Nonlinear subdivision and approximation theory

The purpose of a subdivision schemes is to generate a continuous or smooth object from discrete data by iterative refinement. The mathematics involved ranges from approximation theory and numerical analysis to differential geometry. Prominent applications are e.g. found in geometry processing and computer graphics, and multivariate subdivision is a research topic of much current interest. The available theory for analysis of subdivision schemes can be considered more or less complete for the linear and regular (grid) case, whereas the irregular multivariate case has only recently been solved. In view of the applications present it is natural that subdivision has been extended to nonlinear geometries like surfaces and Riemannian manifolds, or Lie groups and symmetric spaces, or Euclidean space with obstacles. Via the average analogy or the log/exp analogy it is possible to define subdivision rules also in these cases, but analysis, especially smoothness analysis poses new challenges. This research project continues to investigate nonlinear subdivision rules via the method of proximity, which in recent years has been established as a successful way of approaching geometric subdivision. Area to be explored is the multivariate regular case, higher order smoothness, regularity properties of limits and other topics. Further, we consider applications in graphics and image processing: acquisition techniques of increasing sophistication like diffusion tensor imaging produce data which naturally lie in a geometry of higher complexity than just the real numbers or a vector space. Processing of these data has to adapt to the underlying geometry, and it is at this basic level where nonlinear subdivision and corresponding wavelet-type transform come in.