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.