Multivariate nonlinear subdivision schemes

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.

Subdivision means the refinement of discrete data with the purpose of generating a continuous or even smooth limit. This principle is applied for example in Computer Graphics and in particular in 3D animated movies, where shapes are created by a subdivision process from a few control points. Other application areas are found in mathematics, in particular signal processing, because of the close relation between subdivision rules and wavelet transforms. In all these cases, simple subdvision rules`` are the building blocks of procedures which are used to handle and analyse data. This research project studies data of nonstand type, such as diffusion-tensor images, or flight recorder data. Here an instance of the data is an object like a diffusion ellipsoid or the position of a body in space. Such data naturally live in certain manifolds with distinguished geometric structure. Any procedure for their analysis has to respect that structure and cannot simply operate on the bunch of numbers which comprise the result of measurements. It is however fortunate that most subdivision rules can be worded in terms of simple geometric primitives such as midpoints, or centers of mass, and that these concepts can be defined in a natural manner also in the manifolds our data live in (namely Lie groups, or Riemannian manifolds, even general metric spaces). The goal of this research project is the mathematical analysis of the so-modified subdivision processes as regards their convergence, the shape properties (smoothness) of their limits, the stability of subdivision processes, and the application of subdivision rules to the multiresolution analysis of data. The data under consideration are either arranged in a regular manner (such as in image processing) but can also have irregular combinatoris (such as in Computer graphics). We have achieved results in all these areas, which in many places confirm earlier conjectures that manifold-valued subdivision processes enjoy properties similar to their well- studied linear counterparts. For example, one can still read off properties of data from the decay rate of coefficients in a `lazy` multiresolution representation. Other aspects do not carry over unchanged, e.g. multiresolution representations can no longer dispense with redundancy in general. The study of convergence of subdivision processes as such, which must come before all other properties are analyzed, is a separate topic of its own. We achieved results using completely different (stochastic) methods.