Scalar Subdivision Schemes and Box Splines

TL;DR

This paper characterizes scalar multivariate subdivision schemes with specific sum rules as linear combinations of shifted box spline generators, providing a systematic framework for analyzing their structure and smoothness.

Contribution

It establishes a novel characterization of subdivision scheme masks using box spline generators and quotient polynomial ideals, linking sum rules to box spline structure.

Findings

Subdivision schemes are expressible as combinations of box spline generators.

The structure of masks is determined by quotient polynomial ideals.

The approach facilitates systematic analysis of smoothness and polynomial reproduction.

Abstract

We study scalar $d$-variate subdivision schemes, with dilation matrix 2I, satisfying the sum rules of order $k$. Using the results of M\"oller 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 quotient polynomial ideal. The directions of the corresponding box splines are columns of certain unimodular matrices. The quotient 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.