Scalar multivariate subdivision schemes and box splines

Authors:
Maria Charina;Costanza Conti;Kurt Jetter;Georg Zimmermann
Affiliations:
Fakultät für Mathematik, TU Dortmund, D-44221 Dortmund, Germany;Dipartimento di Energetica, Universití di Firenze, Via C. Lombroso 6/17, I-50134 Firenze, Italy;Institut für Angewandte Mathematik und Statistik, Universität Hohenheim, D-70593 Stuttgart, Germany;Institut für Angewandte Mathematik und Statistik, Universität Hohenheim, D-70593 Stuttgart, Germany
Venue:
Computer Aided Geometric Design
Year:
2011

Citing 7
Cited 1

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Polynomial reproduction of multivariate scalar subdivision schemes

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Abstract

We study scalar d-variate subdivision schemes, with dilation matrix 2I, satisfying the sum rules of order k. Using the results of Moller and Sauer, stated for general expanding dilation matrices, we characterize the structure of the mask symbols of such schemes by showing that they must be linear combinations of shifted box spline generators of some polynomial ideal. The directions of the corresponding box splines are columns of certain unimodular matrices. The ideal is determined by the given order of the sum rules or, equivalently, by the order of the zero conditions. The results presented in this paper open a way to a systematic study of subdivision schemes, since box spline subdivisions turn out to be the building blocks of any reasonable multivariate subdivision scheme. As in the univariate case, the characterization we give is the proper way of matching the smoothness of the box spline building blocks with the order of polynomial reproduction of the corresponding subdivision scheme. However, due to the interaction of the building blocks, convergence and smoothness properties may change, if several convergent schemes are combined. The results are illustrated with several examples.