Monday, April 30, 2007
Technische Universität Berlin
Fakultät II, Institut für Mathematik
Str. des 17. Juni 136
10623 Berlin
MA 041
Lecture - 14:15
Abstract:
This is work with Yuanxin (Leo) Liu, PhD candidate at UNC Chapel Hill.
Univariate B-splines are smooth, piecewise polynomial curves that can
be defined on irregularly spaced points (called knots). Because they
are also expressive -- they can reproduce all polynomials up to the
desired degree -- they are often used in computer-aided geometric
design (CAGD) and in function approximation.
All multivariate generalizations of B-splines proposed to date, with
one exception, impose restrictions on knot placement -- to grids,
tensor-product constructions, or multiple knots. The exception is
Neamtu's elegant work, based on higher-order Voronoi diagrams, which
we will describe. Even this has limitations when one wishes to
consider data-dependent surface constructions because once the knot
positions are chosen, their interconnection is fixed.
We observe that the essential property to prove polynomial
reproduction in Neamtu's work is a combinatorial property that is
satisfied by other triangulations. We propose a {\it link
triangulation algorithm}, which dualizes and generalizes Lee's
algorithm for higher order Voronoi diagrams, and guarantees the
combinatorial property (and thus polynomial reproducability) for a
wider class of triangulations.
We prove that the algorithm gives quadratic and cubic splines that include some of the
classical multivariate splines. For example,
we can use our algorithm to reproduce the classic quadratic box spline
basis, the Zwart-Powell element, which means that we can use our splines
to blend patches of quadratic box splines while preserving smoothness
and polynomial reproducibility.
afterwards