Analysis of Quasi-Uniform Subdivision


We study the smoothness of quasi-uniform bivariate subdivision. A quasi-uniform bivariate scheme consists of different uniform rules on each side of the $y$-axis, far enough from the axis, some different rules near the $y$-axis, and is uniform in the $y$ direction. For schemes that generate polynomials up to degree m, we derive a sufficient condition for $C^m$ continuity of the limit function, which is simple enough to be used in practice. It amounts to showing that the joint spectral radius of a certain pair of matrices has to be less than $2^{-m}$. We also relate the H\"older exponent of the $m$-th order derivatives to that joint spectral radius. The main tool is an extension of existing analysis techniques for uniform subdivision schemes, although a different proof is required for the quasi-uniform case. The same idea is also applicable to the analysis of quasi-uniform subdivision processes in higher dimension. Along with the analysis we present a `tri-quad' scheme, which is combined of a scheme on a triangular grid on the half plane $x<0$ and a scheme on a square grid on the other half plane $x>0$ and special rules near the $y$-axis. Using the new analysis tools it is shown that the tri-quad scheme is globally $C^2$.

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BiBTeX entry

author = "A. Levin and D. Levin",
title = "Analysis of quasi uniform subdivision",
journal = "Applied and Computational Harmonic Analysis",
pages = "18-32",
year = 2003,
volume = "15(1)")