smooth limit. The concept goes back to de Rham, and main applications are found in Geometry Processing and Geometric Design, in Signal Processing and Wavelet Analysis, and most prominently in Computer Graphics where subdivision has been used to `model everything that moves`. The mathematics involved ranges from Approximation Theory to Differential Geometry. The appealing properties of subdivision rules come from its multiresolution nature which is especially important for the interactive handling of geometric shapes, and from its capability of representing complex shapes by a small number of control points. Especially well studied are the linear subdivision schemes, where data on the finer level are found as finite linear combinations from data at the coarser level. There is a rather complete theory with regard to limits, smoothness, approximation power, stability, and other properties. Geometric data, however, frequently do not live a linear environment, and are therefore not naturally accessible by the linear theory. Canonical operations on them are nonlinear. For instance, we think of a flight recorder which yields univariate pose data, or diffusion tensor MR images, which represent bivariate data having values in the Riemannian symmetric space of positive definite symmetric matrices. Until quite recently, the available results on nonlinear subdivision processes applied to a collection of rather specific situations. The aim of the present project was to build a systematic theory and to analyze geometric (and necessarily nonlinear) subdivision rules which apply to manifold-valued data in a natural way, i.e., they obey the invariances implied by the geometry. In particular we successfully dealt with well-definedness, existence, and geometric properties such as smoothness of the limit objects generated by nonlinear subdivision processes.