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Institute of Geometry

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Reserach project: Subdivision processes in nonlinear geometries

This project has been funded by the Austrian Science Fund (FWF) under grants P18575, P19870, and the doctoral programme W1230. In this way FWF supported Ph.D. students and Postdocs Philipp Grohs (2006—2010), Esfandiar Navayazdani (2006—2007), Andreas Weinmann (2007—2011), Oliver Ebner (2009—2012), Caroline Moomüller (2013—2017), and Svenja Hüning (2016—).


Subdivision schemes in vector spaces are well understood with respect to their convergence, smoothness of limits, capability of irregular combinatorics, approximation power, and their relation to multiresolution transforms. They are prominently used in computer graphics. It is the long-term purpose of this research project to establish a similar body of knowledge for subdivision processes in nonlinear geometries. The latter occur in all places where the data under consideration do not naturally live in a vector space, but rather in a Lie group (like position data) or a symmetric space (like diffusion tensor image data) or in Euclidean space minus obstacles.

The main idea in dealing with the nonlinear situation is to interpret the basic linear operations basic to subdivision (namely, affine combinations and convex combinations) in ways which permit the transfer to more general situations. This can be done by means of exponential mappings, in a variational way, or by means of a probabilistic interpretation. For instance, a convex combination has an interpretation as the expected value of a random variable. One of the major technical tools in investigating nonlinar subdision processes are proximity inequalities which relate them to their linear counterparts. We have achieved a long list of results. Key ones concern the fact that the limits of certain manifold schemes are as smooth as their linear counterparts (including irregular cases), or that the probabilistic interpretation leads to schemes in CH metric spaces which converge if and only if the linear ones do. We have also successfully investigated stability, approximation power, and the extension to Hermite schemes.