Multivariate Refinement Equations and Convergence of Subdivision Schemes

Abstract

Refinement equations play an important role in computer graphics and wavelet analysis. In this paper we investigate multivariate refinement equations associated with a dilation matrix and a finitely supported refinement mask. We characterize the Lp-convergence of a subdivision scheme in terms of the p-norm joint spectral radius of a collection of matrices associated with the refinement mask. In particular, the 2-norm joint spectral radius can be easily computed by calculating the eigenvalues of a certain linear operator on a finite dimensional linear space. Examples are provided to illustrate the general theory.