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X Analysis of non-uniform subdivision schemes FWF AnONSS 2015.10.01 2018.09.30
This project is devoted to the analysis of non-uniform subdivision schemes.
These are efficient level dependent and spatially variant recursive
algorithms for generation of curves and surfaces. The analysis of
subdivision schemes constitutes a modern and application oriented
branch of approximation theory. Indeed, subdivision schemes play a role in
biological imaging, computer aided geometric design, computer animation,
wavelet frame theory and isogeometric analysis.

The first goal of this project is the development of a general,
computationally efficient method for the analysis of the convergence and
regularity of non-uniform subdivision schemes. In the stationary setting,
the smoothness of subdivision limits is characterized by the joint
spectral radius of a certain finite set of square matrices defined by the
corresponding subdivision rules. Recent results by the applicant
and her co-authors show that, surprisingly, such a link also exists
in the more general non-stationary setting. It is still an open problem,
whether the characterization of the regularity of general
non-uniform subdivision schemes is possible via the joint spectral
radius approach. This project will either give an affirmative
answer to this open problem, thus, demonstrating the full strength
of the joint spectral radius approach, or expose its limitations.

Generation and reproduction properties of subdivision schemes are
particularly relevant for applications. Polynomial generation of
stationary subdivision schemes
determines the efficiency of wavelet frame decomposition and
reconstruction algorithms. Generation and reproduction of exponential
polynomials by non-stationary subdivision schemes link subdivision and
isogeometric analysis. Non-uniform schemes are appreciated for their
capability to reduce unwanted
wobbly artifacts and self-intersections of curves and surfaces of
arbitrary topology. The generation and reproduction properties of
non-uniform subdivision schemes are not completely understood.
The second goal of this project is a characterization
of the variety of shapes and of classes of functions that can be
generated and reproduced by non-uniform schemes.
ANB ... Austrian National Bank
EC ...European Commission
FWF ... Fonds zur Förderung der wissenschaftlichen Forschung
UniVie ... University of Vienna
WWTF ... Wiener Wissenschafts-, Forschungs- und Technologiefonds